Effect of Initial Geometric Imperfections on Nonlinear Vibration of Thin Plate by an Asymptotic Numerical Method

Abstract

In this work, the influence of initial geometric imperfections on geometrically nonlinear free vibrations of thin elastic plates has been investigated by an asymptotic numerical method. The nonlinear strain displacement relationship of von Karman theory is adopted to calculate the elastic strain energy. The harmonic balance approach and Hamilton’s principle are used to convert the equation of motion into an operational formulation. The nonlinear problem is transformed into a sequence of linear ones having the same stiffness matrix, which can be solved by a classical finite-element method. To improve the validity range of the power series, Padé approximants are incorporated and a continuation technique is also used to get the whole solution. Numerical results are discussed and compared to those available in the literature and convergence of the solution is shown for various amplitudes of imperfection.

Introduction

Thin structures such as plates are widely used in various industrial fields such as civil engineering, mechanics, and aeronautics. They are subjected to dynamic loads that can cause vibration amplitude of the order of the thickness of the structure and giving rise to significant nonlinear phenomena. Consequently, the study on nonlinear vibrations of thin plates has assumed considerable importance in recent years. Thin plates are generally object of geometric imperfections that can significantly influence its vibrational behavior. Therefore, attempts to find the vibratory behavior incorporating sensitivity to imperfection have never ceased.

In the literature, several researchers have studied the nonlinear vibrations of plates in the presence of geometric imperfections. Ilanko and Dickinson (1991) studied the effect of geometric imperfection on the vibratory behavior of simply supported plates. Nonlinear vibrations of imperfect plates with different initial conditions are also studied by Liu and Yeh (1993). Ostiguy et al. (1998) used a direct integration method to solve the nonlinear behavior of the initial geometrically imperfect plate. Lin and Chen (1989) presented a set of nonlinear equations to describe the behavior of imperfect isotropic plates. The authors have shown that the nonlinear frequency of the plate is related to the size of the imperfections and that an increase in the size of the imperfection could change the behavior of the nonlinear vibrations of the hard behavior to the soft behavior. Chen et al. (2005) have developed a procedure to analyze nonlinear vibrations of imperfect plates with initial constraints. Their results show that the nonlinear vibration behavior is sensitive to the initial amplitude of the vibrations, the initial stress, and the size of the initial imperfection.

Several other works show that it is very important to consider the initial geometrical imperfections in the design of a mechanical structure as a parameter that could certainly affect several mechanical characteristics of the structure (resonance frequencies, stiffness, buckling limit, etc.) as well as its static and dynamic behavior (Hui 1984; Ilanko and Dickinson 1987; Sassi and Ostiguy 1994a, b). Ostiguy and Sassi (1992) show that, when a plate is imperfect in its unloaded configuration and subjected to an in-plane loading, its response is quite different from the simple case in which the plate is perfectly plane. Sassi et al. (1996) used an approach based on the direct numerical integration of temporal differential motion equations to verify the theoretical predictions of the asymptotic method as well as to advance the understanding of the dynamics of the stability of imperfect plates. Amabili (2006), Amabili and Farhadi (2009), Amabili (2010), and Amabili (2016) included geometric imperfections in their numerical and experimental studies of nonlinear plate vibrations. It has been shown that significant geometric imperfections can be present in real plates and give an initial softening-type nonlinearity, which turns to hardening type for large amplitudes. They also showed that geometric imperfections play a fundamental role in predicting the nonlinear response and these imperfections are generally larger for very thin plates. They concluded that the plates can be considered as shallow shells in the presence of initial geometric imperfections and that this imperfection is one of the reasons for the reduction of nonlinearities of the hard type. In his article, Alijani and Amabili (2013), it is illustrated that large imperfection amplitudes result in significant qualitative changes in the frequency–amplitude response of completely free plates. It is shown that in comparison to perfect plates, imperfect plates have smaller damping ratios and weaker hardening behavior that may turn to softening depending on the shape and amplitude of the imperfection.

The asymptotic numerical method has been successfully used in the literature to study the nonlinear free and forced vibrations of undamped perfect plates (Azrar et al. 1999, 2002) and damped one (Boumediene et al. 2009, 2011) and also to study linear and nonlinear vibrations of buckled perfect plates (Boutyour et al. 2006; Benchouaf and Boutyour 2016). In a recent article (Woiwode et al. 2020), the authors compared two algorithms to establish a harmonic equilibrium equation and numerical pursuit of the solution path for mechanical systems with harmonic drive. They concluded that the algorithm based on asymptotic numerical method and classical harmonic balance is suited for smooth nonlinearities, for instance geometrically nonlinear finite-element models. For non-smooth nonlinearities such as stick–slip friction or unilateral constraints, the algorithm relies on a predictor–corrector scheme and an Alternating Frequency–Time approach is better suited, and convinces with high numerical robustness and efficiency.

The objective of the present work is to study the nonlinear free vibrations of thin elastic plates with initial geometric imperfections using an asymptotic numerical method. The principle of this method is to represent the unknowns (displacements, frequency,$\ldots$) by a power series expansion with respect to a control parameter. By introducing the expansion into the governing equation, the nonlinear problem is transformed into a sequence of simple linear problems which can be solved by a classical finite-element method. At each stage of the proposed procedure, Padé approximants are incorporated to improve the validity range of the power series and to reduce the computational cost. Numerical results are discussed and compared to those available in the literature, and convergence of the solution is shown for various amplitudes of initial imperfection of square and rectangular plates with different types of boundary conditions.

Problem Formulation

Let us consider an elastic thin and homogenous rectangular plaque of thickness h, length L, width l, middle surface $\varOmega$, density mass $\rho$, Young modulus E, and Poisson’s ratio $\nu$. In a rectangular coordinate reference (O; x, y, z), the component displacement of a middle surface point is indicated by u, v, and w in the x, y, and z directions, respectively, as shown in Fig. 1.

Fig. 1

Geometry and coordinate system of a rectangular plate

It is supposed that the structure comprises a geometrical imperfection described by the field of displacement $\mathbf{u} ^* = \left\{ u^*,v^*,w^*\right\} ^t$ compared to the configuration of reference. It was a variable initial imperfection that is introduced into the structure. In order that the additional variable added to the problem equations is a scalar, we decompose $\mathbf{u} ^*$ in the form ($\mathbf{u} ^* = \eta \mathbf{u} ^*_0$) where $\mathbf{u} ^*_0$ is a fixed displacement field which gives the allure of the imperfection, and $\eta$ sets the amplitude. The buckling mode corresponding to the first bifurcation point of the equilibrium curve of the plate was chosen to give the form of defect. The corresponding displacement field has been normalized in order that its largest component is equal to 1. In this way, the additional parameter that is used in the calculations is the true amplitude of the imperfection.

If $\mathbf{u}$ indicates the response of vibration of plate compared to the imperfect natural state, then $\mathbf{u} + \mathbf{u} ^*$ indicates displacement compared to the perfect state of reference, and for simplicity, the assumed initial imperfection is reduced here to its transversal component $w^*$. Based on Kirchhoff hypothesis, the displacement field $\mathbf{u}$ have the following form:

$$\begin{aligned} \left\{ \begin{array}{c} u(x,y) - z \dfrac{\partial w}{\partial x} \\ v(x,y) - z \dfrac{\partial w}{\partial y} \\ w(x,y) + w^* \end{array} \right\}. \end{aligned}$$

(1)

Considering small strains and moderate rotations, the nonlinear strain–displacement relationships are given by:

$$\begin{aligned} \varvec{\varepsilon } = \varvec{\varGamma }(u,v,w,w^*) + z \varvec{\kappa }(u,v,w). \end{aligned}$$

(2)

The membrane strains can be decomposed into a linear and non linear parts $\varvec{\varGamma }(\mathbf{u} ,\mathbf{u} ^*) = \varvec{\varGamma }(\mathbf{u} + \mathbf{u} ^*) - \varvec{\varGamma }(\mathbf{u} ^*) = \varvec{\varGamma }^L(\mathbf{u} ) + \varvec{\varGamma }^{\mathrm{NL}}(\mathbf{u} ,\mathbf{u} ) + 2\varvec{\varGamma }^*(\mathbf{u} ,\mathbf{u} ^*)$ with the following expression:

$$\begin{aligned}&\varvec{\varGamma }^L = \left\{ \begin{array}{c} \dfrac{\partial u}{\partial x} \\ \dfrac{\partial v}{\partial y} \\ \dfrac{\partial u}{\partial y} + \dfrac{\partial v}{\partial x} \end{array} \right\} , \varvec{\varGamma }^{NL} = \left\{ \begin{array}{c} \dfrac{1}{2} \left( \dfrac{\partial w}{\partial x}\right) ^2 \\ \dfrac{1}{2} \left( \dfrac{\partial w}{\partial y}\right) ^2 \\ \left( \dfrac{\partial w}{\partial x}\right) \left( \dfrac{\partial w}{\partial y}\right) \end{array} \right\} , \nonumber \\&\quad 2\varvec{\varGamma }^*(\varvec{u},\varvec{u}^*) = \left\{ \begin{array}{c} \left( \dfrac{\partial w}{\partial x}\right) \left( \dfrac{\partial w^*}{\partial x}\right) \\ \left( \dfrac{\partial w}{\partial y}\right) \left( \dfrac{\partial w^*}{\partial y}\right) \\ \left( \dfrac{\partial w}{\partial x}\right) \left( \dfrac{\partial w^*}{\partial y}\right) + \left( \dfrac{\partial w^*}{\partial x}\right) \left( \dfrac{\partial w}{\partial y}\right) \end{array} \right\} . \end{aligned}$$

(3)

The bending strain is defined by:

$$\begin{aligned} \varvec{\kappa } = \left\{ \begin{array}{c} - \dfrac{\partial ^2 w}{\partial ^2 x} \\ - \dfrac{\partial ^2 w}{\partial ^2 y} \\ -2 \dfrac{\partial ^2 w}{\partial x \partial y} \end{array} \right\} . \end{aligned}$$

(4)

The in-plane forces $\mathbf{N}$ and bending moments $\mathbf{M}$ are assumed to be related to the strain and curvature by the constitutive relations:

$$\begin{aligned}&\mathbf{N} = \left\{ \begin{array}{c} N_x \\ N_y \\ N_{xy} \end{array} \right\} = [\mathbf{C} _m] : \varvec{\varGamma } ,\nonumber \\&\quad \mathbf{M} = \left\{ \begin{array}{c} M_x \\ M_y \\ M_{xy} \end{array} \right\} = [\mathbf{C} _b] : \varvec{\kappa }, \end{aligned}$$

(5)

where the symmetric matrices of material properties $[\mathbf{C} _m]$ and $[\mathbf{C} _b]$ are given by the following relations:

$$\begin{aligned}&{[}{} \mathbf{C} _m] = \frac{Eh}{1-\nu ^2} \begin{bmatrix} 1 &{} \nu &{} 0 \\ \nu &{} 1 &{} 0 \\ 0 &{} 0 &{} \frac{1-\nu }{2} \end{bmatrix} , \nonumber \\&\quad [\mathbf{C} _b] = \frac{Eh^3}{12(1-\nu )^2} \begin{bmatrix} 1 &{} \nu &{} 0 \\ \nu &{} 1 &{} 0 \\ 0 &{} 0 &{} \frac{1-\nu }{2} \end{bmatrix}. \end{aligned}$$

(6)

The total strain energy expression of the plate can be written as:

$$\begin{aligned} V = \frac{1}{2} \int _{\varOmega } \left\{ \varvec{\varGamma }^t:[\mathbf{C} _m]:\varvec{\varGamma } + \varvec{\kappa }^t:[\mathbf{C} _b]:\varvec{\kappa } \right\} \mathrm{d}\varOmega , \end{aligned}$$

(7)

where $^t$ corresponds to transposed form and ":" denotes the ordinary scalar product.

Since $\varvec{\varGamma }$ is quadratic in w, the functional V is of degree 4 with respect to w. As in previous work (Azrar et al. 1999), the nonlinearity is reduced using Hellinger Reissner functional:

$$\begin{aligned} HR(u,v,w,\mathbf{N} ) = \int _{\varOmega } \left\{ \mathbf{N} :\varvec{\varGamma } - \frac{1}{2} \mathbf{N} ^t:[\mathbf{C} _m]^{-1}:\mathbf{N} + \frac{1}{2} \varvec{\kappa }^t:[\mathbf{C} _b]:\varvec{\kappa } \right\} \mathrm{d}\varOmega . \end{aligned}$$

(8)

The kinetic energy of a rectangular plate, by neglecting rotary inertia, is given by:

$$\begin{aligned} T = \frac{1}{2} \rho h \int _{\varOmega } \left\{ \dot{u}^2 + \dot{v}^2 + \dot{w}^2\right\} \mathrm{d}\varOmega . \end{aligned}$$

(9)

Application of the Harmonic Balance Method

In this work, only the periodic vibrations of an undamped plate are considered. To apply the harmonic balance method (Azrar et al. 1999, 2002) and for a simple representation, one considers only the second harmonic to the transverse movements and the second and the third harmonic for the longitudinal displacement as follows:

$$\begin{aligned} \begin{array}{c} \begin{aligned} u(x,y,t) &{} = u^0(x,y) + u^1(x,y) \cos (\omega t) + u^2(x,y) \cos (2 \omega t) \\ v(x,y,t) &{} = v^0(x,y) + v^1(x,y) \cos (\omega t) + v^2(x,y) \cos (2 \omega t) \\ w(x,y,t) &{} = w(x,y) \cos (\omega t), \end{aligned} \end{array} \end{aligned}$$

(10)

where $\omega$ is the natural frequency. Therefore, the principle of the harmonic balance method is to consider the amplitude $u^i (x,y)$, $v^i (x,y)$ and w(x, y) ($i = 0,1,2$) as the main unknowns. The insertion of Eq. (10) in the expressions of the deformations leads to the following formulations:

$$\begin{aligned} \begin{array}{l} \begin{aligned} \varvec{\varGamma }^L(\mathbf{u} ) &{} = \varvec{\gamma }^l(\mathbf{u} ^0) + \varvec{\gamma }^l(\mathbf{u} ^1) \cos (\omega t) + \varvec{\gamma }^l(\mathbf{u} ^2) \cos (2\omega t) \\ \varvec{\varGamma }^{NL}(\mathbf{u} ,\mathbf{u} ) &{} = \dfrac{1}{2} \varvec{\gamma }^{nl}(\mathbf{u} ,\mathbf{u} ) + \dfrac{1}{2} \varvec{\gamma }^{nl}(\mathbf{u} ,\mathbf{u} ) \cos (2\omega t) \\ 2\varvec{\varGamma }^*(\mathbf{u} ,\mathbf{u} ^*) &{} = 2\varvec{\gamma }^*(\mathbf{u} ,\mathbf{u} ^*)\cos (\omega t). \end{aligned} \end{array} \end{aligned}$$

(11)

It is then assumed that the response of the plate in displacements and in stresses can be written in the following form:

$$\begin{aligned} \begin{array}{l} \begin{aligned} \varvec{\varGamma }(\mathbf{u} ) &{} = \varvec{\gamma }^0 + \varvec{\gamma }^1 \cos (\omega t) + \varvec{\gamma }^2 \cos (2\omega t) \\ \mathbf{N} (\mathbf{u} ) &{} = \mathbf{N} ^0 + \mathbf{N} ^1 \cos (\omega t) + \mathbf{N} ^2 \cos (2\omega t). \end{aligned} \end{array} \end{aligned}$$

(12)

To study the history of the solution corresponding to a period, the initial time is set as $t_0 = 0$ and the final time $t_1 = 2\pi /\omega$. Using the expressions indicated by Eq. (11) and applying the Hamilton principle and after integration of Eqs. (8) and (9) as a function of time, we then deduce the following variational formulation:

$$\begin{aligned} \begin{aligned}&\int _{\varOmega } \Big \{ 2\delta \mathbf{N} ^0:(\varvec{\gamma }^l(\mathbf{u} ^0) - [\mathbf{C} _m]^{-1}:\mathbf{N} ^0)\\&\quad + \delta \mathbf{N} ^1:(\varvec{\gamma }^l(\mathbf{u} ^1) - [\mathbf{C} _m]^{-1}:\mathbf{N} ^1)\Big \} \mathrm{d}\varOmega \\&+ \int _{\varOmega } \Big \{ \delta \mathbf{N} ^2:(\varvec{\gamma }^l(\mathbf{u} ^2) - [\mathbf{C} _m]^{-1}:\mathbf{N} ^2) \\&\quad + \varvec{\gamma }^l(\delta \mathbf{u} ):(2\mathbf{N} ^0 + \mathbf{N} ^1 + \mathbf{N} ^2) + \delta \varvec{\kappa }:[\mathbf{C} _b]:\varvec{\kappa }\Big\} \mathrm{d}\varOmega \\&+ \int _{\varOmega } \Big \{(2\delta \mathbf{N} ^0 + \delta \mathbf{N} ^2): \frac{1}{2}\varvec{\gamma }^{nl}(\mathbf{u} ,\mathbf{u} ) \\&\quad + (2\mathbf{N} ^0 + \mathbf{N} ^2): \varvec{\gamma }^{nl}(\mathbf{u} ,\delta \mathbf{u} )\Big \}\mathrm{d}\varOmega \\&+ \int _{\varOmega } \Big \{\delta \mathbf{N} ^1: 2\varvec{\gamma }^*(\mathbf{u} ,\mathbf{u} ^*) \\&\quad + \mathbf{N} ^1: 2\varvec{\gamma }^*(\mathbf{u} ^*,\delta \mathbf{u} )\Big \}\mathrm{d}\varOmega \\&- \omega ^2 \rho h \int _{\varOmega } \Big \{ (u^1 + 4u^2)\delta u \\&\quad + (v^1 + 4v^2)\delta v + w\delta w\Big \} \mathrm{d}\varOmega = 0. \end{aligned} \end{aligned}$$

(13)

The governing equation Eq. (13) can be written in operational form:

$$\begin{aligned} L(\mathbf{U} ) + \widehat{Q}(\mathbf{U} ^*,\mathbf{U} ) + Q(\mathbf{U} ,\mathbf{U} ) - \omega ^2 M(\mathbf{U} ) = 0, \end{aligned}$$

(14)

where $\mathbf{U} = \left\{ u^0,u^1,u^2,v^0,v^1,v^2,w,\mathbf{N} ^0,\mathbf{N} ^1,\mathbf{N} ^2\right\} ^t$ denotes the mixed vector and:

$$\begin{aligned} \langle L(\mathbf{U} ),\delta \mathbf{U} \rangle= & {} \int _{\varOmega } \left\{ 2\delta \mathbf{N} ^0:(\varvec{\gamma }^l(\mathbf{u} ^0) - [\mathbf{C} _m]^{-1}:\mathbf{N} ^0)\right\} \mathrm{d}\varOmega \nonumber \\+ & {} \int _{\varOmega } \left\{ \delta \mathbf{N} ^1:(\varvec{\gamma }^l(\mathbf{u} ^1) - [\mathbf{C} _m]^{-1}:\mathbf{N} ^1)\right\} \mathrm{d}\varOmega \nonumber \\+ & {} \int _{\varOmega } \left\{ \delta \mathbf{N} ^2:(\varvec{\gamma }^l(\mathbf{u} ^2) - [\mathbf{C} _m]^{-1}:\mathbf{N} ^2)\right\} \mathrm{d}\varOmega \nonumber \\+ & {} \int _{\varOmega } \left\{ \varvec{\gamma }^l(\delta \mathbf{u} ):(2\mathbf{N} ^0 + \mathbf{N} ^1 + \mathbf{N} ^2) + \delta \varvec{\kappa }:[\mathbf{C} _b]:\varvec{\kappa }\right\} \mathrm{d}\varOmega \end{aligned}$$

(15)

$$\begin{aligned} \langle Q(\mathbf{U} ,\mathbf{U} ),\delta \mathbf{U} \rangle= & {} \int _{\varOmega } \left\{ (2\delta \mathbf{N} ^0 + \delta \mathbf{N} ^2): \frac{1}{2}\varvec{\gamma }^{nl}(\mathbf{u} ,\mathbf{u} ) + (2\mathbf{N} ^0 + \mathbf{N} ^2): \varvec{\gamma }^{nl}(\mathbf{u} ,\delta \mathbf{u} )\right\} \mathrm{d}\varOmega \end{aligned}$$

(16)

$$\begin{aligned} 2\left\langle \widehat{Q}(\mathbf{U} ^*,\mathbf{U} ),\delta \mathbf{U} \right\rangle= & {} \int _{\varOmega } \left\{ \delta \mathbf{N} ^1: 2\varvec{\gamma }^*(\mathbf{u} ,\mathbf{u} ^*) + \mathbf{N} ^1: 2\varvec{\gamma }^*(\mathbf{u} ^*,\delta \mathbf{u} )\right\} \mathrm{d}\varOmega \end{aligned}$$

(17)

$$\begin{aligned} \langle M(\mathbf{U} ),\delta \mathbf{U} \rangle= & {} \rho h \int _{\varOmega } \left\{ (u^1 + 4u^2)\delta u + (v^1 + 4v^2)\delta v + w\delta w\right\} \mathrm{d}\varOmega . \end{aligned}$$

(18)

The operators L, $\widehat{Q}$, and M are linear, and Q is a quadratic one.

Solution by an Asymptotic Numerical Method

In the present work, the asymptotic developments for nonlinear free vibrations of imperfect plate are done in the vicinity of the bifurcating point and successions of linear problems are numerically solved. However, the validity of the solution obtained is limited. Taking the starting point in the zone of validity and reapplying the ANM, to get the solution branch. This continuation method leads to a very powerful incremental method with an analytical step. The limitation of the validity of the solution has been overcome and the whole backbone curve is analytically obtained in a few steps. The purpose of this section is to solve the problem indicated by Eq. (14) using the asymptotic numerical method. The basic idea consists in searching for the solution branch by power series:

$$\begin{aligned} \begin{array}{c} \mathbf{U} - \mathbf{U} _0 = \sum \limits _{i=1}^{n} a^i \mathbf{U} _i \\ \omega ^2 - \omega _0^2 = \sum \limits _{i=1}^{n} a^i\omega _i. \end{array} \end{aligned}$$

(19)

The backbone curve that determines the free vibration analysis of plate is a branch that bifurcates from the fundamental solution $\mathbf{U} = 0$ at $\omega = \omega _0$, where $\omega _0$ is the linear frequency of vibration. At the bifurcation point ($\mathbf{U} = 0$ and $\omega = \omega _0$), the tangent operator L is singular. Furthermore, it is assumed that the kernel of this tangent operator is one-dimensional, which generally occurs. The unknown vector $\mathbf{U}$ and the frequency parameter $\omega$ can be searched as asymptotic expansions in the vicinity of the bifurcation point and as a function of a control parameter “a”. The control parameter of a development Eq. (19) can be identified as the projection of the displacement increment $(\mathbf{u} - \mathbf{u} _0)$ on the vector $\mathbf{u} _1$:

$$\begin{aligned} a = \langle \mathbf{u} -\mathbf{u} _0,\mathbf{u} _1 \rangle , \end{aligned}$$

(20)

where $\langle \cdot ,\cdot \rangle$ denotes the Euclidean scalar product. By replacing Eq. (19) in Eqs. (14) and (20) taking into account that $\mathbf{U} _0 = 0$, and equating like powers of “’a”, we obtain a series of following linear problems:

order 1:

$$\begin{aligned} \begin{array}{c} L(\mathbf{U} _1) + \widehat{Q}(\mathbf{U} _1,\mathbf{U} ^*) - \omega _0^2 M(\mathbf{U} _1) = 0 \\ \langle \mathbf{u} _1,\mathbf{u} _1 \rangle = 1. \end{array} \end{aligned}$$

(21)

order 2:

$$\begin{aligned} \begin{array}{c} L(\mathbf{U} _2) + \widehat{Q}(\mathbf{U} _2,\mathbf{U} ^*) - \omega _0^2 M(\mathbf{U} _2) = \omega _1 M(\mathbf{U} _1) - Q(\mathbf{U} _1,\mathbf{U} _1) \\ \langle \mathbf{u} _2,\mathbf{u} _1 \rangle = 0. \end{array} \end{aligned}$$

(22)

order p (p > 2):

$$\begin{aligned} \begin{array}{c} L(\mathbf{U} _p) + \widehat{Q}(\mathbf{U} _p,\mathbf{U} ^*) - \omega _0^2 M(\mathbf{U} _p) = \sum \limits _{r=1}^{p-1} (\omega _r M(\mathbf{U} _{p-r}) - Q(\mathbf{U} _r,\mathbf{U} _{p-r})) \\ \langle \mathbf{u} _p,\mathbf{u} _1 \rangle = 0. \end{array} \end{aligned}$$

(23)

To use the classical finite-element method in displacement, it is advisable to return to a formulation in pure displacement (Cochelin 1994). Taking into account relations (16–18), one obtains a series of linear problems in pure displacement at different orders with the operator $\left[ L + \widehat{Q} - \omega _0^2 M\right]$ are of a dimension generated by the vector $\mathbf{U} _1$. This operator is then singular. Using a relaxation technique (Azrar et al. 1993), the numerical solution of the problem at order 1 Eq. (21) gives the linear vibration mode. This makes it possible to determine the linear frequency $\omega _0$ and the corresponding vibration mode $\mathbf{U} _1$. To calculate the vector $\mathbf{U} _p$ of the problem Eq. (23), the only difficulty is to construct the right sides which depend on $\mathbf{U} _q$ and $\omega _q$, ($q < p$). Note that the coefficient $\mathbf{U} _p$ is calculated as a function of the vectors $\mathbf{U} _q$ and $\omega _q$ takes place in the determination of $\mathbf{U} _{p+1}$. It can be seen that all these problems Eqs. (21–23) have the same linear operator $\left[ L + \widehat{Q} - \omega _0^2 M\right]$ which must be reversed once for all. With the classical notation of computational mechanics (Azrar et al. 1999), the discretization of the problem at order 1 Eq. (21) leads to an eigenvalue problem in the form:

$$\begin{aligned} {[}K_e - \omega _0^2 M]\left\{ \mathbf{U} _1\right\} = 0. \end{aligned}$$

(24)

$[K_e]$ is the elastic stiffness matrix and [M] is the mass matrix. The solution of this problem gives the linear modes and linear frequencies of vibration. Denote by $\mathbf{U} _1$ the first mode of vibration and $\omega _0$ its corresponding frequency. The discretization of the problem at order p Eq. (23) reads in which $\mathbf{U} _p$ is the vector of the modal displacement of order p $[K_e - \omega _0^2 M]$ is the tangent stiffness matrix at a bifurcation point which is singular. The orthogonality condition between $\mathbf{U} _p$ and $\mathbf{U} _1$ should be added to Eq. (23) to get an invertible problem. After discretization, this condition reads:

$$\begin{aligned} {[}K_e - \omega _0^2 M]\left\{ \mathbf{U} _p\right\} = \omega _{p-1}[M]\left\{ \mathbf{U} _1\right\} + \left\{ F_p^{nl}\right\} , \end{aligned}$$

(25)

where $\left\{ F_p^{nl}\right\}$ represents the remaining part of the right side of Eq. (23).

The key elements of the computational algorithm are organized as follows:

Continuation Procedure

The part of the response curve obtained previously is sufficiently large for the initial nonlinear vibrations, but the validity of the solution is limited by a convergence radius. For large vibration amplitudes, this limitation must be overcome. This is the main objective of this section. For this, we always consider the developments Eq. (19) with $\mathbf{U} _0 \ne 0$ where $(\mathbf{U} _p, \omega _p)$ is the new unknown parameter to be computed. The path parameter “a” used in the series Eq. (19) can be identified as the projection of the displacement increment $(\mathbf{u} - \mathbf{u} _0)$, and the increase in frequency ($\omega ^2 - \omega _0^2$) on the tangent vector ($\mathbf{u} _1,\omega _1$):

$$\begin{aligned} a = \langle \mathbf{u} - \mathbf{u} _0,\mathbf{u} _1\rangle + (\omega ^2 - \omega _0^2)\omega _1. \end{aligned}$$

(26)

Introducing expressions Eq. (19) into Eqs. (14) and (26) and equating like powers of “a”, one gets the following set of linear problems:

order 1:

$$\begin{aligned} \begin{array}{c} L_t^0(\mathbf{U} _1) - \omega _0^2 M(\mathbf{U} _1) = \omega _1 M(\mathbf{U} _0) \\ \langle \mathbf{u} _1,\mathbf{u} _1\rangle + \omega _1 \omega _1 = 1. \end{array} \end{aligned}$$

(27)

order $p ~ (p > 1)$:

$$\begin{aligned} \begin{array}{c} L_t^0(\mathbf{U} _p) - \omega _0^2 M(\mathbf{U} _p) = \omega _p M(\mathbf{U} _0) + \sum \limits _{r=1}^{p-1} (\omega _r M(\mathbf{U} _{p-r}) - Q(\mathbf{U} _r,\mathbf{U} _{p-r})) \\ \langle \mathbf{u} _p,\mathbf{u} _1\rangle + \omega _p\omega _1 = 0, \end{array} \end{aligned}$$

(28)

in which the tangent operator is defined as $L_t^0(\cdot ) = L(\cdot ) + 2\widehat{Q}(\mathbf{U} ^*,\cdot ) + 2 Q(\mathbf{U} _0,\cdot )$.

To use the finite-element method in motion, returning to the pure displacement formulation; after discretization, one obtains the following matrix problems at each order:

order 1:

$$\begin{aligned} \begin{array}{c} {[}K_{t}^{0} - \omega _0^2 M]\left\{ \mathbf{U} _1\right\} = \omega _1 M \left\{ \mathbf{U} _0\right\} \\ \langle \mathbf{u} _1,\mathbf{u} _1\rangle + \omega _1\omega _1 = 1. \end{array} \end{aligned}$$

(29)

order p (p > 1):

$$\begin{aligned} \begin{array}{c} {[}K_{t}^{0} - \omega _0^2 M]\left\{ \mathbf{U} _p\right\} = \omega _p M \left\{ \mathbf{U} _0\right\} + \left\{ F_{p}^{nl}\right\} \\ \langle \mathbf{u} _p,\mathbf{u} _1 \rangle + \omega _p\omega _1 = 0, \end{array} \end{aligned}$$

(30)

where $\left\{ \widehat{F}_{p}^{nl}\right\}$ represents the remaining part of the right side of Eq. (28).

The key elements of the calculation algorithm are organized as follows:

The polynomial solutions Eq. (19) coincide almost perfectly in the radius of convergence, but they diverge beyond this validity zone. This limit can be calculated automatically using the simple criterion proposed by Cochelin (1994), as follows:

$$\begin{aligned} \begin{array}{c} a_\mathrm{series} = \Big (\zeta \dfrac{{\Vert \mathbf{u} _1\Vert }}{{\Vert \mathbf{u} _n\Vert }}\Big )^{\dfrac{1}{n-1}}, \end{array} \end{aligned}$$

(31)

where $\zeta$ is a small given number. This simple criterion is very useful for defining the range of interest of parameter “a” and has been successfully tested in various studies (Azrar et al. 1999, 2002; Boutyour et al. 2006; Benchouaf and Boutyour 2016). Indeed, at each stage of our procedure, the series of Eq. (19) have a radius of convergence limiting its validity range. However, this can be largely improved using Padé approximants (Hussein et al. 2000):

$$\begin{aligned} \begin{array}{c} \mathbf{U} - \mathbf{U} _0 = \sum \limits _{i=1}^{n} f_i(a)a^i\mathbf{U} _i \\ \omega ^2 - \omega _0^2 = \sum \limits _{i=1}^{n} f_i(a)a^i\omega _i, \end{array} \end{aligned}$$

(32)

where $f_i(a)$ are rational fractions having the same denominator and the vectors $\mathbf{U} _i$ are obtained from $\mathbf{U}$ by the classical Gram–Schmidt orthogonalisation procedure. The validity range of the solution Eq. (32) is defined by the maximal value “$a_\mathrm{max}$” of the control parameter “a” requiring that the relative difference between the displacements at two consecutive orders must be smaller than a given parameter $\delta$:

$$\begin{aligned} \delta = \frac{{\Vert \mathbf{u} _n^p(a_\mathrm{max}) - \mathbf{u} _{n-1}^p(a_\mathrm{max})\Vert }}{{\Vert \mathbf{u} _n^p(a_\mathrm{max}) - \mathbf{u} _0^p\Vert }}. \end{aligned}$$

(33)

Numerical Results

To show the efficiency and the applicability of the proposed algorithm, one considers the nonlinear free vibrations of the isotropic homogeneous rectangular elastic plate of length L, width l, and thickness $h = 1$ mm. The used material is aluminum with properties are: Young’s modulus $E = 70$ GPa, Poisson’s ratio $\nu = 0.3$, and mass density $\rho = 2778$ $\mathrm{kg}/\mathrm{m}^3$. For the discretization of the plate, DKT30 triangular shell element having three nodes and ten degrees of freedom per node ($u^0,u^1,u^2,v^0,v^1,v^2,w, \theta _x,\theta _y,\theta _z$) is adopted. For reasons of symmetry, only a quarter of the plate is discretized. The boundary conditions concerned in the present study are: (i) simply supported with immovable edges ($u = v = w = \theta _x = 0$ at $x = 0$ and $x = L$ and $u = v = w = \theta _y = 0$ at $y = 0$ and $y = l$) and (ii) fully clamped with immovable edges ($u = v = w = \theta _x = \theta _y = 0$) at all edges. The adopted parameters are nondimensionalized as follows: the nonlinear frequency is nondimensionalized with respect to its corresponding linear frequency $\omega$/$\omega _0$ and the maximum amplitude with respect to the thickness $W\mathrm{(center)}/\mathrm{h}$. As it was shown in the previous works (Benchouaf and Boutyour 2016), one takes a truncation order $n = 20$ and an accuracy parameter $\delta = 10^{-4}$. To improve the validity range of the series and to reduce the computational cost, Padé approximants are incorporated at each stage of the procedure.

To verify the validity and the accuracy of the proposed methodology (formulation and computer program), one considers the nonlinear free vibrations of undamped square perfect plate with different types of boundary conditions. In Table 1, one compares the results obtained by the present method and those reported by Azrar et al. (1999) using asymptotic numerical method and considered only the third harmonic for the longitudinal displacement and the results obtained by the FEM and incremental method (Lau et al. 1984). A good agreement is observed between these approaches.

Table 1 Frequency ratio $\omega /\omega _0$ according to the maximum amplitude W(center)/h for a square perfect plate ($L = l = 240$ mm)

The responses of square and rectangular plates taking into account the initial geometric imperfection ($\eta = 1,0$) are given in Figs. 2 and 3 for simply supported and fully clamped boundary conditions, respectively. It is clearly shown that the nonlinearities associated with rectangular plates are more detectable than those of square plates. It also appears that the simply supported boundary conditions yield a larger nonlinear response than the clamped ones.

Fig. 2

Nonlinear free vibrations of simply supported square and rectangular plates with geometric imperfection ($L = 240, 360$ mm, $l = 240$ mm, and $\eta = 1.0$)

Fig. 3

Nonlinear free vibrations of fully clamped square and rectangular plates with geometric imperfection ($L = 240, 360$ mm, $l = 240$ mm, and $\eta = 1.0$)

In Figs. 4 and 5, we present the influence of the initial geometric imperfection on the response for simply supported and fully clamped square plates, respectively. Different initial imperfection amplitudes magnitudes are considered ($\eta = 0.5, 1.0, 1.5,$ and 2.0). It is clearly seen that an increase in the amplitude of the imperfections shifts the curves of the main frequency to the left.

Fig. 4

Effect of imperfection $\eta$ on the frequency amplitude response of simply supported square plate ($L = l = 240$ mm)

Fig. 5

Effect of imperfection $\eta$ on the frequency amplitude response of fully clamped square plate ($L = l = 240$ mm)

Conclusion

In this study, we have presented a methodological approach using the asymptotic numerical method to investigate the nonlinear free vibrations of thin plates with different types of boundary conditions and taken into account the geometric imperfections. Based on von Karman plate theory, the harmonic balance method, and Hamilton’s principle, the nonlinear dynamic problem is transformed into a sequence of linear ones with only one operator to be inverted. The incorporation of Padé approximants at each stage of the algorithm permits large reduction of computational cost. The backbone curves of nonlinear free vibrations of plate are obtained automatically. The application of the continuation procedure permitted one to obtain the nonlinear resonance curves at any desired range of amplitudes. Numerical results for nonlinear frequency and nonlinear displacements are presented and compared for square and rectangular plates. The effect of geometric imperfection in comparison to perfect plates is fully discussed. Presented results agree well with the results available in the literature.