Piezoelectric materials (PZT) are commonly used as actuators and
sensors for vibration suppression in flexible metal or composite
substrates. There are well-established techniques for modeling the
actuation of PZTs when they are bonded to these structures.
However, if the substrate material is much softer than the
piezoelectric actuator/sensor, a higher level of modeling is
needed to predict the local deformations at the interface.
In this research, a finite-length piezoelectric element bonded
perfectly to an infinite elastic strip is modeled. The specific
goal was to quantify the actuation and sensing mechanics of
piezoelectric devices on substrates potentially much softer than
the piezoelectric element. Previous works have addressed
membranes or plates bonded to an elastic half-space subjected to
mechanical or thermal loads.
Euler-Bernoulli beam theory is used to derive equations of
equilibrium for the piezoelectric beam. These equations are then
recast as integral equations for the interface displacement
gradients and equated to the equivalent quantities for an elastic
layer subject to distributed shear and normal tractions. The
resulting singular integral equations are solved by expanding the
interface tractions using a series of Chebyshev polynomials.
First, certain sanity checks are performed to confirm the validity
of the model by choosing a stiff substrate for which
Euler-Bernoulli beam-assumptions holds good.
For certain combinations of geometrical and material parameters,
the substrate has a positive curvature, whereas the piezoelectric
has a negative curvature and vice versa. After analyzing the
forces acting on both piezoelectric and the substrate, the reasons
for this behavior in soft substrates are justified here. Finally,
the range of geometric parameters where the reversal of bending
occurs in the piezoelectric is given.
Actuators based on smart materials generally exhibit a tradeoff between force and stroke. Researchers have surrounded piezoelectric materials (PZT's) with compliant structures to magnify either their geometric or mechanical advantage. Most of these designs are literally built around a particular piezoelectric device, so the design space consists of only the compliant mechanism. Materials scientists researchers have demonstrated the ability to pole a PZT in an arbitrary direction, and some engineers have taken advantage of this to build "shear mode" actuators.
The goal of this work is to determine if the performance of compliant mechanisms improves by the inclusion of the piezoelectric polarization as a design variable. The polarization vector is varied via transformation matrixes, and the compliant actuator is modeled using the SIMP (Solid Isotropic Material with Penalization) or "power-law method." The concept of mutual potential energy is used to form an objective function to measure the piezoelectric actuator's performance. The optimal topology of the compliant mechanism and orientation of the polarization method are determined using a sequential linear programming algorithm. This paper presents a demonstration problem that shows small changes in the polarization vector have a marginal effect on the optimum topology of the mechanism, but improves actuation.
We have developed a detailed model for a piezoelectric patch bonded perfectly to a semi-infinite substrate. There are well-established techniques for representing the effects of piezoelectric actuation on a flexible substrate by equivalent moments, but the accuracy of moments rely on classical beam behavior in both the actuation and substrate layers. The goal of the work presented here is to present a model capable of predicting both the actuation and sensing ability of a smart material on a general substrate. The piezoelectric layer is modeled by classical beam theory, but no kinematic assumptions other than plane strain are imposed on the substrate. Equilibrium is enforced between the piezoelectric patch and the surface tractions over the interface region, and standard Euler-Bernoulli beam theory is then used to form integral equations in terms of the displacement gradients at the interface with the substrate. Green's functions are then derived for a semi-infinite substrate using techniques from contact mechanics. There is no loss of generality in choosing a semi-infinite substrate since the effects of actuation by a patch disappear quickly outside the contact region. Preliminary results that both validate the current model and support the equivalent-moment action models for certain substrates are presented.
It is common to use piezoelectric materials to reduce vibrations or otherwise alter the dynamics of structures made of metal or composite materials. In contrast, this work addresses modeling of piezoelectric patches applied to a rubber substrate. An underlying goal of modeling, however, is to represent the significant physics of a problem with the simplest model possible. There were several simplified approaches to modeling piezoelectric actuation on classical beam and plate elements developed in the late 1980's and early 1990's. Of these, the pin force, extended pin force, and Euler-Bernoulli methods are assessed in this study. The basic concepts of the three approximation methods are developed, and the curvatures predicted by each is compared to predictions from a special-purpose finite element code. The final conclusion is that the constant-strain approaches (pin force and enhanced pin-force) are not accurate for very soft substrates. Future work includes adding the time dependence of rubber materials as well as the possibility of material of geometric nonlinearities.
Proc. SPIE. 4331, Smart Structures and Materials 2001: Damping and Isolation
KEYWORDS: Control systems, Systems modeling, Performance modeling, Matrices, Control systems design, Feedback control, Thermodynamics, Intelligence systems, Filtering (signal processing), Process control
Researchers have been seeking the best combinations of active and passive damping since the mid 1980's. Much of this work has centered around classical control schemes and has assumed real- or complex-valued approximations of the viscoelastic material properties. The time dependence of viscoelastic materials has been addressed effectively by several researchers using internal variable methods. Most of these methods do yield simple entries in existing mass and stiffness matrices. Rather, the time dependence of the system is represented by coupling of the physical and internal degrees of freedom in a set of first-order equations. Just as with many of the physical states, the internal degrees of freedom are not measurable. This work explores the implications of reconstructing these states through an observer for the purpose of feedback control. After reviewing some basic theory for internal variables, the effect on control effort is demonstrated for systems using output feedback versus full state feedback. This work is a preliminary step toward better accounting for the time dependencies of viscoelastic materials in systems with active control.
The chief tool for design of viscoelastic-based damping treatments over the past 20 years has been the modal strain energy (MSE) approach. This approach to damping design traditionally has involved a practitioner to vary placement and stiffness of add-on elements using experience and trial and error so as to maximize the add-on element's share of system MSE in modes of interest. In this paper we develop a new technique for maximizing strain energy as a function of stiffness for add-on structural elements modeled as rank r perturbations to the original stiffness matrix. The techniques is based on a constrained substructure approach allowing us to parameterize strain energy in terms of the eigenvalues of the perturbed structure. An optimality condition is derived that relates the input-output response at the attachment location of the add-on elements to the maximum achievable strain energy. A realizability condition is also derived which indicates whether or not the optimal solution is achievable with passive structural elements. This method has applications in the design of structural treatments for controlling sound and vibration and promises an efficient means of determining the limits of performance of passive structural treatments. An advantage of our approach over existing methods is that the maximum achievable strain energy fraction in the add-on elements is directly computable with the realizability condition then indicating whether the optimal solution is achievable.
Proc. SPIE. 3672, Smart Structures and Materials 1999: Passive Damping and Isolation
KEYWORDS: Chemical elements, Mathematical modeling, Finite element methods, Tumor growth modeling, Metals, Mathematics, Composites, Linear elements, Sun, Intelligence systems
The concept of enhancing energy dissipation in thin beams and panels by adding viscoelastic materials to a structure dates back at least to the early 1950s. Kerwin in 1959 was the first to present a general analysis of viscoelastic material constrained by another metal layer. He made several key simplifying assumptions in the mathematics, as did DiTaranto (1965) and Mead and Markus (1969) in follow-up studies: (1) the constraining layer bends in the transverse direction exactly as the base layer, (2) the viscoelastic layer undergoes pure shear, and (3) the viscoelastic layer does not change its thickness during deformation. While appropriate for damping problems of that time, the role of passive, and now active, damping has expanded in the decades since to the point that many problems of practical engineering interest are no longer represented well by these mathematical models. This paper explores a few pitfalls of simplified modeling through some trade studies using benign-looking sandwich beams. The Mead and Markus assumptions are implemented using finite elements and are compared to a beam comprised entirely of higher order elements. A sandwich beam is also modeled using Euler-Bernoulli beams (acting independently) as facesheets and a linear element for the viscoelastic material, similar to how a sandwich might be modeled using standard elements in a commercial code. The accuracy of damping predictions is inferred from the accuracy of strain energy distributions.
Much of the work done on active and passive constrained layer beams is done with models using kinematic assumptions proposed by Kerwin, Mead and Markus, and others. Typically these analyses use low-order Euler-Bernoulli beams and assume the base and constraining layers undergo identical transverse displacements. These assumptions are reasonable for cases where the middle layer (normally a relatively soft viscoelastic material) is thin and the constraining layer is relatively weak in bending, but many practical cases arise where these assumptions are violated. A few authors over the years have done studies with less restrictive kinematic assumptions, but none have specifically studied the effects of doing so in the context of passive or active damping design. The field of composite structures is rich with techniques for analyzing sandwich structures with and without simplifying assumptions, and it is on this body of work that this paper is based. The percentage of modal strain energy in the viscoelastic core is used as the primary measure of the accuracy of different sets of assumptions. Elasticity solutions are available for selected sets of assumptions and boundary conditions, and these solutions provide a basis for some of the preliminary studies. A zig-zag method is used to construct a piecewise continuous displacement field (C^{1} continuity) that satisfies the appropriate stress continuity between layers in a consistent manner. Finite element analysis provides a versatile way to simulate complicated combinations of boundary conditions, degree of coverage, and kinematic assumptions.
Much of the work done on active and passive constrained-layer beams is done with models using kinematic assumptions proposed by Kerwin, Mead and Markus, and others. The key assumption is that the base and constraining layers undergo identical transverse displacements, which is a reasonable assumption when the middle layer (here a viscoelastic material) is thin and the constraining layer is relatively weak. There are, however, many practical cases where an effective passive damping design requires the stiffness of the constraining layer to be on the order of that of the base layer. If the base structure is stiff to begin with, a constraining layer that will produce good damping is likely to violate the above stated assumption by refusing to follow the base layer exactly. The question arises as to how this affects predictions of damping. In this work the facesheets are treated as independently deforming Timoshenko beams, which results in a more general state of strain in the core material. Expressions for the potential and kinetic energies are developed from basic principles of continuum mechanics, and the assumed modes method is used to predict how levels of strain energy in the core are affected by the assumptions on the relative motions of the facesheets.
Active constrained layer damping consists of either replacing or augmenting the constraining layer for a viscoelastic material with piezoceramic actuators in an attempt to improve vibration suppression properties by synergism between passive and active damping techniques. An important question in such configurations is whether the reduction in actuation ability of the piezoceramic on the beam due to a relatively soft viscoelastic layer is compensated for by enhanced passive damping due to increased shear in the viscoelastic material. Some tradeoffs between pure passive, pure active control, and active constrained layer damping are discussed here. Velocity feedback and LQR are investigated. Several authors have researched and developed formulations for active constrained layer damping techniques. The approach presented here differs from most in that it employs an energy principle for the equations of a beam with partial active/passive constrained layer damping transients. An offshoot of this is the thickness of the viscoelastic layer can be arbitrarily small (even zero), thus opening up the possibility of simulating the realistic design problem where the optimal sizing, length, and thickness of a treatment is subject to a total thickness restriction. The results show that the active constrained layer damping treatment provides better vibration suppression than passive damping treatments, and it even out performs pure active control for low-gain applications.
A Hughes Space Company study was undertaken to (1) acquire the analytical capability to design effective passive damping treatments and to predict the damped dynamic performance with reasonable accuracy; (2) demonstrate reasonable test and analysis agreement for both baseline and damped baseline hardware; and (3) achieve a 75% reduction in peak transmissibility and 50% reduction in rms random vibration response. Hughes Space Company teamed with CSA Engineering to learn how to apply passive damping technology to their products successfully in a cost-effective manner. Existing hardware was selected for the demonstration because (1) previous designs were lightly damped and had difficulty in vibration test; (2) multiple damping concepts could be investigated; (3) the finite element model, hardware, and test fixture would be available; and (4) damping devices could be easily implemented. Bracket, strut, and sandwich panel damping treatments that met the performance goals were developed by analysis. The baseline, baseline with damped bracket, and baseline with damped strut designs were built and tested. The test results were in reasonable agreement with the analytical predictions and demonstrated that the desired reduction in dynamic response could be achieved. Having successfully demonstrated this approach, it can now be used with confidence for future designs as a means for reducing weight and enhancing reliability.
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