Research Article | Open Access
Analysis of Nonlinear Vibrations and Dynamic Responses in a Trapezoidal Cantilever Plate Using the Rayleigh-Ritz Approach Combined with the Affine Transformation
Nonlinear vibrations of a trapezoidal cantilever plate subjected to transverse external excitation are investigated. Based on von Karman large deformation theory, the Rayleigh-Ritz approach combined with the affine transformation is developed to obtain the nonlinear ordinary differential equation of a trapezoidal plate with irregular geometries. With the variation of geometrical parameters, there exists the 1:3 internal resonance for the trapezoidal plate. The amplitude-frequency formulations of the system in three different coupled conditions are derived by using multiple scales method for 1:3 internal resonance analysis. It is found that the strong coupling of two modes can change nonlinear stiffness behaviors of modes from hardening-spring to soft-spring characteristics. The detuning parameter and excitation amplitude have significant influence on nonlinear dynamic responses of the system. The bifurcation diagrams show that there exist the periodic, quasi-periodic, and chaotic motions for the trapezoidal cantilever plate in the 1:3 internal resonance cases and the nonlinear dynamic responses are dependent on the amplitude of excitation. The possible adverse dynamic behaviors and undesired resonance can be avoided by designing appropriate excitation and system parameters.
The linear and nonlinear vibration analysis of plate-like structures has been paid much attention in mechanical, civil, and aerospace engineering fields. The understanding of the vibration behaviors is critical in many aspects, e.g., plate/panel flutter [1–3], energy harvesting of flexible plates , and nonlinear aeroelastic responses of functionally graded materials (FGM) plates, as well as the coupling of these structures with electro-magnetic, acoustic, and thermal fields [5, 6]. Structural nonlinearity exists unavoidably in the engineering structures, which significantly affects the nonlinear oscillation behavior and flutter instability . Consequently, the nonlinear vibration analysis can provide theoretical significance for ground vibration test and flutter analysis of flight vehicle structures.
The nonlinear vibration behaviors of plate-like structures have been investigated by many researchers. The nonlinear vibration of the rectangular plate under the transverse and in-plane excitations was analyzed by Hao et al.  in 2008. It was found that periodic, quasi-period, and chaotic responses for the rectangular plate models take place under certain conditions. Later on, Hao et al.  investigated the nonlinear dynamic behaviors of cantilever FGM rectangular plates under combined transverse excitation force and thermal loads. Their studies demonstrated that the nonlinear dynamic response of the cantilever plate is rather sensitive to transverse external excitation compared with that of a simply supported plate. Zhang  studied the bifurcations and chaotic characteristics of a simply supported rectangular plate under a parametric excitation by using the multiple scales method. The results showed that the rectangular plate can undergo Hopf bifurcation, heteroclinic bifurcations, and Shilnikov-type homoclinic orbit to the saddle focus. Meanwhile, Zhang and Liu  analyzed the local and global bifurcations of the same model subjected to transverse and in-plane excitations simultaneously. The chaotic responses obtained by the numerical simulation are sensitive to the initial conditions. An and Chen  investigated the global bifurcation and chaotic behaviors of a simply supported FGM plate. They also found that the types of the chaotic motions are completely different under different excitations, system parameters, and initial values. Mousa and Reza  studied the nonlinear dynamics and chaotic behaviors of a simply supported plate under in-plane/transverse excitations and aerodynamic loading, and the effects of detuning parameter, excitation amplitude, and dynamic pressure on the nonlinear dynamics of the model were analyzed. It was also found that the variation of control parameters can lead to the multiperiodic and chaotic motions. Sayed et al.  studied the nonlinear vibration of a two degree-of-freedom (DOFs) system with quadratic and cubic nonlinearities subjected to external and parametric excitation forces. Their results showed that all possible resonance could be observed and the system also exhibited chaotic motions under certain conditions.
Generally, in order to solve the nonlinear governing equations of panel/plate efficiently, the Rayleigh-Ritz or Galerkin approach is used for the modal representation of transverse deflections, and the obtained ordinary differential equations (ODEs) can be further solved in modal coordinates analytically and numerically. Ye et al.  used Rayleigh-Ritz approach to obtain the limit cycle oscillation (LCO) of a cantilever plate and discussed the effect of length-to-width ratio on the nonlinear oscillations of the system. Later on, Xie et al.  extended the work of Ye and Dowell and used the same model to examine the chaotic motions and the routes to chaos with the increasing of dynamic pressure. The bifurcation behaviors of a simply supported square plate with the airflow and structural nonlinearities were also studied, and they evaluated the effect of the aerodynamic nonlinearity . Bakhtiari-Nejad and Nazari  analyzed the nonlinear vibration of a cantilever plate with viscoelastic laminate by Rayleigh-Ritz method. The stability and chaotic behavior were obtained by using numerical simulations. Liew et al.  discussed the nonlinear vibration behavior of cantilever plates with constant thickness based on the Mindlin plate theory. Yang et al.  studied the nonlinear response characteristics and experiments of a cantilever hard-coating plate under transverse external excitation. The vibration response experiment agreed well with the numerical results. Dai et al. [21–23] proposed a highly efficient global nonlinear Galerkin method for the analysis of the large-deflection problem of plates. Additionally, a time domain collocation method also has been proposed by Dai et al. [24, 25] to solve nonlinear oscillatory problems, which is promising in the analysis of the von Karman fluttering plate. The aforementioned studies applied the Galerkin/Rayleigh-Ritz approaches to the modelling of nonlinear systems, which could achieve the purposes of DOF reduction and computational cost saving, compared with traditional finite element method.
For nonlinear vibration analysis, due to the complexity for obtaining analytical solutions of nonlinear differential equations, many asymptotic approaches have been developed for this purpose, such as energy balance method , variational approach , multiple scales method [28–32], and homotopy perturbation method. Here, the multiple scales method is a very powerful and well-known approach for analyzing the primary, internal, and super/subharmonic and higher-order resonances of nonlinear dynamic systems by several researchers. Ye et al.  utilized the multiple scales method to obtain the nonlinear oscillations and chaotic behaviors of an antisymmetric cross-ply laminated composite rectangular thin plate under parametric excitation. Their results showed the multipulse orbits exist in the present model. Sayed and Mousa  used this approach to analyze the nonlinear dynamic behaviors of the composite laminated plate with transverse excitations. The investigation demonstrated that the variation of system parameters could avoid worst resonance cases. Zhang et al.  used the multiple scales method to obtain the averaged equation of cantilever rectangular plate under in-plane force and moment for nonlinear vibration analysis and their studies showed that nonlinear dynamic responses could be controlled by changing these two kinds of excitations. Alijani et al.  studied the bifurcations and complicated nonlinear dynamics of FGM rectangular plates based on the von Karman’s nonlinear geometry plate theory and the multiple scales method subjected to the transverse excitation in thermal environment. It was found that the thermally deformed FGM plates have a strong hardening behavior and the effect of volume fraction exponent on nonlinear behavior is not significant. Later on, they analyzed nonlinear parametric instability of FGM plates in thermal environments and showed the complex nonlinear dynamic responses through period-doubling and Neimark-Sacker bifurcations . Xue et al.  analyzed the nonlinear principal resonances of an orthotropic magnetoelastic plate under a transverse magnetic field and a harmonic mechanical force using the multiple scales method. It was found that the nonlinear hardening effect of the principal resonance for the isotropic plate is much stronger than that of the corresponding orthotropic plate.
Due to the simply geometries commonly used in engineering structures and the limitation of the traditional methods, the aforementioned literatures in this field indicate that most of the studies are dealing with nonlinear dynamic problems of rectangle plates. Few studies have been performed on nonlinear dynamic behaviors of the cantilever plates with irregular geometries. Meanwhile, cantilever-like structures are commonly used in a large number of engineering fields such as engine blades, fixed wings, and helicopter rotor blades in aerospace engineering. The studies of nonlinear dynamic behaviors of cantilever plates provide useful insight into ground vibration tests and structure dynamic designs. To take a low-aspect-ratio wing as an example, it can be simplified as a cantilevered trapezoidal wing-like plate. Recently, the nonlinear dynamic characteristics of a trapezoidal wing model under subsonic or hypersonic aerodynamic loads were also investigated by Shokrollahi et al.  and Tian et al. .
Besides, it is also noted that most studies focus on nonlinear vibrations of rectangular plate, and there exist certain resonances with the variations of physical parameters of composite or FGM material. However, to the authors’ knowledge, the studies of parametric resonance and complex dynamic responses of the trapezoidal cantilever plates with irregular geometries have not been explored. Therefore, nonlinear vibration behaviors of a trapezoidal cantilever plate subjected to transverse external excitation are explored in the current study and focused on the 1:3 internal resonance of the plate. The contents of the work are organized as follows. In Section 2, the affine transformation combined with the Rayleigh-Ritz method is used to obtain the nonlinear vibration equations of trapezoidal cantilever plate. Section 3 establishes the averaged equations of the transverse motion by using the multiple scales method. Section 4 is devoted to the studies on the amplitude-frequency characteristics of the system with the 1:3 internal resonance. In Section 5, the nonlinear dynamic responses under certain conditions are obtained numerically from the derived averaged equations. The main conclusions are drawn in Section 6.
2. Mathematical Modeling
A trapezoidal cantilever plate shown in Figure 1(a) is constrained along the edge. The parameters , , , and are defined in Figure 1(a). The nonlinear vibration equations are obtained from Lagrange’s equations based on the von Karman nonlinear strain-displacement relation using kinetic and potential energies.
The kinetic energy and potential energy can be expressed aswhere and are the potential energies induced by stretching and bending, respectively. Their expressions can be given as
The Lagrange’s equation iswhere = is the Lagrangian and is the generalized force.
Substituting the expressions of kinetic and potential energy into (3), one can obtain
The Rayleigh-Ritz method is used to transform the equations of a trapezoidal plate model into modal coordinates, and the ordinary beam mode functions can be used as the mode functions of the trapezoidal plate. Then, the nondimensional deflections in , , and directions are expressed as
Generally, the mode function of a cantilever beam and free-free beam can be used as the mode functions of a traditional rectangular cantilever plate in and directions, respectively , which can be combined to meet the boundary conditions of the model. The mode functions can be expressed aswith , , where and are the mode functions of free-free beam and cantilever beam, respectively.
For the trapezoidal cantilever plate, the mode functions of free-free beam and cantilever beam cannot be used directly. The trapezoidal plate model can be mapped into a nondimensional square plate by using the affine transformation , which is shown in Figure 1(b). The affine transformation can be given as where TR () is defined as the ratio of and and AR is defined in terms of , , and as .
Accordingly, the transformation matrix can be obtained as
For the trapezoidal plate model under transverse excitation, which is shown in Figure 2, the transverse nonlinear vibration is dominant; namely, the effects of the in-plane motions and can be ignored, and only transverse vibration of the trapezoidal cantilever plate is considered for the following analysis.
The present study will focus on the case of 1:3 internal resonance of the trapezoidal cantilever plate. Based on the analysis of the existing studies [24–26], the first two modes of nonlinear oscillations are enough to represent the nonlinear vibration behaviors of the plate, especially used for internal resonance analysis. Thus, the nonlinear oscillations of the trapezoidal cantilever plate in the first two modes are considered only. The expression for transverse deflection is given by
Substituting (9) into (4) and introducing Rayleigh damping, the nondimensional transverse motion equation subjected to transverse harmonic excitation can be obtainedwhereThe element expressions of matrices , are given in (12a) and (12b), and the coefficients of , , , and are given in the Appendix.where , , , , and .
Due to the affine transformation employed for trapezoidal plate model, the two modes of (9) are coupled. The mode decoupling need to be performed for the induced linear system of the nonlinear system, then (10) can be rewritten aswhere and are the amplitudes of the first two modes, are mode shape functions, and are modal damping ratios, and and are the natural frequencies of the first two modes. The expressions of the coefficients in (13a) and (13b) are given as follows:
3. Perturbation Analysis
The condition of the 1:3 internal resonance could exist in the trapezoidal cantilever plate with the variation of geometrical parameters. As = 1, = 0.01 are fixed, the resonance condition can be satisfied by changing TR, , and . Figures 2(a) and 2(b) show the variation of frequency ratio with different TR and for = 1 and , respectively. As shown in Figure 3, several combinations of the parameters TR, , and can satisfy the condition of 1:3 internal resonance.
The multiple scales method is used to obtain the averaged equation of (13a) and (13b) for nonlinear dynamic analysis. Introducing time scales,where is a small perturbation parameter and is the highest order of small parameter. The form of the asymptotic solution can be expressed as The derivatives with respect to are transformed into where and is determined by the requirement of solution precision. Considering the first-order precision solution with , two time scales of and are adopted in the following analysis.
Here, the case of 1:3 internal resonance of the trapezoidal cantilever plate is considered. In the resonance, there existswhere and are the first two natural frequencies of the induced linear system and and are two detuning parameters.
To obtain the averaged equation of (16) by using the multiple scales method, we specify the damping term, nonlinear term, and external excitation term as perturbations. Equations (13a) and (13b) are rewritten as The asymptotic solutions of (19a) and (19b) for and are Substituting (20a) and (20b) into (19a) and (19b) and equating the coefficients of similar power of , we can obtainThe general solution of (21a) can be written as where A1 and A2 are complex functions in .
In order to facilitate the following analysis, set = 1. Eliminating the secular terms from (23a) and (23b), the following averaged equations can be obtained: The functions A1 and A2 are expressed in the polar form:where and are the functions with respect to the amplitude and phase of motion.
4. Analysis of Amplitude-Frequency Response
For the periodic solutions of the system, , , , and in the averaged equations are constants. Hence, set . Eliminating the terms of , the amplitude-frequency response behaviors of the trapezoidal cantilever plate can be obtained
According to the vibration amplitudes of the two modes, the steady-state vibration of the system can be divided into the following three conditions.
(a) When the vibration amplitude of one mode is zero and the amplitude of the other mode changes with the variation of excitation amplitude or frequency, these two modes are noncoupled.
(b) When the vibration amplitude of one mode is constant and the amplitude of the other mode changes with the variation of excitation amplitude or frequency, these two modes are weakly coupled.
(c) When the vibration amplitudes of the two modes both change with the variation of excitation amplitude or frequency, these two modes are strongly coupled.
Based on these three conditions, the nonlinear vibration behaviors of the trapezoidal cantilever plate are discussed as follows. In all cases, the structural parameters of the trapezoidal plate are chosen as = 23°, = 0.81, and = 0.8, and the excitation point located at = 0.25, = 0.75. Thus, the natural frequencies of the corresponding induced linear system are = 10.493 Hz and = 31.492 Hz, respectively.
4.1. Amplitude-Frequency Characteristics of Noncoupled Case
4.2. Amplitude-Frequency Characteristics of Weakly Coupled Case
When the two modes are weakly coupled, setting as constant in (27a) and setting as constant in (27b), to simplify calculation, (27a) and (27b) can be written asIn order to satisfy (28a) and (28b), the following relations can be obtained
4.3. Amplitude-Frequency Characteristics of Strongly Coupled Case
For the stongly coupled case, the motion of the system is dominated by the two modes under a single frequency excitation. When excitation frequency has slight change, the amplitudes of the two modes are influenced to various extents. From (27a), the following equations can be obtained:
5. Results and Discussions
In this section, the amplitude-frequency behaviors and nonlinear dynamic responses of the trapezoidal cantilever plate with 1:3 internal resonance are discussed. In all cases, the structural parameters of the trapezoidal plate are chosen as = 30°, = 0.81, and = 0.8 from Figure 3(b) and the excitation point located at = 0.25, = 0.75. Thus, the natural frequencies of the corresponding induced linear system are = 10.493 Hz and = 31.492 Hz, respectively, which can satisfy the condition of 1:3 internal resonance.
5.1. Amplitude-Frequency Characteristics
5.1.1. Noncoupled Case
For the noncoupled case, it is seen from (29a) that when the excitation frequency approaches the frequency of the second mode, the motion of the first mode will not be excited. The amplitude-frequency curve of the second mode can be obtained from (29b). Figure 4 shows the amplitude-frequency curve of the second mode in the noncoupled case. Here, , = 0.01. Its skeleton curve trends to the direction of frequency increment, which indicates a typical hardening-spring characteristic. The multivalue and jump phenomena exist for the second mode in the resonance region.
5.1.2. Weakly Coupled Case
For the weakly coupled case, the amplitude-frequency curves of these two modes are obtained from (31a) and (31b) shown in Figure 5, where , P0 = 0.01, β2 = π/4. Comparing with Figures 5(b) and 4, the amplitude-frequency curve of the second mode in the noncoupled case can be obtained by shifting the plot of weakly coupled case to right. The results show that the amplitude-frequency curves exhibit hardening-spring behavior. It indicates that weakly coupled effect cannot change the nonlinear stiffness behavior but has a certain influence on resonance frequency of the system. An increscent resonance frequency in Figure 5(b) is shown in the present case. The multivalue phenomenon also exists for the second mode in the resonance region, which can be clarified by the stability analysis of steady-state solutions. According to (26), setting , the phase plot of the second mode can be obtained, which is shown in Figure 6. It is found that the three steady-state solutions , , and correspond to the points of upper, middle, and lower solution branches at in Figure 5(b), respectively. The phase trajectory in Figure 6 demonstrates that ( = 0.04) and ( = 0.2904) are stable focus points and ( = 0.2503) is an unstable saddle point. There exists a line across the saddle point , and the phase-plane is divided into two attraction basins and . Therefore, the present system has two stable solutions and one unstable solution in the multivalue frequency band, but unstable motions can be only obtained in analytical solutions.
Based on the existence condition of steady-state solution, and . The effect of phase on the amplitude-frequency behavior of the second mode is analyzed. As for the small amplitude excitation, hardly affects the amplitude-frequency behavior. As for the large amplitude excitation, can change the shape of amplitude-frequency curve of the second mode but cannot affect the soft/hardening-spring characteristics of the mode. Figure 7 shows the amplitude-frequency curves of the second mode with different under = 2. The results demonstrate that with the increase of excitation amplitude, the effect of phase on the amplitude-frequency behavior becomes remarkable.
5.1.3. Strongly Coupled Case
According to the resonance relation from (18), the difference of σ1 and σ2 is a constant. Substituting into (33a), the amplitude-frequency relation of the first mode can be obtained. Similarly, the relation of the second mode can also be obtained from (33b). Figure 8 shows the amplitude-frequency curves of the system in the strongly coupled case. Here, = , = = 0.01, = 0.01, . The results indicate that the first mode has hardening-spring behavior, but the shape of its amplitude-frequency curve is observably different from that of the weakly coupled case. However, the second mode becomes a soft-spring behavior with the leftward trend of its skeleton curve.
However, it is difficult to reach the accurate multiples of frequencies of the two modes for a practice model. The cases for are discussed in Figure 9. When , the second frequency is slightly more than three times as large as the first frequency. When , the second frequency is slightly less than three times as large as the first frequency. As shown in Figure 9, the values of have little effect on the amplitude-frequency behavior of the first mode. The amplitude-frequency curve of the second mode moves to the right and the corresponding peak increases with the increase of . The results indicate that the rise of the difference of detuning parameters can increase the resonance frequency. Figure 10 shows the amplitude-frequency curves of the two modes for under different excitation amplitudes. The results show that the soft/hardening-spring characteristic of each mode cannot be affected by changing excitation amplitude. The local peak for the first mode slightly increases with the increase of excitation amplitude. On the contrary, the amplitude peak for the second mode decreases and the corresponding resonance frequency increases.
The case for = -5 is shown in Figure 11. It can be seen that the first mode keeps a hardening-spring behavior, and the amplitude-frequency curve of the second mode exhibits a rightward trend, which shows hardening-spring behavior. Furthermore, comparing with the first mode, the effect of amplitude on the second mode is more significant, which shows that the vibration amplitude increases with the increasing of .
5.2. Nonlinear Dynamic Responses
In order to analyze the nonlinear dynamic responses of the system, the averaged equations can be obtained by transforming (24a) and (24b) into the Cartesian form. The functions A1 and A2 can be expressed as where , , , and are the functions with respect to the amplitude and phase of motions. Substituting (34) into (24a) and (24b), the averaged equations in the Cartesian form are obtained as follows:The Jacobi matrix of (35) can be written aswhereThe eigenvalue equation of the Jacobi matrix isand when the real parts of all eigenvalues are negative, the steady solutions are asymptotically stable.
The zero-solution stability of (35) can be determined; setting = = = = 0, the eigenvalues of the Jacobi matrix are