George A. Lesieutre

Damping as a result of piezoelectric energy harvesting

Abstract Systems that harvest or scavenge energy from their environments are of considerable interest for use in remote power supplies. A class of such systems exploits the motion or deformation associated with vibration, converting the mechanical energy to electrical, and storing it for later use; some of these systems use piezoelectric materials for the direct conversion of strain energy to electrical energy. The removal of mechanical energy from a vibrating structure necessarily results in damping. This research addresses the damping associated with a piezoelectric energy harvesting system that consists of a full-bridge rectifier, a filter capacitor, a switching DC–DC step-down converter, and a battery. Under conditions of harmonic forcing, the effective modal loss factor depends on: (1) the electromechanical coupling coefficient of the piezoelectric system; and (2) the ratio of the rectifier output voltage during operation to its maximum open-circuit value. When the DC–DC converter is maximizing power flow to the battery, this voltage ratio is very nearly 1/2, and the loss factor depends only on the coupling coefficient. Experiments on a base-driven piezoelectric cantilever, having a system coupling coefficient of 26%, yielded an effective loss factor for the fundamental vibration mode of 2.2%, in excellent agreement with theory.

Vibration damping and control using shunted piezoelectric materials

Research on shunted piezoelectric materials, conducted mainly over the past decade, provides new options to engineers who are responsible for solving structural vibration control problems. The general method is made possible by the relatively strong electromechanical coupling exhibited by modern piezoelectric materials. If a piezoelectric element is attached to a structure, it is strained as the structure deforms and converts a portion of the energy associated with vibration into electrical energy. The piezoelectric element (which behaves electrically as a capacitor), in combination with a network of electrical elements connected to it (a shunt network), comprises an electrical system that can be configured to accomplish vibration control through its treatment of electrical energy. Four basic kinds of shunt circuits are typically used: resistive, inductive, capacitive, and switched. Each of these kinds of shunts results in characteristically different dynamic behavior: a resistive shunt dissipates energy through Joule heating, which has the effect of structural damping. An inductive shunt results in a resonant LC circuit, the behavior of which is analogous to that of a mechanical vibration absorber (tuned mass damper). A capacitive shunt changes the effective stiffness of the piezoelectric element, which can be used to advantage in, for example, a tunable mechanical vibration absorber. A switched shunt offers the possibilities of controlling the energy transfer to reduce frequency-dependent behavior, or perhaps the conversion of energy to a usable form. This paper reviews recent research related to the use of shunted piezoelectric materials for vibration damping and control.

Time Domain Modeling of Linear Viscoelasticity Using Anelastic Displacement Fields

A time domain model of linear viscoelasticity is developed based on a decomposition of the total displacement field into two parts: one elastic, the other anelastic. The anelastic displacement field is used to describe that part of the strain that is not instantaneously proportional to stress. General coupled constitutive equations for (1) the total and (2) the anelastic stresses are developed in terms of the total and anelastic strains, and specialized to the case of isotropic materials. A key feature of the model is the absence of explicit time dependence in the constitutive equations. Apparent time-dependent behavior is described instead by differential equations that govern (1) the motion of mass particles and (2) the relaxation of the anelastic displacement field. These coupled governing equations are developed in a parallel fashion, involving the divergence of appropriate stress tensors. Boundary conditions are also treated: the anelastic displacement field is effectively an internal field, as it is driven exclusively through coupling to the total displacement, and cannot be directly affected by applied loads. In order to illustrate the use of the method, model parameters for a commonly-used high damping polymer are developed from available complex modulus data.

Finite element modeling of frequency-dependent material damping using augmenting thermodynamic fields

A new method of modeling frequency-dependent material damping in structural dynamics analysis is reported. Motivated by results from materials science, augmenting thermodynamic fields are introduced to interact with the usual mechanical displacement field. The methods of irreversible thermodynamics are used to develop coupled material constitutive relations and partial differential equations of evolution. These equations are implemented for numerical solution within the computational framework of the finite-element method. The method is illustrated using several examples including longitudinal vibration of a rod, transverse vibration of a beam, and vibration of a large space truss structure.

A finite element for beams having segmented active constrained layers with frequency-dependent viscoelastics

A finite element for planar beams with active constrained layer (ACL) damping treatments is presented. Features of this non-shear locking element include a time-domain viscoelastic material model, and the ability to readily accommodate segmented (i.e. non-continuous) constraining layers. These features are potentially important in active control applications: the frequency-dependent stiffness and damping of the viscoelastic material directly affects system modal frequencies and damping; the high local damping of the viscoelastic layer can result in complex vibration modes and differences in the relative phase of vibration between points; and segmentation, an effective means of increasing passive damping in long- wavelength vibration modes, affords multiple control inputs and improved performance in an active constrained layer application. The anelastic displacement fields (ADF) method is used to implement the viscoelastic material model, enabling the straightforward development of time-domain finite elements. The performance of the finite element is verified through several sample modal analyses, including proportional-derivative control based on discrete strain sensing. Because of phasing associated with mode shapes, control using a single continuous ACL can be destabilizing. A segmented ACL is more robust than the continuous treatment, in that the damping of modes at least up to the number of independent patches is increased by control action.

Can a coupling coefficient of a piezoelectric device be higher than those of its active material

An electromechanical coupling coefficient is a measure of the effectiveness with which a piezoelectric material (or a device employing such a material) converts the energy in an imposed electrical signal to mechanical energy, or vice versa. There are different kinds of material and device coupling coefficients, corresponding to different modes of excitation and response. Device coupling coefficients are properties of the device and, although related to the material coupling coefficients, are generally different from them. It is commonly held that a device coupling coefficient cannot be greater than some corresponding coupling coefficient of the active material used in the device. A class of devices was recently identified in which the apparent coupling coefficient can, in principle, approach 1.0, which corresponds to the limit of perfect electromechanical energy conversion. The key feature of this class of devices is the use of destabilizing mechanical pre-loads to counter inherent stiffness. The approach is illustrated for a symmetric piezoelectric bimorph device: theory predicts a smooth increase of the apparent coupling coefficient with pre-load, approaching 1.0 at the buckling load. An experiment verified the trend of increasing coupling with pre-load: a load corresponding to 50% of the buckling load increased the bimorph coupling coefficient by more than 40%. This approach provides a way to simultaneously increase both the operating displacement and force of a piezoelectric device, distinguishing it from alternatives such as motion amplification. and may allow transducer designers to achieve substantial performance gains for some actuator and sensor devices.

Aircraft structural morphing using tendon-actuated compliant cellular trusses

Recently, smoothly-deforming aircraft structures have been investigated for their ability to adapt to varying flight conditions. Researchers aim to achieve large changes in the shape of the wings: area changes of up to 50% and aspect-ratio changes of up to 200% are being pursued. The research described in this paper aims to develop a structural concept capable of achieving continuous stable deformations over a large range of aircraft shapes. The basic concept underlying the approach is a compliant cellular truss, with tendons used as active elements. The truss members of the unit cell are connected through compliant joints such that only modest bending moments may be transmitted from one member to another. Actuation is achieved by pulling on one set of cables while releasing another set. The tendonactuated compliant truss can be made to behave locally, and temporarily, as a nearmechanism, by releasing appropriate cables. As a result, in the absence of aerodynamic forces, the structure can be morphed using relatively low forces. The cables are reeled in or released in a controlled manner while the structure is loaded, hence, the stability of the structure can be maintained in any intermediate position. Highly-distributed actuation also enables the simultaneous achievement of global shape changes as the accumulation of local ones, while the use of compliant joints rather than true rotating joints eliminates binding as a significant concern. A six-noded octahedral cell with diagonal tendon actuation is developed for a bending type deformation in the wing. Initial cell geometry is determined by “strain matching” to the local morphing deformation required by the application. A finite element analysis is performed on a wing made of these unit cells and sized for a representative UAV weighing 3000 lbs. The areas of the individual truss members are sized so that they don’t fail or buckle under the air loads, while deflection at the wing tip is reduced. The octahedral unit cell is capable of achieving smooth deformations of the truss structure. The cell size is dictated by the available space and the morphing strain. The cell sizes are reasonable for strains on the order of 10% to 15% and get smaller for larger strains. Additional cell shapes are being investigated for larger area changes through a process of topology optimization using genetic algorithms. Numerous other technical challenges remain, including the details of actuation and a robust skin.

Time domain modeling of damping using anelastic displacement fields and fractional calculus

Abstract A fractional derivative model of linear viscoelasticity based on the decomposition of the displacement field into an anelastic part and elastic part is developed. The evolution equation for the anelastic part is then a differential equation of fractional order in time. By using a fractional order evolution equation for the anelastic strain the present model becomes very flexible for describing the weak frequency dependence of damping characteristics. To illustrate the modeling capability, the model parameters are fit to available frequency domain data for a high damping polymer. By studying the relaxation modulus and the relaxation spectrum the material parameters of the present viscoelastic model are given physical meaning. The use of this viscoelastic model in structural modeling is discussed and the corresponding finite element equations are outlined, including the treatment of boundary conditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag–Leffler function type. Finally a time step algorithm for solving the finite element equations are developed and some numerical examples are presented.

Finite elements for dynamic modeling of uniaxial rods with frequency-dependent material properties

Abstract New developments in a method of modeling frcquency-depedent material damping and modulus in structural dynamics analysis are reported. The fundamental feature of the general method is the introduction of augmenting thermodynamic fields (ATF) to interact with the mechanical displacement field of continuum mechanics. These ATF are directly motivated by the “internal state variables” of materials science. The coupled partial differential equations which govern the dynamic behavior of a uniaxial rod are numerically solved within the computational framework of the finite element method, resulting in “ATF-damped” finite elements. Previous work in the development of this modeling technique is characterised by the use of a single augmenting field, with application to lightly-damped rods, bourns and truss structures. New developments include : (1) demonstration of the ability to model the behavior of high-damping materials; and (2) the use of multiple augmenting fields to model materials whose behavior departs significantly from that of standard anelastic solids.

Finite element modeling of one-dimensional viscoelastic structures using anelastic displacement fields

A physically motivated approach to modeling the dynamic behavior of viscoelastic structures using augmenting thermodynamic fields was previously reported. Anelastic displacement fields, special kinds of augmenting thermodynamic fields, are now introduced. Instead of addressing physical damping mechanisms directly, as in the earlier approach, their effects on the displacement field are considered. In this approach, the total displacement field comprises two parts: 1) an elastic part and 2) an anelastic part. The material constitutive equations are developed, as well as the governing differential equations and boundary conditions for a one-dimensional structural member, and are compared to the results developed previously using the augmented thermodynamic fields approach. In addition, a physical interpretation of some of the quantities involved is advanced in terms of a classical mechanical analogy. Because the anelastic displacement field(s) and the total displacement field may be treated similarly in analytical or numerical study, the key practical benefit of this anelastic displacement fields approach is that it leads to the straightforward development of time-domain finite element models. Modal analyses and frequency response analyses have been implemented using the matrix manipulation capabilities of a commercial finite element code.