Wicks, Alfred L.
Schiefer, Mark I.
West, Robert L.
| 1. | An apparatus for vibrational analysis of a vibrating structure comprising: geometry file means for storing geometric coordinates of at least one point on the structure; aimable laser means for irradiating the structure with a laser beam; means operably coupled to the geometry file and aimable laser means for aiming the laser beam to irradiate at least one point on the structure; sensing means receiving laser beam light retroreflected from the point on the vibrating structure to which the laser beam is aimed for producing vibrational data which is correlated to the point on the structure. |
| 2. | Apparatus of claim 1 further comprising means for exciting the structure to vibrate. |
| 3. | The apparatus of claim 1 further comprising processing means operably coupled to the sensing means for manipulating the vibrational data obtained to determine the vibrational response of the test struc¬ ture at the point. |
| 4. | The apparatus of claim 3, the processing means including a computer. |
| 5. | The apparatus of claim 4, the geometry file means including a data file stored in the memory of the computer. |
| 6. | The apparatus of claim 3, the vibrational data being timedomain data, and the processing means including analyzer means which transforms the timedomain vibrational data into frequencydomain data. |
| 7. | The apparatus of claim 6, the analyzer means including means for computing the Fourier transform of the frequency domain vibrational data. |
| 8. | The apparatus of claim 3 further comprising output means coupled to the processing means for displaying the vibrational response of the test structure at a point in humanly perceptible form. |
| 9. | The apparatus of claim 3, the exciting means including attaching electromechanical vibrating means for attachment to a structure to apply a force to the structure and induce vibrations in response to an input electrical signal. |
| 10. | The apparatus of claim 8, the exciting means including detector means for measuring the magnitude of the excitation force and outputting a signal to the processing means proportional to the excitation force magnitude, wherein the output signal is used by the processing means to further determine the vibrational response of the structure. |
| 11. | The apparatus of claim 3, the exciting means including acoustic means for projecting sound waves onto a structure to induce vibration of the structure. |
| 12. | The apparatus of claim 1, the geometry file means including generating means for defining a set of geometric coordinates for a point on the test struc¬ ture, whereby the geometric coordinates of the test structure thus defined are stored in the geometry file means. |
| 13. | The apparatus of claim 12, the generating means including a linear measuring device for measur¬ ing the distance to a point on the test structure from a reference point, whereby the distances measured to the point provide geometric coordinates for that point. |
| 14. | The apparatus of claim 12, the generating means including a mechanical gauging device for moving around the test structure to generate a series of signals for at least one point on the test structure, whereby the series of signals correspond to the geometric coordinates for the point to be stored in the geometry file means. |
| 15. | The apparatus of claim 12, the generating means including aimable laser means for aiming a laser beam to at least one point on the structure, the laser means producing signals corresponding to the beam deflection angles necessary to aim the beam at the point, and further including measuring means for determining the distance from the laser means to the point, whereby the deflection angles and measured distance are used in determining the geometric coordinates of the point for storing in the geometry file means. |
| 16. | The apparatus of claim 3, the sensing means including velocimetry means for producing data proportional to the vibrational velocity of a point on the test structure and correlated to the point. |
| 17. | The apparatus of claim 16, the velocimetry means being responsive to the frequency difference between the aimed laser beam and the retroreflected laser light to produce vibrational velocity data. |
| 18. | The apparatus of claim 16, the data from the velocimetry means being proportional to the point vibrational velocity which is along an axis parallel to the line of sight of the laser beam, the processing means including means to manipulate the line of sight velocity data to provide velocity data which is outward normal to the structure surface at the point. |
| 19. | The apparatus of claim l, further comprising a plurality of aimable laser means for irradiating the structure with a plurality of laser beams; and a plurality of sensing means for receiving laser beam light retroreflected from a point on the vibrating structure to produce threedimensional vibrational data which is correlated to the point on the structure. |
| 20. | The apparatus of claim 19, further comprising processing means operably coupled to the plurality of sensing means for manipulating the vibrational data at a point to determine the threedimensional vibrational response of the test structure at the point. |
| 21. | The apparatus of claim 20, the processing means including a computer. |
| 22. | The apparatus of claim 20, the geometry file means including a data file stored in the memory of the computer. |
| 23. | The apparatus of claim 20, the vibrational data being timedomain data, and the processing means including analyzer means which transforms the timedomain vibrational data into frequency domain data. |
| 24. | The apparatus of claim 20, the analyzer means including means for computing the Fourier transform of the frequency domain vibrational data. |
| 25. | The apparatus of claim 20 further comprising output means coupled to the processing means for displaying the vibrational response of the test structure at a point in humanly perceptible form. |
| 26. | The apparatus of claim 20, the multiple sensing means including velocimetry means for producing data proportional to the threedimensional vibrational velocity of a point on the test structure and correlated to the point. |
| 27. | The apparatus of claim 26, the data from the velocimetry means being proportional to the point vibrational velocity which is along an axis parallel to the line of sight of the laser beam, the processing means including means to manipulate the line of sight data to provide velocity data which is point correlated and is along three orthogonal axes of a coordinate system. |
| 28. | An apparatus for vibrational analysis of a vibrating structure comprising: geometry file means for storing geometric coordinates of at least one point on the structure; multiple aimable laser means for irradiating the structure with multiple laser beams; means operably coupled to the geometry file and each of the multiple aimable laser means aimable laser means for aiming the multiple laser beams to irradiate at least one point on the structure; multiple sensing means for receiving the laser beam light of the multiple beams retroreflected from a point on the vibrating structure to produce vibrational data which is correlated to the point on the structure. |
| 29. | The apparatus of claim 28 further comprising processing means operably coupled to the sensing means for manipulating the vibrational data obtained at a point to determine the threedimensional vibrational response of the test structure at the point. go so. |
| 30. | A method for using a laser to obtain point correlated vibrational data from a vibrating structure comprising the steps of: a) obtaining a geometry file containing geometric coordinates for at least one point on the structure; b) aiming a laser beam at the point on the structure using the geometric coordinates of the point from the geometry file; c) sensing the laser light retroreflected from the point on the structure to obtain vibrational data that is correlated to the point. |
| 31. | The method of claim 30 further comprising the step of repeating the laser aiming and sensing for a plurality of points from the geometry file of the structure. |
| 32. | The method of claim 30 further comprising the step of processing the vibrational data to obtain the vibrational response of the structure at a point. |
| 33. | The method of claim 32 wherein the vibrational data is timedomain data wherein the vibrational data is timedomain data further comprising the step of transforming the data from timedomain vibrational data into frequency domain vibrational data. |
| 34. | The method of claim 32 further comprising the step of converting the lineofsight velocity data into velocity data which projects along a line normal to the surface of the vibrating structure at a point on the structure. |
| 35. | The method of claim 32, further comprising the step of displaying the data in a humanly perceptible form. |
| 36. | The method of claim 30, wherein the sensing step produces vibrational velocity data indicating the vibrational velocity at the point on the structure. |
| 37. | The method of claim 36, the sensing step including the steps of: measuring the difference in frequency between the laser beam aimed at a point and the retroreflected laser light from the point; and using the frequency difference to determine the vibrational velocity at the point. |
| 38. | The method of claim 30, the geometry file obtaining step including the steps of: a) determining a set of geometric coordinates for at least one point on the structure; and b) inputting the geometric coordinates into a data file to create the geometry file. |
| 39. | The method of claim 30, the geometry file obtaining step including the step of using a preexisting file containing geometric coordinates of points on the structure. |
| 40. | The method of claim 37, the step of determining geometric coordinates for a point including the step of measuring the distances along three orthogonal axes to at least one point on the structure from a defined origin point wherein the distances corresponds to the geometric coordinates for that point with respect to the origin point. |
| 41. | The method of claim 37, the step of determining geometric coordinates for a point including the steps of: a) defining the coordinates for the origin of a laser beam; b) aiming the laser beam at a point; c) obtaining the horizontal and vertical beam deflection angles necessary to aim at the point; d) measuring the distance from the defined origin of the laser beam to the point; and e) using the measured distance and the deflection angles to compute geometric coordinates of the point for the geometry file. |
| 42. | The method of claim 30, wherein the geometry file is stored in a computer and the computer accesses the geometry file to generate control signals for aiming the laser beam. |
| 43. | The method of claim 30 further comprising the steps of: a) mapping the geometric coordinates of at least one point in the geometry file to a pair of horizontal and vertical deflection angles defined in another coordinate system and corresponding to aiming the beam at the point; and b) aiming the laser beam at a point according to the deflection angles mapped to the point. |
| 44. | The method of claim 43, wherein a computer is used to map the deflection angles to the point and aim the laser at the point. |
| 45. | The method of claim 30 wherein a laser velocimeter is used to produce vibrational velocity data correlated to the point. |
| 46. | The method of claim 43 wherein the mapping step includes the steps of: a) determining the range distance from at least four points on the structure to the origin of the laser beam; b) using the range distances to the points for defining the coordinates of the points as referenced to a laser coordinate system originating at the origin of the laser beam; c) using the point coordinates defined in the laser coordinate system to determine the rotation matrix necessary to rotate the axes of the geometric coordinate system into the axes of the laser coordinate system; d) defining the origin of the laser beam within the geometric coordinate system; and e) combing the geometric origin coordinates of the laser beam and the rotation matrix to yield a transformation equation which maps a set of geometric coordinates to a set of laser coordinates to be used in the laser. |
| 47. | The method of claim 30 further comprising the step of aiming multiple laser beams at a point on the structure using the geometric coordinates of the point from the geometry file. |
| 48. | The method of claim 30 wherein the aiming of the laser beam at a point on the structure is done from multiple independent positions. |
| 49. | The method of claim 47 further comprising the step of processing the vibrational data to obtain the threedimensional vibrational response of the structure at a point. |
| 50. | The method of claim 48 further comprising the step of processing the vibrational data to obtain the threedimensional vibrational response of the structure at a point. |
Appendix
Attached hereto as Appendix A is a source code listing of software for use with the present invention. The contents of Appendix A are incorporated herein by reference. Further, Appendix A contains material which is subject to copyright protection. The owner has no objection to facsimile or microfiche reproduction of the appendix, as they appear in the Patent and Trademark Office patent file or records, but otherwise reserves all rights whatsoever.
Background of the Invention
I. Field of the Invention.
The present invention relates to apparatus and method for dynamic structural testing and experi¬ mental modal analysis of vibrating structures and more specifically to the use of lasers in such analyses.
II. Discussion of Prior Art.
In the area of dynamic structural testing and experimental modal analysis, it is a goal of the
testing community to provide a measurement system which has minimum influence on the object under test. Specifically, it is desirable to provide a measurement system which extracts vibrational information and modal parameters from a vibrating test structure with minimum or no interference to the vibrational modes of the structure.
Conventionally, structural dynamic measure¬ ments are performed using dynamic transducers, such as accelerometers, attached to the vibrating structure. For example, accelerometer output a signal proportional to the acceleration at the point of contact between the vibrating structure and the accelerometer. A typical vibrational testing sequence involves placing these contact transducers at various points of interest on a structure, and vibrating the structure to obtain data about its dynamic response. The data is then processed, such as to obtain the Fast Fourier Transform (FFT) or Frequency response function (FRF) of the velocity at the point. However, contact transducers have mass and, consequently, mass load the test structure in addition to adding rotational inertia, local stiffness and damping to the structure. As a result, the vibrational mode shapes and dynamic responses of a test structure measured using contact transducers are inherently distorted and inaccurate. Attempts have been made to reduce these contact effects by making smaller and lighter transducers;
however, the effects are still existent albeit reduced.
The practicality of using contact trans¬ ducers for structural testing involving high tempera¬ ture or cryogenic conditions is also questionable because these transducers are active components which must be in contact with the test surface, making them susceptible to the conditions of the harsh testing environment. In addition, since contact transducers must be manually placed on a structure, the surface must be prepared and there are possibilities of human error in positioning the transducer and securely attaching on the test structure.
Another problem with a contact transducer measurement system is the reliability of the test set up. Each test point requires a sensor, data lines, and an individual signal conditioning and processing channel. It will be appreciated that the sheer numbers of associated connections, data cables and processing hardware units necessary for a thorough vibrational test, besides being cumbersome, severely reduces the reliability of the test set up. Further¬ more, the tasks of mounting and remounting the trans¬ ducers, calibrating the transducers and associated processing hardware, and routing the cables are extremely labor intensive tasks requiring highly trained engineering talent.
Still another drawback in a dynamic measure¬ ment system using contact transducers is the spatial limitation on the number of test points which can be defined and measured on a test structure due to the finite size of the transducers. The current theoreti¬ cal/computational modeling techniques for developing dynamic models, such as finite element analysis, generate refined dynamic models that have thousands of spatial nodes spaced closely together, each node having up to six degrees of freedom. The degrees of freedom are generally three orthogonal translations, such as translational displacements or velocities, and three orthogonal rotational degrees of freedom. However, the finite size of the contact transducers in combination with testing time constraints restricts experimental test models to often less than a hundred spatial test points when the structure is analyzed. Furthermore, with contact transducers, several of the desired test locations may not be physically accessi¬ ble due to the size of the contact transducer which further reduces the density of test points on the structure. As the density of test points is thus limited, the measurement spatial testing densities are insufficient to allow the rotational degrees of freedom to be determined. Consequently, the measurement results are incompatible to results obtained from theoretical/computational models. Furthermore, the information extracted from each of
these points is limited to the three translational degrees of freedom.
Therefore, there is a need for a non-con¬ tacting vibrational measurement system having no mass loading effects, and which is able to achieve test point densities comparable to or exceeding the point densities available with theoretical/computational modeling techniques allowing an extraction of extraction of up to six degrees of freedom at each test point. Furthermore, the measurement system should be reliable, having a low number of data acquisition channels. Finally, since the data which is measured using contact transducers is associated with a particular point or area on the test structure (i.e., you know where each transducer is physically placed) and since conventional modeling techniques relate dynamic responses and mode shapes to particular theoretical nodes on the structure surface, the non-contact measurement system should maintain a conventional test point-to-data correlation.
Laser technology has offered solutions to some of these problems. More specifically, a laser- based measurement system can make non-contacting, non-loading measurements on a vibrating test structure by measuring the doppler-shifted frequency of an incident laser beam retro-reflected from a vibrating surface on the test structure.
Laser-based measurement techniques have been used to do vibrational analysis of solid vibrating structures. One such technique, termed speckle pattern interferometry or TV-holography, involves directing a laser beam onto a vibrating surface and comparing the reflected or scattered light, the so-called "speckle" picture, to a reference picture symbolizing the non-vibrating surface. The displace¬ ment between the two pictures is used to extract vibrational data about the surface, such as to obtain strains and determine keep in the solid structure. One method of speckle pattern interferometry is disclosed in Vachon U.S. Patent No. 4,591,996. The drawbacks of speckle interferometry are that, firstly, it requires relatively large displacements for accur¬ ate data; secondly, the image must be further inter¬ preted to obtain quantitative results; and thirdly, the optics must be very precise because any distortion in the speckle pattern will be interpreted as motion of the structure. Additionally, the measurement is predominantly qualitative because there is little correlation between the measured data and specific points on the test structure.
A more promising technique for dynamic structural testing and modal analysis of vibrating structures is doppler velocimetry. Laser doppler velocimetry is based on measuring the shift in fre¬ quency (doppler phenomenon) between a reference laser
beam and the retro-reflected beam from a vibrating test surface, which shift is proportional to the relative vibrational velocity of the test surface with respect to the laser. One velocimetry technique, the differential (or dual-beam) technique, focuses two laser beams on a single test point where they con¬ structively and destructively interfere with the fre¬ quency of the illumination of the surface traveling through the interference pattern being proportional to the velocity at the test surface. However, this tech¬ nique requires very accurate alignment of the two laser beams onto the test surface, and therefore, is not desirable for a scanning measurement system in which the beam must constantly move to measure objects having non-continuous, changing surface contours. Additionally, the necessary alignment to aim several laser beams at each test point on a non-continuous surface requires very expensive equipment.
Alternatively, in reference beam veloci¬ metry, a laser beam is split into a reference beam and an incident beam. The incident beam is focused on a vibrating test surface, and the frequency of the doppler-shifted retro-reflected beam returning from that vibrating surface is compared to the reference beam frequency to determine the vibrational velocity at the test surface. In reference beam velocimetry, the sensitive measurement axis is along the line of sight of the retro-reflected laser beam. Therefore, a
point-and-shoot technique can be used to extract the line-of-sight velocity data from a vibrating surface withour srringent alignment requirements.
There are a number of reference beam doppler velocimerers commercially available for taking vibra¬ tional measurements. One of these is a scanning velocimeter in which the beam may be deflected in horizontal and vertical directions to scan a grid on the test structure. The scanning velocimeter, like non-scanning velocimeters, measures the velocity along the line-of-sight of the retro-reflected beam. When the incident beam is not exactly perpendicular to the vibrating test surface, the measured velocity is not the surface normal velocity which is usually desired. Furthermore, the available systems scan the laser beam randomly over the structure, moving in indiscriminate angular increments to provide velocity data which is not specifically correlated to known test points on the structure.
In dynamic structural testing and experi¬ mental modal analysis it is desirable to extract velocity data in a representational form compatible to the form utilized in theoretical/computational model¬ ing techniques, such as finite element modeling. The theoretical/computational representation is conven¬ tionally written as an "outward-normal," orthogonal coordinate system in which the determined velocity is along an axis normal to the vibrating surface of
interest. Directly placed contact transducers accom¬ plish the outward-normal convention by measuring only in their axis of sensitivity. For example, single axial accelerometers attached at a contact point on a vibrating structure measure an outward normal accel¬ eration at the contact point. Therefore, for a system to be compatible with conventional dynamic modeling techniques and to provide measurement data comparable to the data provided by conventional contact trans¬ ducers, the measured data should be extracted in a traditional orthogonal coordinate system yielding outward normal surface velocities. The currently available laser velocimeter systems do not provide surface normal velocities. Their axis of sensitivity is only along the line of sight of the laser beam.
Another drawback to current available laser velocimeter systems is that these systems, because they are limited to line-of-sight measurements, do not provide the three translational degrees of freedom at a point, such as the three orthogonal translational velocities. They provide only a one-dimensional, line-of-sight velocity measurement. This limits the measurement to only one degree of freedom at a mea¬ surement point. ithout the three translation degrees of freedom, the rotational degrees of freedom cannot be found. Furthermore, the single, achievable degree of freedom is for a velocity, displacement or other
translation which is not outward normal to the surface as is desired.
Finally, because current velocimeter systems do not yield velocity data that is correlated to specific points on a test structure, the data is essentially qualitative. Without a velocity data-to- test-point correlation, it is difficult to determine exactly where on the structure a particular dynamic response or vibrational mode data point is being taken. Therefore, the measurement data from available laser measurement systems is not compatible with data from conventional modeling techniques, nor is it compatible with the data available from conventional contact transducer measurements.
Therefore, although currently available laser measurement systems solve some of the problems inherent in conventional, contact-based vibrational measurement systems by providing non-loading, non-contact measurements, the system falls short in other areas. Available laser systems make only line-of-sight measurements, do not produce three-dimensional translational and rotational degrees of freedom, and provide data which does not correspond to specific test points defined on the surface of the non-vibrating structure.
Summary of the Invention
The present invention provides a laser-based measurement system which aims a laser beam to a
plurality of pre-selected test points on a vibrating test structure to determine the test structure's vibrational dynamics and mode shapes correlated to each of these points. Moreover, the correlated velocity data provided by the system is outward normal to the vibrating surface or along the axes of an orthogonal coordinate system. Additionally, the point correlation and the increased spatial testing point densities achievable with the present system allows determination of six degrees of freedom for each test point.
To accomplish aiming of the laser beam of the present invention, a geometrical representation of a test structure is created as an array of test points which are defined in a global orthogonal axis coordi¬ nate system. This geometrical representation or "static" image of the test structure is utilized to aim the laser beam at the desired test points when the test structure is vibrating to extract line-of-sight vibrational velocity. In accordance with a further aspect of the present invention, the line-of-sight velocity data at each point is then manipulated, such as by mathematical transformation, into point-specific velocity data that is either outward normal to the test surface or is along three defined orthogonal axes. The orthogonal or outward normal velocity data for the vibrating structure is thus correlated to the specific test points on the static image, by which
experimental modal analysis or other analysis of the test structure as is possible.
To define the points to be tested, a geomet¬ ric representation or static image of a test structure is created by defining points on the surface of the structure in a global coordinate system and storing the geometrical representation of the structure in a data file referred to as a geometry file. Alterna¬ tively, the geometric representation might already exist, such as in a pre-existing finite element analysis file. If a pre-existing geometric file is not available, the geometric representation is estab¬ lished by a series of linear measurements from a defined global origin to each test point on the structure, and the measured coordinates input to a computer to create a geometry file. An elec¬ tro-mechanical arm may also be manipulated to various points on the structure to create a set of coordinates for each test point which coordinates are input to a geometric file. Still further, the system of the present invention is may be used to generate the geometry file. To this end, the laser beam is sequen¬ tially aimed at a plurality of points on the non-vibrating structure to provide a set of beam deflection angles corresponding to the points. Those deflection angles are then used by the computer to derive a set of point coordinates for the geometry file.
Using the geometry file, a computer then maps each test point to a pair of angular control signals by creating a scan list which assigns each test point to the unique pair of horizontal and vertical angles necessary to aim the beam at that point. The deflection angles correspond to voltage control signals which are input to the velocimeter to aim the laser beam at each test point so as to extract velocity data for each of the points.
The line-of-sight velocity is measured and the velocity data from the velocimeter is input to an analyzer which transforms the time-domain measured data into frequency-domain velocity data. The fre¬ quency-domain data, which is the measured line-of-sight velocity, is corrected so that it is either outward normal, or, in the three dimensional embodiment of the present invention, along three orthogonal axes. The excitation force used to vibrate the test structure is also input to the analyzer where it is transformed into the frequency domain and ratioed with the measured velocity data to determine the frequency response function (FRF) of the struc¬ ture. The FRF and frequency velocity data obtained by the system is thus point correlated, and can be processed to determine the vibrational response of the structure as is conventional. For example, the vibrational data obtained is graphically compared to a wire frame geometry of the static test structure to
illustrate vibrational movement of the structure. Alternatively, the data is output in other forms compatible with known experimental modal analysis techniques.
For essentially flat, two-dimensional test surfaces having one-dimensional vibrational velocities predominantly in the direction outward normal to the surface, a single scanning laser velocimeter system may be used for obtaining the velocity data. For three-dimensional structures having vibrational velocities in directions other than outward normal, it is necessary to aim multiple velocimeters at each test point to extract velocity data. A sufficient multiple velocimeter system utilizes at least three, spaced independent velocimeters or one velocimeter at three independent placement positions to capture the three dimensional velocities of the test surface. The line-of-sight velocities measured at each velocimeter position are transformed into the frequency domain and then corrected so as to lie on the three orthogonal axes of a defined coordinate system.
In experimental modal analysis utilizing contact transducers, it is difficult and sometimes impossible to extract the orthogonal rotational degrees of freedom from the test data, because high spatial testing densities are required for such an extraction. The laser velocimeter of the present in¬ vention has a small effective measurement width which
allows a greater density of test points to be measured than is accomplished utilizing conventional finite size contact transducers. The increased spatial density of the test points meets or exceeds the spatial density that is achieved with theoret¬ ical/computational dynamic models allowing data results compatible with these models. With the increased spatial densities possible using a laser, the multiple velocimeter position embodiment of the present invention allows determination of the rota¬ tional degrees of freedom as well as the three translational degrees of freedom.
Another technical and practical advantage of the present invention is that the velocity measure¬ ments are done serially, with the same velocimeter measuring the velocity at each point. The single channel aspect of the present invention increases the reliability of the system as compared to systems having a multitude of sensors, connectors and process¬ ing channels.
When doing experimental modal analysis, it is desirable and often necessary to correlate a measured parameter to a specific test point. The present invention achieves this correlation which has heretofore not been achievable with available laser measurement systems. The point correlation aspect of the present invention makes the extracted data compat¬ ible with experimental modeling techniques and
conventional measured data. To achieve this correlation, the laser beam must be accurately aimed at each test point, which requires defining the position of each velocimeter within the global coordinate system used to define the geometrical representation of the test structure. One method of defining the velocimeter position is to physically gauge, with a tape measure or other linear measuring device, the distance from the defined global origin of the system or from a point on the test structure to the velocimeter. However, an adequate measurement by this method is difficult to achieve, because of physical measurement inaccuracies and movement of the laser beam origin within the velocimeter housing as the laser beam is moved. Therefore, the present invention provides a unique positioning method which aims the laser at a plurality of reference points and uses the deflection angles necessary to reach those points to define the velocimeter position within the global coordinate system.
Finally, the present invention is extremely compact and mobile and does not require prohibitively precise optical configurations as are necessary with other laser-based measurement techniques.
Therefore, the present invention provides a mobile, reliable measurement system utilizing a scanning laser velocimeter to obtain velocity data which is correlated to defined test points on a
vibrating structure. The laser is non-contacting, eliminating mass loading and similar effects on the test structure, and allowing measurements in physical¬ ly inaccessible spaced and in hostile environments. The scanning capability and increased spatial density increases the reliability of the system, reduces data acquisition time and allows extraction of up to six degrees of freedom from each test point. Finally, the test-point-to-data correlation provides measurement data more closely correlated to the data obtainable from theoretical modeling techniques and conventional measurement systems.
These and other objects and advantages of the present invention will become more apparent from a detailed description of the invention. Brief Description of the Drawings
The accompanying drawings which are incor¬ porated herein constitute a part of this specifica¬ tion, illustrate the invention and, together with the general description of the invention given above and the detailed description given below, serve to explain the principles of the invention.
Fig. 1 is a block diagram of a first embodi¬ ment of a laser velocimeter measurement system in accordance with the present invention;
Fig. 2 is a block diagram of a second embodiment of a laser velocimeter measurement system in accordance with the present invention;
Fig. 3 is a flow chart of the measurement sequence utilized by the system of Figs. 1 and 2;
Fig. 4 is a schematic representation of the scanning laser of Fig. 2 for purposes of explaining use of the system of Fig. 2 to generate the geomet¬ rical representation of a test structure.
Fig. 5 is a geometric diagram illustrating operation of the scanning laser of Fig. 4.
Figs. 6-11 are geometric diagrams for purposes of explaining operation of the system of Fig. 2;
Fig. 12 is a block diagram of a third embodiment of the laser velocimeter measurement system in accordance with the principles of the present invention;
Figs. 13 and 14 are possible configurations for the laser velocimeters of Fig 9.
Detailed Description of a Preferred Embodiment
Referring to Fig. 1, a first embodiment of laser velocimeter measurement system 5 of the present invention comprises a laser velocimeter 10 which irradiates plurality of defined test points 11a, llb-lln on a vibrating test structure 12 with an incident laser beam 14 (shown as solid line) to obtain the vibrational velocity associated with each defined point on structure 12. The incident beam 14 is retro-reflected as beam 16 (shown as dashed line) and
the doppler frequency shift of retro-reflected beam 16 is measured to determine the vibrational velocity at the test points such as point 11. A computer or other controller 18 uses a geometrical representation file 19 of the test structure which is stored in the computer to control velocimeter 10, via lines 20, and aim the incident laser beam 14 at each test point. The velocity data extracted by velocimeter 10 for each test point is line-of-sight velocity data that is measured along the line-of-sight of incident beam 16. The line-of-sight, time-domain velocity from velocimeter 10, is output on line 22 to analyzer 24, where the data is transformed into the frequency domain. The frequency domain, line-of-sight velocity data is then corrected in computer 18 to project the velocity along an axis which is outward normal to the test structure 12, such as the Z-axis of the coordi¬ nate system 34 referenced to structure 12. The data is then further processed or displayed, such as on display 30 connected to computer 18, to provide information about the vibrational response of test structure 12. An additional input to analyzer 24 is the measured excitation force applied to structure 12 by vibrational exciter 26, which is powered on line 25 by analyzer 24. The force applied by vibrator 26 is measured by a detector (not shown) and is input on the excitation force input, on line 27, allows computation of the frequency response function (FRF) of the test
structure 12, defined as H = output velocity/input excitation force. The FRF is a parameter widely measured in experimental modal analysis. Analyzer 24 communicates with computer 18, through lines 28, to allow the data processed by analyzer 24 to be graphically or otherwise represented, such as on display 30 associated with computer 18.
Referring to a second embodiment 50 of the present invention as shown in Fig. 2, a scanning laser velocimeter 52 is positioned to irradiate defined test points, such as point 53, on a vibrating test struc¬ ture 54 and extract velocity data. A scanning veloci¬ meter 52, such as the VPI (Vibrational Pattern Imager) Sensor Series 9000 commercially available from Ometron Limited of London, England, moves its beam both vertically and horizontally to a particular position in response to analog voltage signals at the X and Y inputs of velocimeter 52, respectively. The measure¬ ment system 50 includes a computer 60, such as an HP 9000/300 workstation commercially available from Hewlett Packard. Computer 60 communicates with analyzer 62, such as a Zonic System 7000 analyzer commercially available from Zonic Corporation of Milford, Ohio, through ethernet connection 63.
To measure the vibrational velocity of a test point, such as point 53, on test structure 54, an incident laser beam 68 (shown in solid line) is aimed to irradiate that point. The doppler frequency shift
of the beam 76 (shown in dashed line) retro-reflected from test structure 54 is used to determine the vibrational velocity at the point 53. The incident beam 68 of velocimeter 52 is aimed by supplying analog voltage signals on lines 64 and 66 to mirror galvano¬ meters (not shown) within velocimeter 52, which move internal mirrors (not shown) causing incident laser beam 68 to irradiate a chosen test point, such as point 53. The velocimeter beam control voltages are supplied by digital-to-analog converters (DAC's) 70 and 72. If the range of the output of DAC's 70 and 72 does not scale to the required voltage input range for velocimeter 52, and if noise exists on the input lines 64 and 66, an in-line voltage divider and low pass filter 74, is used as shown in Fig. 2. For example, if the DAC outputs range from 0-10 volts and the velocimeter inputs only require 0-5 volts, the DAC output levels must be scaled down by a factor of 2. The embodiment of Fig. 2 utilizes DAC's and ana¬ log-to-digital converters (ADC's) which are internal to analyzer 62 to handle all of the outputs and inputs of the invention. For example, the Zonic 7000 system has internal Analog Signal Conditioning Inputs (ASCI's) and Analog Signal Conditioning Outputs (ASCO's). In the embodiment of Fig. 2 the ASCO's are controlled internally by analyzer 62 to output partic¬ ular analog voltage signals; however, as will be
appreciated, external DAC's and ADC's can be used instead.
The signal levels supplied to the X and Y inputs of velocimeter 52 are controlled by computer 60, which has a geometrical representation file or geometry file 75 stored in memory containing the coordinates of a plurality of test points defined on test structure 54. The test points are represented by coordinates referenced to a global coordinate system. The computer 60, using the position of velocimeter 52 within the global coordinate system (as is explained below) determines a list of scan angles that corre¬ spond to the horizontal and vertical deflection angles that are necessary to aim incident beam 68 at a particular test point. The scan list actually con¬ tains sets of voltage levels corresponding to particu¬ lar test point deflection angles, which are sequen¬ tially input to the X and Y inputs of velocimeter 52 during a measuring sequence to move the beam 68 from point to point and acquire velocity data at each point.
When velocimeter 52 irradiates a test point, such as point 53, it measures the line-of-sight, time-domain vibrational velocity at that point. An analog voltage signal corresponding to the measured velocity is output by velocimeter 52 on line 78 to an ADC (or Zonic ASCI) 80 which digitizes it for use by computer 60 and analyzer 62. The time-domain.
line-of-sight velocity data is input to analyzer 62, which transforms the velocity signal into the frequen¬ cy-domain. Since the surface normal velocity at a test point is the desired data, the line-of-sight velocity data measured by velocimeter 52 is then corrected by computer 60 to provide surface normal velocity data.
To excite the structure and induce vi¬ bration, the test structure is attached to a vibrator 86, such as the XCite 1105-T/C Exciter head commer¬ cially available from the Zonic Corporation. Alterna¬ tively, the vibrator might be acoustic and project sound waves on the test structure to induce vibration. The vibrator is powered on line 90 by a signal am¬ plified by amplifier 92. A force transducer 94, such as a No. 208 load cell sensor from PCP Peizotronics, Depew, N.Y., senses the force and outputs a corre¬ sponding analog voltage signal to ADC 84 on line 82 which is digitized for use by computer 60 and analyzer 62. With knowledge of the excitation force, analyzer 62 can determine the FRF at a particular point. A third analyzer input, provided by velocimeter 52 via line 94 and ADC 96, is a doppler envelope signal, which is an analog voltage signal indicating when the retro-reflected laser beam signal 76 is too weak to sufficiently be detected by velocimeter 52. In this way, analyzer 62 monitors the quality of the input data to reject an insufficient velocity measurement.
Computer 60 communicates with analyzer 62 and velocimeter 52 so as to control laser beam direc¬ tion, data acquisition, and data processing. When all of the desired velocity data has been obtained, computer 60 communicates with a display monitor 100, such as an HP Color Display from Hewlett Packard, or some other output device so that the experimental modal analysis measurement results may be graphically or otherwise displayed.
Measurement Sequence
To better understand the operation of the measurement system of the present invention, it is helpful to follow through each of the steps of the measurement sequence.
Referring to Fig. 3, in the first step 101 of the measurement sequence, a geometrical representa¬ tion of the test structure is obtained by defining each test point on the structure in the coordinates of an orthogonal global coordinate system. The geomet¬ rical representation can be determined by measuring the distance of each point from the global origin or by using a measurement device such as a robotic arm or the scanning laser velocimeter used in the present invention, to generate a set of coordinates, as will be described in greater detail below. The coordinates are input into a file referred to as a geometry file for use by the system in aiming the laser beam and correlating measured data to specific test points.
Alternatively a geometry file might be an existing finite element analysis file, so as to eliminate the necessity of making geometrical measurements.
When a geometrical representation of the test structure has been determined, step 102 is to choose four reference points from the geometry file for use in determining the transformation matrix used to aim the laser beam, as will be described in greater detail.
At step 103 in the measurement sequence the deflection angles and global coordinates for the four reference points are utilized to determine a transfor¬ mation equation which is used to map each test point to its associated beam deflection angles. In order to aim the laser beam at each test point, it is necessary to transform the global coordinates for each test point into a pair of horizontal and vertical de¬ flection angles which are used by the laser velocimeter to aim its laser beam at the test point. The transformation equation, as found through steps 104-108 of Fig. 3, is necessary for this task. To determine the transformation equation, the range distance to the velocimeter position from each refer¬ ence point is found, step 104. Using the range distance from each reference point and using the corresponding deflection angles necessary to aim the velocimeter beam at each reference point, a set of laser local coordinates are determined for each
reference point as shown in step 105 of Fig. 3. The laser local coordinates of each reference point are then used, step 106, to determine the transformation matrix which is part of the transformation equation. The origin of the laser local coordinate system, which is defined as the laser beam origin, is then defined, step 107, and the transformation equation is complete. As is described in greater detail below, several different three-point combinations of the four chosen reference points are used to determine numerous rotation matrices and laser local origins. The laser positioning method uses an optimization routine, step 108, which determines the best matrix and laser local origin, defined as the "true" rotation matrix and laser local origin. The true rotation matrix and laser local origin then define the transformation equation.
After the transformation equation has been determined, the desired test points to be measured are chosen from the geometry file, step 109. In step 110, utilizing the transformation matrix and the chosen test points, a scan list is determined, which will be used to control the velocimeter to aim the laser beam at each test point. In steps 111-115, the acquisition of data takes place as the system accesses the scan list for a particular test point 112, aims the laser beam at the test point 113, and extracts vibrational velocity data from that point 114. The laser aiming
and data extraction continues 115 until each point of interest has been measured.
The line of sight velocity data is then corrected, step 116, to project onto a surface normal or orthogonal axis. The measured data is processed and output to represent the vibrational dynamics and mode shapes of test structure. Each step in the measurement sequence will now be explained in greater detail below.
I. Geometric Representation of the Test Structure
As previously mentioned, it is first neces¬ sary to accurately define the test object 54 in a geometrical representation of test points which are represented in a three-orthogonal axis global coordi¬ nate system. The geometrical coordinates are then stored in a geometry file. The global coordinate system can be defined arbitrarily around the object to be tested; however, it is preferable to define the global coordinate system so that at least one of the three orthogonal axes of the system is outward normal to the test surface, or at least outward normal to a majority of the surfaces to be tested on structure 54. In this way, the line-of-sight velocity data measured by velocimeter 52 will be outward normal to the test surface when it is projected onto the axes of the global coordinate system.
A pre-existing geometry file of points which was generated in a theoretical modeling or computer aided design (CAD) data base, such as a file created in I-DEAS Test Data Analysis software, Version VI, commercially available from Structural Dynamics Research Corporation (SDRC) of Milford, Ohio, may be loaded into computer 60 without the need to determine a measured geometrical representation of the struc¬ ture. One drawback to using a previously generated geometry representation, however, is that there may be deviations between the actual test structure and the data in the original geometry representation file.
Alternatively, the geometric representation may be created manually by measuring the distance to each test point on the structure from a designated origin of the global coordinate system. Using linear transducers or other similar measuring tools, such as a measuring tape, a fairly accurate geometrical repre¬ sentation is obtained and then input, via a keyboard or mouse (not shown) into a file, such as an I-DEAS file. However, this technique is time consuming, especially where a large number of test points is involved, and it is prone to human measurement errors. Still, absent a more sophisticated technique, manual measurement is a viable means for generating the test structure geometrical representation of the present invention, which is used to create a geometry file 75.
There are also available on the market, commercial gauging systems, such as the Metrecom IND-01 robotic arm commercially available from Faro Technologies of Lake Mary, Florida, or the CMS-2100 system from Chesapeake Laser Systems, Inc. of Lanham, Maryland, which systems may be moved to points on the test structure to generate a series of signals which correspond to coordinates for that point. The coordi¬ nates can then be stored in computer 60 and used to create a geometry file TS, in the I-DEAS software format or some other analysis software format.
Referring to Fig. 4, another means of establishing a geometrical representation is to use scanning velocimeter 52. To this end, the beam is aimed at various parts on the structure, and the beam deflection angles necessary to aim the beam at those test points are obtained. The distance from each of those points to the laser velocimeter is then measured by a linear measuring device, such as a Celesco Model TP 101 Position Transducer commercially available from Celesco Transducer Products, Inc. of Canoga Park, California, which is attached to the velocimeter (see Fig. 4) . Knowing the deflection angles, the distance between the points and the velocimeter, and the global coordinates of the velocimeter, the global coordinates of the measured points may be determined to create a geometry file 75 as will now be explained.
In using velocimeter 52 to determine the geometrical representation of a test structure 54, there are two necessary coordinate transformations to convert from the laser local spherical coordinates of a test point which are defined by the deflection angles of the point, to the desired global Cartesian coordinates of the global coordinate system. First, each horizontal deflection angle, vertical deflection angle, and range measurement (which define the laser local spherical measurement) must be transformed into laser local Cartesian coordinates x, y, and z. Next, the laser local Cartesian coordinates must be convert¬ ed to global Cartesian coordinates.
The initial transformation is affected by two aspects of the velocimeter. Referring to Fig. 4, the origin 0 of the laser beam moves as a function of the beam deflection angle. Also, the transformation to global coordinates must account for the offset origin of the linear transducer C with respect to the beam origin O.
The laser local spherical to laser local
Cartesian coordinate transformation is a five step method. Referring to Fig. 5, point P is a test point in space with angles 0x and 0y being the corresponding deflection angles to aim the beam at point P. In figure 5, the range R is measured with a linear transducer, such as the Celesco Model TP101. The first step in the local transformation is to compute
C , • Equation (1) is an appropriate vector rela¬ tionship which is derived from the vector dot product of two vectors.
Vector v is defined in equation (2) as the vector extending from the origin 0 of the laser beam to the origin C of the linear transducer.
The coordinate equations for the moving origin of the laser beam are obtained from the scanning laser manufacturer, such as Ometron, and are specified in millimeters in equation (3) . That is, equation (3) is unique to the Ometron VPI sensor.
The approximate coordinates for the fixed origin of the linear transducer, mounted to the velocimeter, follow in equation (4) . These coordinates will vary depending upon where the linear transducer is mounted.
Equation (5) shows v. as the unit vector at point O in the direction of point P.
Equations (6) and (7) show the magnitudes of vectors v, and v. w- +vf, &Q ( .
v.=l Q. l
Finally, equation (8) shows the calculation for in terms of the components of vectors v and v..
Step 2 in the location transformation is to compute JO using the law of sines as shown in equation (9) .
The third step is the computation of the range R from point 0 to P. R results from the formulation of the lav of cosines in equation (10) .
*- - hf ÷Ry W R - cos(180 - a "^ £Q *
Step four is the conversion of the θ . θ , and R spherical coordinate values for point P to Cartesian coordinates relative to point 0, shown in equation (11) •
The final step is to translate the relative Cartesian coordinates for point P by the distance from the laser base mount point B of the linear transducer to the laser beam origin 0. This translation in equation (12) places point P in the laser coordinate system defined with the origin at point B (see Fig. 4).
Equation (13) depicts the next transforma¬ tion in homogeneous coordinates needed to convert a point from laser local Cartesian coordinates (L) to global Cartesian coordinates (G) .
The same transformation applies to all points in a single scan. The transformation is composed of the translation and rotation matrices of equation (14) , which can be combined into a single homogeneous 4x4 matrix or simply performed in the proper order (translation, then rotation) .
£c?. li-
-35-
The transformation matrix Q conceptually moves and rotates an orthogonal triad of unit vectors designat¬ ing one coordinate system into the corresponding axes of another coordinate system.
In using the laser velocimeter 52 to extract a geometrical representation of the test structure, the global coordinate system is specified by three points in physical proximity, but independent of the test structure. The points, (P., P 2 and P 3 ) define a global system. Using equation (12) the coordinates for these points are obtained.
The three rows of the upper portion of the 3x3 rotation matrix of equation (14) correspond to unit vectors along the x, y, and z axes of the global coordinate system. Row vector r in equation (15) is defined as the unit vector along P-,P 2 ttlat denotes the positive z axis of the global coordinate system.
i P.P.
' -{ ■ r.. Eq>. 15
Row vector r in equation (16) is the unit vector perpendicular to the plane of P P_P_ that denotes the positive x axis.
→ » r - "- 1 -S e<? - lfa
The cross product of r and r yields the unit vector r in equation (17) that denotes the positive y axis.
^[l, r Z7 rj = r : xr r Q (7
The point t = [t χ t , t χ ] denotes the translation between the laser local origin and a defined global origin. Using equation (14) and the three global reference points P χ , P 2 , and P 3 to define a global coordinate system, each point accessible by the laser beam of velocimeter 52 can then be repre¬ sented in global coordinates. The global coordinates are used to create a geometry file.
II. Selection of Four Reference Points When a sufficient global geometry represen¬ tation of the test structure has been created and input into a geometry file, the system user selects four global reference points from the structure by which to compute the scan list of laser beam de¬ flection angles for each global coordinate test point. Such inputs to the system might be prompted from the user by an input menu, such as a menu created using the X11R3 version of the X-Windows System developed by the Massachusetts Institute of Technology, Cambridge, Massachusetts and Digital Equipment Corp. , Maynard, Massachusetts, and utilizing Version 1.0 of Motif Widgets and the Motif Version of X. Intrinsics from the Open Software Foundation, Cambridge, Massachusetts. The VPI velocimeter from Ometron has an external dedicated keyboard input to manually move the beam; however, the beam can also be positioned by
supplying the appropriate voltage signals to the horizontal and vertical inputs, on lines 66 and 64 of Fig. 2, respectively. Therefore, the beam movement to locate each of the reference points might be facil¬ itated by using a mouse or keyboard input (not shown) to computer 60 to command analyzer 62 to supply the proper voltages on lines 64 and 66 to move the beam until the user sees that it rests on the desired reference point. When the beam reaches the appropri¬ ate reference point, aiming voltages corresponding to the deflection angles for that reference point are recorded and stored in a file in computer 60. III. Determine Transformation Equation
The next step in the measurement sequence of the current invention is to determine a transformation equation which transforms the global coordinates of a point into its corresponding deflection angles, which are utilized to aim the laser beam at that point. The list of deflection angles or "scan" list contains pairs of analog voltage levels, each of which corre¬ sponds to either the vertical or horizontal beam deflection angle necessary to reach a particular test point.
As explained above, it is necessary to know the position of the velocimeter in the global coordi¬ nate system in order to correctly aim the laser at each test point. This position might be determined through a series of linear measurements; however, the
inaccuracy of such measurements reduces the accuracy of the aimed beam and consequently, the accuracy of the data. Therefore, in accordance with another aspect of this invention, the measurement system determines the position of the laser in the global coordinate system by utilizing a unique positioning method. Once .the position of the laser is determined, a rotation equation is found which converts each test point from the global coordinate system into de¬ flection angle pairs which make up the scan list and are used to the laser beam. In computing the scan list, once the position of the velocimeter is known in the global coordinate system, a laser local coordinate system is defined which has the origin of the laser beam, as placed on one of the scanning mirrors, as its coordinate origin. From this laser local coordinate system, a scan list of appropriate deflection angles is computed so that the beam can be aimed at each test point.
The first step in the positioning method is to choose reference points and record their corre¬ sponding deflection angles, as illustrated in step 102 of Fig. 3. Next, the system determines the range distance from each reference point to the laser velocimeter using the global coordinates and de¬ flection angles of the reference points, 104 of Fig. 3. Determining the range distance to each reference point requires an iterative method as described below.
Knowing the existence of a point in one coordinate system (the global coordinate system) and determining the position of that point in another coordinate system (the series of scan angles) involves a coordinate transformation. In reference to Fig. 6, a point P, defined in the global coordinate system of XYZ, can be defined in the coordinate system of XYZ, origin at 0, by the following equation:
R„ =« 0 +* ; £ ( ?. 18
where R_ is the range vector between the origins of each coordinate system, R_ is the range vector to point P from the global coordinate origin o, and R p is the range vector to point P from the origin, 0, of the laser local coordinate system. The laser local coordinate system is defined as a three orthogonal axis coordinate system which has its origin at the origin of the laser beam (see Fig. 4). Projecting equation onto the three axes, yields:
where [a, b, c] is the laser local origin coordinate
(O's coordinate) position in the global coordinate system and where the matrix [RT] is the rotation matrix to rotate one coordinate system into the other.
I. x I. •
[RT] is the rotation matrix. [RT\ = ffl- ' TTl- /7I- - &Q. n. n.
From equation 19, if the global coordinates of point P are known, along with the laser local origin's coordi¬ nates and the rotation matrix of the laser local coordinate system, the laser local Cartesian coordi¬ nates for that point can be found.
Since equation 19 yields the laser local Cartesian coordinates of a test point and since the input to the velocimeter must be in the form of deflection angles in the horizontal and vertical direction, the laser local Cartesian coordinates (x, y, z) must be transformed into laser local scanning coordinates for that point. This can be done by equation 21:
&?. 2.1
Conversely, knowing the horizontal and vertical deflection angles and the range to a particular point from the laser local origin, the laser local Cartesian coordinates can be determined by equation 22:
£Q 7J-
where :
φ-φ'+θ, a ' dt Φ'= « ' Λ — sinQ j £Q. l dt - separation distance
J
Separation distance d is defined as the distance between the internal horizontal and vertical scan mirrors of the particular scanning velocimeter. For the Ometron VPI device, this distance is 46mm.
Again, examining equation 19, it is appreci¬ ated that the origin of the velocimeter in the global coordinate system [a, b, c] and the rotation matrix [RT] must be determined in order to compute a scan list to control the velocimeter beam. However, there is no way of guessing the rotation matrix and the velocimeter laser beam origin.
The determination of the rotation matrix and the laser head position can be accomplished through the following methodology. First, three reference points, whose global coordinates are known, are chosen on the structure and the laser beam is aimed at these
points. Second, the scan angles (or rather, appropri¬ ate input voltages) needed to reach each of these points are recorded. Next, the range to each point from the laser beam origin of the velocimeter is determined. Finally, utilizing the global coordinates for each point and the laser local coordinates for each point as determined from the scan angles, the rotation matrix and the laser local origin coordinates for equation 2 are found.
The following method initially utilizes three reference point coordinates to derive the necessary equations. However, as will be illustrated below, a fourth reference point is necessary to find the true velocimeter ranges, and therefore, step 102 of Fig. 3 requires four points and their respective deflection angles. Returning now to equation 19, having three reference points which are designated as A, B, and C, yields three sets of equations:
£ ?. 74
There are fifteen unknowns in equation 24. Three describe the laser local origin coordinates, nine describe the rotation matrix (see equation 19) , and three describe the unknown ranges R &/ R fa , R c (recall from equations 21, 22 and 23 that knowing the range to a particular point and the horizontal and vertical deflection angles needed to reach that point, the laser local Cartesian coordinates for that point) can be derived. The rotation matrix of equation 19 is an orthogonal matrix and, therefore, there are six orthogonality/unitary vector constraints among its nine elements. These are given by equation 25:
ll +mi ÷ n: =1 ii÷ ÷πi = I l +m; +n =1 q. Z-5
^.. +m-m- +Λ.Λ- =0 l-l τ ÷ m-m- -t-Λ./Ϊ I =0
With equations 24 and 25, there is a total of fifteen equations with fifteen unknowns. The fifteen equ¬ ations are nonlinear, and therefore, are extremely difficult to solve simultaneously or by other numer¬ ical methods. Instead, a geometrical method is utilized. First, the three ranges from the laser head to the reference points are determined. Second, the rotation matrix is determined. Finally, the laser local origin coordinate is found. The following table, Table 1, defines the symbol conventions:
TABLE 1. Various Coordinate System
A. Determining the Ranges to the Velocimeter Positions
Referring to Fig. 7, three arbitrary refer¬ ence points A, B and C are defined. From the global coordinates of these reference points, the distances between A and B, between B and C, and between C and A, marked as D., D_, and D_, respectively, are known from the structural geometry file. Angles .AOB, . BOC f and -£-- COA are designated as ., φ , and φ _. Applying the cosine law to triangles AOB, BOC, and COA, yields equation 26:
Knowing ψ . , φ _, and , and the distances D.,
D, -., D- -t, it is possible to solve for R.o, and Rc_ to obtain the desired ranges. Equation 26 is a simulta¬ neous quadratic equation having eight solutions. Since the ranges are always going to be real and positive, a search method is used to find all real and positive solutions to equation 26. From the third equation of equation 26, solving for R ^ as:
R JU = R.OWΦ, ± j(R c cosy, y -iR^ - D ) &Q.27
For the second equation of equation 26, solving for R.
R bll = R : cos9 z x j(R c cosv, -<R - Di ) &Q . --?>
Substituting R and R, into the first equation of equation 26 yields an equation having only one un¬ known, R . The root of f(R_) = 0 gives us the range R , and substituting range R into equations 27 and 28 gives the ranges R and R..
As stated above, the true range to a refer¬ ence point is real and positive. Therefore, it is desired to ensure the solutions for R a , R^, and R to be real and positive.
To ensure that the range solution for R a, is real and positive, the following inequalities
(R t cosO t f-fRΪ-DD≥Q £Q. Z°\
R c cost*. - ^(R c cos . ) 2 -(R -D*)≥Q &Q. 3^
yield:
R C ≤D. / ino,
R C ≥D, &?. l
To ensure that the range solution for R, is real and positive, the following inequalities
(R c cos9,) l -(R -DD≥Q £Q $2.
R,cosQ z --j(R e cos 1 Y -{R t l -Dl) 0 ^? 33 yield:
-47-
rι ** •
To ensure that both ranges R and R are real and positive, R %___» must satisfy:
Max{D,D, )<R : < Min(D, IΛΛΦ, ,D, I sin®, ) &Q.35
This is the search root interval for f(R c ) =o. Rewriting the root search interval yields:
B≤R.≤E &q. -bio where:
B m Max(D 2 .D,) Q. v,η
E = Min{D z I sin®. ,D, I sinv. )
Referring to equations 27 and 28, it is appreciated that there are four cases for f(R )=o, which are the following: plus-plus case:
R b = R : cos Φ, + -ItR c CosQif-'RS-D;) ζ ^ Q ?f f(R c ) = R* + R -2R Λ R > cost), - D. 1
plus-minus case:
R 3 = R. cos®, +/(R ff cσ.sΦ,) i -fR f i -D j i J
R b f(R c ) = R* + Rt -2R.R t cos*, - D, α
minus-plus case:
R. = R. cos®, -^(R c cos®,r- ( R 1 -Di )
R b = R. cos® z + • l(R e cos i )1 - ~ <Rc 1 -D Q.40 f(R c ) = R + R - 2R,R h cos 9, - D*
minus-minus case:
R, = R.cos®, - (R,cos®,)' -<R.~ -Dϊ)
R b = R. cos®- - (R. cos® z r -(R, 1 -Dt) &Q. 1 f(R. ) = R;÷ Rϊ - 2RR. cos®, - D?
Because f(R ) is also a quadratic function, it at most has two real and positive solutions. Therefore, the total number of solutions are at most eight.
Utilizing equations 35-41, f(R )=0 is solved numerically. First, the interval of (B, E) is evenly divided into small sub-intervals. Next, the value of f(Rc) at each of the sub-interval boundaries is found.
Then, the sub-interval values are multiplied to find a sign change in f(R ) . During the search, the minimum absolute value of f(Rc„) (called Fmm. is stored. If there are sign changes, a bisection method or equiva¬ lent root finding routine is used to find the root. If there is no sign change, there may be multiple roots. In the multiple root case, [Fmm. I is compared
with [f(R )] at boundarv B and E. If [F . j is less than rhe absolute value of both f(B) and f(E), there exists an minimum value for f(R c ) . For this condi¬ tion, a one-dimensional optimization method is utilized to find the point at which [f(R c )] is mini¬ mum. Multiple roots can also exist within search boundary [B, E, ] . In the case, where the multiple roots occur within the search boundary, [F m ^ n J will occur at B or E. Mathematically, for the multiple root condition, the minimum absolute value of f( c ) or
[Fm. ] is zero. However, since zero is difficult to reach for real problems using experimental data, an accuracy condition is utilized which, when satisfied, is accepted as a root for f(R c )=0. The software implementing the bisection method is included in appendix A in the SEARCH subroutine accessed by the TMATRIX subroutine.
Having outlined a method for solving equation 27, the angles , - , and^ 3 must be found. These angles are a function of the laser local scan¬ ning coordinates. For example, for point A, if R is known, the laser local scanning coordinate can be transformed into the laser local Cartesian coordinates by equation 21. The direction cosine for vector OA is defined by:
_ = J R. q. Z-
I . = * R.
Determining the same direction cosines for vectors OB and OC, the angles φ χ , φ 2 , and φ 3 are obtained by the following equation:
cost φ, ) = l ZA l B T m λ m a + n 7jA n UB cost®. ) = £- . c - m 7a m. ÷ n-^ <?. cost ®, )= i. . Λ ÷ m- c - λ ÷Λ_ C Λ- A
Referring back to equations 35-41, it is seen that the ranges R . R. , and R are given in terms of φ , φ , and φ . However, equations 42 and 43 define the angles ψ , O 2 , and ψ 3 as functions of the ranges R , R. , and R . This problem cannot be solved in one step. The ranges must be solved using an iterative process. To begin the iterative process, an initial set of values for R &/ R^, and R c are assumed. Then, the local Cartesian coordinates for these respective ranges are determined from equation 21. Next, these initial values are substituted into equations 42 and 43 to yield a set of angles, ψ ^, φ , and φ _.. These angles are then substituted into equations 35-41 which are solved to produce an updated range set R , R, , R . Continuing this
substitution process yields a converged solution for the ranges.
To accomplish the iterative process, an initial set of range values, R , R fa , and R , must be assumed. This initial guess is accomplished by setting d^.=0, i.e., assume the distance between velocimeter mirrors is zero which allows calculation of the direction cosines in equation 42 without specific range information for each reference point. For example, for point A, equation 22 yields vector OA'ε laser local Cartesian coordinates as:
x Λ = R, sinQ 7λ y A = R j C osd 7A sinQ^ &? ■ ^ z λ = R_, cosθ- λ cosθ^
Then , from equation 19 , the direction cosines are:
l jA = x A / R j = ιnQ 7A m 0A = - 7 Λ R * = ∞sd 7A sind^ ^ 45
n oA - ~- \ ! R j- = COS Q -A cos -A
Therefore, the range information is not required to obtain the direction cosines when the separation distance, d , is set to zero. With this initial guess for the ranges, equation 42 and 43 are utilized to obtain φ , φ , and φ _. These angles are, in turn, substituted into equations 35-41 to obtain a second set of ranges R , R. , and R_ from the initial guess.
From simulated and experimental testing, it was determined that generally the convergence of the ranges will occur after about five iterations.
Since equations 35-41 are quadratic equa¬ tions, they yield several solutions which are real and positive, one of these solutions is the correct range solution; however, with only three test points, it is impossible for a computer to determine which one solution is the correct range solution. It was found that through the use of a fourth point D, equations 35-41 can be solved for four three-point groups. For example, group 1 can be designed as as points A, B, and C; group 2 as points A, B, and D; group 3 as points A, C, and D; and group 4 as points B, C, and D. In this way, the true range can be determined because the true range will be common to the solution sets for each three-point group. This is because the true range is the same no matter which three-point group is used. For example, the true R will be the same for groups 1, 2 and . Based on these facts, the true range can be chosen and used for the next iteration.
Since there will always be errors in the experimental data for the coordinate positions of the reference points, each three-point set will utilize a slightly different version of the true range solution. The ranges can be averaged using equation 24:
-53-
Where N + / referto the updatedrange by using the range from N th iterations
As illustrated below, both unaveraged and averaged range solutions are used to determine the best trans¬ formation matrix for compute the scan list.
B. Determination of the Rotation Matrix Having found the ranges for all four refer¬ ence points A, B, C, and D, and knowing the horizontal and vertical deflection angles to reach these points, the global Cartesian coordinates and their correspond¬ ing laser local Cartesian coordinates are now known for each of these points using equations 21-23. From this information, a rotation matrix can be found. Referring again to equation 19 , it is seen that transforming a point from one coordinate system to another coordinate system requires translatinq the origin and then rotating the three orthogonal axes. The axis rotation, as symbolized by rotation matrix [RT] , is defined as rotation of the coordinate axes through an angle Θ around a predefined axis k. Therefore, when determining the rotation matrix [RT] , axis k is first determined and then angle is derived.
Re erring to Fig. 7, and equation 24, and eliminating the laser local origin [a, b, c] for the present (it will be found later in the method, as shown below) , yields:
The geometric meaning of equation 47 is that the coordinate system OXYZ and OXYZ are being merged together with their common origin being point A. Their relative angular orientation does not change. Equation 47 can be written as:
where fRT] is the rotation matrix and,
The vectors [d ] and [d χ ] are the same vectors in space but expressed in different coordinate systems, as are [d_] and [d_].
The coordinate system OXYZ now coincides with coordinate system OXYZ. Next, the coordinate system OXYZ must be rotated by an angle © about axis k, which is defined by the direction cosines 1^, m. , and n.. Therefore, the rotation matrix in this case is:
I- -r l k t k vQ + cd i t m t υθ- n t 5θ t. t n k vQ + m t S Q
[RT} = m 7 m- m- l.m.vQ + n.SB m k m k vβ + C Q 4 n t υθ- / t 5θ n n- n τ l k n k vβ - m t S m t n k vQ+ έ t SQ n k n k v Q + C Q J
where CQ = cosQ SQ = sinQ vβ = \ - cosQ ~nά k = {l t .m l ,n k )
The vectors [d ] , [dl] and [ 2 ], [d " 2 ] are used to determine axis, k, and angle, θ , and to ultimately obtain the rotation matrix.
In reference to Fig. 3, by rotation of a two-dimensional coordinate system, a vector P has a representation (x, y) in one coordinate system and a representation (x, 7) in the rotated coordinate system, as depicted by angle θ . In Fig. 9, it is seen that if P is rotated at an angle θ to give vector P' ; P' will have the same representation in the (x, y) coordinate system as vector P has in the (x, y) coordinate system. Therefore, numerically, ' equals x and y' equals y.
Now, applying the same approach for a three-dimensional problem, to obtain the axis of rotation, k, and the rotation angle, Θ , information is needed from two vectors. From the discussion above, it is known that vectors d. and d_ in frame OXYZ become ά. ± and " d_ 2 in frame OXYZ. It is further known from the two-dimensional frame rotations, for example in Figs. 8 and 9, that if d. and d- are rotated an equal angle Θ about the axis k to yield d ' and d_ ' , then the expression of d ' and d ' in OXYZ will be exactly the same as vectors d. and dL in frame OXYZ. This yields:
-=J ' e ~ . 5i d = d-
Therefore, the following conditions about these vectors can be defined:
1) the angle between d. and k and the angle between d ' and k are the same;
2) the angle between d_ and k and the angle between d ' and k are the same.
Defining the direction cosines of these vectors as:
a. t b x .c f τ d. 1 ay,bZ,cZ ϊoτd J
Condition 1) and condition 2) , yields:
-57-
a.t k ÷b.m t +c.n t -a ' k +b m k +c.'n k ,t k ÷ b.m k + cn k = a 't k +b z 'm k +c z 'n k &*. 53
Equation 53 can be rewritten as:
where :
A = ,
A. = α.
Making direction axis k a unit vector, yields:
Equations 54 and 55 can then be solved to determine vector k.
Solving equations 54 and 55 requires ex¬ pressing two of the three direction cosines for axis k in terms of a third. In order to eliminate errors associated with solving for axis k from equations 54 and 55, two of the three direction cosines should be expressed in terms of the third direction cosine whose absolute value is maximum, that is, where translation of a particular vector onto one axis is greater than
-58-
the projections onto the other two axes. For example, if n, is the maximum component, equation 54 yields:
m k = kn k
£Q 5b
_-C,B +C z B, _GA,-C.A.
A,B Z - Λ.B, A,3 j - .B.
and substitution into equation 56 and into equation 55 yields:
n t =l/- l + k; +k: £Q ^7
So the solution for axis k is:
Similarly, an m. that is maximum, yields:
m k =\ll + r+k: n =icm t
£<?.5 < ]
Finally, maximum 1,. yields:
t k - i / jϊ+kF kJ m k = k.t k n k = k £ , ^°
-AC ÷ AC, k, = B, C j — B j
To make an initial determination as to which direction cosine is maximum, the coefficients of equation 54 can be examined. Referring now to equa¬ tions 58, 59 and 60, if a particular direction cosine is maximum, then k. and k_ will be minimum. For k. and to be minimum, their denominators must be maximum. The denominators of equations 58, 59 and 60 are denoted by the following determinants:
Therefore, if PZ is maximum, n. will be that maximum. If PY is maximum, m. will be the maximum. Finally, if PX is maximum, 1. will be the maximum. If PZ, PY and PZ are zero, frame OXYZ makes no rotation and the rotation matrix is a unit matrix.
After axis k has been determined, the rotation angle Q can be found, using either a d χ , d ' d„ , or d ' Referring to Fig. 10, if d and d ' l m. 2 are use d , the angle Q between axis k and d χ is ;
cosi®) = έ t a x + m k b + n k c,
and the angle Q ' between d χ and d 1 / is
cos(Q') = a, . + b.b, + c,c, £^>. [?3 >
Referring now to Fig. 11, if χ is the projec t ion of d in a plane perpendicular to axis k, and d ' is the projection of d χ ' onto the same plane, the angle b etween d and d^' is the desired rotation angle Θ . The law of cosine yields the distance between the tips of d- and d^' :
S imilariy, the distance between the tips of d^ and d χ ' can be found.
d~ = | 2 sin 1 ® sin z ® - Q . ϋ->
These two distances are the same because the projec¬ tion onto a plane perpendicular to axis k will not change the distance between the tips of the two vectors. C ombining equations 64 and 65, yields:
Q Uo
Then, the rotated angle is found to be:
cos(θ)= i-{1- cosQ') I sin 1 φ
£Q bl sin(Q) = τ-Jl - cos z β Determining which pairs of vectors (d χ d^ or d 2 d' 2 ) are to be used to find Θ , is based on their angular displacement from axis k. The vector pair with the greatest angle from axis k should be chosen to reduce the error associated with the computation.
Referring to equation 67, it is seen that a sign must be chosen for sin ( Θ ) . Also, the sense of axis k may not be correct. Utilizing both the k axis and θ , the correct rotation matrix can be determined by utilizing both a positive and negative sign for equation 67 and calculating two rotation matrices, RT+ and RT-. Utilizing these rotation matrices and equation 68:
the correct answer can be determined. If both mat¬ rices, [RT+] and [RT-] satisfy equation 68, the same condition is checked for d_.
{d z } = [RT]{d z } ?. tø
If both matrices then satisfy equations 68 and 69, then [RT+]=[RT-]. This means that the angle of rotation, Q , is 0" or 180".
c. Determination of the Origin Coordinate For the Laser Local Origin
After determining the rotation matrix, it is the origin coordinate for the laser local system is determined by using any of the reference points A, B,
C, or D. When point A is used, for example, solving for the origin coordinate from equation 2 yields:
D. Generation of the Scanning List Knowing the rotation matrix and the origin coordinate, equation 19 can be written as
Using equation 71, all of the points to be scanned are mapped from the global coordinates to the laser local Cartesian coordinate system. Then, using equation 21, the laser local coordinates can be transformed into laser local scanning coordinates and the scan list is created. In this transformation, it is not necessary to know the range, because only the horizontal and vertical scanning angles are needed to aim the laser beam at a test point.
E. Optimization in Finding the Rotation Matrix and Laser Local Origin Coordinate
As shown above, four reference points are used to obtain a true range from the laser head to a reference point. Equation 24 produces four unaveraged range solutions and a fifth averaged solution. However, to obtain the rotation matrix and the laser local origin coordinate, only three points need to be utilized. It is difficult to determine which set of three points yields the best rotation matrix and origin coordinate. However, it can be appreciated that the true rotation matrix and origin coordinate will generate scanning coordinates for the four reference points which are the same as what were measured when the laser was physically aimed at each point. However, because of the errors associated with the numerical calculations, the scan angles calculated by the rotation matrices determined from different three-point groups will always differ.
In order to determine the "best" group of three points to be used to calculate the "best" rotation matrix and coordinate origin, eight rotation matrices and eight origin coordinates are found for the following cases: for average ranges 1) point A, B and C; 2) point A, B and D; 3) A, C and D; and 4) B, C and D; for unaveraged ranges: 1) point A, B and C; 2) point A, B and D; 3) A, C and D; 4) B, C and D. Each of these transformation matrices and origin
coordinates will yield scanning coordinates for a particular reference point. The rotation matrix and origin which gives the minimum difference between the measured scan coordinates (measured in step 102 of Fig. 3) and the calculated scan coordinates is chosen as the "true" rotation matrix and laser local coordi¬ nate origin. The error difference is determined by:
where: ZQ. -
The "true" rotation matrix and origin coordinate from this error equation is utilized to generate the scanning list. A Fortran language software program for determining the transformation equation using the above method is attached in software appendix A as subroutine TMATRIX.
IV. Selection of Geometry Points for a Scan List When the transformation equation has been determined for transforming coordinates in the global coordinate system into scanning coordinates in the laser local coordinate system, the measurement system of the present invention is able to create a scan list of deflection angles from the geometry of the
structure. Each set of deflection angles is repre¬ sented as a set of output voltages that is necessary to aim the laser beam at the test point.
In the current embodiment of the measurement system, the scan list is not created using every test point in the geometry file. Rather, the system prompts the user with a menu allowing the user to choose the points of interest (109 of Fig 3). For example, a particular group within the geometry file of the test structure can be chosen by the user, and the scan list is computed for only this subset of the geometry file.
V. Create the Scan List
Referring again to Fig. 3, when the points of interest are chosen, a scan list is created (110 of Fig. 3.). The scan list is created utilizing the test structure geometry file and the transformation matrix formed in steps 102-108 of the measurement sequence of Fig. 3. As discussed above, the scan list comprises sets of voltage command signals which correspond to the beam deflection angles needed to reach each chosen test point. Once an appropriate scan list has been generated, the system begins data acquisition.
VI. Data Acquisition
In reference to Fig. 2, the laser beam 68 is aimed at each of the test points in the scan list and the vibrational velocity is measured at the point. Computer 60 accesses the scan list for each test point
and commands DAC's 70 and 72 to output voltage signals corresponding to the horizontal and vertical angles associated with the point. For example, for the Zonic 7000 analyzer, this command might be accomplished using the ZETA command language of the Zonic system, manuals explaining the ZETA language being commercial¬ ly available from Zonic, which commands the ASCO to output a particular analog voltage.
Before data acquisition begins, however, the appropriate equipment should be calibrated. Since the scanning accuracy of the velocimeter is important to the measurement system of the present invention, it is necessary that the beam control voltages output from the analyzer or from an external DAC are exact. Due to equipment inaccuracies, a DAC output level, or an ASCO level in the case of a Zonic 7000 analyzer, will not always correspond to the voltage level that is commanded by the input command to the DAC or ASCO. For example, the ASCO may be commanded in a ZETA command to output a 5 volt analog signal and will respond by outputting 5.1 volts. This causes scan inaccuracies. Consequently, the DAC's or ASCO's should be calibrated so that command input will produce the desired signal level output. One way of achieving this calibration is to output the highest and lowest voltage levels possible from the DAC's or ASCO's corresponding to specific upper and lower inputs and assume a linear regression of the input to
the output. In this way, it can be determined linear¬ ly exactly what input will give a desired output from the DAC or ASCO. Referring back to Fig. 2, a digital voltmeter 77 is attached between the velocimeter inputs 64 and 66 and computer 60 to accomplish this calibration task by measuring the actual voltage output by the DAC's with respect to the particular output voltage commanded by the computer.
Another source of scan error is the voltage- -to-degree ratio of the scanning velocimeter. This ratio will vary from velocimeter to velocimeter, and may possibly be different than the manufacturer's specification. A method for calibrating the velocimeter to determine the true voltage-to-degree ratio is to change the input signal in incremental voltage steps and determine the angular degrees deflected. Then from a slope-intercept linear equation, the voltage-to-degree relationship can be determined. For example, from this calibration technique, it was determined that for a particular Ometron VPI velocimeter, the horizontal deflection voltage-to-degree calibration curve was substantially linear and was defined by the equation Y = 2.398 * X + 9.7E - 3 were X in the input voltage and Y is the corresponding horizontal deflection angle. Similarly, for the same scanning device, the equation for the vertical deflection angle was determined as Y = 2.519 * X - 16.IE - 3. All of the angular and voltage
calibrations should be accomplished before the beam begins to aim and acquire velocity data.
Another parameter which must be taken into account with respect to the scanning velocimeter is the amount of hysteresis that may be inherent in the system as it moves from right to left and from top to bottom in its scan. It was empirically determined that if the user places the scanning velocimeter close to the structure so that the full angular scanning ranges of the beam are utilized in measuring the entire test structure, then the effects of the hysteresis on the measured velocities is minimized.
Finally, the scanning velocimeter should be sufficiently warmed up and properly ventilated so that there is no thermal drift of the deflection angles taking place during data acquisition.
When the output control channels, the DAC's and the scanning laser apparatus have been calibrated, the system is ready for proper data acquisition. The measurement of the present invention creates an output file in a format such as that of SDRC I-DEAS, Version VI, Test Data Analysis Software, to receive the measured velocity data. Referring to Fig. 2, a power signal is sent on line 90 to a vibration shaker 86 to excite test structure 54 to induce vibration. As the velocimeter 52 moves from test point to test point, it extracts the line-of-sight, time-domain vibrational velocity at that point.
VII. Transform Data to Outward Normal Representation
As stated previously, the velocity measure¬ ment taken by the velocimeter is along the line-of- sight of the incident laser beam; however, surface normal velocities are desired. Therefore, the present system corrects the velocity data to give velocity data that is along the axis that is outward normal to the test surface. Assuming that the "home" position (i.e., 0* horizontal and 0 * vertical deflections) of the laser beam defines the surface normal direction, the line-of-sight data can be converted into surface normal orthogonal velocities utilizing the following equation:
V'
V. = LOS
COSθr COSθ.- &?. 73
where,
V z = true outward noππai velocity
V a- = relative line-of-sight velocity (no laser-head motion) θ ,Q- = scan coordinates
Equation 73 is utilized in the present system for test structures having vibrational veloc¬ ities predominantly in one direction, i.e., outward normal to the test structure surface. The measured velocity is output as an analog signal to ADC 80 on line 73, where it is digitized for use by analyzer 62. The force applied to the test structure 54 is sensed by force transducer 94 and is also input to analyzer 62 through ADC 84. Analyzer 62 utilizes the excita¬ tion force input signal to find the Frequency Response Function (FRF) at a point on the vibrating structure, defined as the ratio of the output measured velocity to the input excitation force. Analyzer 62, such as the Zonic 7000 Fast Fourier Transform (FFT) analyzer, transforms the time-domain velocity and force measure¬ ments into the frequency domain so that the data can be further processed. The frequency domain data output from analyzer 62 is then corrected for the line-of-sight discrepancy and stored in an appropriate output file or memory location in computer 60 to be further processed or displayed.
In accordance with another feature of the present invention, the data acquisition time is reduced as compared to the time associated with conventional contact transducer system measurements, and the spatial testing density of points is in¬ creased. The effective measurement width of a scan¬ ning laser beam is small in comparison to the size of conventional contact transducers. For example, the Ometron VPI velocimeter can have an effective measure¬ ment beam width as small as 1 mm versus the 6.25 mm of a typical accelerometer. Using the Ometron device, this represents approximately a 6.2:1 increase in the spatial resolution of measurement test points for a laser measurement versus an accelerometer measurement. Under test, the improvement is even more significant, as typical accelerometer measurements are made in spacings of 125 mm or greater to minimize the number of measurement points and the data acquisition time. The apparatus of the present invention, allows moving the beam point-to-point, such that test points approx¬ imately 1.5 mm apart may be measured giving an in¬ creased measurement spatial density of approximately 83:1.
By virtue of the present invention data acquisition time is reduced a considerably even though the spatial density of test points is increased. The test point travel time for a laser velocimeter (ap¬ proximately 60 msec/point for the ometron VPI sensor
is driven by the Zonic System 7000) and the scanning capability allows measurements of velocities at many points rapidly without the need for multiple sensors, multiple output lines and multiple signal conditioning units. Furthermore, because the data is measured serially, the reliability of the velocimeter measure¬ ment system is increased.
VIII. Process and Display Data
Finally, when all the velocity data has been measured, the present invention graphically displays or otherwise processes the data as required by the application for which the measurement system is being utilized. For example, utilizing I-DEAS Test Data Analysis software, wire frame geometries of the non-vibrating test structure can be created utilizing the original geometrical representation file. Then, the powerful geometry display and animation routines of I-DEAS Test Data Analysis can be implemented using the same software to show the dynamic response of the test structure, other graphical representations, such as graphs of the test structure's frequency point correlated response functions, are possible with commercially available software packages such as SDRC I-DEAS. Additionally, the Zonic System 7000 contains software titled Zeta Graphics, commercially available with the Zonic 7000 analyzer system, which can be utilized to graphically display the measured test
-73 -
results and corresponding frequency response functions.
Multiple Position Laser Velocimeter System- Three Dimensional Structural Imaging
As mentioned previously, the single position velocimeter embodiment of the present invention does not fully measure three dimensional structures having more than one predominant vibrational velocity along their surfaces. Referring to Fig. 12, showing another embodiment of the present invention, multiple veloci¬ meters 150, 152, and 154, are placed around a test structure 156 to obtain three dimensional velocities, along each of the three orthogonal axes in the global coordinate system. Alternatively, the same result can be achieved by moving a single velocimeter to multiple different positions and having it extract data at each of the positions. The multiple (minimum of three) positions of the velocimeters around the test struc¬ ture ensure that all three translational degrees of freedom, such as the three translational velocities, can be adequately measured.
The multiple velocimeter embodiment system of the present invention operates in essentially the same way as the single velocimeter embodiment in its scanning and data acquisition sequences, following the test sequence shown in Fig. 3. Each velocimeter laser beam is aimed to irradiate a chosen test point on the
structure and the extracted velocity data is trans¬ formed, corrected, analyzed and displayed. However, with multiple velocimeter positions, additional parameters must be examined.
When multiple velocimeter positions are utilized, it is necessary that each of the velocimeter positions have a substantially different view of the test surface. When positioning multiple velocimeters, or determining multiple positions for a single veloci¬ meter, the position of one velocimeter relative the other positions becomes important. The data extracted in the multiple velocimeter embodiment of the current invention must be translated into a three axis coordi¬ nate system so that a three dimensional analysis of the structure can be achieved. Therefore in convert¬ ing the data to an orthogonal representation, equation 73, utilized for the single velocimeter position embodiment of the present invention, cannot be used. Instead, the equation for projecting the line-of-sight data measured by a three velocimeter position embodi¬ ment onto the orthogonal axes of a coordinate system is given in matrix form as:
-75-
where range vectors to the multiple velocimeter positions:
having magnitudes:
define each velocimeter position, A, B and C in the global coordinate system. As illustrated above, the position of each velocimeter is defined in the global coordinate system using the method which computes the scan list for the test points. Therefore, knowing the line-of-sight velocities to a particular test point, and the position of each velocimeter position in the global coordinate system, the three orthogonal trans¬ lational velocities at the test point can be found. Furthermore, with the measurement of three dimensional translational velocities in combination with the increased spatial testing densities of the laser measurement system of the present invention, the rotational velocities may be derived form the measured data. Referring to equation 74, to determine the orthogonal velocities, the transformation matrix of the equation must be inverted. Equation 74 is now written as:
This inversion only exists if none of the velocimeter positions are colinear, and if any one particular velocimeter position is not coplanar with respect to the two other velocimeter positions. This is the first criteria that must be met when determining multiple velocimeter positions (see Fig. 12) . An additional criteria involves defining the velocimeter positions so that no single position is such that the incident beam of the velocimeter is striking the vibrational test surface essentially tangential to that surface.
It will be appreciated that at any given test point, one velocimeter position may provide a more accurate measurement than a second position with respect to a particular orthogonal axis; while, with respect to one of the other orthogonal axes, the second position may provide a more accurate velocity measurement. By way of example, utilizing two veloci¬ meter positions, if one axis of the global coordinate system at a point, such as the z-axis, is essentially parallel to the incident beam of a velocimeter at position one, the z-axis velocity measurement made by this velocimeter will be highly accurate. However,
the velocities measured along the x and/or y axes by the velocimeter at position one will not be as accu¬ rate. Alternatively, if a velocimeter at position two is spaced so that its incident beam is essentially perpendicular to the z-axis at the test point, its velocity measurements along the z-axis will not be as accurate as that provided by the velocimeter of position one; however, its measurements along the x and/or y axes will be more accurate than those made by the velocimeter at position one. This is due to the transformation of the line-of-sight velocity onto the orthogonal axes of a coordinate system. Large angles between the axis of interest and the line-of-sight of the laser beam result in errors in the transformation of the data onto that axis. Therefore, when determin¬ ing multiple laser positions for the three-dimensional embodiment of the present invention, no one velocimeter should have an incident beam that falls essentially tangential to a test point on the vibrat¬ ing surface.
As seen by the transformation matrix equ¬ ation 74, errors in the determination of a particular velocimeter position translate into orthogonal veloci¬ ty transformation errors because the coordinates of the velocimeter position are used in transforming the data from line-of-sight data to orthogonal data. In addition to these positional errors, there is a certain amount of inherent error in the system is
measurement of velocity due to noise. Both positional errors and noise errors are affected by how each position in a multiple position velocimeter system is determined with respect to the test structure and each other velocimeter position.
For determining an optimal configuration for multiple velocimeter positions, each position should be examined with respect to a reference axis. In accordance with this aspect of the present invention, the reference axis might be defined as a central axis which is located essentially in the middle of the test structure and normal to the surface to be measured. Referring to Fig. 13, the angle 0 between a central axis 120 and the position vectors 122, 124 and 126 for each velocimeter position is defined as the separation angle. Therefore, it will be appreciated that, depending upon the test point to be irradiated, a large separation angle for a particular velocimeter position might be desirable for velocity measurements along one axis while a small separation angle might be preferable to measure the velocity along the other orthogonal axes. However, use of a reference axes, while yielding better results, is not an absolute necessity,
It has been analytically ascertained that, with respect to measuring velocities along all axes in the global coordinate system, each velocimeter posi¬ tion preferably should have a separation angle in a
range between 25 β and 75' from the defined reference axis. This range is optimal to reduce measurement errors resulting from system noise and laser position errors, as well as being optimal to measure velocities which are not subject to large angular transformation errors when the line-of-sight velocity data is pro¬ jected into an orthogonal coordinate system.
Although the exact positions of the veloci¬ meters are arbitrary once the separation angle crite¬ ria is accounted for in accordance with one aspect of the present invention, a configuration that may be used is shown in Fig. 13, where the velocimeters 130, 132, 134 are equally spaced around a circle 136 and the central axis 120 extends normal to circle 136 through its center. A more practical configuration might be that shown in Fig. 14 where two velocimeters 140 and 142 are positioned equally spaced apart from each other and a third velocimeter 144 is equidistant between velocimeters 140 and 142 and vertically above them. In this configuration, the central axis lies along line 146. Whatever the laser velocimeter placement in the multiple position embodiment of the present invention, their position in space relative to the global coordinate system must be accurately determined such as by the positioning of the present system or a similar method.
By virtue of the present invention struc¬ tural testing and modal analysis using a scanning
laser velocimeter to measure the velocity at particu¬ lar test points on a vibrating test structure. The measured vibrational data is thus correlated to defined points on a geometrical representation of the test structure.
The measurement system operates without contacting the test structure, and therefore the measurements do not have any of the errors associated with mass loading and damping that occur with conven¬ tional contact transducer-based vibrational measure¬ ment systems. The small effective measuring size of laser beams greatly increases the spatial density of the measurement points, while the reduced data acqui¬ sition measurement time of the velocimeter and its scanning capabilities reduces the total acquisition time for testing a structure. The correlation and the increased spatial density allowed by the present invention makes the results comparable to computation¬ al/theoretical modeling techniques such as finite element analysis.
While the present invention has been illus¬ trated by description of a preferred embodiment and while the preferred embodiment has been described in detail, it is not the intention of the applicants to restrict or any way limit the scope of the appended claims to such detail. Additional advantages and modifications will readily appear to those skilled in the art. For example. Appendix A contains software
code for programs written in the Fortran and C lan¬ guages to implement the present invention using an HP 9000 Series engineering workstation and a Zonic 7000 Analyzer. However, the present invention is not limited to the use of this commercially available hardware. In another example of a possible modifica¬ tion, position configurations for the multi- velocimeter system differing from those shown in Figs. 10 and 11 may be utilized. Furthermore, a scanning laser device which measures translations other than velocity, such as a laser device which measures displacement, may be utilized without departing from the scope of the present invention. Additionally, the apparati and methods for extracting a geometrical representation of a test structure disclosed herein are not limiting upon the present invention. Accord¬ ingly, departures may be made from the preferred embodiment of the invention without departing from the spirit or scope of applicant's general inventive concept.
Having described the invention, what is claimed is:
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