modeling of active and passive damping patches with
TRANSCRIPT
INTRODUCTION Current lightweighting trends in vehicle design
improve fuel economy, but as a result vehicle components can become
more susceptible to unwanted vibrations and structure-borne noise.
Additionally, modern lightweight materials (such as aluminum) often
have less inherent structural damping, resulting in increased
vibration amplitudes. One example is a transmission casing cover, a
vehicle component that is often excited by structure-borne
vibration in the drivetrain system and that is frequently an
efficient radiator of sound [1]. Furthermore, the geometry of such
a cover may include irregularities that increase its radiation
efficiency. Gear mesh frequencies common in the mid-frequency range
(200-1500 Hz) are responsible for exciting modes of these covers.
Damping solutions to reduce noise radiated by large-amplitude
vibration are needed that are both lightweight-keeping with the
lightweighting trends discussed-as well as effective in terms of
both size and cost.
Passive constrained layer damping patches have been shown to be an effective damping solution for these types of problems [1-2]. Such patches typically use a viscoelastic core with a large material loss factor (tuned to a given temperature range) and dissipate vibrational energy via shear deformation. Certain active patches (in particular piezoelectrics) have also been studied for vibration reduction applications both passively, in a shunted-circuit configuration [3], as well as actively, in a destructive interference sense [4]. The latter method introduces vibration to the system via a control signal intended to interfere with unwanted vibration from a disturbance (in
this case, structure-borne vibration from the powertrain). Combined active and passive methods have been proposed previously in the form of “active constrained layer damping” [5] with active materials embedded in traditional passive damping patches. However, little research has been done on concurrent “side-by-side” active and passive damping to find the effect of passive damping patches on control from an active patch. As such, improved modeling methods are needed to determine the potential benefits of such patch configurations, as well as to provide insight to the design process.
While both active and passive patch methods have been applied to a variety of simple structures including classical beams, plates, and shells [3, 4, 5, 6], realistic components are more difficult to model as they often include curvature, features such as fillets and varying thickness, and complex boundaries such as bolted joints. Accordingly, this paper seeks to address this void in terms of application-based modeling with the following objectives. (1) Perform an experiment to determine the dynamic behavior of a transmission casing cover. (2) Propose a design-oriented analytical method for modeling of active and passive patches (in the “side-by-side” configuration) to be implemented on the plate-like cover. (3) Quantify the efficacy of active and passive patches for reducing vibration in terms of modal loss factor and insertion loss, and evaluate other qualitative benefits of combined active and passive methods. (4) Validate the model with an experiment. (5) Propose design guidelines in terms of patch location and placement for given operating conditions.
Modeling of Active and Passive Damping Patches with Application to a Transmission Casing Cover
Joseph Plattenburg, Jason Dreyer, and Rajendra Singh Ohio State University
ABSTRACT Combined active and passive damping is a recent trend that can be an effective solution to challenging NVH problems, especially for lightweight vehicle components that demand advanced noise and vibration treatments. Compact patches are of particular interest due to their small size and cost, however, improved modeling techniques are needed at the design stage for such methods. This paper presents a refined modeling procedure for side-by-side active and passive damping patches applied to thin, plate-like, powertrain casing structures. As an example, a plate with fixed boundaries is modeled as this is representative of real-life transmission covers which often require damping treatments. The proposed model is then utilized to examine several cases of active and passive patch location, and vibration reduction is determined in terms of insertion loss for each case. Results are compared to an experiment with an actual transmission casing for validation, using piezoelectric active patches and constrained-layer passive patches with a viscoelastic core. Conclusions are drawn about patch size and location in terms of NVH reduction capability, and guidelines are suggested for the dynamic design process.
CITATION: Plattenburg, J., Dreyer, J., and Singh, R., "Modeling of Active and Passive Damping Patches with Application to a Transmission Casing Cover," SAE Int. J. Passeng. Cars - Mech. Syst. 8(3):2015, doi:10.4271/2015-01-2261.
2015-01-2261 Published 06/15/2015
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Figure 1.
Figure 1. (cont.) Transmission casing and cover: (a) production sample; (b) simplified version for experimental study; (c) detail of production cover; (d) detail of simplified cover
Accelerance measurements, defined as acceleration per unit force, are made of the cover in the in-situ condition (bolted to the casing) with an impulse hammer. A schematic of this procedure, as well as force and measurement locations, are shown in Figure 2a.
Figure 2. Schematic of experimental procedure on casing cover: (a) impact hammer with driving point location shown (xd, yd) ≈ (0.44 Lx, 0.71 Ly); (b) shaker input for driving point measurement
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Figure 2. (cont.) Schematic of experimental procedure on casing cover: (a) impact hammer with driving point location shown (xd, yd) ≈ (0.44 Lx, 0.71 Ly); (b) shaker input for driving point measurement
Driving-point and cross-point measurements are made at 28 different locations (using roving-hammer type measurement methods) in order to extract mode shapes. A least-squares parameter estimation technique is used to obtain natural frequencies, modal loss factors, and mode shapes. The accelerance spectra (driving point) for both cover thicknesses are plotted in Figure 3. Due to better experimental results and modal correlation for the thin cover, this case will be used for the remainder of the paper. Furthermore, the second, or (2, 1) mode, will be considered as the “mode of interest” because it is well isolated, has low damping and a well-defined mode shape, and it is near the middle of the targeted frequency range. Natural frequencies for each mode, along with modal damping ratio and modal index are given in Table 1. Modal indices are defined as (m, n), where m and n are the total number of local extrema in the long (x) and short (y) dimensions, respectively. As an example, the measured flexural displacement of the (2, 1) mode is shown in Figure 4a.
Figure 3.
Figure 3. (cont.)Experimental driving-point accelerance magnitude spectra of casing cover: (a) thin cover, thickness = 1.42 mm; (b) thick cover, thickness = 2.92 mm
Table 1. Selected modes of thin casing cover
Figure 4. Flexural displacement magnitude at (2, 1) mode: (a) measured, isometric and side views; (b) finite element model
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FINITE ELEMENT MODEL OF CASING COVER In order to characterize the nature of the boundaries for the in-situ configuration of the cover (and to determine if fixed boundaries are an appropriate approximation to the bolted connection), a finite element model (FEM) is developed in Abaqus (shown in Figure 5b). The computational model is assumed to be a thin plate (modeled with 2D shell elements) fixed at the boundaries. The plate boundary is approximated as the inside of the contact region between the casing and the cover (illustrated in Figure 5a). Material properties used for the computational model are given in Table 2, where hi is plate thickness, Ei is the Young's modulus, ρi is the density, ηi is the loss factor, νi is the Poisson's ratio, and i refers to layers 1 (passive constraining layer), 2 (viscoelastic layer), and 3 (base structure).
Table 2. Material properties of steel cover and passive patches
The natural frequencies of the finite element model (no patches, layer 3 only) are computed from the eigensolution and are listed in Table 1, and the deformation of the (2, 1) mode is given in Figure 4b. These natural frequencies are found to be significantly larger than those measured from the experiment. This is because while the computational model assumes perfectly fixed boundaries (i.e. infinite stiffness), the bolted boundaries of the experiment have a finite stiffness associated with them. A more accurate finite element model could be developed including bolt stiffness and other contact effects as demonstrated in [1]. Nevertheless, using the current model the modes are well correlated, as seen in Figure 4, so the assumed fixed-boundary approximation will suffice for the scope of this work. Such a classical boundary approximation will make the analytical model more tractable and easier to implement.
Figure 5.
Figure 5. (cont.) Fixed-boundary approximation for the cover: (a) casing- cover interface ( ); (b) finite element approximation of fixed boundary
ANALYTICAL MODEL The analytical model proposed here is based on the work of Kung and Singh [6] and extended by Plattenburg et al. [7] (where a more detailed analysis is given). Consider a thin rectangular steel plate of dimensions Lx × Ly with fixed boundaries and a number of active and passive patches. A schematic of the structure is shown in Figure 6.
Figure 6. Thin fixed rectangular plate with active and passive patches: example case for analytical study
Note that passive patches consist of two layers (steel constraining layer and viscoelastic core) and active patches consist of a patch on the top and bottom surface of the plate (assumed to not affect ρ3 or E3). The ith layer has thickness hi, as well as material properties Ei, ρi, ηi, and νi, where i = 1, …, 3. Material properties are listed in Table 2 (note that the viscoelastic material loss factor and Young's modulus are dependent on frequency, f). A disturbance force, Fd and measurement, w, are also included in the Figure 6 schematic.
The motion of the base layer (3) is written as a summation of shape functions similar to modal expansion [8]. The shape functions, (x, y), must satisfy the fixed boundary conditions, so they are assumed to be separable (i.e. m,n(x, y) = Xm(x)Yn(y)), and to take the form of standard fixed-fixed beam modes (illustrated in Figure 7a):
(1)
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where βm is the wave number that satisfies cosh(βmLx) cos(βmLx) = 1, Cm is a normalization constant such that Xm(x) has unity mean square value on [0, Lx], m and n are the same modal indices defined previously, and Yn(y), Cn, and βn follow analogous definitions [8].
Figure 7. Shape functions for the example of Figure 6: (a) sample functions for fixed boundaries with m = 1 ( ), m = 2 ( ), m = 3 ( ); (b) arbitrary displacement decomposed into weighted summation of shape functions
It is assumed that any flexural displacement profile of the plate, w(x, y), may be written as a weighted summation of N of these shape functions (depicted in Figure 7b):
(2)
where Φ is a 1 × N vector of shape functions and q is a N × 1 vector of weighting coefficients. Furthermore, assume that (i) all three layers have the same flexural displacement, (ii) shear in layers 1 and 3 is negligible, and (iii) all other motions (in-plane, rotation, and layer 2 shear) can be derived from w(x, y) via kinematic relationships as described in [6]. Now we have all motions of all layers (where flexure is of the most interest) with the weighting coefficients as the only unknowns.
The kinetic and potential energy ( and , respectively) of the system (neglecting rotary inertia) is written in the form:
(3a-b)
where M and K are equivalent mass and stiffness matrices defined as surface integrals of the shape function vectors (over the area of the ith layer, Ai) scaled by inertia (H) and elasticity (E) matrices, respectively, and summed over all layers (i = 1, …, 3). These take the form:
(4a)
(4b)
where D is a spatial differential operator on the shape functions. Note if frequency-dependent properties are used for the viscoelastic material (E(f) and/or η(f), as in Table 2), the stiffness matrix will be frequency-dependent, K(f), as well.
External, non-conservative forcing exists from the active patches and from the disturbance forces, in the form of line moments at the patch boundaries [4] and point loading from the disturbance. These are written as N × 1 forcing vectors Qc and Qd by defining a functional form as follows:
(5a)
(5b)
Here, F is the total distributed non-conservative force, including effects from disturbance and control inputs. In the case of the disturbance, F is simply the transverse point force, and for the active patches, it is the equivalent force-couple induced by the line moment. Qc and Qd are typically phase-linked.
Applying Lagrange's equation, we have , and assuming harmonic excitation and response (at angular frequency ω= 2πf), the equation of motion is written as:
(6)
where tildes (∼) refer to complex-valued quantities (due to loss factor and phase linked inputs). The complex solution vector q is computed from Eq. (6) provided that the system matrix on the left-hand side is invertible. The unforced eigensolution can be computed for [ ] and mode shapes, natural frequencies (ωi), and modal loss factors (ηi) are computed from the complex eigenvalues and eigenvectors. The flexural displacement at any point, w(x, y), is computed from q using Eq. (2).
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VERIFICATION OF ANALYTICAL MODEL While the analytical model derived previously is valid for a rectangular plate, most applications (including the casing cover) have more complicated geometries. To simplify the shape slightly from the one shown in Figure 5, a representative geometry is chosen with two semicircular arcs connected by straight lines, as denoted in Figure 8a. In order to account for this curved boundary, a spatially-varying Young's modulus, E (x, y), can be used for the base plate (layer 3). If regions of the rectangular plate near the fixed boundary are significantly stiffened, they will behave approximately like fixed boundaries. This procedure is illustrated in Figure 8b, where the corners are assumed to have E = 10 Esteel.
Natural frequencies are computed from the finite element model and the analytical formulation for both a rectangular fixed boundary (from Figure 6) and the curved fixed boundary from Figure 8a and are listed in Table 3. The analytical model agrees with FEM within 1% for the rectangular boundary and within 5% for the curved boundary. Mode shapes of the curved boundary geometry from both FEM and theory are displayed for reference in Figure 8c and 8d for the (2, 1) mode, where good agreement is again observed. This gives confidence in the spatially-varying material property method, which will be implemented for the remainder of the paper.
Table 3. Natural frequencies of two plates from analytical model and FEM
Figure 8.
Figure 8. (cont.) Modified cover geometry: (a) simplified outline; (b) stiffening of rectangular plate to approximate curved boundaries; (c) (2, 1) mode shape from finite element model; (d) (2, 1) mode shape from analytical model
PATCH CASE STUDIES USING ANALYTICAL MODEL To investigate the effect of active and passive damping patches for noise and vibration reduction, three cases will be studied and compared to the base case of an undamped plate: (I) one passive patch, (II) one active patch, and (III) one passive patch with one
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active patch (combination of cases I and II). The locations of the patches for the three cases are shown pictorially in Figure 9. Disturbance force and measurement locations are also labeled in Figure 9.
Figure 9. Patch, disturbance, and measurement locations for 3 cases, passive patches ( ), active patches ( ), disturbance ( ), and measurement (+): (a) case I; (b) case II; (c) case III
For case I, a disturbance force is applied to the model and the accelerance spectra are determined with and without a passive patch. The passive patch location is chosen to target the (2, 1) mode, since from [6], locations of large strain correspond to good damping patch locations. The accelerance magnitude spectra are plotted in Figure 10, where significant attenuation (approximately 17 dB) is seen at the
second mode. This gives confidence in the chosen patch location. Modal loss factors can also be computed from the analytical model. Assuming a structural loss factor for steel of 0.005, η2 = 0.005 with no damping patch. Using the viscoelastic loss factor from Table 2 (obtained from a bench test), with the addition of the case I damping patch the model predicts η2 = 0.017 (> 200% increase). Finally, additional, albeit less significant attenuation is observed at other, non-targeted modes, such as the (3, 1) mode.
Figure 10. Simulated accelerance magnitudes for plate: undamped ( ); case I ( )
For case II, the disturbance force is applied at a single frequency corresponding to the (2, 1) mode, at 599 Hz. The active patch is used to induce destructive interference by introducing vibration at 599 Hz and out of phase from the disturbance. This is equivalent to ensuring that the acceleration due to the disturbance at some point (x0, y0), ad, is equal to and opposite the acceleration at (x0, y0) due to the control, ac. The flexural displacement, w(x, y), can be computed from Eqs. (2) and (6) and acceleration is related to displacement (for harmonic motion) as: a(ω) = −ω2w(ω). Therefore, a transfer function relating the required control parameters (magnitude and phase) to the disturbance input is derived such that ad (x0, y0) = −ac (x0, y0) is satisfied.
This transfer function, relating control strain (εc = Vc d31, where Vc is control voltage and d31 is piezoelectric constant) to disturbance force is plotted in Figure 11a. The magnitude of this transfer function has sharp peaks and a dynamic range on the order of 80 dB. The phase of the transfer function is typically close to either 0° or ±180°. Then, using the calculated control amplitude and phase for a given frequency (here, 599 Hz), the active control is applied at a single frequency. The achieved acceleration attenuation from the model is shown in Figure 11b, where the control phase is also swept from −180° to 180° for reference. At the optimal control phase, approximately 40 dB attenuation is observed, whereas 180° opposite, 6 dB increase (consistent with amplitude doubling by constructive interference) is seen.
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Figure 11. Case 2 active patch model results: (a) control input transfer function; (b) single frequency acceleration given different inputs: disturbance alone ( ), active patch alone ( ), combined ( )
Finally, case III is investigated, in which the passive patch from case I and the active patch from case II are combined. The same control transfer function (as from Figure 11a) is computed and both the case II and case III transfer functions are plotted in Figure 12a for comparison. Similarly, the single frequency active control simulation is performed for the (2, 1) mode and the case II and III results are shown in Figure 12b.
The transfer function has a similar shape, but the sharp peaks in the amplitude are significantly decreased and the range is reduced to around 60 dB. This means by adding sufficient damping, large control amplitudes are not required, improving controllability that would be inhibited by voltage limitations. Lower overall control amplitudes could also improve controller stability in the case of simple proportional feedback control. As for the single frequency simulation, the results are similar to case II except that the entire curve is shifted down by approximately 17 dB, consistent with the reduction observed to the (2, 1) mode in case I. The optimal phase angle for active control is also shifted by around 10°. This suggests that while there are some complicated interactions between control and disturbance induced by the presence of the passive patches (especially seen in the phase of the transfer function in Figure 12b), the reduction effects are approximately additive in some cases.
Figure 12. Combined active and passive patch results using the analytical model, case II ( ), case III ( ): (a) control input transfer function, effects of increasing damping denoted by arrows ( ); (b) single frequency acceleration
EXPERIMENTAL STUDIES WITH PASSIVE AND ACTIVE PATCHES The cases from the preceding section are experimentally implemented with the setup shown in Figures 2a and 2b. An impulse hammer (case I) or electrodynamic shaker (cases II and III) is used to introduce the disturbance input. A passive patch consisting of a steel constraining layer and a viscoelastic adhesive core (detailed in Table 2) [9] is applied in the case I location and a macro-fiber composite PZT piezoelectric active patch [10] is applied to the case II location.
For the case I comparison, accelerance magnitude spectra measured with and without the passive damping patch are plotted in Figure 13a, where the (2, 1) mode experiences attenuation of 8 dB insertion loss, comparable to the analytical prediction. Also, as found with the model, modes other than the (2, 1) mode are not attenuated as significantly. Furthermore, the modal loss factor, η2 increases from 0.002 to approximately 0.012. For the case II and III comparisons, the single frequency active control is performed at 530 Hz, near the (2, 1) mode. The shaker is used to excite the casing cover at the disturbance location, the active patch is excited at the optimal amplitude, with phase swept from −180° to 180°, and results are
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plotted in Figure 13b. For the undamped case (case II), approximately 40 dB attenuation is observed at −168° (slightly different from the expected −180° due to phase lag from capacitance in the patch). For the damped case (case III), similar attenuation is seen to case II, but the whole curve is shifted down by approximately 17 dB, showing good agreement to the model results of Figure 12b. A slight shift in the optimal phase is also observed.
Figure 13. Experimental results with passive and active patches: (a) accelerance spectra, undamped ( ) vs. case I ( ); (b) single frequency active control, case II ( ) vs. case III ( )
PASSIVE AND ACTIVE PATCH GUIDELINES Based on the analytical and experimental results presented in the previous sections, there is a clear benefit from combining active and passive damping patches. Passive damping alone is capable of inducing significant damping at certain targeted modes and in some cases multiple modes if patch location is chosen properly. The attenuation can be limited, however, when considering only partial coverage (in case I, approximately 5% coverage by the passive patch is considered). Active patches on the other hand can provide as much as 40 dB insertion loss at a single frequency, however they are more expensive, require a power supply and controller, and have only been shown to provide narrow-band attenuation. The conjunction of active and passive patches, however, not only adds additional attenuation to the active control, it has been shown to improve the ability of control as well. Control amplitudes necessary for control can be reduced while also increasing controller stability for simple control schemes. Passive damping should also improve vibration behavior away from the single frequency targeted by the active patch with proper choice of location.
General guidelines for patch placement are derived by Kung and Singh [11] and Crawley and de Luis [12] for passive and active patches, respectively. These results are found to be consistent with the model presented here for the combined patch case. The guidelines are summarized as follows. Passive damping patches induce the most damping (and correspondingly the highest modal loss factor) at locations of maximum surface strain for a given mode shape. For a fixed (or approximately fixed) plate, these will occur near boundaries or at anti-nodes. Since classical boundaries are rarely encountered in practice, anti-nodes away from boundaries are recommended if the application permits. If anti-nodes common to multiple modes of interest exist, these are optimal locations. Active patches introduce a line moment at patch boundaries. Since internal moment is proportional to normal strain, they are thus able to introduce the greatest motion to the structure when patch boundaries are near points of large strain. Again, if anti-nodes common to multiple modes of interest exist, these should be good active patch locations. Active and passive patch locations may be chosen to target the same mode, so as to maximize the attenuation at one frequency of interest, or they may be chosen for different modes to achieve a more broadband damping effect.
CONCLUSION This article proposes a design-oriented analytical model for studying the effects of active and passive damping patches on thin plate-like structures, using a transmission casing cover as an example. Casing covers are efficient radiators of sound, in particular in the mid- frequency range when excited by gear mesh frequencies via structural path interactions. With the growing trend of lightweighting in vehicle components, unwanted noise and vibration will continue to be a major issue. While small damping patches are ideal candidates for such applications, prior research is sparse on the concurrent side-by- side active and passive damping patch approach. This model provides an accurate and efficient method to investigate patch behavior from a design perspective and to study the effects of combined active and passive patches. The formulation allows for parametric design studies or patch location case studies to be performed computationally much more quickly than traditional finite element methods.
While the model presented here has the potential to be quite effective for certain applications, it does have some limitations. For instance, complicated plate shapes can be handled with spatially varying material properties, however very complex or 3-dimensional components cannot be studied. Furthermore, assumptions (made by most models) such as classical boundaries, perfect patch adhesion, and linearity have been made which will pose problems with some applications. Nevertheless, when used in conjunction with other tools such as FEM, the proposed model provides valuable insight into the physics of patch damping and vibration control problems.
REFERENCES 1. Crimaldi, D. and Singh, R., “Vibro-Acoustic Studies of Transmission
Casing Structures,” ASME Design Engineering Technical Conference, Atlanta, GA, September 13-16, 1998, Paper No. DETC98/PTG-5788.
2. Kim, J. and Singh, R., “Effect of Viscoelastic Patch Damping on Casing Cover Dynamics,” SAE Technical Paper 2001-01-1463, 2001, doi:10.4271/2001-01-1463.
3. Hosberg, J. and Le Coent, A., “Explicit Solution Format for Complex- Valued Natural Frequency of Beam with R-Shunted Piezoelectric Laminate Transducer,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228(1):31-44, 2014, doi:10.1177/0954406213480615
Plattenburg et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 8, Issue 3 (September 2015)
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5. Benjeddou A., “Advances in Hybrid Active-Passive Vibration and Noise Control via Piezoelectric and Viscoelastic Constrained Layer Treatments,” Journal of Vibration and Control 7:565-602, 2001, doi:10.1177/107754630100700406
6. Kung, S. W., and Singh, R., “Complex Eigensolutions of Rectangular Plates with Damping Patches,” Journal of Sound and Vibration 216(1):1-28, doi:10.1006/jsvi.1998.1644
7. Plattenburg, J., Dreyer, J. T., and Singh, R., “Active and Passive Damping Patches on a Thin Rectangular Plate: A Refined Analytical Model with Experimental Validation,” Journal of Sound and Vibration, accepted 2015 subject to minor revisions.
8. Meirovitch, L., “Fundamentals of Vibrations,” (New York, McGraw Hill, 2001), 383-408, ISBN: 0-07-288180
9. 3M Scotch, http://www.scotchbrand.com/, accessed Dec. 4, 2014 10. Smart Material Corp., http://www.smart-material.com/, accessed Dec. 4,
2014 11. Kung, S. W., and Singh, R., “Development of Approximate Methods for
the Analysis of Patch Damping Design Concepts,” Journal of Sound and Vibration 219(5):785-812, doi:10.1006/jsvi.1998.1876
12. Crawley, E. F., and de Luis, J., “Use of Piezoelectric Actuators as Elements of Intelligent Structures,” AIAA Journal 25(10):1373-1385, doi:10.2514/3.9792
CONTACT INFORMATION Professor Rajendra Singh Acoustics and Dynamics Laboratory NSF I/UCRC Smart Vehicle Concepts Center Dept. of Mechanical and Aerospace Engineering The Ohio State University [email protected] Phone: 614-292-9044 www.AutoNVH.org http://svc.engineering.osu.edu/
ACKNOWLEDGMENTS The authors would like to thank the OSU Graduate School, the Ohio Space Grant Consortium, the Smart Vehicle Concepts Center (www. SmartVehicleCenter.org), and the National Science Foundation Industry/University Cooperative Research Centers program (www. nsf.gov/eng/iip/iucrc) for supporting this work through graduate fellowships and financial assistance.
DEFINITIONS a - acceleration
E - Young's modulus
E - elasticity matrix
q - shape function weighting coefficient
q - shape function weighting vector
Q - generalized force vector
X, Y - one-dimensional shape function
β - wave number
ε - normal strain
η - loss factor
ν - Poisson's ratio
- shape function
d - disturbance input
i - layer index
K - kinetic (energy)
P - potential or strain (energy)
0 - measurement location
T - matrix transpose
- functional form for generalized forcing
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Passive constrained layer damping patches have been shown to be an effective damping solution for these types of problems [1-2]. Such patches typically use a viscoelastic core with a large material loss factor (tuned to a given temperature range) and dissipate vibrational energy via shear deformation. Certain active patches (in particular piezoelectrics) have also been studied for vibration reduction applications both passively, in a shunted-circuit configuration [3], as well as actively, in a destructive interference sense [4]. The latter method introduces vibration to the system via a control signal intended to interfere with unwanted vibration from a disturbance (in
this case, structure-borne vibration from the powertrain). Combined active and passive methods have been proposed previously in the form of “active constrained layer damping” [5] with active materials embedded in traditional passive damping patches. However, little research has been done on concurrent “side-by-side” active and passive damping to find the effect of passive damping patches on control from an active patch. As such, improved modeling methods are needed to determine the potential benefits of such patch configurations, as well as to provide insight to the design process.
While both active and passive patch methods have been applied to a variety of simple structures including classical beams, plates, and shells [3, 4, 5, 6], realistic components are more difficult to model as they often include curvature, features such as fillets and varying thickness, and complex boundaries such as bolted joints. Accordingly, this paper seeks to address this void in terms of application-based modeling with the following objectives. (1) Perform an experiment to determine the dynamic behavior of a transmission casing cover. (2) Propose a design-oriented analytical method for modeling of active and passive patches (in the “side-by-side” configuration) to be implemented on the plate-like cover. (3) Quantify the efficacy of active and passive patches for reducing vibration in terms of modal loss factor and insertion loss, and evaluate other qualitative benefits of combined active and passive methods. (4) Validate the model with an experiment. (5) Propose design guidelines in terms of patch location and placement for given operating conditions.
Modeling of Active and Passive Damping Patches with Application to a Transmission Casing Cover
Joseph Plattenburg, Jason Dreyer, and Rajendra Singh Ohio State University
ABSTRACT Combined active and passive damping is a recent trend that can be an effective solution to challenging NVH problems, especially for lightweight vehicle components that demand advanced noise and vibration treatments. Compact patches are of particular interest due to their small size and cost, however, improved modeling techniques are needed at the design stage for such methods. This paper presents a refined modeling procedure for side-by-side active and passive damping patches applied to thin, plate-like, powertrain casing structures. As an example, a plate with fixed boundaries is modeled as this is representative of real-life transmission covers which often require damping treatments. The proposed model is then utilized to examine several cases of active and passive patch location, and vibration reduction is determined in terms of insertion loss for each case. Results are compared to an experiment with an actual transmission casing for validation, using piezoelectric active patches and constrained-layer passive patches with a viscoelastic core. Conclusions are drawn about patch size and location in terms of NVH reduction capability, and guidelines are suggested for the dynamic design process.
CITATION: Plattenburg, J., Dreyer, J., and Singh, R., "Modeling of Active and Passive Damping Patches with Application to a Transmission Casing Cover," SAE Int. J. Passeng. Cars - Mech. Syst. 8(3):2015, doi:10.4271/2015-01-2261.
2015-01-2261 Published 06/15/2015
Copyright © 2015 SAE International doi:10.4271/2015-01-2261 saepcmech.saejournals.org
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Figure 1.
Figure 1. (cont.) Transmission casing and cover: (a) production sample; (b) simplified version for experimental study; (c) detail of production cover; (d) detail of simplified cover
Accelerance measurements, defined as acceleration per unit force, are made of the cover in the in-situ condition (bolted to the casing) with an impulse hammer. A schematic of this procedure, as well as force and measurement locations, are shown in Figure 2a.
Figure 2. Schematic of experimental procedure on casing cover: (a) impact hammer with driving point location shown (xd, yd) ≈ (0.44 Lx, 0.71 Ly); (b) shaker input for driving point measurement
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Figure 2. (cont.) Schematic of experimental procedure on casing cover: (a) impact hammer with driving point location shown (xd, yd) ≈ (0.44 Lx, 0.71 Ly); (b) shaker input for driving point measurement
Driving-point and cross-point measurements are made at 28 different locations (using roving-hammer type measurement methods) in order to extract mode shapes. A least-squares parameter estimation technique is used to obtain natural frequencies, modal loss factors, and mode shapes. The accelerance spectra (driving point) for both cover thicknesses are plotted in Figure 3. Due to better experimental results and modal correlation for the thin cover, this case will be used for the remainder of the paper. Furthermore, the second, or (2, 1) mode, will be considered as the “mode of interest” because it is well isolated, has low damping and a well-defined mode shape, and it is near the middle of the targeted frequency range. Natural frequencies for each mode, along with modal damping ratio and modal index are given in Table 1. Modal indices are defined as (m, n), where m and n are the total number of local extrema in the long (x) and short (y) dimensions, respectively. As an example, the measured flexural displacement of the (2, 1) mode is shown in Figure 4a.
Figure 3.
Figure 3. (cont.)Experimental driving-point accelerance magnitude spectra of casing cover: (a) thin cover, thickness = 1.42 mm; (b) thick cover, thickness = 2.92 mm
Table 1. Selected modes of thin casing cover
Figure 4. Flexural displacement magnitude at (2, 1) mode: (a) measured, isometric and side views; (b) finite element model
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FINITE ELEMENT MODEL OF CASING COVER In order to characterize the nature of the boundaries for the in-situ configuration of the cover (and to determine if fixed boundaries are an appropriate approximation to the bolted connection), a finite element model (FEM) is developed in Abaqus (shown in Figure 5b). The computational model is assumed to be a thin plate (modeled with 2D shell elements) fixed at the boundaries. The plate boundary is approximated as the inside of the contact region between the casing and the cover (illustrated in Figure 5a). Material properties used for the computational model are given in Table 2, where hi is plate thickness, Ei is the Young's modulus, ρi is the density, ηi is the loss factor, νi is the Poisson's ratio, and i refers to layers 1 (passive constraining layer), 2 (viscoelastic layer), and 3 (base structure).
Table 2. Material properties of steel cover and passive patches
The natural frequencies of the finite element model (no patches, layer 3 only) are computed from the eigensolution and are listed in Table 1, and the deformation of the (2, 1) mode is given in Figure 4b. These natural frequencies are found to be significantly larger than those measured from the experiment. This is because while the computational model assumes perfectly fixed boundaries (i.e. infinite stiffness), the bolted boundaries of the experiment have a finite stiffness associated with them. A more accurate finite element model could be developed including bolt stiffness and other contact effects as demonstrated in [1]. Nevertheless, using the current model the modes are well correlated, as seen in Figure 4, so the assumed fixed-boundary approximation will suffice for the scope of this work. Such a classical boundary approximation will make the analytical model more tractable and easier to implement.
Figure 5.
Figure 5. (cont.) Fixed-boundary approximation for the cover: (a) casing- cover interface ( ); (b) finite element approximation of fixed boundary
ANALYTICAL MODEL The analytical model proposed here is based on the work of Kung and Singh [6] and extended by Plattenburg et al. [7] (where a more detailed analysis is given). Consider a thin rectangular steel plate of dimensions Lx × Ly with fixed boundaries and a number of active and passive patches. A schematic of the structure is shown in Figure 6.
Figure 6. Thin fixed rectangular plate with active and passive patches: example case for analytical study
Note that passive patches consist of two layers (steel constraining layer and viscoelastic core) and active patches consist of a patch on the top and bottom surface of the plate (assumed to not affect ρ3 or E3). The ith layer has thickness hi, as well as material properties Ei, ρi, ηi, and νi, where i = 1, …, 3. Material properties are listed in Table 2 (note that the viscoelastic material loss factor and Young's modulus are dependent on frequency, f). A disturbance force, Fd and measurement, w, are also included in the Figure 6 schematic.
The motion of the base layer (3) is written as a summation of shape functions similar to modal expansion [8]. The shape functions, (x, y), must satisfy the fixed boundary conditions, so they are assumed to be separable (i.e. m,n(x, y) = Xm(x)Yn(y)), and to take the form of standard fixed-fixed beam modes (illustrated in Figure 7a):
(1)
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where βm is the wave number that satisfies cosh(βmLx) cos(βmLx) = 1, Cm is a normalization constant such that Xm(x) has unity mean square value on [0, Lx], m and n are the same modal indices defined previously, and Yn(y), Cn, and βn follow analogous definitions [8].
Figure 7. Shape functions for the example of Figure 6: (a) sample functions for fixed boundaries with m = 1 ( ), m = 2 ( ), m = 3 ( ); (b) arbitrary displacement decomposed into weighted summation of shape functions
It is assumed that any flexural displacement profile of the plate, w(x, y), may be written as a weighted summation of N of these shape functions (depicted in Figure 7b):
(2)
where Φ is a 1 × N vector of shape functions and q is a N × 1 vector of weighting coefficients. Furthermore, assume that (i) all three layers have the same flexural displacement, (ii) shear in layers 1 and 3 is negligible, and (iii) all other motions (in-plane, rotation, and layer 2 shear) can be derived from w(x, y) via kinematic relationships as described in [6]. Now we have all motions of all layers (where flexure is of the most interest) with the weighting coefficients as the only unknowns.
The kinetic and potential energy ( and , respectively) of the system (neglecting rotary inertia) is written in the form:
(3a-b)
where M and K are equivalent mass and stiffness matrices defined as surface integrals of the shape function vectors (over the area of the ith layer, Ai) scaled by inertia (H) and elasticity (E) matrices, respectively, and summed over all layers (i = 1, …, 3). These take the form:
(4a)
(4b)
where D is a spatial differential operator on the shape functions. Note if frequency-dependent properties are used for the viscoelastic material (E(f) and/or η(f), as in Table 2), the stiffness matrix will be frequency-dependent, K(f), as well.
External, non-conservative forcing exists from the active patches and from the disturbance forces, in the form of line moments at the patch boundaries [4] and point loading from the disturbance. These are written as N × 1 forcing vectors Qc and Qd by defining a functional form as follows:
(5a)
(5b)
Here, F is the total distributed non-conservative force, including effects from disturbance and control inputs. In the case of the disturbance, F is simply the transverse point force, and for the active patches, it is the equivalent force-couple induced by the line moment. Qc and Qd are typically phase-linked.
Applying Lagrange's equation, we have , and assuming harmonic excitation and response (at angular frequency ω= 2πf), the equation of motion is written as:
(6)
where tildes (∼) refer to complex-valued quantities (due to loss factor and phase linked inputs). The complex solution vector q is computed from Eq. (6) provided that the system matrix on the left-hand side is invertible. The unforced eigensolution can be computed for [ ] and mode shapes, natural frequencies (ωi), and modal loss factors (ηi) are computed from the complex eigenvalues and eigenvectors. The flexural displacement at any point, w(x, y), is computed from q using Eq. (2).
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VERIFICATION OF ANALYTICAL MODEL While the analytical model derived previously is valid for a rectangular plate, most applications (including the casing cover) have more complicated geometries. To simplify the shape slightly from the one shown in Figure 5, a representative geometry is chosen with two semicircular arcs connected by straight lines, as denoted in Figure 8a. In order to account for this curved boundary, a spatially-varying Young's modulus, E (x, y), can be used for the base plate (layer 3). If regions of the rectangular plate near the fixed boundary are significantly stiffened, they will behave approximately like fixed boundaries. This procedure is illustrated in Figure 8b, where the corners are assumed to have E = 10 Esteel.
Natural frequencies are computed from the finite element model and the analytical formulation for both a rectangular fixed boundary (from Figure 6) and the curved fixed boundary from Figure 8a and are listed in Table 3. The analytical model agrees with FEM within 1% for the rectangular boundary and within 5% for the curved boundary. Mode shapes of the curved boundary geometry from both FEM and theory are displayed for reference in Figure 8c and 8d for the (2, 1) mode, where good agreement is again observed. This gives confidence in the spatially-varying material property method, which will be implemented for the remainder of the paper.
Table 3. Natural frequencies of two plates from analytical model and FEM
Figure 8.
Figure 8. (cont.) Modified cover geometry: (a) simplified outline; (b) stiffening of rectangular plate to approximate curved boundaries; (c) (2, 1) mode shape from finite element model; (d) (2, 1) mode shape from analytical model
PATCH CASE STUDIES USING ANALYTICAL MODEL To investigate the effect of active and passive damping patches for noise and vibration reduction, three cases will be studied and compared to the base case of an undamped plate: (I) one passive patch, (II) one active patch, and (III) one passive patch with one
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active patch (combination of cases I and II). The locations of the patches for the three cases are shown pictorially in Figure 9. Disturbance force and measurement locations are also labeled in Figure 9.
Figure 9. Patch, disturbance, and measurement locations for 3 cases, passive patches ( ), active patches ( ), disturbance ( ), and measurement (+): (a) case I; (b) case II; (c) case III
For case I, a disturbance force is applied to the model and the accelerance spectra are determined with and without a passive patch. The passive patch location is chosen to target the (2, 1) mode, since from [6], locations of large strain correspond to good damping patch locations. The accelerance magnitude spectra are plotted in Figure 10, where significant attenuation (approximately 17 dB) is seen at the
second mode. This gives confidence in the chosen patch location. Modal loss factors can also be computed from the analytical model. Assuming a structural loss factor for steel of 0.005, η2 = 0.005 with no damping patch. Using the viscoelastic loss factor from Table 2 (obtained from a bench test), with the addition of the case I damping patch the model predicts η2 = 0.017 (> 200% increase). Finally, additional, albeit less significant attenuation is observed at other, non-targeted modes, such as the (3, 1) mode.
Figure 10. Simulated accelerance magnitudes for plate: undamped ( ); case I ( )
For case II, the disturbance force is applied at a single frequency corresponding to the (2, 1) mode, at 599 Hz. The active patch is used to induce destructive interference by introducing vibration at 599 Hz and out of phase from the disturbance. This is equivalent to ensuring that the acceleration due to the disturbance at some point (x0, y0), ad, is equal to and opposite the acceleration at (x0, y0) due to the control, ac. The flexural displacement, w(x, y), can be computed from Eqs. (2) and (6) and acceleration is related to displacement (for harmonic motion) as: a(ω) = −ω2w(ω). Therefore, a transfer function relating the required control parameters (magnitude and phase) to the disturbance input is derived such that ad (x0, y0) = −ac (x0, y0) is satisfied.
This transfer function, relating control strain (εc = Vc d31, where Vc is control voltage and d31 is piezoelectric constant) to disturbance force is plotted in Figure 11a. The magnitude of this transfer function has sharp peaks and a dynamic range on the order of 80 dB. The phase of the transfer function is typically close to either 0° or ±180°. Then, using the calculated control amplitude and phase for a given frequency (here, 599 Hz), the active control is applied at a single frequency. The achieved acceleration attenuation from the model is shown in Figure 11b, where the control phase is also swept from −180° to 180° for reference. At the optimal control phase, approximately 40 dB attenuation is observed, whereas 180° opposite, 6 dB increase (consistent with amplitude doubling by constructive interference) is seen.
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Figure 11. Case 2 active patch model results: (a) control input transfer function; (b) single frequency acceleration given different inputs: disturbance alone ( ), active patch alone ( ), combined ( )
Finally, case III is investigated, in which the passive patch from case I and the active patch from case II are combined. The same control transfer function (as from Figure 11a) is computed and both the case II and case III transfer functions are plotted in Figure 12a for comparison. Similarly, the single frequency active control simulation is performed for the (2, 1) mode and the case II and III results are shown in Figure 12b.
The transfer function has a similar shape, but the sharp peaks in the amplitude are significantly decreased and the range is reduced to around 60 dB. This means by adding sufficient damping, large control amplitudes are not required, improving controllability that would be inhibited by voltage limitations. Lower overall control amplitudes could also improve controller stability in the case of simple proportional feedback control. As for the single frequency simulation, the results are similar to case II except that the entire curve is shifted down by approximately 17 dB, consistent with the reduction observed to the (2, 1) mode in case I. The optimal phase angle for active control is also shifted by around 10°. This suggests that while there are some complicated interactions between control and disturbance induced by the presence of the passive patches (especially seen in the phase of the transfer function in Figure 12b), the reduction effects are approximately additive in some cases.
Figure 12. Combined active and passive patch results using the analytical model, case II ( ), case III ( ): (a) control input transfer function, effects of increasing damping denoted by arrows ( ); (b) single frequency acceleration
EXPERIMENTAL STUDIES WITH PASSIVE AND ACTIVE PATCHES The cases from the preceding section are experimentally implemented with the setup shown in Figures 2a and 2b. An impulse hammer (case I) or electrodynamic shaker (cases II and III) is used to introduce the disturbance input. A passive patch consisting of a steel constraining layer and a viscoelastic adhesive core (detailed in Table 2) [9] is applied in the case I location and a macro-fiber composite PZT piezoelectric active patch [10] is applied to the case II location.
For the case I comparison, accelerance magnitude spectra measured with and without the passive damping patch are plotted in Figure 13a, where the (2, 1) mode experiences attenuation of 8 dB insertion loss, comparable to the analytical prediction. Also, as found with the model, modes other than the (2, 1) mode are not attenuated as significantly. Furthermore, the modal loss factor, η2 increases from 0.002 to approximately 0.012. For the case II and III comparisons, the single frequency active control is performed at 530 Hz, near the (2, 1) mode. The shaker is used to excite the casing cover at the disturbance location, the active patch is excited at the optimal amplitude, with phase swept from −180° to 180°, and results are
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plotted in Figure 13b. For the undamped case (case II), approximately 40 dB attenuation is observed at −168° (slightly different from the expected −180° due to phase lag from capacitance in the patch). For the damped case (case III), similar attenuation is seen to case II, but the whole curve is shifted down by approximately 17 dB, showing good agreement to the model results of Figure 12b. A slight shift in the optimal phase is also observed.
Figure 13. Experimental results with passive and active patches: (a) accelerance spectra, undamped ( ) vs. case I ( ); (b) single frequency active control, case II ( ) vs. case III ( )
PASSIVE AND ACTIVE PATCH GUIDELINES Based on the analytical and experimental results presented in the previous sections, there is a clear benefit from combining active and passive damping patches. Passive damping alone is capable of inducing significant damping at certain targeted modes and in some cases multiple modes if patch location is chosen properly. The attenuation can be limited, however, when considering only partial coverage (in case I, approximately 5% coverage by the passive patch is considered). Active patches on the other hand can provide as much as 40 dB insertion loss at a single frequency, however they are more expensive, require a power supply and controller, and have only been shown to provide narrow-band attenuation. The conjunction of active and passive patches, however, not only adds additional attenuation to the active control, it has been shown to improve the ability of control as well. Control amplitudes necessary for control can be reduced while also increasing controller stability for simple control schemes. Passive damping should also improve vibration behavior away from the single frequency targeted by the active patch with proper choice of location.
General guidelines for patch placement are derived by Kung and Singh [11] and Crawley and de Luis [12] for passive and active patches, respectively. These results are found to be consistent with the model presented here for the combined patch case. The guidelines are summarized as follows. Passive damping patches induce the most damping (and correspondingly the highest modal loss factor) at locations of maximum surface strain for a given mode shape. For a fixed (or approximately fixed) plate, these will occur near boundaries or at anti-nodes. Since classical boundaries are rarely encountered in practice, anti-nodes away from boundaries are recommended if the application permits. If anti-nodes common to multiple modes of interest exist, these are optimal locations. Active patches introduce a line moment at patch boundaries. Since internal moment is proportional to normal strain, they are thus able to introduce the greatest motion to the structure when patch boundaries are near points of large strain. Again, if anti-nodes common to multiple modes of interest exist, these should be good active patch locations. Active and passive patch locations may be chosen to target the same mode, so as to maximize the attenuation at one frequency of interest, or they may be chosen for different modes to achieve a more broadband damping effect.
CONCLUSION This article proposes a design-oriented analytical model for studying the effects of active and passive damping patches on thin plate-like structures, using a transmission casing cover as an example. Casing covers are efficient radiators of sound, in particular in the mid- frequency range when excited by gear mesh frequencies via structural path interactions. With the growing trend of lightweighting in vehicle components, unwanted noise and vibration will continue to be a major issue. While small damping patches are ideal candidates for such applications, prior research is sparse on the concurrent side-by- side active and passive damping patch approach. This model provides an accurate and efficient method to investigate patch behavior from a design perspective and to study the effects of combined active and passive patches. The formulation allows for parametric design studies or patch location case studies to be performed computationally much more quickly than traditional finite element methods.
While the model presented here has the potential to be quite effective for certain applications, it does have some limitations. For instance, complicated plate shapes can be handled with spatially varying material properties, however very complex or 3-dimensional components cannot be studied. Furthermore, assumptions (made by most models) such as classical boundaries, perfect patch adhesion, and linearity have been made which will pose problems with some applications. Nevertheless, when used in conjunction with other tools such as FEM, the proposed model provides valuable insight into the physics of patch damping and vibration control problems.
REFERENCES 1. Crimaldi, D. and Singh, R., “Vibro-Acoustic Studies of Transmission
Casing Structures,” ASME Design Engineering Technical Conference, Atlanta, GA, September 13-16, 1998, Paper No. DETC98/PTG-5788.
2. Kim, J. and Singh, R., “Effect of Viscoelastic Patch Damping on Casing Cover Dynamics,” SAE Technical Paper 2001-01-1463, 2001, doi:10.4271/2001-01-1463.
3. Hosberg, J. and Le Coent, A., “Explicit Solution Format for Complex- Valued Natural Frequency of Beam with R-Shunted Piezoelectric Laminate Transducer,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228(1):31-44, 2014, doi:10.1177/0954406213480615
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5. Benjeddou A., “Advances in Hybrid Active-Passive Vibration and Noise Control via Piezoelectric and Viscoelastic Constrained Layer Treatments,” Journal of Vibration and Control 7:565-602, 2001, doi:10.1177/107754630100700406
6. Kung, S. W., and Singh, R., “Complex Eigensolutions of Rectangular Plates with Damping Patches,” Journal of Sound and Vibration 216(1):1-28, doi:10.1006/jsvi.1998.1644
7. Plattenburg, J., Dreyer, J. T., and Singh, R., “Active and Passive Damping Patches on a Thin Rectangular Plate: A Refined Analytical Model with Experimental Validation,” Journal of Sound and Vibration, accepted 2015 subject to minor revisions.
8. Meirovitch, L., “Fundamentals of Vibrations,” (New York, McGraw Hill, 2001), 383-408, ISBN: 0-07-288180
9. 3M Scotch, http://www.scotchbrand.com/, accessed Dec. 4, 2014 10. Smart Material Corp., http://www.smart-material.com/, accessed Dec. 4,
2014 11. Kung, S. W., and Singh, R., “Development of Approximate Methods for
the Analysis of Patch Damping Design Concepts,” Journal of Sound and Vibration 219(5):785-812, doi:10.1006/jsvi.1998.1876
12. Crawley, E. F., and de Luis, J., “Use of Piezoelectric Actuators as Elements of Intelligent Structures,” AIAA Journal 25(10):1373-1385, doi:10.2514/3.9792
CONTACT INFORMATION Professor Rajendra Singh Acoustics and Dynamics Laboratory NSF I/UCRC Smart Vehicle Concepts Center Dept. of Mechanical and Aerospace Engineering The Ohio State University [email protected] Phone: 614-292-9044 www.AutoNVH.org http://svc.engineering.osu.edu/
ACKNOWLEDGMENTS The authors would like to thank the OSU Graduate School, the Ohio Space Grant Consortium, the Smart Vehicle Concepts Center (www. SmartVehicleCenter.org), and the National Science Foundation Industry/University Cooperative Research Centers program (www. nsf.gov/eng/iip/iucrc) for supporting this work through graduate fellowships and financial assistance.
DEFINITIONS a - acceleration
E - Young's modulus
E - elasticity matrix
q - shape function weighting coefficient
q - shape function weighting vector
Q - generalized force vector
X, Y - one-dimensional shape function
β - wave number
ε - normal strain
η - loss factor
ν - Poisson's ratio
- shape function
d - disturbance input
i - layer index
K - kinetic (energy)
P - potential or strain (energy)
0 - measurement location
T - matrix transpose
- functional form for generalized forcing
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