Exponential Stability for Piezoelectric Beam With Memory, Magnetic Effect, and Time-Varying Delay

Stabilization of nonlinear non-uniform piezoelectric beam with time-varying delay in distributed control input

Models for piezoelectric beams and structures with piezoelectric patches generally ignore magnetic effects. This is because the magnetic energy has a relatively small effect on the overall dynamics. Piezoelectric beam models are known to be exactly observable, and can be exponentially stabilized in the energy space by using a mechanical feedback controller. In this paper, a variational approach is used to derive a model for a piezoelectric beam that includes magnetic effects. It is proven that the partial differential equation model is well-posed. Magnetic effects have a strong effect on the stabilizability of the control system. For almost all system parameters the piezoelectric beam can be strongly stabilized, but is not exponentially stabilizable in the energy space. Strong stabilization is achieved using only electrical feedback. Furthermore, using the same electrical feedback, an exponentially stable closed-loop system can be obtained for a set of system parameters of zero Lebesgue measure. These results are compared to those of a beam without magnetic effects.

In this work, we focus on a one-dimensional initial-boundary value problem of a thermoelastic piezoelectric beam in the presence of magnetic effects with distributed delay acting in the mechanical equation. The well-posedness of the system is initially demonstrated by applying semigroup theory (Hille-Yosida theorem). Also, based on the construction of a Lyapunov functional, which is equivalent to the energy functional of the problem through multiplier techniques, we show that a unique dissipation through frictional damping is strong enough to ensure exponential stabilization of the model. Next, the results are compared to those of the electrostatic or quasi-static approaches. Moreover, we consider the fully dynamic and electrostatic or quasi-static piezoelectric beams with thermal effects, boundary feedback controllers and boundary distributed delays and using the multipliers technique, we establish an exponential stability result of the solution. Our results are related to the distributed delay weights. Finally, we give some numerical tests to illustrate the theoretical result.

Existing smart composite piezoelectric beam models in the literature mostly ignore the electro-magnetic interactions and adopt the linear elasticity theory. However, these interactions substantially change the controllability and stabilizability at the high frequencies, and linear models fail to represent and predict the governing dynamics since mechanical nonlinearities are pronounced in certain applications such as energy harvesting. In this paper, first, a consistent variational approach is used by considering nonlinear elasticity theory to derive equations of motion for a single-layer piezoelectric beam with and without the electromagnetic interactions (fully dynamic and electrostatic). This modeling strategy is extended for the three-layer piezoelectric smart composites by adopting the two widely-accepted sandwich beam theories. For both single-layer and three-layer models, the resulting infinite dimensional equations of motion can be formulated in the state-space form. It is observed that the fully dynamic nonlinear models are unbounded boundary control systems (same in linear theory) ${\bf \dot y}(t)=(\mathcal A +\mathcal N) {\bf y} (t) + \mathcal B u(t)$, the electrostatic nonlinear models are unbounded bilinear control systems ${\bf \dot y}(t)=(\mathcal A +\mathcal N){\bf y} (t) + (\mathcal B_1+ \mathcal B_2 {\bf y}) u(t)$ in sharp contrast to the linear theory. Finally, we propose $\mathcal B^*-$type feedback controllers to stabilize the single piezoelectric beam models. The filtered semi-discrete Finite Difference approximations is adopted to illustrate the findings.

To predict electrical generation in piezoelectric small-scale beam energy harvesting devices, it is important to have a complete mathematical model that captures the different associated phenomena. In the literature, some authors propose several alternatives of non-linear mathematical formulations, with non-linearities coming from different physical aspects. All these formulations present good aptitudes to predict the nonlinear behavior of the system under different values of accelerations, geometry and boundary conditions. At the same time, they do not represent a unified general proposal for modeling multimodal energy harvesting devices of any type of mode generation and boundary conditions at large excitations. In this sense, this paper presents a mathematical description of inextensional nonlinear Euler-Bernoulli piezoelectric beams that combines the best contributions of the literature to the voltage generation of multimodal nonlinear piezoelectric energy harvesters (geometric, material and damping non-linearities). The developed analytical model yields a total set of N+ 1 ordinary differential equations for the first N modes and for the output voltage. However, direct solution of this ordinary nonlinear differential system of N equations is computationally costly. Instead, a reduced algebraic system of 2() algebraic equations is proposed applying the method of averaging. Its main advantage is that it makes more suitable and computationally economical for the implementation of a parameter identification process involving any number of piezoelectric inserts (unimorph or bimorph) and mode of generation (d33 or d31). Two types of validations are presented for some selected physical systems to test the validity of the assumptions: a numerical one, by the direct integration of the equations of motion and an experimental one. A final comparison between the results demonstrates the importance of the having a unified nonlinear model to predict the generated voltage in multimodal energy harvesters.

It is widely accepted in the literature that magnetic effects in the piezoelectric beams is relatively small, and does not change the overall dynamics. Therefore, most models for piezoelectric beams completely ignore the magnetic energy. These models are known to be exponentially stabilizable by a mechanical feedback controller in the energy space. In this paper, we use a variational approach to derive the differential equations and boundary conditions that model a single piezoelectric beam with magnetic effects. Next, we show that the resulting control system can be formulated as a port-Hamiltonian system and is hence well-posed. Finally, by using only an electrical feedback controller (the current flowing through the electrodes), we show that the closed-loop system is strongly stable in the energy space for a dense set of system parameters. The difference between this result and that for models that neglect magnetic effects is discussed.

In this paper, we consider a one-dimensional dissipative system of piezoelectric beams with magnetic effect, inspired by the model studied by Morris and Özer (Proc. of 52nd IEEE Conference on Decision & Control (2013) 3014–3019). Our main interest is to analyze the issues relating to exponential stability of the total energy of the continuous problem and reproduce a numerical counterpart in a totally discrete domain, which preserves the important decay property of the numerical energy.

In this paper, we consider a fully‐dynamic piezoelectric beam model subjected to a magnetic effect, where the heat flux is given by Cattaneo's law. It is well known that, in the absence of delay, the dissipation produced by the heat conduction is strong enough to make the piezoelectric beams exponentially stable. However, time delay effects may destroy this behavior. Here, we show the existence and uniqueness of solutions through the semigroup theory. Furthermore, under a smallness condition on the delay, we prove an exponential stability result via establishing the appropriate Lyapunov functional. Finally, we numerically illustrate the asymptotic behavior of the solution.

In this paper, a piezoelectric beam is investigated under fully‐dynamic magnetic effects and long‐range dielectric and strain memories. The mathematical model under consideration is for the type of piezoelectric beams which strongly couple the electromagnetic effects with mechanical vibrations. This particular model is highly preferred for complex systems for which the vibrational profile is under the strong influence of the electromagnetic interactions. By a careful analysis, an optimal energy decay result is explicitly established under a wide class of kernel functions of the memory terms. The results are based on the multipliers technique. These results are compared to the widely‐used electrostatic/quasi‐static models which entirely exclude magnetic effects.

Considering the stiffness characteristics of piezoelectric layer, the bending stiffness of piezoelectric cantilever beam is obtained by applying the first-order shear deformation theory. The finite element model of piezoelectric cantilever beam is established by Hamilton variation principle, and the modal superposition method is employed to reduce the order of the finite element model. At the maximum strain point, the sensors/actuators are equipped in pairs. Based on the uncertain dynamic model of piezoelectric cantilever beam, the independent modal space control method based on LQR (linear quadratic regulator) control is employed for the active control of the smart beam structure, and the weighted matrices Q and R are selected according to the energy criterion. The numerical simulations and experiments verify the effectiveness of the proposed finite element model and the active vibration optimal control.

It is well known that magnetic energy of the piezoelectric beam is relatively small, and it does not change the overall dynamics. Therefore, the models, relying on electrostatic or quasi-static approaches, completely ignore the magnetic energy stored/produced in the beam. A single piezoelectric beam model without the magnetic effects is known to be exactly observable and exponentially stabilizable in the energy space. However, the model with the magnetic effects is proved to be not exactly observable / exponentially stabilizable in the energy space for almost all choices of material parameters. Moreover, even strong stability is not achievable for many values of the material parameters. In this paper, it is shown that the uncontrolled system is exactly observable in a space larger than the energy space. Then, by using a $B^*-$type feedback controller, explicit polynomial decay estimates are obtained for more regular initial data. Unlike the classical counterparts, this choice of feedback corresponds to the current flowing through the electrodes, and it matches better with the physics of the model. The results obtained in this manuscript have direct implications on the controllability/stabilizability of smart structures such as elastic beams/plates with piezoelectric patches and the active constrained layer (ACL) damped beams/plates.

Hysteresis is highly undesired for the vibration control of piezoelectric beams especially in high-precision applications. Current-controlled piezoelectric beams cope with hysteresis substantially in comparison to the voltage-controlled counterparts. However, the existing low fidelity current-controlled beam models are finite dimensional, and they are either heuristic or mathematically over-simplified differential equations. In this paper, novel infinite-dimensional models, by a thorough variational approach, are introduced to describe vibrations on a piezoelectric beam. Electro-magnetic effects due to Maxwell’s equations factor in the models via the electric and magnetic potentials. Both models are written in the standard state-space formulation (A, B, C), and are shown to be well-posed in the energy space by fixing the so-called Coulomb and Lorenz gauges. Different from the voltage-actuated counterparts, the control operator B is compact in the energy space, i.e. the exponential stabilizability is not possible. Considering the compact $$C=B^*-$$type state feedback controller (induced voltage), both models fail to be asymptotically stable if the material parameters satisfy certain conditions. To achieve at least asymptotic stability, we propose an additional controller. Finally, the stabilizability of infinite-dimensional electrostatic/quasi-static model (no magnetic effects) is analyzed for comparison. The biggest contrast is that the asymptotic stability is achieved by an electro-mechanical state feedback controller for all material parameters. Our findings are simulated by the filtered semi-discrete Finite Difference Method.

This paper presents a dynamic model of a piezoelectric bimorph beam with a tip mass for low level power harvesting. The piezoelectric bimorph beam is modelled as an Euler-Bernoulli beam with two input transversal and longitudinal base excitations. The strain field due to the longitudinal base input excitation can affect the piezoelectric response parameters although the transverse bending field has most often been considered in the use of the cantilevered piezoelectric bimorph in stimulating polarity and electric field for the energy harvester. The piezoelectric bimorph beam with centre brass shim can be analysed using series and parallel connections depending on the piezoelectric coupling and electric field parameters. The extracted power from the piezoelectric bimorph beam can be used for the powering of electronic storage devices, electronic media and wireless sensors. In this paper, we propose analytical methods for developing constitutive energy field differential equations using virtual work concepts (Weak form) from the interlayer elements of the piezoelectric bimorph beam. Analytical solutions of the constitutive dynamic equations from longitudinal extension, transverse bending and electrostatic fields are solved using Laplace transforms to obtain transfer functions between their relationships.

Obtaining energy from human body activities to power electronic equipment has broad application prospects. In this study, a piezoelectric energy harvesting system is designed to harvest energy from human finger rapping. The harvested energy can drive a laser lamp group to operate. The piezoelectric energy harvester uses a piezoelectric cantilever beam with a spring-mass block at the free end to obtain the kinetic energy of the finger operating a laser pen. The structure can produce considerable bending deformation under finger movement. At the same time, a dual-input synchronized switch harvesting on inductor circuit is designed to extract the charge generated by the piezoelectric beam when the spring-mass block at the free end is rapped and store it in the energy storage element. The charge and discharge of the energy storage element are controlled by a DC-DC management circuit. Experimental results show that the average harvesting power of the piezoelectric energy acquisition system is 169 μW, and its power density is 388.5 μW/cm3. When the spring-mass block is rapped for 8 s, the energy generated by the piezoelectric energy harvester lights the laser pen for 0.35 s.

In this paper, a structure for a piezoelectric beam vibrator driven by a groove cam is analysed. The vibrator takes the simplified form of a piezoelectric beam model, where one end of the beam is clamped and the other end is simply supported, and is thus named the clamped/simply supported piezoelectric beam model (CSPBM). Mathematical models of the damped forced vibration and electromechanical energy conversion processes of the CSPBM are established based on the harmonic displacement excitation of the simply supported end. Factors that affect the energy generated by the CSPBM are analyzed theoretically and are simulated separately using both the MATLAB software and ANSYS software. Theoretical analysis results indicate that there is an optimal value of the ratio of the base plate thickness to the beam thickness (α) at which the energy generated by the CSPBM is maximum. In addition, the optimal α value of a unimorph beam is about 0.3 and is irrelevant to the material parameters of the beam. The voltage and energy generated by the CSPBM are measured on an experimental bench and the results show that the maximum generated voltage increases with increasing the first natural frequency when the α value of piezoelectric beam and the amplitude of displacement excitation are constant. In addition, the theoretical results of the generated voltage are coincident with experimental results, which confirms the validity of the theoretical model. The vibrator driven by a groove cam provides a practical form of the piezoelectric beam excited by the displacement.