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Title:
COMPONENTS AND CORRESPONDING METHODS OF MANUFACTURE PROVIDING VIBRATION-DAMPING PROPERTIES
Document Type and Number:
WIPO Patent Application WO/2018/163147
Kind Code:
A1
Abstract:
A manufacturing process and corresponding mechanical components are formed by additive manufacturing so as to include fully encapsulated cavities containing powder, typically deployed in distributed locations within the component, so as to increase the damping properties of the component compared to a similar component without such cavities. Since the filled cavities are completely enclosed, they do not impinge on the overall performance characteristics of the component and do not require maintenance.

Inventors:
SETTER EYAL (IL)
PRIVMAN NETA (IL)
Application Number:
PCT/IL2018/050126
Publication Date:
September 13, 2018
Filing Date:
February 05, 2018
Export Citation:
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Assignee:
RAFAEL ADVANCED DEFENSE SYSTEMS LTD (IL)
International Classes:
F16F9/30; B28B1/00; B28B5/00; B28B7/04
Foreign References:
US20150052898A12015-02-26
US20160222796A12016-08-04
US20150034604A12015-02-05
Attorney, Agent or Firm:
FRIEDMAN, Mark (Moshe Aviv Tower 54th floo, 7 Jabotinsky Street 07 Ramat Gan, IL)
Download PDF:
Claims:
WHAT IS CLAIMED IS:

1. A process for producing a structural component with vibration- damping properties, the process comprising the steps of:

(a) deploying a layer of a granular composition;

(b) selectively amalgamating at least one region of said layer to form a solid-layer profile of the structural component;

(c) repeating the steps of deploying and selectively amalgamating so as to form successive solid-layer profiles of the structural component, each solid-layer profile being united with a preceding one of said solid-layer profiles, until the structural component is complete,

wherein successive of said solid-layer profiles are formed such that the final structaral component fully encloses at least one internal cavity within which a quantity of said granular composition remains trapped.

2. The process of claim 1, wherein said granular composition comprises a metal or metal alloy.

3. The process of claim I, wherein each of said layers comprises a uniform layer of a uniform granular composition.

4. The process of claim 1, wherein at least a subset of said layers comprise a layer containing distinct regions of distinct granular compositions.

5. The process of claim 1, wherein said selectively amalgamating is performed by a directed beam of electromagnetic energy.

6. The process of claim 1, wherein said selectively amalgamating is performed by selective application of a binding composition.

7. The process of claim 1, wherein said final structural component fully encloses a plurality of said internal cavities within which a quantity of said granular composition remains trapped, said plurality of said internal cavities being distributed so as to be deployed symmetrically about an axis of symmetry or a plane of symmetry of said final structural component.

8. The process of claim 1, further comprising providing a system, said stractural component being configured to operate as a part of the system, wherein operation of the system induces vibration of said stractural component in at least one oscillatory mode defining at least one region of primary oscillatory displacement, and wherein said at least one internal cavity is located in said at least one region of primary oscillatory displacement.

9. The process of claim 1, further comprising, prior to steps (a)-(c):

(a) defining a mathematical model of the mechanical properties and dynamic response of the structural component including modelling of multiple modes of vibration;

(b) defining a vibratory excitation delivered to the component, and boundary conditions dictated, by a system within which the component is to be deployed;

(c) performing a mathematical optimization process employing said mathematical model, said excitation and said boundary- conditions so as to optimize a set of parameters relating to said stractural component,

wherein said set of parameters comprises parameters defining location, shape and dimensions of said at least one internal cavity containing trapped granular material.

10. The process of claim 1, wherein said at least one internal cavity is subdivided by at least one partition.

11. The product of claim 1, wherein said trapped quantity of said granular material is non-contiguously amalgamated such that said trapped quantity of granular material includes multiple particles of unequal sizes.

12. A product comprising a structural component formed by an additive manufacturing process by amalgamation of a granular precursor material, wherein said structural component is formed with at least one fully enclosed internal cavity within which is trapped a quantity of said granular precursor material in granular form.

13. The product of claim 12, wherein said internal cavity is subdivided by at least one partition.

14. The product of claim 12, wherein said trapped quantity of said granular precursor material is non-contiguously amalgamated such that said trapped quantity of granular precursor material includes multiple particles of unequal sizes.

15. The product of claim 12, wherein said structural component contains a plurality of said internal cavities containing said trapped granular precursor material.

Description:
Components and Corresponding Methods of Manufacture Providing Vibration-Damping Properties

FIELD AND BACKGROUND OF THE IN VENTION

The present invention relates to damping of vibrations and, specifically, to a method of manufacturing mechanical components with vibration damping properties, and corresponding components.

Vibration in mechanical systems is a common phenomenon that may be derived from external environmental conditions, e.g. flow effects of an aircraft wing, or from internal operation, such as the effect of imbalance in a rotating engine shaft. Vibration suppression is a crucial property in most mechanical systems and takes a great deal of effort in the design, in order to maintain structural integrity' and accuracy of performance. The most common manner of achieving vibration attenuation is by adding dampers or shock-absorbers to the system. However, dampers may not always be added, especially where confined spaces or harsh environmental conditions are present. Commercial vibration dampers may be based on elastomers, on fluid viscous effects, or electrodynamic forces. Indeed, external dampers increase the damping ratios substantially. However, these types of dampers suffer from relatively low static stiffness, limited resistance to temperature, and may require maintenance, particularly where elastomers or seals are employed. Moreover, when using external dampers, the design is limited to a discrete set of positions and damper parameters, thus the outcome would probably be a compromise rather than an optimal solution, or it may divert the system dynamical and mechanical properties (mass/inertia/natural frequencies) from its original design point.

An alternative mechanism for vibration reduction discussed in the literature harnesses friction based effects occurring in granular media. Preliminary investigation has been done by Booty, C. (2014) Experimental investigation of damping flexural vibrations using granular materials Loughborough University's Institutional Repository pp. 547-558 and Saeki M. (2001) IMPACT Damping With Granular Materials In A Horizontally Vibrating System Journal of Sound and Vibration pp. 153-161, to explore impact suppression and ambient vibration reduction effects, respectively, with the use of friction in granular media. Each party offers an analytical model based on a series of experiments conducted on particles 1-7 mm in diameter, placed in a box and excited by a shaker. Such an approach is very limiting in its industrial applications, due to the structural complexity of adding a granular media container to a structure. SUMMARY OF THE INVENTION

The present invention is a method of manufacturing mechanical components with vibration damping properties, and corresponding components.

According to the teachings of the present invention there is provided, a process for producing a structural component with vibration-damping properties, the process comprising the steps of: (a) deploying a layer of a granular composition; (b) selectively amalgamating at least one region of said layer to form a solid-layer profile of the structural component; (c) repeating the steps of deploying and selectively amalgamating so as to form successive solid- layer profiles of the structural component, each solid-layer profile being united with a preceding one of said solid-layer profiles, until the structural component is complete, wherein successive of said solid-layer profiles are formed such that the final structural component fully encloses at least one internal cavity within which a quantity of said granular composition remains trapped.

According to a further feature of an embodiment of the present invention, the granular composition comprises a metal or metal alloy.

According to a further feature of an embodiment of the present invention, wherein each of the layers comprises a uniform layer of a uniform granular composition. According to a further feature of an embodiment of the present invention, wherein at least a subset of said layers comprise a layer containing distinct regions of distinct granular compositions.

According to a further feature of an embodiment of the present invention, wherein the selectively amalgamating is performed by a directed beam of electromagnetic energy.

According to a further feature of an embodiment of the present invention, wherein the selectively amalgamating is performed by selective application of a binding composition.

According to a further feature of an embodiment of the present invention, wherein the final stractural component fully encloses a plurality of said internal cavities within which a quantity of said granular composition remains trapped, said plurality of said internal cavities being distributed so as to be deployed symmetrically about an axis of symmetry or a plane of symmetry of said final stractural component.

According to a further feature of an embodiment of the present invention, wherein the process further comprised providing a system, said stractural component being configured to operate as a part of the system, wherein operation of the system induces vibration of said stractural component in at least one oscillatory mode defining at least one region of primary- oscillatory displacement, and wherein said at least one internal cavity is located in said at least one region of primary oscillatory displacement.

According to a further feature of an embodiment of the present invention, the process comprises, prior to steps (a)-(c): (a) defining a mathematical model of the mechanical properties and dynamic response of the structural component including modelling of multiple modes of vibration; (b) defining a vibratory excitation delivered to the component, and boundary conditions dictated, by a system within which the component is to be deployed; (c) performing a mathematical optimization process employing said mathematical model, said excitation and said boundary conditions so as to optimize a set of parameters relating to said structural component, wherein said set of parameters comprises parameters defining location, shape and dimensions of said at least one internal cavity containing trapped granular material.

According to a further feature of an embodiment of the present invention, wherein at least one internal cavity is subdivided by at least one partition.

According to a further feature of an embodiment of the present invention, wherein the trapped quantity of said granular material is non- contiguously amalgamated such that said trapped quantity of granular material includes multiple particles of unequal sizes.

There is also provided according to the teachings of an embodiment of the present invention, a product comprising a structural component formed by an additive manufacturing process by amalgamation of a granular precursor material, wherein said structural component is formed with at least one fully enclosed internal cavity within which is trapped a quantity of said granular precursor material in granular form.

According to a further feature of an embodiment of the present invention, wherein the internal cavity is subdivided by at least one partition.

According to a further feature of an embodiment of the present invention, wherein the granular precursor material is non-contiguously amalgamated such that said trapped quantity of granular precursor material includes multiple particles of unequal sizes.

According to a further feature of an embodiment of the present invention, wherein the structural component contains a plurality of said internal cavities containing said trapped granular precursor material.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is herein described, by way of example only, with reference to the accompanying drawings, wherein: FIG. 1A-C are schematic isometric views of the structural components used to demonstrate the vibration damping properties of the invention.

FIGS. 2A and 2B are isometric depictions of a first flexible mode of the components represented in FIGS. IB and 1C, respectively.

FIG. 3 is a summary table of the mechanical properties of the components represented in FIGS. 1A-1C.

FIGS. 4(a)-(f) are partial side and top view X-Ray image pairs of the components represented in FIGS. IB and 1C, respectively, and a top view X- ray image of the component of FIG. 1A.

FIGS. 5 A and 5B are experimental frequency response results of the components represented in FIGS. 1A-1C with differing frequency scales.

FIG. 6A is a graph showing experimental time response results of the components represented in FIGS. IB and 1C.

FIG. 6B is a region of the graph of FIG. 6A showing the time response results for the component of FIG. IB only, with an expanded time scale.

FIG. 7 is a graph showing a theoretical calculation of expected time response of a component exhibiting Coulomb dry friction.

FIG. 8 is a flow chart of a typical optimization process for designing structural components with vibration-damping properties.

FIGS. 9 A and 9B are schematic depictions of a set of constraints that are fed into the optimization process of FIG. 8, and a hypothetical outcome of the optimization process, respectively.

FIGS. 10(a)-(d) are schematic representations of a blade from a rotor of a turbine or jet engine illustrating possible distributions of granular material to achieve damping of various modes of vibration.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention is a method to incorporate Embedded Distributed Granular Damping (EDGD) in a structure by exploiting heretofore unexploited capabilities of Additive Manufacturing (AM) to integrate granule-filled cavities into a component, and corresponding components.

The principles and operation of the present invention may be better understood with reference to the drawings and the accompanying description.

In a typical solid AM process, a granular composition, e.g., a powder, is deployed in a sequence of thin layers, each of which is amalgamated in selective regions. This amalgamation may be achieved using any technique or technology known for solid additive manufacturing, as further detailed below. Successive amalgamated regions bond with each other to gradually build the component, layer by layer. At the end of the process, the component is extracted and cleaned from the surrounding powder. According to conventional techniques, in order to avoid powder becoming trapped inside any internal cavities, escape holes are incorporated into each design, connecting any internal voids to the outermost surface of the component, so as to allow residual powder within the voids to be removed in a post-processing cleaning step. In contrast, the method of certain preferred embodiments of the present invention deliberately creates fully encapsulated cavities containing pow r der, typically deployed in distributed locations within the component, so as to increase the damping properties of the component compared to a similar component without such cavities. Since the filled cavities are completely enclosed, they do not impinge on the overall performance characteristics of the component, and may, for example, maintain high stiffness, high resistance to harsh environmental conditions, require no maintenance, provide high durability and long service life, and offer almost unlimited design configurations.

The method is of special interest where the design is bound to complex geometries, confined volumes, or where standard dampers cannot be applied, e.g. in rotating blades. As a result, integration of granular damping volumes via AM is highly advantageous. The capability of additive manufacturing to generate complex structures that combine solid and encapsulated powder provides valuable opportunities to optimize modal design, where target values may be set for both the stiffness and damping, or maximal allowed response of the structure, under a given input Power Spectral Density (PSD). The resulting components typically display one or more of: attenuation of vibrations resulting from oscillations or impacts, noise isolation, and/or ambient vibration reduction.

The present invention may be implemented using any powder-based additive manufacturing technology, and employing materials suitable for implementing such technology. Particularly, the invention is preferably implemented using AM techniques generally referred to as "powder bed" techniques in which thin layers of granular material are successively laid down, and selective regions of the layer amalgamated to form a solid layer profile of the final component being constructed. Relevant techniques for amalgamating the granular material include, but are not limited to, fusing, melting or sintering by application of energy via a beam of electromagnetic radiation, such as a laser beam or an electron beam, by heating via other heating technologies, or by application of a "binder", which may be an adhesive composition or a chemical which reacts with, or catalyzes, a chemical reaction involving the granular material in order to achieve amalgamation of the granular material. Details of the process parameters, energy levels, control of atmospheric conditions etc. will all be selected according to the type of process and materials to be used, all as will be clearly understood by a person having ordinary skill in the art, and are typically unchanged from the standard operating parameters of conventional AM processes.

The component configurations and corresponding methods can be further diversified by deploying a plurality of granular compositions as described by Subramanian. et al. in US Patent Application Publication No. US20150034604A1. Under this embodiment the regions of the component to be amalgamated are layered with one material, for example a metal or a metal alloy, while the granular material that is deposited in a region which will become a granular filling within a cavity is wholly or partially composed of a different material, for example a ceramic. According to a further option, at least part of the granular material deposited in the regions of a layer which are to become the filled cavity may be chosen to be materials which do not undergo amalgamation under the normal working conditions used to achieve amalgamation of the solid regions. The non-amalgamating material may for example be material having a higher sintering temperature and/or melting point, or that is less reactive, all according to the AM technology used.

In another embodiment, the component has a plurality of internal cavities. Each section of the component could contain an unamalgamated portion. Alternatively, a single section could contain a series of such portions subdivided or separated by bridging regions of solid (amalgamated) material, thereb}^ preserving the component's external shape and providing the required mechanical strength while improving its damping properties. Most conventional components have regions of relatively low stress, in which the entire thickness of the material is not required in order to provide the required mechanical properties. Such regions provide candidate locations for inclusion of granular-filled cavities without compromising the required mechanical properties. A similar result is achieved by dividing a cavity with one or more partitions or scaffolding. Such supports contribute to the component's mechanical strength while also increasing the surface area of the component with the powder and thus influence the damping properties.

According to another optional feature, applicable to all embodiments of the present invention, the powder within a cavity is non-contiguously amalgamated to produce particles of varying sizes. Thus, for example, the amalgamation process may be used to generate elongated rods surrounded by powder, aligned in a predefined direction, so as to provide anisotropic (directional) damping properties, for example, as illustrated in FIG. 10(d). Without in any way limiting the scope of the present invention, it is believed that the use of elongated granules (rods) generates enhanced damping for motion in a direction parallel to the rods, due to increased surface area experiencing shear-friction with the surrounding granules. N on- directional enlarged particles, such as spheres, may also be generated within the cavity. Example 1

In order to demonstrate the proposed method, simple beam models were designed and fabricated. Generally, dry friction is a non-linear phenomenon.

Here we use a linearized approach using a damping ratio ζ (a linear property).

Since the damping ratio ζ depends on the natural frequency as well as on the damping, it was important to compare structures with similar natural frequencies. Three beam models are presented here:

IA. "Uniform beam" - a uniform rectangular cross-section fully solid beam (FIG. 1 A),

1B. "5-Chamber Beam" - a beam with 5 fullv-enclosed cavities filled with granules to provide an EDGD structure (FIG. IB)

IC. "Stiffness-equivalent beam" - a fully solid beam, with varying cross section (FIG. 1C) so as to yield a close approximation to the same natural frequency as the beam of FIG. IB.

Since the chambers of the powder contribute to the total mass but do not contribute to the stiffness, the cross section of beam 1C was designed to yield the same mass distribution and the same bending stiffness (and therefore also fiexural moment of mertia) in a primary flexion direction in which the beam, flexes parallel to its smallest dimension. By satisfying the above criteria, beams IB and 1C are formed so as to exhibit a similar natural frequency.

In FIG. IB the chambers containing the powder 10 are shown. A beam. 1C of varying cross-section yielding the same natural frequency as the beam IB with powder chambers is shown. It will be noted that, corresponding to the positions in beam IB where the power chamber 10 exists, a corresponding wider and lower cross- section 11 is provided in beam 1C.

The beams were manufactured from Ti-6A1-4V alloy by AM. The natural frequencies of the designed beam model were validated numerically and experimentally. The nominal mechanical properties of the solid metal used for the numerical analysis are: Modulus of Elasticity E = 140.8GPa , Poisson's ratio v = 0.3 l , and mass density ρ = 4428.8 kg/m 3 . FIGS. 2A-2B show the first flexible mode shape of vibration of the two free beam models. It can be noted that the mode shapes are very similar, and the natural frequencies, as measured in the experiments are as close as 1.3%, as given in the table in FIG. 3. It should be noted however, that the mass of the two beams are about 9% apart. This difference may cause a change in the damping ratio of about 4% (or a factor of 1.04). It will be shown later that this is well within the acceptable level of error, with comparison to the measured data. The fully solid beam of uniform cross-section 1A has, as expected, a higher natural frequency due to its higher stiffness.

In order to demonstrate the powder chambers inside the beams, the fabricated parts were scanned by X-Ray imaging. Partial view r s of the beams are depicted in FIG. 4. The two figures on the left show (a) side and (b) face views of the beam with EDGD IB, and powder chambers given in lighter gray shades due to their slightly lower density (the triangulated tips of the chambers are used for up-growing of the part in the AM process to prevent ceiling collapse). The figures in the middle show (c) side and (d) face views of the beam of varying cross-section and no powder 1C. The figure on the right (f) shows a scan of the fully solid beam of uniform cross-section 1 A.

The damping of the fabricated beams was measured experimentally using two experimental systems. In the first, low level acoustic excitation was used, and in the second, the impact response to higher forcing levels was recorded. Vibration measurements were conducted via Laser Doppler Vibrometer (LDV).

In the first experiment with the low level acoustic excitation, the beams were excited by a loudspeaker playing chirp signals (a signal of time varying frequency). The middle of each beam was placed on a flexible mount while the vibrations of the beam's end were being measured by LDV. The input and output signals were transformed to the frequency domain to yield the system's frequency response function (FRF). The results are plotted in FIGS. 5A and 5B. It can be noted that the beam with granular damping 12 and the fully solid beam with equivalent natural frequency 13 have a very close natural frequency (1.3% difference), and as expected, the fully solid beam of uniform cross section 14 has a higher natural frequency due to its higher stiffness. It can also be noted that the peak vibration amplitude of the two solid beams with no damping (13 and 14) are very close. FIG. 5B shows the peaks of the beam with granular damping 12 and the fully solid beam with equivalent natural frequency 13 more clearly, with an expanded frequency scale. More significantly for the purposes of the present invention, it can be seen that the vibration amplitude of the damped beam is attenuated. Calculating the damping ratio using the half-power method (dashed lines 15 in FIG. 5B), yields a damping ratio ζ for the damped beam that is increased by 70% compared to the reference equivalent beam. (The mass difference as given in the table of FIG. 3 would only account for a change in damping of about 4%.)

In a second set of experiments, the beam was impacted with a modal hammer, while hung loosely by two flexible strings. This type of excitation introduces larger forces into the structure. The beam was measured and impacted in its center. The experimental measurements of the time response curves are plotted in FIG. 6A. In this case, the linearized damping ratio can be estimated from the measured data by curve-fitting the envelope of the time response signals. Using this method it was found that the beam with embedded granular damping (solid line) showed a damping ratio of approximately ζ= 0. 2%

which is about 50 times greater than the value of the undamped beam of equivalent stiffness (dashed line). Comparing these results to the results from acoustic excitation, it was demonstrated that higher levels of excitations yield higher damping levels for the beam with embedded granular damping.

While examining the curve for the beam with embedded granular damping as shown with an enlarged time scale in in FIG. 6B, it is noted that the decay envelope is triangular, which differs from the common exponential envelope as found for the solid beam 1A, which is commonly encountered in physical and mechanical systems governed by viscous damping. The distinctive triangular decay corresponds to the form observed according to theoretical calculations, as shown in FIG. 7, for the case of the dry (Coulomb) friction mechanism, which is believed to be the dominant mechanism of granular damping with dense packing.

Optimization

The capability of additive manufacturing to generate complex structures that combine solid and powder to implement EDGD allows an optimal modal or spectral design. In the optimization process, both the stiffness and damping, or maximal allowed response of the structure under a given temporal input or PSD (Power Spectral Density), are set as the objectives. A non-limiting exemplary flow chart for the proposed optimization scheme is given in FIG, 8. First, the modal or spectral objectives of the structure are to be determined, that is the natural frequencies and damping ratios, or alternatively the response limits are defined. Next, the expected tempo-spatial or spectral profile of the excitation is formulated. Later on, the physical constraints, such as volumetric constraints, or fixations are expressed. Additional constraints on the model may be imposed by design limitations of the AM process, and in particular, limitations on abrupt overhangs in the direction of additive layering. The data is then fed into an optimization platform for processing using the appropriate physical model. A possible mathematical formulation is a Finite Element Model (FEM), where the structure is meshed and solved for given nodes. This can be achieved semi-analytically via software such as MATLAB™ or numerically via software such as ANSYS IM . The model is then used for modal or harmonic analysis so as to calculate the natural frequencies, damping ratios and mode-shapes, or alternatively the temporal or spectral response of the stracture can be estimated from transient or spectral analyses. The outcome of the optimization will typically be a family of possible solutions, a pareto, from which the designer may choose. The optimal design may now be fabricated using AM and tested experimentally for performance. A schematic representation of the process described above is illustrated in FIGS, 9 A and 9B. FIG, 9 A depicts an illustration of the constraints and objectives. The box boundaries express the maximal allowed volume, in which the system is to be contained, and the limits for the modal parameters (frequencies a, h and damping ratio c) are also stated. The forcing magnitude as a function of time or frequency is fi(co), f2(i), its position and direction is marked by the vector arrow. FIG. 9B shows a schematic representation of an outcome of the optimization process, which is an optimal design of the structure, where some domains are solid and some are enclosed powder chambers (dotted area) that provide damping. The design is ready for AM and satisfies the dictated objectives and constraints.

Example 2

Schematic qualitative representations of designs that provide a desired amount of damping and stiffness under various conditions are demonstrated for a blade of a rotor, for example, suitable for use in a jet engine or turbine, in Fig 10. The vibration of such blades may be primarily characterized by bending fiexural modes, torsional modes, or edgewise modes. Each oscillatory mode defines at least one region of primary oscillatory displacement, defined for this purpose as a region in which an amplitude of vibration due to the mode in question is above half of amplitude at local maximum of the corresponding mode, and typically also at least one node at which little or no displacement occurs. According to the operational speed of the engine, and/or other operating parameters of the system as a whole, it is possible to identify which modes of vibration are most likely to be excited, and to incorporate regions of granular damping in the region(s) of primaiy oscillatory displacement for that mode, thereby providing damping which is tailored for those primaiy vibrational modes.

Thus, in our example, a nominal blade, corresponding to the volumetric constraint, is given in FIG. 10(a). Where a primaiy vibration mode of concern is a fiexural bending (flapping) mode, depicted in FIG. 10(b) by curved arrows representing the vibration direction, a suitable example of deployment of a region of granular damping is the dotted volume of FIG. 10(b). Such deployment would also be highly effective to damp any cantilever flapping mode occumng as a cantilever relative to a boundary condition of being fixed to a central rotor hub (not shown). In FIG. 10(c) a blade designed for torsional damping is presented, where the powder volumes (dotted) are dispersed in the areas of predicted high displacements and velocities, yet not too close to the edges in order to maintain stiffness. In FIG. 10(d) a blade with an edgewise mode damping is presented. In this non-limiting exemplary case, as mentioned earlier, the powder volumes contain internal solidified rod shaped small structures. These solidified bodies, immersed in the surrounding pow r der, are aimed to increase the directional friction with the powder with respect to the edgewise vibration direction, as indicated by the arrows. Needless to say, any combination of the abovementioned solid-powder configurations can be realized. Furthermore, in most practical situations, the design process will seek solutions to provide an acceptable level of damping in multiple different modes, and various multi-variable numerical optimization techniques are typically preferred over the above intuitive approach.

To the extent that the appended claims have been drafted without multiple dependencies, this has been done only to accommodate formal, requirements in jurisdictions wiiich do not allow such multiple dependencies. It should be noted that all possible combinations of features which would be implied by rendering the claims multiply dependent are explicitly envisaged and should be considered part of the invention. It will be appreciated that the above descriptions are intended only to serve as examples, and that many other embodiments are possible within the scope of the present invention as defined in the appended claims.