Exploiting the Internal Resonance for the Vibration Suppression of Beams via Piezoelectric Shunt Circuits

Abstract

This paper investigates the vibration suppression of cantilevered beams using nonlinear shunted piezoelectric circuits. The beam’s inertia and geometric nonlinearities are considered. A quadratic nonlinear piezoelectric capacitance is used such that there exists a two-to-one internal resonance between the mechanical and electrical modes. The internal resonance coupling is exploited to trigger the saturation phenomenon such that the beam’s vibration reaches a limit beyond an excitation amplitude threshold. The equations governing the nonlinear vibration of the beam coupled with the shunt circuit are derived, and modal analysis is used to obtain a system of two nonlinearly coupled modal equations. The equations are then numerically integrated to obtain the results. A parametric study is performed to assess the significance of system parameters, such as the location of the piezoelectric patch, its size, circuit resistance, and nonlinear gain, on the effectiveness of vibration suppression. The results show that the proposed design effectively suppresses the linear and nonlinear vibrations of the beam. The proposed absorber is space-efficient and does not add mass to the primary system, and hence, it has the potential in systems where the weight matters, such as aerospace applications.

1. Introduction

Structural vibration suppression has been of common interest as an essential component in the design of systems for performance and integrity. In many engineering applications, vibration is undesirable and even harmful, which can eventually lead to failure [1]. As a result, efficiently mitigating vibration has always been a goal across a wide range of engineering applications. Traditional methods such as dynamic vibration absorbers (DVAs) and tuned mass dampers (TMDs) [2] have long been reliable instruments for reducing vibration and providing effective solutions to practical problems. However, they often exhibit limitations in terms of adaptability, accuracy, and efficiency. Nonetheless, innovation always intends to push boundaries in the dynamic realm of vibration control. Piezoelectric shunt circuits are one of the innovative treatments that have drawn growing interest as vibration absorbers. By utilizing piezoelectric transducers coupled to an electrical impedance, Forward [2] suggested a vibration control method by transforming strain energy from the structure into electrical energy. Based on their methods of control, researchers categorize piezoelectric shunt circuits into three groups: hybrid, active, and passive. The electrical components in the shunt circuit and the existence of any external energy source determine which of those types is used. Using the circuit’s intrinsic resistance to absorb vibrational energy, the passive approach dissipates energy passively without the need for an external power source. Conversely, active techniques use sensors and actuators to detect vibration and activate real-time vibration suppression [3]. From straightforward beams to more intricate space constructions, the active control approach has effectively decreased vibration in a variety of applications. In order to operate actuators, an active control system usually needs high-performance digital signal processors and power amplifiers, which are not suitable for space applications. Passive control techniques are less resilient to system disruptions even though they are easier to build than active control systems [4]. Consequently, the possibility of hybrid control systems to effectively provide vibration attenuation through piezoelectric shunt circuits has raised interest in them. These systems offer a flexible method of lowering vibration in a range of applications by combining the advantages of energy efficiency and customization. It is important to remember that passive techniques can be quite successful in reducing high-frequency excitations, whilst active techniques can be utilized to restrict low-frequency vibrations. Furthermore, hybrid approaches combine the best features of active and passive methods, making them appropriate for wide-frequency bands [5,6]. When two modes of a nonlinear system are mutually synchronized [7], meaning that their frequencies are in a simple integer ratio, internal resonances occur. Saturation phenomena, a special interaction between these modes, can be produced by using this condition. When an external stimulation is applied to the first mode, the saturation phenomenon, which was initially identified by Nayfeh et al. [8], creates a threshold for one of the modes and transfers the energy to the other mode, resulting in an energy exchange. The system behavior and vibration amplitude can be greatly affected by this method. Internal resonance type in the system is determined by setting the frequency ratio between the modes and the type of nonlinearity coupling between them. A quadratic nonlinearity in the system is necessary to attain a 2:1 internal resonance if the frequency ratio between the modes is doubled. Moreover, cubic nonlinearity results in a different kind of internal resonance, known as 3:1, where the frequency ratio of the two modes is three [9].

Passive vibration techniques using piezoelectric transducers have been used by many researchers. Hagood and Flotow [10] were the first to use a piezoelectric transducer connected to a passive electrical circuit to dissipate mechanical energy. Moreover, they added a series resistor–inductor circuit and adjusted the electric circuit’s impedance frequency to match the structural resonant frequency to mitigate the maximum vibration amplitude. However, large inductors are needed to achieve the low electrical resonance needed to diminish low-frequency vibration. This drawback was addressed in the work of [11]. Yamada [12] used a multi-layer piezoelectric transducer connected to an (RL) circuit. The purpose of multi-layering the piezoelectric element was to increase its capacitance, reducing the requirement for a high inductance value at low frequencies. Lossouarn et al. [13] investigated how to improve the inductance values using closed magnetic cores composed of materials with high permeability. They showed that customized designs can enable the use of passive resonant shunt techniques at lower frequencies.

Researchers have explored multimode piezoelectric shunt circuit vibration control. A method for suppressing several modes with a single piezoelectric device was created by Holkamp [14]. To create a single piezoelectric component, additional LRC circuit branches are connected in parallel to an RL branch. There were as many RLC branches as there were modes to take into account. In order to regulate the branches, alternative circuit layouts were also suggested. Using a current-blocking circuit, Wu [15,16] created a multi-mode shunt. At other branch frequencies, the LC anti-resonant circuits’ electrical resistance is intended to reach infinity and function as an open circuit, while the targeted branch that corresponds to the structure’s resonance frequency functions and reduces vibration. However, Behrens et al. [17,18] employed shunt circuits with current-flowing branches. The current-flowing shunt and the current-blocking shunt serve the same purpose. Instead of halting the current at its normal frequency, the current-flowing shunt allows it to continue. Since current blocking necessitates an equal number of inductors for each mode, this approach is superior [19]. A novel tuning technique for a simpler current-blocking shunt circuit was presented by Raze et al. [20] and verified both experimentally and numerically. Their novel approach seeks to enhance multimode damping without the use of any optimization algorithms by concentrating on the effects of nonresonant modes and the electromechanical coupling between the structure and the shunt branches. This strategy differs from others that employ optimization algorithms to adjust circuit parameters, including ant colony optimization [21], particle swarm optimization [22,23], and other tuning methods [24,25].

The use of internal resonances for vibration suppression was first proposed in [26] by introducing a nonlinear auto-parametric absorber that is based on internal resonance. Their approach relies on connecting a single-degree-of-freedom system to a cantilever beam with a tip mass. After forcing the system to resonate, they discovered that the absorber could efficiently limit and create a threshold on the amplitude of oscillations of the main mass by tuning the beam and other system frequencies. Moreover, most of the researchers have used internal resonance in active control systems [27,28]. Oueini et al. [29] introduced a new active nonlinear vibration absorber for flexible constructions that makes use of quadratic nonlinearities and two-to-one internal resonances. The technique connects second-order controllers to a cantilever beam via sensors and actuators.

Taşkıran and Özer [30] suggested a passive nonlinear piezoelectric vibration isolation, and the induced structural nonlinearity was demonstrated experimentally. A passive nonlinear component is suggested to be a hardening capacitor obtained using a P–N junction. An analytical model was derived for a parallel-connected macro-fiber composite (MFC) piezoelectric material in a bimorph configuration attached to a cantilever beam, and the model was solved numerically. Obaidullah and Erturk [31] presented a shunt with a nonlinear impedance equivalent to a positive or negative cubic inductor connected in parallel with a linear inductor and a linear resistor. This shunt, along with the inherent piezoelectric capacitance, behaves as a synthetic Duffing circuit, providing a digital analog of a hardening- or softening-type nonlinear stiffness, linear stiffness, linear damping, and mass. Numerical simulations were performed using single-term and multi-term harmonic balance solutions to predict the periodic response. These approximate analytical predictions were compared against time-domain numerical simulations and were experimentally validated across various mechanical excitation amplitudes and nonlinear coefficients. Rui et al. [32] introduced a semi-active control method based on a piezoelectric smart cantilever beam with a passive shunt circuit for vibration control. The contribution is that the proposed semi-active control method is used to achieve energy dissipation switching by changing the piezoelectric electrical boundary conditions. A prototype was manufactured, and the measurement system was constructed. The correctness and feasibility of the semi-active vibration control method were verified through experiments.

Shami et al. [33] provide a hybrid nonlinear vibration attenuation method. To achieve a 2:1 internal resonance between the mechanical and electrical resonances, the shunt circuit was tuned using a quadratic nonlinearity. Above a threshold, this allowed the mechanical amplitude to be independent of the forcing amplitude by producing a nonlinear antiresonance and a saturation phenomenon. It has been researched and optimized to tune the shunt circuit’s nonlinear properties.

The objective of this work is to exploit the internal resonance to effectively suppress the nonlinear vibration of cantilever beams equipped with a nonlinear piezoelectric shunt circuit. The coupled system consists of a cantilever beam as the structure, a piezoelectric patch as the transducer, and a shunt circuit as the nonlinear absorber. A 2:1 internal resonance is created between the mechanical natural and electrical natural frequencies by tuning the elements of the electrical circuit. A parametric study is performed to find the optimal circuit parameters for the maximum vibration suppression while preserving the 2:1 internal resonance. The performance of the proposed absorber is investigated under resonant harmonic excitation of the structure, and it is found to significantly reduce the vibration amplitude over a substantial frequency range around resonance. Moreover, the absorber is found to function equally well for the linear vibration of the beam and the nonlinear vibration when inertia and geometric nonlinearities are taken into consideration. The distinctive contribution of this work is that it employs nonlinear capacitance in passive circuits to trigger internal resonance for the saturation phenomenon, performs a parametric tuning strategy, and lastly applies the control strategy on geometrically nonlinear beam dynamics. Challenges and limitations in real-world applications, e.g., aerospace applications, include the risk of affecting structural integrity, inefficient performance for systems that often vibrate across a wide spectrum, and control of multiple vibration modes at once. The rest of this paper is organized as follows: Section 2 presents the mathematical model of the nonlinear vibration of a cantilever beam coupled to a piezoelectric shunt circuit, Section 3 presents the results and discussions, Section 4 presents some conclusions, and finally outlines the references cited in this paper.

2. Problem Formulation

We consider the lateral vibration of a cantilever beam attached to a piezoelectric shunt circuit, as shown in Figure 1. The shunt circuit parameters, which are typically resistance, capacitance, and inductance, are selected such that the resulting coupled response minimizes the lateral vibration of the beam as a primary system. Namely, the shunt circuit includes a nonlinear capacitance that is tuned to induce a 2:1 internal resonance between the mechanical and the electrical natural frequencies. An experimental setup for the linear vibration of a cantilever beam coupled with a piezoelectric shunt was presented by Shami et al. [33].

The proposed nonlinear piezoelectric shunt circuit is used to mitigate the lateral vibration of a cantilever beam that is harmonically excited at its fundamental resonant frequency. The equation governing the nonlinear lateral vibration of an Euler–Bernoulli cantilever beam, and the associated boundary conditions are given using [34]

$$m\ddot{v}+{EIv}^{⁗}+{\displaystyle \frac{m}{2}}{\left\{ v^{\prime }{\int }_L^x{\displaystyle \frac{{\partial }^2}{\partial t^2}} \left[{\int }_0^x{{(v}^{\prime })}^2 dx\right]dx\right\}}^{\prime }+{EI\left[v^{\prime }{\left(v^{\prime }{v}^{″}\right)}^{\prime }\right]}^{\prime }=F\mathit{cos}\left(\Omega t\right) ,$$

(1)

$$v=0, v^{\prime }=0 at x=0,$$

(2)

and

$$v^{\prime \prime }=0, v^{\prime \prime \prime }=0 at x=L,$$

(3)

$v(x,t)$ is the lateral vibration; $E$, $I$, $m$, and $L$ are the modulus of elasticity, the second moment of area, the mass per unit length, and the beam’s length respectively; $F$ is the amplitude of the applied harmonic force, and $\mathsf{\Omega }$ is its frequency; $x$ is the spatial coordinate, and $t$ is time. The dot and prime denote derivatives with respect to time and length, respectively. Solving the linear free vibration problem yields the following mode shapes and natural frequencies:

$${\varphi }_i\left(\xi \right) = \mathit{sin}{k}_i\xi -\mathit{sinh}{k}_i\xi -{\sigma }_i(\mathit{cos}{k}_i\xi -\mathit{cosh}{k}_i\xi )$$

(4)

and

$${\omega }_i=\sqrt{{\displaystyle \frac{EI}{{mL}^4}}}{k}_i^2 ,$$

(5)

where $\xi$ is the nondimensional spatial coordinate such that $\xi = x/L$, and the constants $k$ and $\sigma$ are defined as

$$\mathit{cos}{k}_i\mathit{cosh}{k}_i+1=0$$

(6)

and

$${\sigma }_i ={\displaystyle \frac{\mathit{sinh}{\mathit{k}}_i+\mathit{sin}{k}_i}{\mathit{cosh} k_i+\mathit{cos} k_i}}.$$

(7)

The orthogonality conditions are defined as

$${\int }_0^1{\varphi }_i\left(\xi \right){\varphi }_j\left(\xi \right) d\xi =\left\{\begin{array}{c}1, i=j\\ 0, i\ne j\end{array}\right.$$

(8)

and

$${\int }_0^1{\varphi }_i\left(\xi \right) {\varphi }_i\prime \prime \prime \prime \left(\xi \right) d\xi =\left\{\begin{array}{c}{k}_i^4, i=j\\ 0, i\ne j\end{array}\right. .$$

(9)

Let the displacement $u(\xi ,t)$ be discretized as follows:

$$v\left(\xi ,t\right) = {\sum }_{i = 1}^N{\phi }_i\left(\xi \right) q_i\left(t\right),$$

(10)

where ${\varphi }_i$ is the linear vibration mode shapes, $q_i$ is unknown generalized coordinates, and $N$ is the number of modes retained in the discretization. Substituting Equation (10) into Equation (1), multiplying the resulting equation by the mode shape ${\varphi }_n$, integrating over the domain and noting the orthogonality conditions, the discretized equations are obtained as follows:

$${\ddot{q}}_n+{\omega }_n^2{q}_n+\sum _{i,j,k}^N{a}_{ijkn}{ q}_i {\displaystyle \frac{{\partial }^2}{\partial t^2}}\left(q_j{q}_k\right)+\sum _{i,j,k}^N{b}_{ijkn} q_i{q}_j{q}_k = f_n\mathit{cos}\Omega t, n = 1, \dots ,N.$$

(11a)

Equation (11) is written in a compact index form that can be expanded into N differential equations if N modes are used in the discretization. It includes the inertia nonlinearity in the first term and the geometric nonlinearity due to the relatively large deformation in the second term where both are of cubic type. The constants $a_{ijkn}$ and $b_{ijkn}$ are mode-shape dependent and defined as

$$a_{ijkn}={\displaystyle \frac{1}{2L^2}}{\int }_0^1 {\left\{\sum _i^N{\varphi }_i^{\prime }\left(\xi \right){\int }_1^{\xi }\left[{\int }_0^{\xi }\sum _j^N\sum _k^N{\varphi }_j^{\prime }\left(\xi \right){\varphi }_k^{\prime }\left(\xi \right)d\xi \right]d\xi \right\}}^{\prime }{\phi }_n\left(\xi \right) d\xi$$

(11b)

and

$$b_{ijkn}={\displaystyle \frac{EI}{m{L}^6}} {\int }_0^1{\left[{\sum }_i^N{\varphi }_i^{\prime }\left(\xi \right) {\left({\sum }_j^N{\sum }_k^N{\phi }_j^{\prime }\left(\zeta \right){\phi }_k^{″}\left(\zeta \right)\right)}^{\prime }\right]}^{\prime }{\phi }_n\left(\xi \right) .$$

(11c)

The forcing term $f_n$ is the projection of the excitation force onto the $n$th mode that is defined as

$$f_n = {\displaystyle \frac{1}{m}}{\int }_0^{1}F\left(\xi \right) {\phi }_n\left(\xi \right) d\xi .$$

(11d)

Assuming that only the first mode significantly contributes to the response, the inertia and geometric nonlinear terms are calculated as follows

$$\begin{gathered} {\displaystyle \frac{1}{2L^2}}{\int }_0^1{\left\{{\phi }^{\prime }\left(\xi \right)q\left(t\right){\int }_1^{\xi }{\displaystyle \frac{{\partial }^2}{\partial t^2}} \left[{\int }_0^{\xi }{\left({\phi }^{\prime }\left(\zeta \right)q\left(t\right)\right)}^{2}d\xi \right]d\xi \right\}}^{\prime }\phi \left(\xi \right) d\xi \\ ={\displaystyle \frac{4.597}{L^2}} (\ddot{q}{q}^2+{\dot{q}}^{2}q) \end{gathered}$$

(12)

and

$${\displaystyle \frac{EI}{m{L}^6}}{\int }_0^1{\left[{ \phi }^{\prime }\left(\xi \right)q\left(t\right) {\left({\phi }^{\prime }\left(\xi \right){\phi }^{″}\left(\xi \right)q^2\left(t\right)\right)}^{\prime }\right]}^{\prime }\phi \left(\xi \right)d\xi ={\displaystyle \frac{40.441EI}{m{L}^6}}{q}^3=a q^3,$$

(13)

where $a$ is the coefficient of cubic nonlinearity. Therefore, we neglect the effect of inertia nonlinearity for the case where the first mode is directly excited and keep the geometric nonlinearity. The governing equation that represents an uncoupled nonlinear cantilever beam is given as

$$\ddot{q}+{\omega }_1^{2}q+a q^3=f\mathit{cos}\Omega t .$$

(14)

In accordance with Shami et al. [33] and Lossouarn et al. [13], the set of coupled equations describing the interaction between a passive shunt circuit and a cantilever beam is derived. The beam, piezoelectric patch, and circuit are represented by the following equations, respectively:

$$\ddot{q}+2{\zeta }_1{\omega }_1\dot{q}+{\omega }_1^{2}q +a q^3 +K_{c}V = f\mathit{cos}\left(\Omega t\right) ,$$

(15)

$$C_{p}V-Q-K_{c}q =0 ,$$

(16)

and

$$L\ddot{Q}+R\dot{Q}+V+V_{nl}=0 ,$$

(17)

where L and R are the inductance and resistance values of the RL shunt circuit, $K_c$ is the piezoelectric coupling coefficient, $C_p$ is the capacitance of the piezoelectric patch, and V and Q represent the voltage and electrical charges, respectively. By eliminating the voltage from Equations (16) and (17), the coupled equations reduce to

$$\ddot{q}++2{\zeta }_1{\omega }_1\dot{q}+\left({\omega }_1^2+{\displaystyle \frac{K_c^2}{C_p}}\right)q+{\displaystyle \frac{K_c}{C_p}}Q+a q^3=f cos\left(\Omega t\right)$$

(18)

and

$$\ddot{Q}+2{\zeta }_e{\omega }_e\dot{Q}+{\omega }_e^{2}Q+{\displaystyle \frac{K_c}{L{C}_p}}q+{\displaystyle \frac{1}{L}}{V}_{nl}=0 ,$$

(19)

where the electrical natural frequency and electrical damping ratio are defined as

$${\omega }_e={\displaystyle \frac{1}{\sqrt{L{C}_p}}}$$

(20)

and

$${\zeta }_e={\displaystyle \frac{R}{2}}\sqrt{{\displaystyle \frac{C_p}{L}}} .$$

(21)

Investigating Equation (18), the effective natural frequency of the beam is defined as

$${\widehat{\omega }}_1={\omega }_1^2+{\displaystyle \frac{K_c^2}{C_p}} .$$

(22)

Inducing an Internal Resonance into the System

Nonlinear coupling between two modes whose natural frequencies are connected by an integer is known as internal resonance. Quadratic nonlinearity is necessary to introduce a 2:1 internal resonance. As proposed in [28], quadratic feedback and control signals can produce nonlinearity in active systems. In the meantime, nonlinear electrical components, like a nonlinear capacitor whose capacitance inversely varies inversely with the amplitude of the charge, can also be used to create nonlinearity in passive and hybrid systems. Operational amplifiers and varactor diodes can be used to physically construct this [35]. The nonlinear capacitance and its voltage can be represented as

$$C_{nl}={\displaystyle \frac{1}{Q}}$$

(23)

and

$$V_{nl}=N{\displaystyle \frac{Q}{C_{nl}}}=N{Q}^2.$$

(24)

The constant N acts as a nonlinear gain or proportionality factor that dictates how the voltage scales with the square of the charge. Nonlinear capacitors arise from dielectric materials whose permittivity varies with the electric field. The form given in Equation (24) is not standard for common nonlinear capacitors like varactors [36]. Equation (19) becomes

$$\ddot{Q}+2{\zeta }_e{\omega }_e\dot{Q}+{\omega }_e^{2}Q+{\displaystyle \frac{K_c}{L{C}_p}}q+{\displaystyle \frac{N}{L}}{Q}^2=0 .$$

(25)

Now, Equations (18) and (25) govern the dynamics of the coupled system; however, the coupling terms in both equations are not of the same form. Therefore, we introduce the following dimensionless parameters to unify the coupling terms:

$$\tau = {\widehat{\omega }}_1 t, \widetilde{ q} = q , \tilde{Q} = \sqrt{L} Q, \tilde{\Omega } = {\displaystyle \frac{\Omega }{ {\widehat{\omega }}_1}} .$$

(26)

As result, the nondimensional governing equations are defined as follows:

$$\tilde{\ddot{q}}+2{\zeta }_1\tilde{\dot{q}}+\tilde{q}+\alpha \tilde{Q} +{\displaystyle \frac{a}{{\widehat{\omega }}_1^2}} {\tilde{q}}^3 = \overline{f} cos\left(\tilde{\Omega }\tau \right)$$

(27)

and

$$\ddot{\tilde{Q}}+2{\zeta }_e{\omega }_r \dot{\tilde{Q}}+{\omega }_r^2\tilde{Q}+\alpha \tilde{q}+{\displaystyle \frac{N}{L^{\frac{3}{2}}{\widehat{\omega }}_1^2}}{\tilde{Q}}^2=0$$

(28)

where the electromechanical frequency ratio, ${\omega }_r$, the coupling coefficient, $\alpha$, the electromechanical coupling factor (EMCF), $k_1$, and the forcing amplitude, $\overline{f}$, are defined as

$${\omega }_r={\displaystyle \frac{{\omega }_e}{{\widehat{\omega }}_1}}, \alpha ={\displaystyle \frac{ K_c}{{{\widehat{\omega }}_1^{2}C}_p\sqrt{L}}}, k_1={\displaystyle \frac{K_c}{{\widehat{\omega }}_1\sqrt{C_p}}}, \overline{f}={\displaystyle \frac{f}{{\widehat{\omega }}_1^2}}$$

(29)

For simplicity, the hat ($~)$ notation will be omitted in the following equations, and the dot refers to differentiation with respect to the nondimensional time $\tau$. One may note that Equations (27) and (28) govern the coupled electromechanical model of a cantilever beam equipped with a nonlinear shunt circuit using the physical coordinate $q\left(t\right)$ for the beam and $Q\left(t\right)$ for the circuit. Yet, in terms of the physical coordinates, the system of equations is linearly coupled.

To introduce the nonlinear coupling that makes the internal resonance possible, we use modal analysis technique to transform the governing equations into a new set of modal coordinates. The nonlinearly coupled modal equations can be derived as

$$\begin{gathered} {\ddot{x}}_1+{\xi }_1{\dot{x}}_1+{\xi }_2{\dot{x}}_2+{\omega }_r^2{x}_1+{\displaystyle \frac{N}{\sqrt{L^3} {\widehat{\omega }}_1^2}} x_1^2+{\displaystyle \frac{2N\theta }{\sqrt{L^3} {\widehat{\omega }}_1^2}} x_1{x}_2+{\displaystyle \frac{N{\theta }^2}{\sqrt{L^3} {\widehat{\omega }}_1^2}} x_2^2+{\displaystyle \frac{a{\theta }^4}{{\widehat{\omega }}_1^2}}{x}_1^3 \\ +{\displaystyle \frac{3a{\theta }^3}{{\widehat{\omega }}_1^2}}{x}_1^2{x}_2+{\displaystyle \frac{3a{\theta }^2}{{\widehat{\omega }}_1^2}}{x_{1}x}_2^2+{\displaystyle \frac{a\theta }{{\widehat{\omega }}_1^2}}{x}_2^3 =-\theta \overline{f}\mathit{cos}\Omega \tau \end{gathered}$$

(30)

and

$$\begin{gathered} {\ddot{x}}_2+{\xi }_2{\dot{x}}_1+{\xi }_3{\dot{x}}_2+x_2+{\displaystyle \frac{N\theta }{\sqrt{L^3} {\widehat{\omega }}_1^2}} x_1^2+{\displaystyle \frac{2N{\theta }^2}{\sqrt{L^3} {\widehat{\omega }}_1^2}} x_1{x}_2+{\displaystyle \frac{N{\theta }^3}{\sqrt{L^3} {\widehat{\omega }}_1^2}} x_2^2+{\displaystyle \frac{a{\theta }^3}{{\widehat{\omega }}_1^2}}{x}_1^3 \\ +{\displaystyle \frac{3a{\theta }^2}{{\widehat{\omega }}_1^2}}{x}_1^2{x}_2+{\displaystyle \frac{3a\theta }{{\widehat{\omega }}_1^2}}{x_{1}x}_2^2+{\displaystyle \frac{a}{{\widehat{\omega }}_1^2}} x_2^3 =\overline{f} , \end{gathered}$$

(31)

where

$${\xi }_1=2{\zeta }_1{\theta }^2+2{\zeta }_e{\omega }_r, {\xi }_2=-2{\zeta }_1\theta +2{\zeta }_e{\omega }_r\theta ,{\xi }_3=2{\zeta }_1+2{\zeta }_e{\omega }_r{\theta }^2.$$

For details on the derivation of the modal equations, the reader is referred to Appendix A. It can be observed from Equations (30) and (31) that the modal equations exhibit quadratic and cubic nonlinearities. The system parameters can be tuned such that there exists a 2:1 internal resonance between the modal coordinates. This demands that ${\omega }_r$ be set equal to 1/2, which means that the electric frequency, ${\omega }_e,$ is half the effective mechanical frequency, ${\widehat{\omega }}_1,$ as per Equation (29).

The material properties of the beam material and the PZT patch are given in Table 1. The numerical value of the PZT capacitance is calculated as follows:

$$C_p ={\displaystyle \frac{2{ϵ}^T{ϵ}^o{L}_p{b}_p}{t_p}} =193.16 \mathrm{nF} .$$

(32)

The piezoelectric coupling coefficient $K_c$is defined as [37]

$$K_c ={\int }_0^{L_p/L}2E_p{d}_{31}{b}_p{\displaystyle \frac{\left(t+t_p\right)}{2}}{\varphi }_1^{″}\left(\xi \right)\Delta H d\xi ,$$

(33)

where $\mathsf{\Delta }H = H\left(\xi -{\xi }_1\right)-H(\xi -{\xi }_2)$ is the Heaviside function, and $E_p$ is the elastic modulus for the piezoelectric material.

Table 1. Material characteristics of the structure and the piezoelectric patch [38].

Parameter	Material
	Structure	PZT-P (ZrTi) O3 C-82
Length (mm)	260	100
Width (mm)	15	15
Thickness (mm)	1	0.5
Elastic modulus (GPa)	210	--
Density (kg/${\mathrm{m}}^3)$	7850	--
Permittivity	--	3650
Piezoelectric coefficient (C/N)	--	266$\times {10}^{-12}$

Table 1. Material characteristics of the structure and the piezoelectric patch [38].

Parameter	Material
	Structure	PZT-P (ZrTi) O3 C-82
Length (mm)	260	100
Width (mm)	15	15
Thickness (mm)	1	0.5
Elastic modulus (GPa)	210	--
Density (kg/${\mathrm{m}}^3)$	7850	--
Permittivity	--	3650
Piezoelectric coefficient (C/N)	--	266$\times {10}^{-12}$

3. Results and Discussion

3.1. Response of the Linear Structure Coupled with the Shunt Circuit

In this section, we investigate the linear vibration of the beam where geometric and inertia nonlinearities are assumed to be negligible. Nevertheless, the quadratic nonlinearity of the piezoelectric capacitance is preserved to maintain the 2:1 internal resonance between the structure and the circuit active. The cantilever beam is subjected to a resonant harmonic lateral excitation that is tuned to the first natural frequency of the beam. The circuit parameters are $L = 3470$ H, $R = 3500$ Ω, and $N = {10}^{10}$. These parameters are selected such that a 2:1 internal resonance between the structure’s first natural frequency, ${\widehat{\omega }}_1$, and the electrical frequency, ${\omega }_e$, is maintained. By tuning the excitation frequency tot the first natural frequency of the beam and increasing the excitation amplitude, the amplitude–force plots for the modal displacements $x_1$ and $x_2$ are shown in Figure 2. The figure shows the saturation phenomenon as the force amplitude reaches a threshold of 0.16 N. Beyond this level of excitation; the input energy is transferred to the modal displacement $x_1$ while the amplitude of the modal displacement $x_2$ remains constant. The time history of the modal displacements $x_1$ and $x_2$ before and after saturation is shown in Figure 3. As the figures show, there is a significant reduction in the vibration amplitude of modal coordinate $x_2$, which contributes most to the beam’s actual vibration coordinate. According to Equations (10) and (A7), the beam’s deflection is defined as

$$u\left(\xi ,t\right)=\varphi \left(\xi \right)q\left(t\right)=\varphi \left(\xi \right)\left(-\theta x_1+x_2\right).$$

(34)

The numerical value for the constant $\theta$ is 0.042 for the set of parameters used in this study. This means that the actual vibration amplitude is mostly contributed by the modal coordinate $x_2$. The saturation amplitude and the forcing threshold depend on the circuit’s parameters, the modal damping of the structure, and the piezoelectric patch size and configuration. The significance of these parameters will be further investigated in the next section. The effectiveness of the shunt circuit in attenuating the beam’s vibration amplitude is identified by presenting the beam’s response without the absorber, which is called the uncoupled system, and the beam’s response with the absorber, which is called the coupled system, as shown in Figure 4. As the figure shows, there is a significant reduction in the resulting vibration amplitude due to the coupling between the beam and the shunt circuit, where the energy transfer is directed to the circuit after the force has reached its threshold. Figure 4b also shows the linear behavior of the beam since the inertia and geometric nonlinearities have not been considered in this simulation.

3.2. Selection of the Design Parameters

In this section, a parametric study is performed to assess the influence of each shunt-circuit parameter on the performance of the proposed absorber. Namely, the parameters of the circuit, damping, and the piezoelectric patch are considered. The objective is to determine the optimum set of parameters for maximum vibration suppression. First, the coupled system’s response is studied with variation in the circuit resistance, R. Figure 5 presents the force-response plot for the beam’s vibration with three values of the circuit resistance: 3, 3.5, and 4 k$\mathsf{\Omega }$. The figure indicates that decreasing resistance results in decreased saturation threshold force and decreased vibration amplitude. Namely, the force saturation threshold reduces from 0.18 N to 0.14 N, and the vibration modal amplitude reduced from 0.05 to 0.04 as the resistance reduced from 4 k$\mathsf{\Omega }$ to 3 k$\mathsf{\Omega }$, respectively, as shown in the figure. This result agrees with the results reported in [39].

The influence of the gain, the constant N in the model, is considered. We examined the effect of the nonlinear gain on the absorber’s performance at three values: ${10}^{10}, 2\times {10}^{10}, 3\times {10}^{10}$ $V/C^2$, as shown in Figure 6. As the figure shows, as the gain increases, the vibration amplitude and the force saturation threshold decrease. The governing equations, Equations (30) and (31), show that increasing the gain N strengthens the coupling between the electrical and mechanical modes, and hence increases energy transfer.

The piezoelectric patch, which couples the beam with the absorber, the electric circuit, plays an important role in vibration suppression. The influence of the piezoelectric patch parameters, including its location on the beam and the geometrical dimensions (surface area and thickness), is investigated. First, we consider three key locations for the piezoelectric patch: at the beam’s root, middle, and tip, as shown in Figure 7.

According to Equations (29) and (32), the location of the piezoelectric patch on the beam matters when it comes to the piezoelectric material coefficient, $K_c$, and the electromechanical coupling factor (EMCF), $k_1$. The EMCF values for the piezoelectric patch locations shown in Figure 7a–c are calculated as $6.31\%$, $2.97\%$, and $0.06\%$, respectively. It is worth mentioning that the 2:1 internal resonance ratio has been preserved throughout parametric study. The force-response plot for the beam’s vibration with the three PZT patch locations is shown in Figure 8. As the figure illustrates, placing the PZT patch at the beam’s root is the most efficient in terms of vibration suppression due to the maximum strain generated at the beam’s root. The PZT-patch location not only reduces the steady-state vibration amplitude, but it also reduces the force saturation threshold, as Figure 8 implies. These results emphasize that the location of the PZT patch strongly influences vibration suppression performance due to variation in strain energy distribution along the beam.

Next, we consider the significance of the size of the piezoelectric patch on the performance of the absorber. Two parameters depend on the PZT patch size: the electromechanical coupling coefficient (EMCF) and the capacitance of the patch, as shown in

Table 2. Size variation and properties for five different piezoelectric patches.

Patch.	$\mathbf{Dimensions} ({\mathit{L}}_{\mathit{p}}\times {\mathit{b}}_{\mathit{p}})$	Surface Area	$\mathbf{EMFC} ({\mathit{k}}_1)$	PZT Capacitance
A	0.09 m × 0.014 m	0.00126 m2	2.71%	81.13 $\mathrm{nF}$
B	0.12 m × 0.0117 m	0.00140 m2	2.90%	90.17 $\mathrm{nF}$
C	0.10 m × 0.015 m	0.00150 m2	2.97%	96.58 $\mathrm{nF}$
D	0.11 m × 0.016 m	0.00176 m2	3.23%	113.3 $\mathrm{nF}$

. Figure 9 presents the force-response plot for the beam’s vibration for various patch sizes. The figure shows that as the patch size increases, the coupling coefficient increases, and the capacitance increases. Regarding the absorber’s effectiveness, it is found that a larger PZT patch size results in more effective absorption, as it is evident from the lower saturation force threshold and the reduced vibration amplitude as the patch size increases.

Finally, we consider the effect of the thickness of the patch on the absorber’s performance. The thickness of the piezoelectric patch was varied from 0.3 mm to 0.7 mm, and the coupling coefficient and the PZT capacitance were calculated, as shown in

Table 3. Thickness variation and properties for three different piezoelectric patches.

Patch	$\mathbf{Thickness}\text{:} {\mathit{t}}_{\mathit{p}}$ (mm)	Electromechanical Factor$\text{:} {\mathit{k}}_1$	PZT Capacitance: ${\mathit{C}}_{\mathit{p}}$ (nF)
A	0.3	2.00%	161.0
B	0.5	2.97%	96.58
C	0.7	3.98%	68.98

. It can be noticed that as the patch thickness increases, the coupling coefficient increases, and the capacitance decreases. This is similar to the effect of the patch surface area, and hence, a similar trend is expected in terms of vibration suppression effectiveness, as shown in Figure 10.

3.3. Response of the Nonlinear Structure Coupled with the Shunt Circuit

The results of the previous section showed that the suggested vibration control methodology efficiently suppresses the beam’s linear vibration via the 2:1 internal resonance between the electrical and mechanical natural frequencies. In this section, the beam’s cubic geometric nonlinearity is taken into consideration. The goal is to examine the effectiveness of the absorber in suppressing the nonlinear vibration of the beam. Figure 11 presents the frequency response curves near resonance of the uncoupled system, the beam’s response without the absorber, for various values of the excitation amplitudes. Each frequency-response plot has two segments that result from forward and backward sweeps around the resonant frequency, which is the frequency of the first mode, to capture the upper and lower stable branches. The beam shows a hardening nonlinear response and ranges of frequency of multiple solutions and the possibility of jumping from one stable branch to the other.

The steady-state response of the nonlinear vibration of the uncoupled beam, without the shunt circuit, and the coupled beam, with the shunt circuit, at an excitation amplitude of 0.5 N is shown in Figure 12a. The PZT patch was fixed at the beam’s root, and the nonlinear gain $N$ was set to $3\times {10}^{10} V/C^2$. The rest of the circuit parameters are given in row C in Table 1 for the patch surface area, and row B in Table 2 for the patch thickness. The force-response plot of the beam’s response is shown in Figure 12b where the saturation phenomena holds and the nonlinear response of the beam without the shunt is evident. Frequency-response plots of the nonlinear vibration for the baseline beam without the absorber and the coupled beam with the shunt circuit are shown in Figure 13 for a variety of force amplitudes. The absorber shows an excellent ability to mitigate the nonlinear vibration and achieve significant reductions of about 63%, 73%, and 75% for forcing amplitudes of 0.1 N, 0.3 N, and 0.5 N, respectively. The performance of the proposed absorber proves to be robust for both the linear and nonlinear vibration of the cantilever beam. Moreover, Figure 13 shows that the absorber’s performance is effective over a significant bandwidth of excitation frequencies. The beam shows a strong hardening nonlinear behavior due to geometric nonlinearity. The implication of this behavior is that the nonlinear natural frequency shifts from the linear one, which may impact the robustness of the proposed control strategy that is based on employing the saturation phenomenon due to the 2:1 internal resonance. Nevertheless, the proposed suppression methodology shows an excellent ability to transfer the energy from the beam to the circuit at relatively high forcing amplitude, as shown in Figure 12 and Figure 13.

4. Conclusions

The nonlinear vibration suppression of cantilever beams using shunted piezoelectric elements is investigated. The piezoelectric shunt circuit exhibits a quadratic nonlinear capacitance that is tuned with the beam’s natural frequency to achieve a two-to-one internal resonance. The internal resonance coupling is exploited to trigger the saturation phenomenon, where the beam’s vibration reaches an almost constant value beyond a force excitation threshold. A mathematical model for the nonlinear vibration of a cantilever beam coupled with the piezoelectric shunt circuit considering the inertia and geometric nonlinearity was developed. A reduced-order model based on Galerkin’s discretization method was developed and used to numerically simulate the performance of coupled and uncoupled systems. First, the response of the linear structure to resonant harmonic excitations was examined with and without the shunt circuit. It was found that the shunt circuit performs effectively as a lightweight, easy-to-use, absorber around the resonant frequency. The vibration intensity of the beam is reduced significantly due to the saturation phenomenon after the force amplitude reaches its threshold. A parametric analysis was performed to serve as a design guideline for optimizing system parameters including the circuit parameters and piezoelectric patch size and location. It was found that the larger the piezoelectric patch surface area, the lower the vibration amplitude after the saturation. It was also found that attaching the patch at the beam’s root is more effective in vibration suppression than moving it toward the tip. The cantilever beam’s nonlinearity was examined, and the geometric cubic nonlinearity was found dominant over the inertia nonlinearity for the primary resonance considered in this study. The shunt circuit absorber was tested against resonant harmonic excitation for the nonlinear structure, and its performance was found to be robust and highly effective. Future research directions include, but are not limited to, exploring multi-mode internal resonances, experimental prototyping, and nonlinear vibration suppression in wide-band applications.