# Singularity Analysis of Composite Laminated Piezoelectric Rectangular Plate Structure with 1 : 2 Internal Resonance

Authors: Yuhong Guo; Yuhua Guo; Wei Zhang; Ruiping Wen
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This study investigates the dynamical behavior of the composite laminated piezoelectric rectangular plate with 1 : 2 internal resonance near the singularity using the extended singularity theory method. Based on the previous four-dimensional averaged equations of polar coordinates where the partial derivative terms are not equal to zero, the universal unfolding with codimension 3 of the proposed system is given. The main material parameters that affect the dynamic behavior of the laminated piezoelectric rectangular composite plate near the singularity under transverse excitation are revealed by the transition set of universal unfolding with codimension 3. In addition, the plots of the transition set in three bifurcation parameters space are discussed. These numerical results can show that the stability near the singularity of the proposed system is better when period ratio is less than zero.


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## Body

## 1. Introduction

Singularity theory has been widely applied as a qualitative bifurcation analysis method. It can use the uniform and clear method to deal with various complex bifurcation problems and establish comprehensive links between system dynamic behavior and system parameters with singularity theory. But so far, the application of singularity theory in the chaotic dynamics of dynamical systems faces challenges.The introduction of the singularity theory to the single-degree-of-freedom system with a single parameter can lead to complex and interesting bifurcation phenomena, such as the pitchfork bifurcation. Schaeffer and Golubitsky [1] investigated the bifurcation phenomenon of a model under the chemical reaction. Martinet [2] discussed the open fold of smooth mapping buds under the strong equivalence and gave various forms of open fold theorem. Chen and Langford [3] gave the C-L method, by combining the L-S method and singularity theory, and studied the bifurcation of the periodic solution for nonlinear dynamical equation undergoing parametric excitation. Jin and Zou [4] studied a restrained pipe conveying fluid and obtained dynamical behaviors by singularity theory in different persistent regions. Chen et al. [5] gave the bifurcation analysis of an arch structure with the parametric and forced excitations in codimension 5. Nigol et al. [6, 7] analyzed the torsional galloping mechanism under a galloping excitation. Yu et al. [8] also studied the torsional feedback mechanism under a galloping excitation. Then, with the gradual adding of bifurcation parameters, the singularity theory has been applied to a single-degree-of-freedom system with multiparameters. Futer et al. [9] and Sitta [10] gave classification of bifurcation problems under codimension not greater than 1, considering the bifurcation parameter symmetry. Lari-Lavassani and Lu [11] gave a full and accessible account of unfolding and finite determinacy and stability theorems of the multiparameter bifurcation problem. Seyranian and Mailybaev [12] studied the multiparameter stability theory and analyzed the influence of system parameters on the stability. Qin and Chen [13] analyzed a bifurcation system with two parameters. Zhang and Chen [14] studied stability and bifurcation behaviors for a model of simply supported functionally graded materials rectangular plate subjected to the transversal and in-plane excitations by means of combination of analytical and numerical methods. Guo and Zhang [15] studied the bifurcation problems of the composite laminated piezoelectric rectangular plate structure with three bifurcation parameters by singularity theory in the case of 1 : 2 internal resonance, the sign function is employed to the universal unfolding of bifurcation equations in this system, the proposed approach can ensure the nondegenerate conditions of the universal unfolding of bifurcation equations in this system to be satisfied, the study presents the proposed system with three bifurcation parameters is a high codimensional bifurcation problem with codimension 4, and 6 forms of universal unfolding are given.The current piezoelectric materials, which include piezoelectric lead-zirconate-titanate (PZT) and piezoelectric polyvinylidene fluoride (PVDF), are new type of functional materials in engineering fields. Piezoelectric materials can be used as the actuators and sensors in engineering structures. Tauchert et al. [16] reviewed the theoretical research progress of the piezoelectric intelligent composite structure. Halim and Reza Moheimani [17] studied that piezoelectric materials were glued or embedded into controlled structures as sensing elements to achieve vibration suppression by consuming or diverting the mechanical energy of controlled structures. Zhang et al. [18] investigated the nonlinear oscillations and chaotic dynamics of a parametrically excited simply supported symmetric crossply laminated composite rectangular thin plate with the geometric nonlinearity and nonlinear damping. Della and Shu [19] reviewed various models and numerical analysis results of piezoelectric composite laminated plates. Hao et al. [20] analyzed the nonlinear oscillations, bifurcations, and chaos of a functionally graded materials plate. Karnaukhov and Tkachenko Ya [21] studied the active damping of nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators by the dynamic-programming method. Liu et al. [22] studied the nonlinear forced vibrations of functionally graded material (FGM) sandwich cylindrical shells with porosities on an elastic substrate and analyzed the effects of the core-to-thickness ratio, porosity volume fraction, power-law exponent, and external excitation on nonlinear forced vibration characteristics of FGM sandwich shells with porosities. Liu et al. [23] developed a new solution approach to solve nonlinear forced vibrations of functionally graded (FG) piezoelectric shells in multiphysics fields; the novel feature of this approach is that it can efficiently obtain the unstable solution and tackle the difficult problems in mathematics encountered during formulation. Zhang et al. [24] established the governing equations of motion for the nonlinear oscillations of a simply supported symmetric crossply composite laminated piezoelectric plate subjected to the transverse, in-plane excitations, and the excitation loaded by piezoelectric layers and studied the periodic and chaotic dynamics of the composite laminated piezoelectric plate.The singularity theory for two-degree-of-asymmetry freedom bifurcation systems with a single parameter has been well developed by Golubitsky and Schaeffer [25]. The universal unfolding of bifurcation equations was obtained, and the conclusion showed that the number of codimension was equal to the number of auxiliary parameters. Present, there is limited research about the singularity theory for two-degree-of-asymmetry freedom bifurcation systems with multiparameters. In two-degree-of asymmetry freedom bifurcation systems with multiparameters, the number of auxiliary parameters is less than codimension, which is contradiction with the previous conclusion. In this work, the sign function is introduced to solve this problem and ensure nondegenerate conditions of universal unfolding of the bifurcation equations are satisfied, and determining the summation of the number of sign functions and the number of auxiliary parameters is equal to codimension. Then, for the dynamical equations of the composite laminated piezoelectric rectangular plate structure in the case of principal parametric resonance and 1/2 subharmonic resonance for the first-order mode and primary resonance for the second-order mode because only transverse nonlinear oscillations of the composite laminated piezoelectric rectangular plate are considered, the governing equations of motion can be reduced to a two-degree-freedom nonlinear system under combined parametric and external excitations by using Galerkin’s method [24]. By employing sign function, the universal unfolding of the 1 : 2 internal resonance bifurcation equations is derived. The transition sets in the parameters plane are calculated, and the bifurcation diagrams are depicted. The main material parameters that affect the dynamic behavior of the laminated piezoelectric rectangular composite plate near the singularity under transverse excitation are revealed by universally unfolding with codimension 3. The motivation of this work is to provide theoretical basis for the materials selection in the product design stage.


## 2. Bifurcation Equations of the Composite Laminated Piezoelectric Rectangular Plate with 1 : 2 Internal Resonance

The nondimensional motion equations of the laminated piezoelectric composite rectangular plate as shown in (1) are taken into consideration [24]:(1)w¨1+ω12w1+μ1w˙1+α2cosΩ1t+α3cosΩ2t+α4cosΩ4tw1+α5w12w2+α6w22w1+α7w13+α8w23=F1cosΩ3t,w¨2+ω22w2+μ2w˙2+β2cosΩ1t+β3cosΩ2t+β4cosΩ4tw1+β5w22w1+β6w12w2+β7w23+β8w13=F2cosΩ3t,where w1 and w2 are the amplitudes of the first-order and second-order modes for the composite laminated piezoelectric rectangular plate vibration, F1=F2=−16q/3π2 denotes the transverse excitation, μ1and μ2 are the structure damping coefficients, and other coefficients in (1) are presented in Appendix.Consider the case of principal parametric resonance and 1 : 2 internal resonance. In this resonant case, there are the following resonant relations:(2)ω12=14ω2+εσ1,ω22=ω2+εσ2,Ω1=ω,Ω2=Ω3=Ω4=ω,ω2≈2ω1,where ω1 and ω2 are the two different linear natural frequencies, and σ1 and σ2 are the two detuning parameters.By using the multiple scale method [24], it can obtain the averaging equations:(3)a˙1=−12μ1a1−14α2+α3+α4a1sin2ϕ1,a1ϕ˙1=12σ1a1−14α6a1a22−38α7a13−14α2+α3+α4a1cos2ϕ1,a˙2=−12μ2a2−14F2sinϕ2,a2ϕ˙2=14σ2a2−18a12a2−316β7a23−14F2cosϕ2.Using the relations between trigonometric functions, the bifurcation equations can be determined from (3) as follows:(4)a˙1+12μ1a12+a1ϕ˙1−12σ1a1+14α6a1a22+38α7a132=14α2+α3+α4a12,a˙2+12μ2a22+a2ϕ˙2−14σ2a2+18a12a2+316β7a232=14F22.Expanding (4) leads to the following bifurcation equations:(5)g1=k11a16+k12a14a22+k13a12a24+k14σ1a14+k15σ1a12a22+σ12a12+k16a12+τ1=0,g2=k21a26+14a14a22+k22a12a24+k23σ2a24−σ2a12a22+σ22a22+4μ22a22−F22+τ2=0,where(6)k11=916α72,k12=34α6α7,k13=14α62,k14=−32α7,k15=−α6,k16=μ12−14α2+α3+α42,τ1=4a˙1a˙1+μ1a1+ϕ˙14ϕ˙1+3α7a12+2α6a22−4σ1a12,k21=916β72,k22=34β7,k23=−32β7,τ2=16a˙2a˙2+μ2a2+ϕ˙216ϕ˙2+4a12+6β7a22−8σ2a22.In (6), τ1 and τ2 refer to the combination of the gradients of displacement and phase angle. It can be referred to as the periodic parameter in (5). Studies on the dynamical behavior near the singularity have been carried out in the literature for the cases where τ1=0 and τ2=0 [15]. Therefore, when τ1≠0, τ2≠0, and σ1=σ2, the study on the dynamical behavior near the singularity is focused in this work.


## 3. Singularity Analysis for 1 : 2 Internal Resonance

Proposition 1.
The universal unfolding of (5) is given as(7)G=G1,G2=a12+2ε1σ1a2+2ε3F2+τ1,a22−F22+2ε2F2a1+τ2,where εii=1,2,3 is the sign function. The proof process of Proposition 1 will be given later in this work.Whenτ1≠0 and τ2≠0, (7) is further written as(8)G=G1,G2=a12+2ε1σ1a2+2ε3F2+τ1,a22−F22+2ε2F2a1+κτ1,where κ=τ2/τ1 is called the period ratio in this work.In the following analysis, the transition set of the universal unfolding (8) is discussed.Equation (8) for bifurcation points is obtained as(9a)a12+2ε1σ1a2+2ε3F2+τ1=0,(9b)a22−F22+2ε2F2a1+κτ1=0,(9c)a1ε2F2=ε1σ1a2=ε3ε2a1−F2.Multiplying−κ in equation (9a) and then substituting it into equation (9b), obtain(10)κa12+2ε1σ1a2+2ε3F2−a22+F22−2ε2F2a1=0.By equation (9c), it can get(11)a1=F2±F22+4ε2ε3F22ε2,a2=−ε1σ1F2±ε1σ1F22+4ε2ε3F22ε2ε3.Substituting (11) into (10), the bifurcation occurs when(12)κε2ε3F2±F22+4ε2ε3F22±4σ12F22+4ε2ε3F2+8ε2−4σ12F2−ε2ε3−ε1σ1F2±ε1σ1F22+4ε2ε3F22∓4ε2ε3F2F22+4ε2ε3F2=0.Equation (8) for hysteresis points is obtained as(13a)a12+2ε1σ1a2+2ε3F2+τ1=0,(13b)a22−F22+2ε2F2a1+κτ1=0,(13c)a1a2−ε1ε2σ1F2=0,(13d)v12,v22∈rangedG,where v=v1,v2. Now assume σ1≠0, F2≠0, and(14)v=a1,ε1σ1,and it is noted that v12,v22∈rangedG if and only if(15)v12,v22⋅ε1σ1,a2=0.Equation (15) implies that(16)a12+ε1σ1a2=0,multiplying by a1 and substituting it into equation (13c), obtain(17)a1=−ε2σ12/3F21/3,and using equations (16) and (17), get(18)a2=−ε1σ11/3F22/3.Substituting equations (17) and (18) into (10), the hysteresis occurs when(19)F22+σ12/3F24/3+κ2ε3F2−σ14/3F22/3=0.Equation (8) for the double limit points is(20a)a12+2ε1σ1a2+2ε3F2+τ1=0,(20b)a22−F22+2ε2F2a1+κτ1=0,(20c)a1a2−ε1ε2σ1F2=0,(20d)a11,a21≠a12,a22,whenF2=const.Using (12), double limit points occur when(21)σ1=0,κε2ε3F2±F22+4ε2ε3F22+8ε2F2∓4ε2ε3F2F22+4ε2ε3F2=0.B, H, and D have equations as follows:(22a)B:κε2ε3F2±F22+4ε2ε3F22±4σ12F22+4ε2ε3F2+8ε2−4σ12F2−ε2ε3−ε1σ1F2±ε1σ1F22+4ε2ε3F22∓4ε2ε3F2F22+4ε2ε3F2=0,(22b)H:F22+σ12/3F24/3+κ2ε3F2−σ14/3F22/3=0,(22c)D:σ1=0,κε2ε3F2±F22+4ε2ε3F22+8ε2F2∓4ε2ε3F2F22+4ε2ε3F2=0.From the above analysis, (22a)–(22c) represent the relationship among transverse excitation F2, period ratio κ, and detuning parameter σ1 when the composite laminated piezoelectric rectangular plate structure subjected to small perturbations exists the bifurcation, hysteresis, and double limit points.


## 4. Numerical Results

The system is simulated by using MAPLE. It take amplitudesa1≥0, a2≥0, and transverse excitation F2≥0. Three-dimensional sketch of transition variety of (8) is obtained by applying (22a)–(22c), in Figure 1, where B implies the bifurcation set, H implies the hysteresis set, and D implies the double limit points. From Figure 1, it can see that the whole parametric space is divided into three kinds of different persistent regions by (22a)–(22c). It is observed that the amplitudes a1 and a2 have a solution in region ①, corresponding to the original system frequency response equations that have only one solution, the composite laminated piezoelectric rectangular plate structure subjected to small perturbations is in stable state. Having multiple solutions in region ②, corresponding to the multiple solutions band of the original system frequency response equations, the composite laminated piezoelectric rectangular plate structure subjected to small perturbations is in stable state. Having nonsolution in the region ③, corresponding to the original system frequency response equations have no solution, the composite laminated piezoelectric rectangular plate structure with small perturbations is in stable state. However, entering region ① from region ②, jumping will occur, and the hysteresis will occur when entering region ② from region ③; either way, the composite laminated piezoelectric rectangular plate structure subjected to small perturbations is in unstable state.Figure 1
Three-dimensional sketch of transition variety of (8) when ε1=1, ε2=1, and ε3=1, B represents the bifurcation set, and H represents the hysteresis set.The bifurcation diagrams of the three kinds of different persistent regions are given in Table1. From Table 1, it is observed that when the period ratio κ changes in region ①, there is a steady solution, whereas when the period ratio κ translates into region ②, there are multiple solutions. Furthermore, it is observed that when the period ratio κ changes in region ③, there is no periodic solution because the amplitude a1 is negative.Table 1
Case of amplitudea1 in three kinds of different persistent regions.
①②③No solutionIn Figure2, it is observed that entering region ① from region ② causes jumping. Furthermore, it is observed that hysteresis occurs while entering region ② from region ③. From Figure 2, it is further found that the system is stable and the motion is periodic whether period ratio κ is chosen in region ① or ②. However, the amplitudes of the periodic motions are not the same for different initial values. Furthermore, it is observed that the system is unstable when period ratio κ is chosen in regions ① and ②, or regions ①, ②, and ③, or regions ② and ③.Figure 2
Plane sketch of amplitudea1 when ε1=1, ε2=1, ε3=1, τ1=−6, F2=0.2, and σ1=−1. B represents the bifurcation set, and H represents the hysteresis.Case 1.
The period ratioκ is chosen in regions ① and ② as shown in Figure 3. Entering region ① from region ② causes jumping in the motion state of amplitude a1; thus, the system is unstable and bifurcation occurs. Due to the periodicity of the period ratio κ, the case of the value of the amplitude a1 is given as … ⟶ single value ⟶ more values ⟶ single value ⟶ ⋯.Figure 3
Three-dimensional sketch of amplitudea1 when κ is chosen in the regions ① and ②.Case 2.
The period ratioκ is chosen in regions ①, ②, and ③ as shown in Figure 4. Entering region ① from region ②, jumping will occur, the hysteresis will occur when entering region ② from region ③, so jumping and the hysteresis will occur in the motion state of amplitude a1; thus, the system is unstable and bifurcation occurs. Due to the periodicity of the period ratio κ, the case of the value of the amplitude a1 is given as … ⟶ single value ⟶ more values ⟶ no solution ⟶ more values ⟶ single value ⟶ ⋯.
From Figure5, it is observed that when the excitation F2 changes, (8) is unable to compute the analytic solutions for amplitude a1. This is due to the uncertainty of the time point from region ③ to region ②. Therefore, (8) may produce chaotic solutions.Figure 4
Three-dimensional sketch of amplitudea1 when κ is chosen in the regions ①, ②, and ③.Figure 5
Plane sketch of amplitudea1 when.τ1=−6.Case 3.
The period ratioκ is chosen in regions ② and ③ as shown in Figure 6. Entering region ② from region ③, the hysteresis will occur in the motion state of the amplitude a1; thus, the system is unstable and bifurcation occurs. Due to the periodicity of period ratio κ, the case of the value of the amplitude a1 is given as …no solution ⟶ more values ⟶ no solution ⟶ ⋯.
It is observed that when the excitationF2 changes, (8) may produce chaotic solutions. This is due to the uncertainty of the time point from region ③ to region ②.
In (6), the expressions of τ1 and τ2 indicate that the initial value of parameters α6, α7, and β7 have an important influence on the range of value of period ratio κ. In other words, in addition to the transverse excitation F2 in (8), the initial values of parameters α6, α7, and β7 can change the range of value of period ratio κ. From the above analysis, a set of appropriate initial values for parameters α6, α7, and β7 can be chosen, so that the period ratio κ can be chosen in region ① or ②; thus, the system is stable and the motion is periodic.
By the changing of bifurcation parametersF2 and σ1, understand bifurcation properties of the composite laminated piezoelectric rectangular plate structure subjected to small perturbations. Figure 7 is the bifurcation diagrams of (9a) and (9b) when parameters F2 and κ change and the other parameters are fixed. In Figure 7(a), the parameters are chosen as F2=0.4, κ=0.3, and τ1=−6, in which the bifurcation and hysteresis occur. In Figure 7(b), the parameters are chosen as F2=0.4, κ=−0.3, and τ1=−6, in which the bifurcation and hysteresis cannot occur. In Figure 7(c), the parameters are chosen as F2=0.5, κ=0.3, and τ1=−6, in which the bifurcation and hysteresis occur. In Figure 7(d), the parameters are chosen as F2=0.5, κ=−0.3, and τ1=−6, in which the bifurcation and hysteresis cannot occur. In Figure 7(e), the parameters are chosen as F2=1.9, κ=0.3, and τ1=−6, in which the bifurcation and hysteresis occur. In Figure 7(f), the parameters are chosen as F2=1.9, κ=−0.3, and τ1=−6, in which the bifurcation and hysteresis cannot occur. Figure 8 is also the bifurcation diagrams of (9a) and (9b) when the parameters σ1 and κ change and the other parameters are fixed. In Figure 8(a), the parameters are chosen as σ1=0.5, κ=0.3, and τ1=−6, in which the bifurcation and hysteresis occur. In Figure 8(b), the parameters are chosen as σ1=0.5, κ=−0.3, and τ1=−6, in which the bifurcation and hysteresis cannot occur. In Figure 8(c), the parameters are chosen as σ1=1, κ=0.3, and τ1=−6, in which the bifurcation and hysteresis occur. In Figure 8(d), the parameters are chosen as σ1=1, κ=−0.3, and τ1=−6, in which the bifurcation and hysteresis cannot occur. In Figure 8(e), the parameters are chosen as σ1=2, κ=0.3, and τ1=−6, in which the bifurcation and hysteresis occur. In Figure 8(f), the parameters are chosen as σ1=2, κ=−0.3, and τ1=−6, in which the bifurcation and hysteresis cannot occur. It is observed that the jumping phenomenon and hilltop bifurcation occur at the critical point from region (1) to region (2).
Meanwhile, by choosing different bifurcation parametersF2 and σ1, displacement portraits of the amplitude a1 are simulated. Figure 9 is displacement portraits of the amplitude a1 of (9a) and (9b) when the parameter F2 change and the other parameters are fixed. In Figure 9(a), the parameters are chosen as F2=0.1 and σ1=1, in which the jump phenomenon cannot occur. In Figure 9(b), the parameters are chosen as F2=0.5 and σ1=1, in which the bifurcation and hysteresis occur. In Figure 9(c), the parameters are chosen as F2=1.5 and σ1=1, in which the bifurcation and hysteresis occur. In Figure 9(d), the parameters are chosen as F2=2 and σ1=1, in which the bifurcation and hysteresis cannot occur. Figure 10 is also displacement portraits of the amplitude a1 of (9a) and (9b) when the parameter σ1 change and the other parameters are fixed. In Figure 10(a), the parameters are chosen as F2=0.5 and σ1=0.2, in which the jump phenomenon occurs. In Figure 8(b), the parameters are chosen as F2=0.5 and σ1=0.6, in which the jump phenomenon occurs. In Figure 8(c), the parameters are chosen as F2=0.5 and σ1=1.2, in which the jump phenomenon occurs. In Figure 8(d), the parameters are chosen as F2=0.5 and σ1=1.4, in which the jump phenomenon cannot occur.Figure 6
Three-dimensional sketch of amplitudea1 when κ is chosen in the regions ② and ③.Figure 7
The bifurcation diagrams of (9a) and (9b) when ε1=1, ε2=1, and ε3=1 (left, κ=0.3; right, κ=−0.3).
(a)(b)(c)(d)(e)(f)Figure 8
The bifurcation diagrams of (9a) and (9b) when ε1=1, ε2=1, and ε3=1 (left, κ=0.3; right, κ=−0.3).
(a)(b)(c)(d)(e)(f)Figure 9
Displacement portraits of the amplitudea1 of (9a) and (9b) when ε1=1, ε2=1, and ε3=1.
(a)(b)(c)(d)Figure 10
Displacement portraits of the amplitudea1 of (9a) and (9b) when ε1=1, ε2=1, and ε3=1.
(a)(b)(c)(d)


## 5. The Proof of Proposition 1

The proof process of Proposition1 is briefly described as follows. The proof process is divided into five steps. The first step is to calculate the restricted tangent space as per equation (5). The second step is to obtain a strong equivalent normal form of equation (5) using the restricted tangent space. The third step is to establish the necessary and sufficient conditions that should be satisfied by the universal unfolding of the normal form. The fourth step is to calculate the polynomial space of the normal form. Finally, the fifth step is to prove the universal unfolding of the normal form.Step 1.
Calculating of the restricted tangent space of (5).Proposition 2.
The restricted tangent spaceRTg,1 of the germ gz,λ is given as(23)RTg,1=M2+Mσ1,F2+F22+τ1+τ2E⟶z,λ.Proof.
According to Proposition 1.4 (page 169 in Golubitsky and Schaeffer II, 1985),RTg,1 is generated (as module over Ez,λ) by the sixteen mappings:(24)g1,0,g2,0,0,g1,0,g2,a1g1,a1,g2,a1,a2g1,a1,g2,a1,σ1g1,a1,g2,a1,F2g1,a1,g2,a1,τ1g1,a1,g2,a1,τ2g1,a1,g2,a1,a1g1,a2,g2,a2,a2g1,a2,g2,a2,σ1g1,a2,g2,a2,F2g1,a2,g2,a2,τ1g1,a2,g2,a2,τ2g1,a2,g2,a2.
M2+Mσ1,F2+F22+τ1+τ2E⟶z,λ is generated by the following twenty mappings:(25)a12,0,a1a2,0,a22,0,σ1a1,0,σ1a2,0,F2a1,0,F2a2,0,τ1,0,τ2,0,F22,0,0,a12,0,a1a2,0,a22,0,σ1a1,0,σ1a2,0,F2a1,0,F2a2,0,τ1,0,τ2,0,F22.
There exists a reversible matrixA between (24) and (25) when a1,a2,σ1,F2=0,0,0,0:(26)u1u2=Av1v2=A1A2A3A4v1v2,where(27a)u1=g1,0g2,00,g10,g2a1g1,a1,g2,a1a2g1,a1,g2,a1σ1g1,a1,g2,a1F2g1,a1,g2,a11×8Τ,(27b)u2=τ1g1,a1,g2,a1τ2g1,a1,g2,a1a1g1,a2,g2,a2a2g1,a2,g2,a2σ1g1,a2,g2,a2F2g1,a2,g2,a2τ1g1,a2,g2,a2τ2g1,a2,g2,a21×8T,(27c)v1=a12,0a1a2,0a22,0σ1a1,0σ1a2,0F2a1,0F2a2,0F22,0τ1,0τ2,01×10Τ,(27d)v2=0,a120,a1a20,a220,σ1a10,σ1a20,F2a10,F2a20,F220,τ10,τ21×10Τ,(28a)g1,a1=6k11a14+4k12a12a22+2k13a24+4k14σ1a12+2k15σ1a22+2σ12+k16a1,(28b)g2,a1=a12a2+2k22a23−2σ1a2a1a2,(28c)g1,a2=2k12a13+4k13a1a22+2k15σ1a1a1a2,(28d)g2,a2=6k21a24+12a14+4k22a12a22+4k23σ1a22−2σ1a12+2σ12+4μ22a2,(28e)A1=c100c200001000c30c400−10100000000000000000000c50000000000c500000000000c500000000000c50000,A2=00000000000000000000c100c200001000c30c400−101c60000000000c600000000000c600000000000c60000,A3=00000000c5a10000000000c5a10c70000000000c700000000000c700000000000c700000000000c7a20000000000c7a2,A4=00000000c6a10000000000c6a10c80000000000c800000000000c800000000000c800000000000c8a20000000000c8a2,(29a)c1=k11a14+k12a12a22+k13a24+k16,(29b)c2=k14a13+k15a1a22+σ1a1,(29c)c3=k21a24+14a14+k22a12a22+4μ22,(29d)c4=k23a23−a12a2+σ2a2,(29e)c5=6k11a14+4k12a12a22+2k13a24+4k14σ1a12+2k15σ1a22+2σ12+k16,(29f)c6=a12a2+2k22a23−2σ2a2a2,(29g)c7=2k12a13+4k13a1a22+2k15σ1a1a1,(29h)c8=6k21a24+12a14+4k22a12a22+4k23σ2a22−2σ2a12+2σ22+4μ22.
In (5), if k16=0, bifurcation analysis is meaningless, so k16>0 will be studied in this study; similarly, k16<0 can also be analyzed. When a1=a2=σ1=F2=0, substituting μ2>0 into (29a)–(29h), it can obtain(30)c1=k16≠0,c2=0,c3=4μ22>0,c4=0,c5=2k16≠0,c6=0,c7=0,c8=8μ22>0.
This matrixA equals(31)A1′A2′A3′A4′,where(32a)A1=k16000000010004μ220000−101000000000000000000002k1600000000002k16000000000002k16000000000002k160000,(32b)A2=00000000000000000000k16000000010004μ220000−1010000000000000000000000000000000000000000,(32c)A3=00000000000000000000000000000000000000000000000000000000000000000000000000000000,(32d)A4=0000000000000000000008μ2200000000008μ22000000000008μ22000000000008μ2200000000000000000000000.
It can be seen that there are the columns with the linear correlation in the16×20 matrix A, namely, the 3th, 8th, and 10th columns, the 11th and 19th columns, and the 13th, 18th, and 20th columns. The 9th, 10th, 15th, and 16th rows and the 5th, 7th, 14th, and 16th columns are identically zero. Therefore, the 9th, 10th, 15th, and 16th rows and the 5th, 7th, 8th, 10th, 14th, 16th, 18th, and 20th columns can be eliminated. Thus, the original 16×20 matrix has rank 12. The proof is completed.Step 2.
The strong equivalent normal form of (5).Proposition 3.
Let(33)gz,λ=hz,λ+qz,λ,where(34a)h=k16a12+τ1,4μ22a22−F22+τ2,(34b)q=k11a16+k12a14a22+k13a12a24+k14σ1a14+k15σ1a12a22,k21a26+14a14a22+k22a12a24+k23σ1a24−σ1a12a22.
Ifq∈M3+M2σ1,F2+MF22+Mτ1+Mτ2E⟶z,λ is a higher-order term, then g is strongly equivalent to h.Proof.
By Proposition2, a higher order term of M2+MF2+F22+τ1+τ2 is(35)M3+M2F2+MF22+Mτ1+Mτ2.
For polynomialg, it is observed that(36)k11a16,0,k12a14a22,0,k13a12a24,0,0,k21a26,0,14a14a22,0,k22a12a24belong toM3E⟶z,λ,k14σ1a14,0,k15σ1a12a22,0,σ12a12,0,0,k23σ2a24,0,σ1a12a22,0,σ1a22belong toM2σ1E⟶z,λ.
Thus,(37)q∈M3+M2σ1,F2+MF22+Mτ1+Mτ2E⟶z,λ.
By applying Theorem 1.3 (page 168 in Golubitsky and Schaeffer II, 1985), it can getg is strongly equivalent to h. The proof is completed.
According to Proposition3, polynomial g will be replaced by polynomial h in the following discussion.Proposition 4.
The nondegenerate conditions ofh satisfy(38)D>0,τ1,τ2×k16,0≠0,τ1,τ2×0,4μ22≠0,and h is equivalent to(39)a12+τ1,a22−F22+τ2.Proof.
Let(40)hz,λ=fz+F220,−1+τ1,τ2,where fz=k16a12,4μ22a22; according to (2.6), (2.7), and (2.8) (page 402 in Golubitsky and Schaeffer I, 1985),(41)Qh=2k16a1008μ22a2=16k16μ22a1a2,D=b2−4ac=256k162μ24.
Sincek16≠0, μ2>0,(42)D>0.
Applying Theorem 2.2 (page 403 in Golubitsky and Schaeffer I, 1985), choose nonzero vectorsz1=1,0 and z2=0,1, one on each line, and let wi=fzi,i=1,2. It can obtain(43)w1=k16,0,w2=0,4μ22.
Sinceτ1≠0 and τ2≠0,(44)τ1,τ2×k16,0≠0,τ1,τ2×0,4μ22≠0,and it can be known that τ1,τ2 lies in quadrant 1or quadrant 2 (page 403 in Golubitsky and Schaeffer I, 1985).
The normal form of the germh can be expressed as(45)a12+τ1,a22−F22+τ2.
The proof is completed. It is observed from (45) that its universal unfolding needs complementary linear terms about the state variables and constant terms.Step 3.
The necessary and sufficient conditions that the universal unfolding of the normal form ofh should satisfy.
Improving Theorem 3.1 (page 409 in Golubitsky and Schaeffer I, 1985), by introducing a sign functionε in (Guo and Zhang, 2019), Proposition 5 is given as follows.Proposition 5.
Let(46)Ha1,a2,σ1,F2,τ1,τ2=h1a1,a2,σ1,F2,τ1,τ2,h2a1,a2,σ1,F2,τ1,τ2,be the unfolding of a bifurcation problem h in two state variables which satisfies the nondegeneracy conditions of Proposition 4. Then, H is a universal unfolding of h if and only if(47)Ha1,a2,0,F2,τ1,τ2=h1a1,a2,0,F2,τ1,τ2,h2a1,a2,0,F2,τ1,τ2,(48)detP=det0h1,a1a1h1,a1a20h2,a1a1h2,a1a20h1,a2a1h1,a2a20h2,a2a1h2,a2a2h1,σ1h1,σ1a1h1,σ1a2h2,σ1h2,σ1a1h2,σ1a2h1,F2h1,F2a1h1,F2a2h2,F2h2,F2a1h2,F2a2h1,τ1h1,τ1a1h1,τ1a2h2,τ1h2,τ1a1h2,τ1a2h1,τ2h1,τ2a1h1,τ2a2h2,τ2h2,τ2a1h2,τ2a2≠0,where it is evaluated at a1,a2,σ1,F2,τ1,τ2=0,0,0,0,τ1,τ2.Proof.
MatrixP can be expressed by six vectors:(49)P=χ1,χ2,χ3,χ4,χ5,χ6T,in which(50a)χ1=0,h1,a1a1,h1,a1a2,0,h2,a1a1,h2,a1a2,(50b)χ2=0,h1,a2a1,h1,a2a2,0,h2,a2a1,h2,a2a2,(50c)χ3=h1,σ1,h1,σ1a1,h1,σ1a2,h2,σ1,h2,σ1a1,h2,σ1a2,(50d)χ4=h1,F2,h1,F2a1,h1,F2a2,h2,F2,h2,F2a1,h2,F2a2,(50e)χ5=h1,τ1,h1,τ1a1,h1,τ1a2,h2,τ1,h2,τ1a1,h2,τ1a2,(50f)χ6=h1,τ2,h1,τ2a1,h1,τ2a2,h2,τ2,h2,τ2a1,h2,τ2a2.
Substituting (45) into (50a)–(50f),(51)χ1=0,2,0,0,0,0,χ2=0,0,0,0,0,2,χ3=0,0,0,0,0,0,χ4=0,0,0,0,0,0,χ5=1,0,0,0,0,0,χ6=0,0,0,1,0,0.
In (51), only four vectors χ1, χ2, χ5, and χ6 are linearly independent because H is the universal unfolding of h, and there exist two complementary linearly independent vectors. It is easy to find a complementary subspace to h, spanned by(52)ε1σ1a2,0,0,ε2F2a1,where εi is a sign function, εi=+1,0,−1i=1,2.
The introducedε can satisfy (47) and (48) (Guo and Zhang, 2019). From (52), it is obtained that(53)Ha1,a2,0,F2,τ1,τ2=h1a1,a2,0,F2,τ1,τ2,h2a1,a2,0,F2,τ1,τ2.
Obvious, whenεi=0,(54)Ha1,a2,F2,τ1,τ2,0=h1a1,a2,F2,τ1,τ2,0,h2a1,a2,F2,τ1,τ2,0.
Substituting (53) into (48), when εi≠0,(55)detP≠0.
Conversely, in (51), there are only four vectors χ1, χ2, χ5, and χ6, which are linearly independent because detP≠0, and the linearly independent vectors complementary to h need two. It may be adopted that(56)ε1σ1a2,0,0,ε2F2a1.
Thus,H=a12+2ε1σ1a2+τ1,a22−F22+2ε2F2a1+τ2 will be obtained. The proof is completed.Step 4.
Calculating of the polynomial space of the normal form ofh.Proposition 6.
The polynomial space ofh is given by(57)Rdhz,λY1,…,dhz,λYm,hλ,λhλ,λ2hλ,…=Ra1,0,0,a2,0,F2,1,0,0,1.Proof.
To derive the above equality, it should be noted that(58)u1′u2′=Bv1′v2′=B1B2B3B4v1′v2′,where(59a)u1′=h1,a1,0h1,a2,0h1,F2,0h1,τ1,0h1,τ2,01×5Τ,(59b)u2′=0,h2,a10,h2,a20,h2,F20,h2,τ10,h2,τ21×5Τ,(59c)v1′=a1,0a2,0F2,01,01,01×5Τ,(59d)v2′=0,a10,a20,F20,10,11×5Τ,(59e)B1=2000000000000000001000000,B2=0000000000000000000000000,B3=0000000000000000000000000,B4=000000200000−2000000000001.
It is found thatu1′Tu2′T can be expressed by four vectors a1,0, 0,a2, 0,F2, and 1,±1. Hence, it can be obtained that(60)Rdhz,λY1,…,dhz,λYm,hλ,λhλ,λ2hλ,…=Ra1,0,0,a2,0,F2,1,0,0,1.
The proof is completed.Step 5.
Proving of the universal unfolding of the normal form ofh.Proposition 7.
The codimension ofh in E⟶z,λ is 3, and the universal unfolding of (39) is given as(61)G=a12+2ε1σ1a2+2ε3F2+τ1,a22−F22+2ε2F2a1+τ2,where εii=1,2,3 is a sign function.Proof.
According to Theorem 2.1 and equation 2.7 (page 211 in Golubitsky and Schaeffer II, 1985), it can be obtained that(62)Th,1=M2+MF2+σ1,F22+τ1+τ2E⟶z,λ+Ra1,0,0,a2,0,F2,1,0,0,1.
It is not difficult to see that the largest intrinsic idealItrTh,1 contained in Th,1 is just(63)ItrTh,1=M2+Mσ1,F2+F22+τ1+τ2E⟶z,λ,ItrTh,1⊥=Ra1,0,a2,0,σ1,0,F2,0,0,a1,0,a2,0,F2,0,σ1,1,0,0,1,whose dimension is 10.
Clearly, the dimension ofRa1,0,0,a2,0,F2,1,0,0,1 is 5; hence, there exists a basis for a subspace of ItrTh,1⊥, which is complementary to Ra1,0,0,a2,0,F2,1,0,0,1.
Consequently, it is easy to find a complementary subspace toTh,1, spanned by(64)a2,0,σ1,0,F2,0,0,a1,0,σ1.
According to Proposition5, in (64), for a2,0 and 0,a1, there exist two sign functions ε1 and ε2, such that ε1σ1a2,0 and 0,ε2F2a1 will exist in universal unfolding of h. For F2,0, there exist one sign function ε3, such that ε3F2,0 will exist in universal unfolding of h. For σ1,0 and0,σ1, there exists one sign function ε3, such that ε1F2,0 will exist in universal unfolding of h.
According to the above analysis, the universal unfolding ofh is given by(65)G=a12+2ε1σ1a2+2ε3F2+τ1,a22−F22+2ε2F2a1+τ2.
Using Proposition5, when εi=0, (65) can be obtained that(66)Ga1,a2,σ1,F2,τ1,τ2=a12+τ1,a22−F22+τ2
Whenεi≠0, it is obtained from (65) that(67)det020000000002002ε10002ε30002ε20000100100000≠0.
Obviously, Proposition1 and Proposition 7 are the same.


## 6. Conclusions

The bifurcation of the laminated piezoelectric composite rectangular plate structure with 1 : 2 internal resonance is a high codimensional problem. The universal unfolding with codimension 3 of the bifurcation equations is carried out using the extended singularity theory method. Furthermore, the main material parametersα6, α7, and β7 that affect the dynamic behavior of the laminated piezoelectric composite rectangular plate structure near the singularity under transverse excitation are revealed by the transition set of universal unfolding with codimension 3.The transition set divides the parameters space into three kinds of different persistent regions, the results indicate that entering region ① from region ②, jumping will occur, the hysteresis will occur when entering region ② from region ③, and the composite laminated piezoelectric rectangular plate structure subjected to small perturbations is in unstable state. The numerical results from the above analysis show that the stability of the proposed system is better whenκ<0. It can pick a set of appropriate initial value for parameters α6, α7, and β7, such that period ratio κ is chosen in region ① or ②; thus, the system is stable and the motion is periodic.For the composite laminated piezoelectric rectangular plate structure, it is demonstrated that there exists abundant dynamic bifurcation patterns in a small perturbation sense. Some new chaotic dynamics are also presented. Clearly, these results provide some inspiration and guidance for the analysis and dynamic designs of this structure.



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*Source: 5552304-2021-08-17.xml*