Free vibration analysis of a spinning piezoelectric beam with geometric nonlinearities

Wei Li · Xiao?Dong Yang · Wei Zhang · Yuan Ren · Tian?Zhi Yang

Abstract The linear and non-linear free vibrations of a spinning piezoelectric beam are studied by considering geometric nonlinearities and electromechanical coupling effect. The non-linear differential equations of the spinning piezoelectric beam governing two transverse vibrations are derived by using transformation of two Euler angles and the extended Hamilton principle,wherein an additional piezoelectric coupling term and different linear terms are present in contrast to the traditional shaft model. Linear frequencies are obtained by solving the standard eigenvalues of the linearized system directly, and the nonlinear frequencies and non-linear complex modes are achieved by using the method of multiple scales. For free vibrations analysis of a spinning piezoelectric beam, the non-linear modal motions are investigated as forward and backward precession with different spinning speeds. The responses to the initial conditions for this gyroscopic system are studied and a beat phenomenon is found, which are the n validated by numerical simulation. The influences of some parameters such as electrical resistance, rotary inertia and spinning speeds to the non-linear dynamics of a spinning piezoelectric beam are investigated.

Keywords Spinning piezoelectric beam· Free vibrations· Non-linear frequencies· Complex modes

1 Introduction

A spinning piezoelectric beam can often be used to make a piezoelectric vibratory gyroscope, which has several applications such as in mobile phones, high-grade cars, intelligent robotics, military weapons and aerospace systems. In the past,the piezoelectric vibratory gyroscope has become one of the best essential electromechanical system sensors, especially in the field of inertial navigation systems [1]. To simplify the analysis of the piezoelectric gyroscope, traditionally, most investigations are confined to the linear system with piezoelectric excitation and piezoelectric detection [2, 3]. Based on the linear approximation and non-linear “slow” system, Lajimietal. [4] investigated the non-linear estimate of the mechanical thermal noise for an electrostatic gyroscope. The applications of non-linear analysis are also significant for flexible components, and the characteristics of the non-linear gyroscopic system have attracted much attention in a field of rotating shaft [5,6]. By using the fractional calculus and the Gurtin–Murdoch theory, Oskouie et al. [7] investigated the nonlinear vibration of viscoelastic Euler–Bernoulli nanobeam. By considering three types of boundary conditions, Zhao et al. [8] studied the natural frequencies of the Timoshenko beam with surface effects. Recently, the re has been a growing research interest to investigate piezoelectric materials on energy harvesters [9–11] and microstructures [12–15] for rotational motion. As a conclusion, the non-linear characteristics of gyroscopes modeled by a spinning piezoelectric beam should be investigated, so that the guidance to improve the performance of piezoelectric vibratory gyroscope can be proposed.

Modal analysis of gyroscopic system is an effective tool to investigate the dynamic responses and mode interactions [16,17]. However, when coping with the gyroscopic continuum,the modal theories become difficult because the complex modes should be considered [18, 19]. To this end, Rosenburg[20] firstly presented the non-linear normal modes, which exp and ed the modal motions from the linear non-gyroscopic systems to non-linear non-gyroscopic systems. The non-linear normal mode concept is utilized in the field of non-linear systems by many researchers. By using the multiple scales method, Nayfeh and Nayfeh [21, 22] studied the non-linear normal modes with internal resonance and geometric nonlinearity of a one-dimensional continuous system. To solve the modal motions of gyroscopic systems, Shaw and Pierre[23, 24] used the invariant manifold method to exp and the non-linear normal modes to the gyroscope coupling systems.The work of Carlos et al. [25] made a comparison for the nonlinear normal modes of an axially loaded beam by both the invariant manifold method and multiple scales method. For a linear system, Uspensky and Avramov [26] studied the nonlinear normal mode under a forced excitation by using the invariant manifold method and Rauscher method. To analyze the free vibration of a gyroscopic system, Arvin and Nejad[27] described the complex dynamical characteristics of nonlinear normal modes. In their work, Qian et al. [28] studied parametric instability analysis of a linear gyroscopic system based on the traditional coupled gyroscopic system and decoupled gyroscopic modes decoupling method. Recently, the re are several valuable research towards non-linear normal modes of undamped systems [29–31] and damped systems [32, 33] by using numerical calculation. The study by Pan et al. [34] used the complex modal technique to evaluate the natural frequencies and complex modes of serpentine belt drives.

The gyroscopic effect caused by spinning motion appears comprehensively in the rotor dynamics systems [35, 36]. By deriving the closed polynomial of frequency equations and integral forms under an ordinary forcing function, Sturla and Argento [37] studied the free and forced vibrations of a viscoelastic rotating Rayleigh beam. The work Ishida and Inoue [38] considered the effect of internal resonance of a non-linear rotor. It is always hard to gain the physical model of nonlinear rotor-bearing system, thus Ma etal. [39] identified a data-driven non-linear auto-regressive network with exogenous inputs (NARX) model to solve this problem. By considering r and om excitations, Hosseini and Khadem [40]investigated the vibration and stability of a spinning beam with r and om characteristics subject to white noise by using the finite element method. In order to guide the design of distorted model, Luo et al. [41] provided a new dynamic scaling law of geometrically distorted model in predicting the dynamic characteristics. Moreover, many researchers analyzed the free vibrations dynamic properties of shafts by different methods [5, 6, 42, 43].

In this paper, the piezoelectric coupling governing differential expressions with non-linearities in curvature and inertia of a spinning piezoelectric beam are obtained, and the natural frequencies, as well as gyroscopic complex modes, are analyzed. By using the multiple scales method,the non-linear modal motions and non-linear frequencies are investigated. The responses to the initial values are discussed for the gyroscopic system by multiple scales method, and numerical simulation validates the results. The piezoelectric coupling effect and non-linear features of the gyroscopic continuum are also investigated in detail. The contribution of the electrical resistance, rotary inertia, electromechanical coupling coefficient and spinning speeds to the forward and backward natural frequencies of the spinning piezoelectric beam are studied, which prompts possible optimizations in the design of piezoelectric vibratory gyroscopes.

Fig. 1 Spinning beam with surrounded four piezoelectric films

2 Governing equations of a spinning piezoelectric beam

Figure1 shows the structure of a spinning beam which is surrounded with four piezoelectric films. The length (L1, L2,L) and width (wb, wp) of the beam and piezoelectric films are shown in the figure. The beam displacement is made of three components, u(s, t), v(s, t), and w(s, t), along the inertial frame x, y, and z directions, respectively, where s denotes the undeformed arclength along the x-axis from the root of the beam to the observed reference point, t denotes time. The x–y–z coordinate system denotes the inertial frame.

The transformation of two Euler angles by which an arbitrary beam cross section can be expressed with three coordinate systems is shown in Fig.2. The x0–y0–z0system is a

For an in-extensional beam, u′ ? ?(v′2+ w′2)/2, exp anding ? with a Taylor series, cos ? = 1??2/2, sin? = ???3/6, the transformation matrix P can be attained as

spinning frame around the x-axis with constant speed Ω of the undeformed beam; the x1–y1–z1and x2–y2–z2systems are orthogonal coordinate frames associated with Euler angle transformation. Moreover, we let (ix, iy, iz),and (i1,i2, i3) represent the unit vectors of the x0–y0–z0, x1–y1–z1, and x2–y2–z2coordinate frames, respectively.

From the undeformed plane, the cross section first spins by α degree about n axis from x0–y0–z0to x1–y1–z1

The transformation matrix B(α) can be expressed by the displacements [44]

Here, the primes of (u, v, w) denote the derivatives with respect to s, respectively.

Further, the cross-section spins ? by about x1axis from x1–y1–z1to x2–y2–z2, thus the transformation from x0–y0–z0to x2–y2–z2is

Using the concept of continuity, one can obtain the deformed curvatures ρi(i = 1, 2, 3)

The relative angular velocity vector ω to the inertial frame are the n obtained as

Fig. 2 Sequence of Euler angle transformation

Here, the dots of (u, v, w) denote the derivatives with respect to t, respectively.

The deformation of any point on the beam can be denoted by the position vector R as

Using directional time derivatives,in the coordinate system of x0–y0–z0is expressed as

with

The kinetic energy of a spinning Rayleigh beam can be obtained by substituting Eqs.(6), (8), and (9) into the following expression

According to the assumption of an inextensional beam and using the geometric boundary condition u(0,t) = 0, one can obtain

Hence, the kinetic energy can be obtained as

with

In Eq.(13), H(s) is the Heaviside function, A is the cross sectional area of the beam, m, ρ(s), and j are the total mass,total density, and total rotary inertia of the beam and the piezoelectric films, respectively, and ρb,p, wb,p, and hb,p(wb/hb= 1) are the volumetric mass density, width and thickness of the beam and the piezoelectric film, respectively. For all the parameters used in this paper, subscript b denotes the beam material and p denotes piezoelectric film.

The mechanical properties of piezoelectric films are coupled with their electric properties. For the configuration and non-linear strain considered here, the electrical displacement is one dimensional and the stress–strain relations for these materials are known as follows [45, 46]

where Epis the stiffness coefficient, ing33is the dielectric permittivity, Tpis the stress, S is the strain, e31is the piezoelectric strain constant, D is the electrical displacement, E3is the electrical field, and Vv,wis the voltage. The relation between voltage and current is

where Z is electrical resistance, Q is charge quantity.

An isotropic beam is considered, and the stress–strain satisfies the relation

where Ebst and s for stiffness coefficient.

The total potential energy for spinning beam can be derived as follows

Substituting Eq.(15) into Eq.(14), then further inserting the results and Eq.(16) into Eq.(17), we can obtain

w here EI=EbIb+S(s)EpIpis bending stiffness,Cp=wpσ33∕hpis piezoelectric film capacitor,? =wpe31(hb∕2+hp)is electromechanical coupling coefficient, and

The virtual work done by mechanical damping and electrical resistance is

The next step is to substitute the results of kinetic, potential energy and virtual work into the Hamilton principle

Because the torsional frequency is larger than the flexural frequency, so the twist angle ? can be neglected [42]. By using voltage and current relation and exp anding the results up to three orders, we can obtain the following four partial governing equations with electromechanical coupling

Su b s t i t u t i n g p e r i o d i c v o l t a g e sinto the last two equations of Eq.(22), removing the overbar above the displacements and voltages for simplification, we obtain

with Z0= 1/(iωCp).

Hence, substituting Eq.(23) into Eq.(22), the last two equations of (22) can be eliminated by using displacements to replace voltages [47], and the n the equations can be reduced to partial differential equations of two degrees of freedom

Introducing dimensionless variables and parameters

Substituting Eq.(25) into Eq.(24), the dimensionless form of the governing equations becomes

By neglecting the terms of the rotary inertia and geometric non-linearity, our governing Eq.(26) can recover those equations in Refs. [1, 2] that focused on the linear counterpart of the piezoelectric gyroscope. In contrast to the nonlinear shaft models [42, 48], which studied only a rotating beam without piezoelectric materials, two additional piezoelectric coupling terms (κS(s)″2v/(1 + Z0/Z) and κS(s)″2w/(1 + Z0/Z)) and different linear terms such as gyroscopic coupling terms and centrifugal force terms are presented in the current formulation for an investigation.

In this study, considering the simply-supported boundary condition (v = w=0 and v″ = w″ = 0) at both ends, we use Galerkin method and the appropriate sine function

where n is the mode number, p and q are generalized temporal coordinates and coupled to each other. Substituting Eq.(27) into Eq.(26), letting L1= 0, L2= L, and assuming a constant base angular speed Ω, we can obtain two non-linear ordinary differential equations

3 Analysis of the linear frequency and responses to initial conditions

In this section, the free vibration of the linear part of Eq.(28)is studied first. Neglecting damping and non-linear terms,one can obtain two second-order linear ordinary differential equations with respect to t

Substituting Q = μeiωtinto Eq.(29), and according to the boundary conditions, the natural frequencies based on the linear system can be obtained as

Fig. 3 Natural frequencies versus spinning speed (J = 0.002, n = 1,κ = 0.5, and c = 0)

The n, the corresponding column vector μ can be obtained by substituting ωfand ωbof Eq.(31) back into Eq.(29). The column vector μ is different when Ω greater or smaller than a critical value

where the first column [?i 1]Tcorresponding to ωfrepresents forward precession and the second column [i 1]Tcorresponding to ωbrepresents backward precession in the sub-critical case. On the critical value the re exists a switch point of the first mode from backward precession to forward precession. Beyond the critical point, both modes are forward precession.

The two scalar equations of Eq.(29) are only linearly gyroscopic coupled by neglecting the non-linear terms and damping. In Fig.3, the first two natural frequencies versus spinning speeds are plotted for electrical resistance Z0/Z = 10 and Z0/Z = ∞ with parameters J = 0.002, n = 1, κ = 0.5, and c = 0. For each value of Z, two natural frequencies correspond to the two gyroscopic modes of the spinning piezoelectric beam. Figure3 shows that the forward frequency ωfincreases and the backward frequency ωbdecreases, followed by an increase after the critical point. In the local amplified plot, the re are no electric fields in both directions for the case of shortened electrodes (Z0/Z = ∞). In this case, the piezoelectric coupling effect related to electric fields does not exist. When the electrodes are not shortened (Z0/Z = 10), as presented in Fig.3, the re are electric energy transfers in the two directions, which causes stiffening of the spinning piezoelectric beam, and higher frequencies are found. Figure3 also shows that the frequency ωbis decreasing first, and the n increasing, and the re exists a critical switch value. When the electrodes are not shortened (Z0/Z = 10), the point of critical value is pushed back compared with the shortened case(Z0/Z = ∞), although the effect of the electrodes is weak.Similarly, Fig.4 shows the second two natural frequencies versus spinning speeds, which are plotted for Z0/Z = 10 and Z0/Z = ∞ with parameters J = 0.002, n = 2, κ = 0.5, and c = 0.When shortened electrodes (Z0/Z = ∞) are considered, the re exist neither electric fields nor piezoelectric coupling effect.When the electrodes are not shortened (Z0/Z = 10), electric field appears causing higher frequencies. We will further explain the dynamics of the four typical points Af,b, Bf,b, Cf,b, and Df,bon Figs.3 and 4 in the analysis of modal motions in the next section.

Fig. 4 Natural frequencies versus spinning speed (J = 0.002, n = 2,κ = 0.5, and c = 0)

4 Application of multiple scales method for the non?linear system

The method of multiple scales is extensively used in this study of gyroscopic system. Now we treat the non-linear gyroscopic system Eq.(28) by using the procedure of the multiple scales method. The n, the solutions of Eq.(28) are assumed as

where ing is a bookkeeping device denoting small parameter, and the fast and slow time scale T0= t and T2= ing2t are introduced. Damping c is scaled with cing2since it is usually very weak. The time derivatives can be written as

with D0= ?/?T0, D2= ?/?T2.

Substituting Eqs.(33) and (34) into Eq.(28) and equating the coefficient of different orders of ing yields

Substituting Eq.(37) into Eq.(36), one can obtain

where “nst” denotes non-secular terms, “cc” denotes the conjugate of the proceeding terms, and

To determine the solvability conditions of gyroscopic ordinary differential Eq.(38), q3and p3are expressed as

Substituting Eq.(40) into Eq.(39) and equating the coefficient of eiwfT0, we obtain

Similarly, for coefficient of eiwbT0, the following equations are obtained

Equations(41) and (42) are composed of algebraic equations with respect to A11(T2), A21(T2) and A12(T2), A22(T2).The nontrivial condition can be expressed as [42]

After some mathematical manipulations, the solvability conditions can be written as

with

For the super-critical case,