Robust algorithms for experimental modal analysis

FIELD OF THE INVENTION

The present invention relates to experimental modal vibration analysis. More specific, the invention relates to a method for testing an object with respect to its vibration modes, wherein a number of subsequent vibration excitations are provided at a selected excitation location on the object, and the response from the excitation is measured by a number of vibration sensitive sensors in order to obtain dataset from each sensor quantitative for the vibration of the object at the location of the sensors.

BACKGROUND OF THE INVENTION

Modal identification is the process of estimating modal parameters from vibration measurements obtained from different locations of a structure. The modal parameters of a structure include the mode shapes, natural (or resonance) frequencies and the damping properties of each mode that influence the response of the structure in a frequency range of interest. Another — yet not so important modal parameter - is the modal mass used to secure the right scaling between loading and responses.

Modal parameters are important because they describe the inherent dynamic properties of the structure. Since these dynamics properties are directly related to the mass and the stiffness, experimentally obtained modal parameters provide information about these two physical properties of a structure. The modal parameters constitute a unique information that can be used for model validation, model updating, quality control and health monitoring.

In traditional modal analysis, the modal parameters are found by fitting a model to the Frequency Response Function (FRF) relating excitation forces and vibration response. For larger structures, natural influences, like wind loads, are used as a quasi white noise excitation for which the response is measured, as described in European patent EP 1 250 579 by Brincker. However, the latter corresponds to operational modal

analysis (OMA) in contrast to experimental modal analysis (EMA), where the response of an object of interest is observed after excitation by a controlled and measured force at a selected excitation point, for example by hitting the object with a hammer or by shaking the object.

hi the most simple modal test only one input and one observation point is used. This is called single input, single output (SISO). If several observation points are used - for instance by using more sensors, or by moving one sensor to several points, but remaining with only one loading point - the case is termed single input, multiple output (SIMO). If several input points are used - for instance by using more loading devices or by moving the loading device around on the structure, but remaining with only one observation point - the term is a multiple input, single output case (MISO). If several input points and several observation points are used, the term used is a multiple input, multiple output (MIMO) case.

In vibration testing, it is often important to be able to distinguish and estimate closely spaced vibration modes. This is only possible in the MIMO case. The reason for this is that the rank of the Frequency Response Function (FRF) matrix is reduced to one in the SISO, the SIMO and the MISO case. In the MMO case the ranki? of the FRF (Frequency Response Function) matrix is given by

(1) R = min{N _x ,N _Y }

where N _x is the number of inputs and N _y is the number of outputs. The number of closely spaced modes to be distinguished and estimated is limited to the rank of the FRF matrix.

In a typical test case, a limited number of sensors are available, and thus, the number of sensors will typically be smaller than the required number of observation points. This problem is solved by moving the sensors around on the object for the above mentioned SIMO or MIMO case.

Every time, a new measurement is taken with a fixed location of the sensors, a new data set is defined. The best solution is to cover all observation points using only one data set, because then, all mode shape information is obtained at once. If the sensors are moved around, uncertainty on the mode shape is introduced due to individual er- rors on each data set. This uncertainty also influences the estimation of the modal mass.

The simplest loading device is of the hammer type. The structure is given an impulse loading, and the response is observed. An ideal excitation pulse would be in the form of a delta function. However, a hit with a hammer is never an ideal impulse. Rather, the loading by the hammer gradually builds up and fades out again. Thus, a certain triggering criterion must be applied to define an approximate time of the excitation impulse. The corresponding responses to the excitation are typically defined as

Thus, both the input signal x(t) and the output signal y(t) are shifted in time defining a new time scale τ in order to discard data prior to the event of the pulse, the time of the event being defined as the time where the loading signal crosses through a certain triggering level x(t) = a corresponding to τ — 0. A small amount of data might be used for negative times τ < 0 so that all information about the pulse is included in the subsequent analysis. It is characteristic for the classical triggering that only one triggering point is used.

A limiting factor for the accuracy of the method is the uncertainty in the excitation due to variations in the excitation loading. If a hammer is used for the excitation, there may be a slight variation in the applied force and speed, leading to unknown variations in the response. An improvement can be achieved by implementing a force sensor in the hammer itself, which is the normal case. The applied force can than be used for scaling the response, if a linear scaling is assumed.

However, there may still remain uncertainties in the direction and excact location of the hammer excitation, which introduces uncertainties. Also, the hammer may bounce on the surface and lead to several excitations. The problems with hammer testing is discussed on page 119 in DJ. Ewins: Modal Testing: Theory and Practice, John Wiley & Sons Inc. , 1995 (reprint). In the same reference on pages 153-159, the classical signal processing of the measured signals is explained, wherein among different kinds of transient signals, the hammer excitation is the most common. As disclosed, for each hammer excitation, the response must have died out within the measurement time of each record, which is necessary in order to avoid the so-called leakage error. There- fore, in order to have enough information and reduce the noise content in the signal, usually, several responses have to be averaged. This is normally done by estimating the FRF (Frequency Response Function) and then averaging in the frequency domain.

Alternatively, in order to reduce the noise, the process of excitation can be repeated a number N _t of times to perform an averaging in the time domain as given in equation (3).

Here, several signals of the type as described by Eq. (2) are averaged. It is important to note that the triggering condition is introduced in order to line up the different events so that each event correspond to similar conditions at time τ = 0 , thus only one triggering point is used for each event.

After each single excitation, the response has to die out, before a new excitation is performed. However, as Ewins points out, the averaging of the signals (and having to deal with the problems of the double-hits) might diminish the advantage of the simple hammer excitation.

Thus, there is still need for improvements in the art.

DESCRIPTION / SUMMARY OF THE INVENTION

It is therefore the object of the invention to improve existing techniques. Especially, it is the purpose of the invention to provide a method for vibration measurements, where excitation uncertainties are minimised.

This purpose is achieved by a method for testing an object with respect to its vibration modes, wherein a number of subsequent vibration excitations are provided at a selected excitation location on the object, and the response from the excitation is meas- ured by a number of vibration sensitive sensors in order to obtain dataset from each sensor quantitative for the vibration of the object at the location of the sensors. According to the invention, at least one sensor is provided as a reference sensor stationary at the same measuring location on the object during the number of subsequent vibration excitations. Further, at least one mobile sensor is provided, which is moved to various locations on the object during the subsequent excitations.

When the number of detectors is less than the desired number of locations for measurements, it is customary to move sensors around on the object, as also explained above. According to the invention, it is better not to move all sensors around, but to keep some of the sensors in the same place during all data sets. The stationary sensors are denoted reference sensors that are used for better control of the measurement process.

For example, the measurement signal from the reference sensor may be used for scal- ing of the measurement signal from the mobile sensor. This eliminates to a high degree the influence on the data sets by variations in the excitation, for example different directions of excitation by a hammer.

An alternative use of the measurement signal from the reference sensor is for quality control of the vibration excitation. For example, the signal from the reference sensor may be used to check, whether hammer excitations are within acceptable variations or whether the excitation varies more than acceptable for the measurement.

Thus, invention relates to experimental modal analysis, where the excitation of the object maybe controlled in strength, time and location on the object.

hi operational modal analysis (where the inputs are unknown) reference sensors are always used, but in classical experimental modal analysis, reference sensors are not used because the loading as measures by a sensor in a hammer is used as reference.

The loading reference however is a bad reference due to the uncertainties introduced by to measurement errors (loading conditions like distribution, directions etc.), and thus, it is better to use certain observation points as references, because the measure- ment conditions are exactly the same in all data sets.

Using several reference sensors makes the mathematical problem of the mode shape estimation over-determined, and thus, from over determination, better mode shape estimates and better modal mass estimates are obtained.

In this case the rank R of the FRF matrix is still given by Eq. (1).

(1) R = min{N _x ,N _Y }

where N _x is the number of input points and N _y is the number of observations points. Thus, the rank of the FRF matrix is not diminished by keeping some of the sensors as reference sensors, and therefore the capability of the test for estimating closely spaced modes is not diminished. Actually, the typical result is that the capability of estimating closely spaced modes is significantly improved. The reason is the following. When moving sensors around on a structure, it might occur, that in a certain data set, the mobile sensors are placed in locations where one or more of the modes show weak responses. In such case, the corresponding mode - and especially the mode shape - would be badly estimated if the reference were not be present, which ensures a significant response from all modes of interest.

In a further embodiment, the excitation location is point like, which is typically the case when using a hammer. When using a shaker, the excitation location may be point like but need not be point like, as the connection between the shaker and the object

could be over a larger area. Also, when using shakers, the object may be connected to shakers at more than one location, which leads to a higher rank of the FRF matrix and to a better signal-to-noise ratio.

Preferably, the object is configured such that the selected excitation location is arbitrarily selectable in order to be able to place the excitation at a location which is beneficial for excitation of certain vibration modes or for excitation of as many vibration modes as possible. However, in contrast to excitation with white noise by wind or traffic noise, as it typically is used in buildings when exerting Operational Modal Analy- sis, the number of excitation locations in Experimental Modal Analysis is limited, countable and can be chosen arbitrarily.

According to prior art, data sets for vibration measurements are analysed to find the vibration modes. In this analysis, the Frequency Response Function is used, which is the Fourier transform of the response function normalised by division with the Fourier transform of the excitation input function. The latter is typically found in prior art modal tests by including a sensor in the hammer for excitation. The sensor in the hammer implies that subsequent excitations in different directions nevertheless yield the same input function, as the sensor in the hammer cannot distinguish between variations in the excitation direction.

Thus, it would be desirable to be independent from the measurements of a sensor in the hammer. This purpose solved if an accelerometer sensor is provided at the location of excitation. The accelerometer takes the place of the sensor in the hammer and the measurement signal from the accelerometer can be used as denominator in the FRF for a reference sensor or for a mobile sensor or for both of them. As the accelerometer is connected to the object and not to the hammer, the direction of the hit with the hammer - or likewise hitting tools - can be measured and yield a higher precision in the analysis.

hi addition, there is a different way to avoid the sensor, such as a force sensor, in a hammer. This is achieved by a further embodiment, where a Frequency Response Function (FRF) is determined for the reference sensor and a Frequency Response

Function (FRF) is determined for the mobile sensor with the excitation signal as a variable denominator in the FRF. As both FRF have the same denominator, this denominator may be eliminated by dividing the FRF of the mobile sensor by the FRF of the reference sensor. In other words, as the signal from the reference sensor can be used for scaling instead of the input signal. There is no necessity for calculating the FRF itself by using the Fourier transform of the input function, is suffices to find the scaled response function of the mobile sensor using the signal from the reference sensor as a scaling function.

This implies as well that a hammer without internal sensor may be used. Taking into regard the fact that hammers with sensors are expensive, this is a great advantage, as any hitting object may be used for the method according to the invention.

It has been recognised during development of the invention that it is advantageous, if reference observation points are used also as input points. As described above, this can be done by using an accelerometer that for excitation of the object is hit by a hammer for transferring the impact of the hammer to the object through the accelerometer.

When the loading is applied directly in a point where the response is observed, the point of loading can be identified from the time series. Thus the test planning is easier, and the risk of making a mistake when performing the test (like hitting the structure in a wrong point according to the test planning) is removed.

Alternatively, different excitation points may be selected subsequently, where a certain input point is used as location for a reference sensor during those excitations where the certain input point is not excited. Therefore, the method according to the invention comprises selecting a first location for excitation of the object and selecting a second location for first reference measurements by a reference sensor, subsequently selecting a different location for the excitation and using the first location for second reference measurements by a reference sensor. For example, the input point and the location for the reference sensor may be swapped such that the above mentioned different location is the second location.

Primarily, this part of the invention is based on the fact that the loading points and the points where the reference sensors are placed are chosen based on the same criteria. In both cases, the optimal points are the points where all mode shapes have large components. Thus, taking the points for the loading and the reference points to be the same simplifies the testing and optimizes the information acquired.

In this case the rank of the FRF matrix is given by

(4) R ~N _yr

Where JV^ is the number of reference sensors. Thus, using several reference points leads in this case automatically to a MIMO case (rank larger than unity) that allows for estimation of closely spaced modes.

A special advantage is achieved by using Random Decrement technique for transfor- mation of the measured response signals before estimation of the FRF. The RD technique may use an estimation of a Random Decrement function as a sum of time segments for which the input excitation at the triggering time is equal to a triggering level. This is typical for RD techniques.

An introduction to Random Decrement signal processing with a short literature review is found on pages 5-12 in John Christian Asmussen: Modal Analysis Based on the Random Decrement Technique, Application to civil engineering Structures. PhD thesis, Department of Building Technology and Structural Engineering, University of Aalborg, 1997. Asmussen reads paragraph 1.2.1. "The random response of a structure at the time t _o +t is composed of three parts: 1) The step response from the initial displacements at the time t ₀ . 2) The impulse response from the initial velocity at the time t ₀ . 3) A random part which is due to the load applied to the structure in the period t ₀ to to+t." and as continued, "What happens if a time segment is picked out every time the random response, x(t) has an initial displacement, say x(t)=a, and these time segments are averaged?", with an answer in the reference, "As the number of averages increase the random part due to the random load will eventually average out and be negligible. Furthermore, the sign of the initial velocity is expected to vary randomly with time so

the resulting initial velocity will be zero. The only part left is the free decay response from the initial displacement, a."

In the same reference on pages 141-15O ₃ Asmussen shows how the Random Decrement technique can be applied for estimation of the FRF function. The principle is illustrated in the first example where a system with one degree of freedom (one mode) is driven by white noise. In this case the Random decrement technique turns the random input into a perfect pulse, and the corresponding response into the corresponding impulse response function, Fig 7.1, 7.2 and 7.5. The corresponding results of the FRF is shown in Fig 7.3, 7.4 7.6 and 7.7 of the Asmussen reference.

Generalizing the above triggering condition of the classical test triggering gives according to the RD

N,

⁽⁵⁾ ^ ⁾ 4 ^+r fc

Where the triggering criterion T _x can be any criterion on the input, for instance

(6) T _x : x = a

T _x : x > a T _x : x > v,x = a

where the level a is selected with the aim to have sufficient input measurements.

The difference to the classical test triggering is the general triggering conditions given by Eq. (6) and - which is more important - many triggering points from the time series are used to form an average based on a large number of averaged data segments.

The nomenclature in connection with these equations and the equations in the following is given by the table of nomenclatures as shown below.

Nomenclature table

Equation (4) and (5) is an implementation of the Random Decrement (RD) technique, and it is also known, that for any triggering criterion,

(7) Y _r (ω) = H(ω)X,.(ω)

expresses the relationship between the Fourier transformed X(ώ) of the input, the Fourier transformed Y(ω) of the output and the Frequency Response Function H(ω) . This allows the exact estimation of the FRF component

(8) H(ω) = X _r (ω) 7»

Implementation of the RDD technique gives the advantages, that the impulse response information can be extracted with higher accuracy (since more triggering points are used), that all the information can be extracted from a single time series (there is no reason to restart the measurement engine for every new hit), and finally there are no constraints on the loading (not double hit problem). Finally, defining the triggering to select a certain input level opens the possibility to investigate non-linearity on the same data set by adjusting the triggering level. By finding such non-linearity, corresponding measures can be taking into account when scaling the response with respect

to the excitation input. This is in contrast to prior art, where linearity is assumed and used for scaling purposes as discussed in the introduction above.

In a further embodiment, the RD technique uses estimating a Random Decrement functions as a sum of time segments for which the input excitation is equal to a triggering level and for which the time differential of the input excitation is larger than a predetermined level. This way, a more precise dataset is achieved by combining the criterion where the input level is equal to a trigger level with a criterion on the velocity of the object due to the input excitation. This corresponds to the third option in equa- tion (6).

However, an advantageously larger set of data may be achieved, if the Random Decrement function implies a sum of time segments for which the input excitation larger than a triggering level. This corresponds to the second option in equation (6).

In connection with the RD technique, the excitation by hitting the object with a hammer is preferred over shaker excitation, as the method seems to have especially many advantages if the excitations are time spaced. This is due to the fact that in prior art measurements, the hammer method implies that the excitation has to fade out before the next excitation, which introduces additional noise. With the method according to the invention, this is not any more necessary, and influence by noise is drastically reduced. The object can be hit by the hammer many times within a relatively short time span without having to take into regard any necessary fading. Each further hit by the hammer increases the amount of useful data in the measurements.

As a consequence, the invention removes both problems pointed out by Ewins. The double hit problem is removed by using the Random Decrement technique, which is able to transform an irregular input signal (like double hits) back to something that is close to an ideal pulse. Further, the Random Decrement results in an averaging in the time domain, and therefore can process one long time series of hits. Therefore, the invention does not imply the necessity of taking many individual time series including the fading out of the excitation with subsequent averaging in the frequency domain.

As the user does not have to wait for the response signal to die out, the results is that the user can hit as much as convenient, take one long record, let the Random decrement transform the signal to an ideal pulse and the corresponding response, and then finally estimate the FRF. Because the signal does not have to die out before a new excitation is performed, noise in the signals are drastically reduced. And the influence of hitting a little different concerning location and direction is removed by using the reference sensors for mode shape scaling and merging.

SHORT DESCRIPTION OF THE DRAWINGS

The invention will be explained in more detail with reference to the drawing, where

FIG, 1 illustrates a first embodiment of the invention,

FIG. 2 illustrates a second embodiment of the invention,

FIG. 3 illustrates a third embodiment of the invention, FIG. 4 illustrates the triggering criterion,

FIG. 5 illustrates sampling criteria.

DETAILED DESCRIPTION / PREFERRED EMBODIMENT

The present invention is a generalization of the classical procedure concerning the way vibration testing is performed, and the way the excitation is performed.

FIG. 1 illustrates an embodiment of the invention. An object in the form of a plate 1 is excited at excitation point 2 by a hammer 3. A reference sensor 4 is placed stationary on the object 1, whereas three other sensors 5, 5', 5" are located in a first row of points and are to be moved to other rows of points 6 in order to investigate the vibration modes of the object 1. The excitation point 2 and the location of the stationary sensor 4 are fixed.

FIG. 2 illustrates a second step for measurement in a further embodiment of the invention. In this further embodiment for the method, a first step of measurements have been performed according to FIG. 1, after which the positions of the sensor 4 and the

excitation point 2 have been swapped. This implies that the vibration at the point of excitation can be examined as well.

FIG. 3 illustrates a further embodiment, where sensor 4 is an accelerometer placed at the point of excitation 2 and is hit directly by the hammer 3. This implies that the excitation in point 2 can be measured directly and be used for scaling of the measurements from the detectors 5, 5', 5".

FIG. 4 illustrates the excitation of the object. Ideally, the excitation should be a delta function at time τ. However, as the excitation pulse causes a rising and fading of the vibration, a criterion for the pulse is found. According to this criterion, the excitation is defined to have occurred around the time t where the excitation x(t) is equal to the level a . This reflects equation (2) above.

The RD sampling according to equation (5) above is illustrated in FIG. 5. Measurements are sampled each time t _1? t _% ...,t ₈ the criterion is fulfilled that x(t)=α. This corresponds to the first criterion in equation (6) above.

In order to achieve more data points, a number of measurements may be sampled within the time spans ti to t ₂ , t ₃ to t ₄ , t ₅ to t ₆ , and t ₇ to tg, which is illustrated as points 8 on the curve in a time span 7 within t ₃ to t ₄ , though it applies equally well to the other three time spans ti to t ₂ , t ₅ to t ₆ , and t ₇ to t ₈ . This corresponds to the second criterion in equation (6) above.