Nonlinear dynamic response of a stiffened plate with four edges clamped under primary resonance excitation

Abstract

An approach is presented to study the nonlinear forced vibration of a stiffened plate. The stiffened plate is divided into one plate and some stiffeners, with the plate considered to be geometrically nonlinear, and the stiffeners taken as geometrically nonlinear Euler beams. Assuming the displacement of the stiffened plate, Lagrange equation and modal superposition method are used to derive the dynamic equilibrium equations of the stiffened plate according to energy of the system. A stiffened plate with four clamped edges subjected to harmonic excitation is studied by means of the method of multiple scales; the first approximation solutions of the double-modal motion of the system are obtained. Numerical examples for different stiffened plates are presented to discuss the steady response of the primary resonance and the amplitude–frequency relationship; and some nonlinear forced vibration characteristics of the stiffened plate are obtained, which are useful for engineering design.

1 Introduction

A thin plate stiffened by a set of stiffeners can achieve greater strength with relatively less material. Eccentrically stiffened plates have been widely used in engineering of bridges, ship hulls, aircrafts, etc. To make full use of the stiffness provided by stiffeners, they are often attached to plates along the main load-carrying directions. Since they are often subjected to severe dynamic loads, the vibration characteristics of stiffened plates are of considerable importance to mechanical and structural engineers.

Among the known solution techniques to the vibration of stiffened plates, there are grillage model [1], Rayleigh–Ritz method [2, 3], finite element method [4, 5], finite difference method [6, 7], differential quadrature method [8], meshless method [9] and other methods [10–12]. The occurrence of any arbitrary time-varying loads on such structures is very common.The magnitude of such loads may be so great that it may lead to large deformations of the structure, as plates and stiffened plates belong to the category of thin-walled structures. At large defection level, membrane stresses are produced which give additional stiffness to the structure. Moreover, the strain–displacement relationship becomes nonlinear in this range, which is rather the basic source of nonlinearity of the present problem. Geometrically nonlinear vibration of plates has long been a subject receiving considerable research efforts. Numerous studies have been reported in open literature: see, for example, those by Haterbouch and Benamar [13], Nerantzaki and Katsikadelis [14], Celep and Guler [15]. Presently, the common analytical methods for the nonlinear vibration of plates include Ritz method, Galerkin method, perturbation method, successive approximation method, finite difference method, Runge–Kutta integration and so on [16–22].

The geometrical nonlinearity may lead to energy exchange between different modes. This is the well-known internal resonance phenomenon [23–26]. There have been also many publications on the internal resonance for plates. Chang et al. [27] investigated nonlinear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance. Abe et al. [28] studied two-mode responses of thin rectangular laminated plates subjected to a harmonic excitation with internal resonance by using the method of multiple scales. Ribeiro and Petyt [29] investigated nonlinear free vibration of isotropic plates with internal resonance. However, the investigations on the internal resonance for stiffened plates are very limited. Thus, it is of significant importance to investigate the dynamic characteristics of stiffened plates under the condition of internal resonance.

In the recent literature, most investigations on the vibration of stiffened plates do not take the damping into account, which cannot reflect the real dynamic characteristics of stiffened plates. Only a few works combine both geometrical nonlinearity and damping: Amabili made an experimental and numerical study of plates with viscous damping and subjected to harmonic excitation [16, 30]. This author studied also numerically circular cylindrical panels with viscous damping [31]. An amplitude equation, based on an approximated harmonic balance method and Galerkin’s procedure, was proposed by Daya et al. [32] to study sandwich beams and plates with central viscoelastic layers. Touze and Amabili built reduced-order models for damped geometrically nonlinear systems [33]. They considered a modal viscous damping. Ribeiro and Petyt [34–36] used the hierarchical finite element method in an in-depth investigation of the nonlinear response of clamped rectangular plates. Boumediene et al. investigated nonlinear forced vibration of damped plates by an asymptotic numerical method [37].

The objective of this paper is to analyze the geometrically nonlinear forced vibration of a stiffened plate with four edges clamped under primary resonance excitation. In the paper, the damping of the plate taken into account is of a viscous type. Even if this structural damping is relatively simple, it represents a large part of engineering applications. The governing equations of motion, which are derived by using the Lagrange equation, are reduced to a two-degree-of-freedom nonlinear system by assuming mode shapes. Considering the case of three-to-one internal resonance, a parametric study is then carried out to gain an insight into the dynamic behavior of such a stiffened plate.

2 Derivation of the governing equation

In order to simplify the problem, material nonlinearity is not considered in this paper, and the material of stiffened plate is isotropic. Consider a stiffened plate in Fig. 1, which is composed of x-stiffeners, y-stiffeners and a plate. The generally used orthotropic plate theory assumes both x-stiffeners and y-stiffeners to be smeared over the plate. However, this theory has three disadvantages: first, the dynamical differential equation is difficult to formulate because of the non-uniform mass per unit area; second, the neutral surfaces may not coincide in the orthogonally stiffened directions; third, this theory is inaccurate for the computation of the local effect. This paper deals with the stiffeners and the plate separately, formulates the energy equations respectively, and then substitutes the energy equations into Lagrange equation.

Fig. 1

Structure of the stiffened plate

In Fig. 1, u and v denote the displacements of the middle surface of the plate along the x-direction and the y-direction, respectively, and w denotes the deflection of the plate. Taking geometric nonlinearity of the plate into consideration, the strain energy of the plate is composed of flexural strain energy U _pb and membrane strain energy U _pm , which are given by

(1-1)

(1-2)

where $D(= \frac{Eh^{3}}{12( 1 - \mu^{2} )})$ is the flexural rigidity of the plate, E and μ are the Young’s modulus and Poisson ratio of the plate material, respectively.

The kinetic energy of the plate is

(2)

where ρ is the mass per unit volume of the plate material and h is the thickness of the plate.

Take x-stiffeners and y-stiffeners as Euler beams with geometric nonlinearity considered. Besides, the torsional rigidities of the stiffeners can be neglected [38]. Thus the strain energies include the axial strain energy and the bending strain energy. In the following, d _x and d _y denote the distance between the neutral plane of the x-stiffeners, y-stiffeners and the two bending neutral planes of the compound sections, respectively; N _x and N _y denote the numbers of x-stiffeners and y-stiffeners, respectively; the coordinates of the ith x-stiffener and y-stiffener are y=y _i and x=x _i , respectively.

The strain energies of the ith x-stiffener and y-stiffener are given as follows:

(3-1)

(3-2)

where EA _x and EA _y are the axial rigidities of x-stiffeners and y-stiffeners, respectively; EI _x and EI _y are the flexural rigidities of x-stiffeners and y-stiffeners with respect to the neutral surface, respectively.

Only the z-directional motion of stiffeners is considered, and the kinetic energy of the ith x-stiffener and y-stiffener is given as follows:

(4-1)

(4-2)

where A _x and A _y are the sectional areas of x-stiffeners and y-stiffeners, respectively.

The total strain energy U of the stiffened plate is then given by

$$ U = U_{p} + U_{s} $$

(5)

where U _p =U _pb +U _pm , $U_{s} = \sum_{i = 1}^{N_{x}} U_{sxi} + \sum_{i = 1}^{N_{y}} U_{syi}$.

The total kinetic energy T of the stiffened plate is then given by

$$ T = T_{p} + T_{s} $$

(6)

where $T_{s} = \sum_{i = 1}^{N_{x}} T_{sxi} + \sum_{i = 1}^{N_{y}} T_{syi}$.

The displacement of the stiffened plate is assumed to be the product of time function and space function; then it can be expressed in terms of tensors as follows:

(7-1)

(7-2)

(7-3)

where $u_{ij}^{t}( t )$, $v_{ij}^{t}( t )$ and $w_{ij}^{t}( t )$ are the generalized coordinates in x,y and z directions, respectively; $u_{ij}^{d}( x,y )$, $v_{ij}^{d}( x,y )$ and $w_{ij}^{d}( x,y )$ are the mode functions in x,y and z directions, respectively; $\overline{M}$ and $\overline{N}$ indicate the terms necessary in the expansion of the in-plane displacements; M and N indicate the terms necessary in the expansion of the out-of-plane displacements.

Substituting Eqs. (7-1)–(7-3) into Eqs. (5) and (6), the strain energy and the kinetic energy can be expressed in terms of generalized coordinates as follows:

(8-1)

(8-2)

where each tensor is given by

(9-1)

(9-2)

(9-3)

(9-4)

(9-5)

(9-6)

(9-7)

(9-8)

(9-9)

(9-10)

(9-11)

(9-12)

(9-13)

(9-14)

where the subscripts ij and mn of $d_{ijmn}^{1}, d_{ijmn}^{3}, e_{ijmn}^{1},\allowbreak f_{ijmn}^{1}, f_{ijmn}^{2}, f_{ijmn}^{3}$ and g _ijmn are symmetric; the subscripts mn and kl of $e_{ijmnkl}^{2}$ are symmetric; the subscripts ij, mn, kl and pq of $e_{ijmnklpq}^{3}$ are symmetric; and the subscripts ij and mn, kl and pq of $d_{ijmnklpq}^{6}$ are symmetric.

The Lagrange equation is given by

$$ \frac{d}{dt}\biggl( \frac{\partial T}{\partial \dot{q}_{j}} \biggr) - \frac{\partial T}{\partial q_{j}} + \frac{\partial U}{\partial q_{j}} = Q_{j} $$

(10)

where q _j is the generalized coordinate of the jth degree of freedom; $\dot{q}_{j}$ is the derivative of the generalized coordinate with respect to t of the jth degree of freedom; and Q _j is the non-conservative generalized force with respect to q _j .

Assume that W _j is the virtual work with respect to q _j , then Q _j can be expressed as

$$ Q_{j} = \frac{\delta W_{j}}{\delta q_{j}}. $$

(11)

Substituting Eqs. (8-1) and (8-2) into Eq. (10), q _j is expressed by $u_{ij}^{t}, v_{ij}^{t}$ and $w_{ij}^{t}$, respectively. Then the following nonlinear dynamic equations can be obtained:

(12-1)

(12-2)

(12-3)

Dimensionless formulation is introduced by putting

(13)

Substituting Eq. (13) into Eqs. (12-1)–(12-3), q _j is expressed by $u_{ij}^{*}, v_{ij}^{*}$ and $w_{ij}^{*}$, respectively. Then Eqs. (12-1)–(12-3) can be written in dimensionless form as

(14-1)

(14-2)

(14-3)

where each tensor is given by

(15-1)

(15-2)

(15-3)

(15-4)

(15-5)

(15-6)

(15-7)

(15-8)

(15-9)

(15-10)

(15-11)

(15-12)

(15-13)

(15-14)

(15-15)

(15-16)

(15-17)

Upon neglecting the in-plane inertia, which is an acceptable assumption in most engineering applications of thin plates [39], Eqs. (14-1) and (14-2) can be reduced to

(16-1)

(16-2)

By solving Eqs. (16-1) and (16-2), the in-plane generalized coordinates $u_{ij}^{*}$ and $v_{ij}^{*}$ can be obtained as follows:

(17)

where [k] is the coefficient matrix in Eqs. (16-1) and (16-2); {d} is the constant matrix in Eqs. (16-1) and (16-2).

Substitute $u_{ij}^{*}$ and $v_{ij}^{*}$ into Eq. (14-3) and obtain the nonlinear differential equation with respect to $w_{ij}^{*}$. Then, it can be solved according to mathematical methods.

3 Solution procedure

3.1 Double-mode equation of motion of the stiffened plate

A stiffened plate with four edges clamped is investigated in this paper. The boundary conditions are: u=v=w=∂w/∂x=0 at x=0,a; u=v=w=∂w/∂y=0 at y=0,b. In order to investigate the relationship between two modes, this paper only deals with the double-mode transverse vibration of a stiffened plate. Then, the displacement is assumed as

(18)

(19)

(20)

The damping of the plate taken into consideration is of a viscous type [39], and a non-conservative viscous damping force may be derived from a potential function. The potential function for viscous forces is called the Rayleigh dissipation function, and is given by

$$ R = \frac{1}{2}\hat{c}\int_{0}^{a} \int _{0}^{b} \bigl( \dot{u}^{2} + \dot{v}^{2} + \dot{w}^{2} \bigr) \,dx\,dy $$

(21)

where $\hat{c}$ has a different value for each term of the mode expansion.

According to Eqs. (18)–(20), Eq. (21) can be written as

(22)

where $\hat{c}_{11}$ and $\hat{c}_{13}$ are the damping constants, which can be evaluated from experiments.

The generalized force Q _j with respect to viscoelastic damping can be expressed as

$$ Q_{j} = - \frac{\delta R}{\delta \dot{q}_{j}} $$

(23)

where $\dot{q}_{j}$ is the derivative of the generalized coordinate with respect to time of the jth degree of freedom.

The transverse excitation is expressed as

$$ q( x,y,t ) = F\cos( \varOmega t ) $$

(24)

where F is the amplitude of the transverse excitation and Ω is the frequency of the transverse excitation.

Replacing the frequency Ω and time t with dimensionless quantities Ω ^∗ and τ in Eq. (24), respectively, one obtains:

$$ q( x,y,\tau) = F\cos\bigl( \varOmega^{*}\tau\bigr) $$

(25)

where $\varOmega^{*} = \varOmega a^{2}\sqrt{\frac{\rho h}{D}}$.

Upon neglecting the in-plane damping force, which is an acceptable assumption in most engineering applications of thin plates [39], taking the transverse damping force into account and according to Eqs. (11) and (23), the generalized forces with respect to $w_{11}^{t}$ and $w_{13}^{t}$ are expressed as:

(26-1)

(26-2)

Replacing the generalized coordinates $w_{11}^{t}, w_{13}^{t}, \varOmega$ and time t with dimensionless quantities $w_{11}^{*}, w_{13}^{*}, \varOmega^{*}$ and τ in Eqs. (26-1) and (26-2), respectively, one obtains

(27-1)

(27-2)

Substituting Eqs. (18)–(20), (27-1) and (27-2) into (14-3), and using Eqs. (13), (16-1), (16-2) and (17), the nonlinear dynamic differential equations of the stiffened plate are obtained as

(28-1)

(28-2)

where the expression of each tensor is not given in this paper due to space limitations.

3.2 Approximate solution of the primary resonance

Both quadratic and cubic nonlinear terms are included in Eqs. (28-1) and (28-2). The quadratic nonlinear terms are caused by the stiffeners, while the cubic nonlinear terms are caused by both the plate and the stiffeners. Hence, the method of multiple scales can be used to solve the nonlinear dynamic differential equations. A small parameter ε should be introduced so that the excitation, damping and nonlinear terms appear in the same perturbation equation. Suppose

(29)

Substituting Eq. (29) into Eqs. (28-1) and (28-2) yields

(30-1)

(30-2)

Introducing the time scales

$$ T_{0} = \tau,\qquad T_{1} = \varepsilon\tau. $$

(31)

an approximate solution can be given by

(32)

(33)

Substituting Eqs. (32) and (33) into (30-1) and (30-2), and equating the coefficients of ε ⁰ and ε ¹ in both sides of the equation, we obtain

(34)

(35)

(36)

(37)

where D _i =∂/∂T _i .

The complex solution of Eqs. (34) and (35) can be written as

(38)

(39)

where A ₁(T ₂) and B ₁(T ₂) are unknown complex functions, and cn is the complex conjugate part to the preceding terms. According to the linear vibration theory, we get

(40)

(41)

(42)

(43)

Substituting Eqs. (38) and (39) into Eqs. (36) and (37) yields

(44)

(45)

When the frequency of excitation is close to the natural frequency of the derived system, primary resonance may occur. In order to simplify the problem, this paper only deals with the condition that the frequency of excitation is close to the fundamental frequency ($\varOmega^{*} \approx\omega_{1}^{*}$). When there is no internal resonance, the steady solution is uncoupled. Hence, the internal resonance is considered in this paper. The primary nonlinear terms are cubic. Hence, the 3:1 internal resonance $\omega_{2}^{*} \approx 3\omega_{1}^{*}$ is analyzed in the following. Introducing a detuning parameter σ ₁, suppose

$$ \omega_{2}^{*} = 3\omega_{1}^{*} + \varepsilon\sigma_{1}. $$

(46)

Introducing a detuning parameter σ ₂, suppose

$$ \varOmega^{*} = \omega_{1}^{*} + \varepsilon \sigma_{2}. $$

(47)

Suppose the particular solution of Eqs. (44) and (45) has the following forms [24]:

(48)

(49)

Substituting Eqs. (46)–(49) into Eqs. (44) and (45), and $\hat{p}_{1,12}\cos( \varOmega^{*}\tau)$ and $\hat{p}_{3,12}\cos( \varOmega^{*}\tau)$ taking the complex form $\frac{1}{2}\hat{p}_{1,12}e^{i( \omega _{1}^{*}T_{0} + \sigma _{2}T_{1} )} + cn$ and $\frac{1}{2}\hat{p}_{3,12}\times e^{i( \omega _{1}^{*}T_{0} + \sigma _{2}T_{1} )} + cn$, respectively, we then get

(50)

(51)

where NST stands for non-secular terms.

The solvability condition demands

(52)

(53)

(54)

(55)

To make it clear, the right terms of Eqs. (52)–(55) are expressed as R ₁,R ₂,R ₃ and R ₄, respectively. According to the theory of linear algebra, the rank of coefficient matrix of Eqs. (54)–(57) should equal to that of augmented matrix to ensure that A ₂,A ₃,B ₂ and B ₃ have non-zero solution. It can be deduced from Eqs. (40) and (41) that both determinants of Eqs. (52) and (53), Eqs. (54) and (55) are zero. Hence, in order to ensure that A ₂,A ₃,B ₂ and B ₃ have non-zero solution, it requires

(56)

(57)

The complex functions A ₁ and B ₁ can be expressed as

(58)

(59)

where $\hat{a}_{1}, \hat{b}_{1}, \theta_{1}$ and θ ₂ are real functions with respect to T ₁. Substituting Eqs. (58) and (59) into the expressions of R ₁,R ₂,R ₃ and R ₄, and by employing Eqs. (56) and (57) setting the real and imaginary parts to be zero, respectively, we then obtain

(60)

(61)

(62)

(63)

where $\lambda_{1} = p_{34} - ( \omega_{1}^{*} )^{2}, \lambda_{2} = p_{34} - ( \omega_{2}^{*} )^{2}$, γ ₁=θ ₂−3θ ₁+σ ₁ T ₂,γ ₂=σ ₂ T ₂−θ ₁.

For the steady response, it demands

$$ D_{1}\hat{a}_{1} = D_{1}\hat{b}_{1} = D_{1}\gamma_{1} = D_{1}\gamma_{2} = 0. $$

(64)

According to the relationship between γ ₁,γ ₂ and θ ₁,θ ₂, we get

(65)

(66)

Eliminating γ ₁ and γ ₂ from Eqs. (60)–(63), and employing Eqs. (64)–(66), the equations about $\hat{a}_{1}$ and $\hat{b}_{1}$ can be obtained:

(67)

(68)

where

(69-1)

(69-2)

(69-3)

(69-4)

(69-5)

(69-6)

(69-7)

(69-8)

(69-9)

According to Eqs. (67) and (68), the amplitude–frequency curves can be obtained. Substituting Eqs. (38) and (39) into Eqs. (32) and (33), the first approximation solutions of steady response are obtained as

(70)

(71)

From Eqs. (70) and (71) it can be observed that the steady response is the superposition of two periodic vibration solutions under primary resonance excitation when taking the internal resonance into consideration.

4 Numerical results and discussion

Stiffened plates with four clamped edges are analyzed. The parameters are as follows: a=0.6 m, b=0.4 m, E=2.1×10¹¹ Pa, μ=0.3, ρ=7.85×10³ kg/m³, h=0.004 m, A _x =A _y =1.92×10⁻⁴ m², I _x =I _y =9.216×10⁻⁹ m⁴, $\hat{c}_{11}=0.15$ kg/(s⋅m²), $\hat{c}_{13}=0.1$ kg/(s⋅m²). In order to investigate the influence of the stiffeners, six cases are analyzed including N _x =1,3,5 and N _x =N _y =1,3,5, where N _x and N _y are the numbers of the x-stiffeners and the y-stiffeners. The following work includes two aspects: (1) The steady response of the primary resonance of the stiffened plates under the amplitude of the transverse excitation F=100 N/m² and the frequency of excitation $\varOmega^{*}=0.99\omega_{1}^{*}$; (2) Considering F=50,100,150 N/m², respectively, the amplitude–frequency response of the primary resonance of the stiffened plates.

4.1 The steady response of the primary resonance

Suppose $\omega_{2}^{*}=3\omega_{1}^{*}$; it means that the internal resonance is detuned and completed. According to the above parameters, the double-modal steady response of different stiffened plates under the primary resonance excitation ($\varOmega^{*}=0.99\omega_{1}^{*}$) is shown in Figs. 2–7, where Δ ₁ and Δ ₃ are the first two terms in Eqs. (70) and (71), which respectively are given by:

(72)

(73)

Fig. 2

Double-modal steady response of the stiffened plate with four clamped edges (N _x =1)

From Figs. 2–7 the following can be observed: that the internal resonance takes effect when the system vibrates, and both the low-frequency and high-frequency vibrations are excited; that all the amplitudes of these steady response curves decrease with the increase of the number of the stiffeners, this being because the stiffeners increase the flexural rigidity of the stiffened plate; that the amplitude of (1,3)th mode is close to that of (1,1)th mode, which indicates that the deformation of higher-order mode cannot be neglected when the lower-order mode is excited for the influence of internal resonance.

Figures 2, 3, and 4 are the steady response curves of the primary resonance of the stiffened plates when respectively N _x =1, N _x =3 and N _x =5. From these three figures, it can be observed that the amplitudes of the steady response curves are very close when N _x =3 and N _x =5. Thus, it shows the stiffeners contribute little to the flexural rigidity of the stiffened plate when the number of the x-stiffeners exceeds 3.

Fig. 3

Double-modal steady response of the stiffened plate with four clamped edges (N _x =3)

Fig. 4

Double-modal steady response of the stiffened plate with four clamped edges (N _x =5)

Figures 5, 6, and 7 are the steady response curves of the non-resonance of the stiffened plates when respectively N _x =N _y =1, N _x =N _y =3 and N _x =N _y =5. From these three figures it can be observed that the amplitudes of the steady response curves are very close when N _x =N _y =3 and N _x =N _y =5. Thus, it shows that the stiffeners contribute little to the flexural rigidity of the stiffened plate when the numbers of both the x-stiffeners and the y-stiffeners exceed 3.

Fig. 5

Double-modal steady response of the stiffened plate with four clamped edges (N _x =N _y =1)

Fig. 6

Double-modal steady response of the stiffened plate with four clamped edges (N _x =N _y =3)

Fig. 7

Double-modal steady response of the stiffened plate with four clamped edges (N _x =N _y =5)

Combining Figs. 3 and 5 one observes that the curves for the amplitudes of the steady response when N _x =N _y =1 are smaller than those when N _x =3; therefore setting the stiffeners in two directions can increase the flexural rigidity of the stiffened plate more efficiently than setting them in only one direction.

4.2 The amplitude–frequency response of the primary resonance

Similarly as in the previous section, suppose $\omega_{2}^{*}=3\omega_{1}^{*}$. According to Eqs. (67) and (68), the amplitude–frequency response curves for the six different stiffened plates (N _x =1, N _x =3, N _x =5, N _x =N _y =1, N _x =N _y =3, N _x =N _y =5) under different transverse excitations can be obtained as shown in Figs. 8–19. In these figures, $a_{1}=\hat{a}_{1}$, $b_{1}=\hat{b}_{1}$.

Fig. 8

Amplitude–frequency response a ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =1)

Fig. 9

Amplitude–frequency response b ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =1)

Fig. 10

Amplitude–frequency response a ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =3)

Fig. 11

Amplitude–frequency response b ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =3)

Fig. 12

Amplitude–frequency response a ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =5)

In Figs. 8–19, the curve with respect to F=0 is a backbone curve. From these twelve figures, the following can be observed: first, that the resonant region tends to tilt to the right, thus the amplitude–frequency response curve having a hardened spring characteristic; second, that the flexural vibration amplitude increases with the approach of the excitation frequency to the linear flexural vibration frequency of the stiffened plate; third, that in the right branch of each resonant curve, the solution is not stable when the amplitude increases with σ ₂ increasing. Besides, the nonlinear flexural vibration amplitude increases slowly, since the structural damping exists and the internal resonance takes effect.

With the increase of the amplitude of the transverse excitation, the amplitudes of both a ₁ and b ₁ increase as well. Besides, the spacing between each pair of the resonant curves becomes larger, and the primary resonant curves exhibit both the jumping and the lagging phenomena. Under constant amplitude of excitation, the amplitude of the response curve increases as σ ₂ approaches zero, and several equilibrium points appear simultaneously when σ ₂ is greater than zero.

From Figs. 8–13 it can be observed that the amplitudes of both a ₁ and b ₁ decrease and the spacing between each pair of resonant curves becomes smaller as the number of the x-stiffeners increases, but the change is not obvious after the number of the x-stiffeners exceeds 3. From Figs. 14–19 it can be observed that the amplitudes of both a ₁ and b ₁ decrease and the spacing between each pair of the resonant curves becomes smaller as the numbers of both the x-stiffeners and the y-stiffeners increase, but the change is not obvious after the numbers of both the x-stiffeners and the y-stiffeners exceed 3. It can be concluded that the primary resonant region is influenced most obviously when only one stiffener is set in each direction.

Fig. 13

Amplitude–frequency response b ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =5)

Fig. 14

Amplitude–frequency response a ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =N _y =1)

Fig. 15

Amplitude–frequency response b ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =N _y =1)

Fig. 16

Amplitude–frequency response a ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =N _y =3)

Fig. 17

Amplitude–frequency response b ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =N _y =3)

Fig. 18

Amplitude–frequency response a ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =N _y =5)

Fig. 19

Amplitude–frequency response b ₁–σ ₂ of the stiffened plate with four clamped edges (N _x =N _y =5)

5 Conclusions

In this investigation, the nonlinear forced vibration of the stiffened plates with four clamped edges has been investigated based on the Lagrange equation and the energy principle. The method of multiple scales method is employed to analyze the double-modal motion of the stiffened plate. Besides, numerical examples for different stiffened plates are presented in order to discuss the steady response of the primary resonance and the amplitude–frequency relationship of the stiffened plates, which yields the following conclusions:

• (1) For the stiffened plate with four clamped edges, there is a 3:1 internal resonance between the (1,3)th and (1,1)th modes. Numerical simulation about the internal resonance shows that the coupling effect between two modes is an important problem during structural design when the parameters of the stiffened plate are in the range of the internal resonances.

• (2) When the number of the stiffeners in one direction exceeds 3, the stiffener’s contribution to the flexural rigidity of the stiffened plate becomes not obvious.

• (3) The flexural rigidity of the stiffened plate can be increased more efficiently by setting the stiffeners in two directions than in only one direction.

• (4) For the existing structural damping and the internal resonance which takes effect, the nonlinear flexural vibration amplitude increases slowly, and the amplitude–frequency response curve has a hardened spring feature. Besides, in the right branch of each resonant curve, the solution is not stable when the amplitude increases with σ ₂ increasing.

• (5) With the increase of the amplitude of the transverse excitation, both amplitudes of the two coupling modes increase. Besides, the spacing between each pair of the resonant curves becomes larger, and the primary resonant curves exhibit both the jumping and the lagging phenomena. These are the major differences between the linear and the nonlinear systems.

• (6) The primary resonant region is influenced most obviously when only one stiffener is set in each direction.