We develop an approach to reduce the governing equation of motion for the nonlinear vibration of a clamped laminated composite to the Duffing equation in a decoupled modal form. The method of weighted residuals enables such a reduction for laminates with clamped boundary conditions. Both rigidly clamped and loosely clamped boundary conditions are analyzed using this method. The reduction method conserves the total energy of the system. The decoupled modal form Duffing equation has constant modal parameters in terms of the laminated composite material's properties and geometries. The numerical computations illustrate the individual modal response with an emphasis of the transitional phenomena to chaos caused by the large load.

The nonlinear dynamic response of a thin laminate occurs with a large curvature deformation. A common threshold for nonlinear phenomena is a deflection greater than half of the plate thickness. Large deformation associated with high-order strain can be described by a second-order von Karman strain field or a strain field of higher order. Nonlinear vibrations often arise when structures and membranes are subject to high pressure or intense thermal fields. This is typical of microstructures subject to aerodynamic forces, such as those as analyzed by Suhir [

One approach to analyzing the dynamic response of plates and laminates in nonlinear elastic deformation is to use a Galerkin-type method, or the energy method, to reduce the governing partial differential equation to an ordinary differential equation for the time-dependent variable. Such reduction procedures are complicated by the nonlinear strain compatibility equation. This makes it very difficult to obtain a decoupled modal form ordinary differential equation for the vibration of laminates; see Bloom and Coffin [

Thermal vibration of functionally graded materials has been studied using a method that bears similar characteristics to that for the plate thermal vibrations. For example, Hao et al. [

For the laminate vibration in clamped boundary conditions, two different forms of the deflection approximating function have been used previously, for both linear and nonlinear deformations. Sundara Raja Iyengar and Naqvi [

Motivated by finding a unified approach to obtaining a decoupled modal form equation while preserving the total energy, we have found that the method of weighted residuals (MWR), as described by Crandall [

In this paper, the MWR is applied to study the deflection of symmetric thin isotropic laminates for both rigidly clamped and loosely clamped boundary conditions. This method leads to a decoupled modal form Duffing equation for each mode, with constant coefficients expressed in terms of the material properties and laminate geometries. However, our reduction approach requires a particular choice of the approximating function, as we will demonstrate. In the following, we first obtain the Duffing equation in a decoupled modal form. Our key interest in the Duffing system here lies in the transitional phenomena associated with the loading and initial conditions, instead of parametric analyses as we did earlier [

The nonlinear von-Karman strain field is defined as

Note that the approximating functions in (

Define the residual of the equilibrium equation, (

The general characteristics of the Duffing equation have been studied extensively for various behaviors that are relevant for laminate response analysis. The Duffing equation with damping in a normalized form is

For our Duffing equation (

We will use the fourth-order Runge-Kutta method to illustrate the fundamental mode response of the Duffing equation for the chaotic and periodic behaviors in relation to the loading and initial conditions. The computation also generates the Lyapunov exponents to verify the transition to chaos. The case study examines a thin laminate of

Physical parameters of a PWB.

Layer | Material | Specific weight (g/ | Young's module(MPa) | Poisson ratio | Thickness (mm) |
---|---|---|---|---|---|

1 | Cu/FR-4 | 7.599 | 0.205 | 0.030 | |

2 | FR-4 | 1.200 | 0.190 | 0.200 | |

3 | Cu | 8.310 | 0.400 | 0.035 | |

4 | FR-4 | 1.200 | 0.190 | 1.000 |

Modal parameters of the Duffing equation.

Mode | Natural frequency | Natural frequency ratio | Stiffness |
---|---|---|---|

M1: | 6.7839 | 7.1888 | |

M2: | 2.5381 | 1.2548 | |

M3: | 4.2430 | 4.3085 | |

M4. | 6.1008 | 5.8140 |

A symmetric PWB (

Figures

Mode-1 Poincare map in finite periodic orbits subject to the harmonic load

Map with connected periodic points

Map with disconnected periodic points

Mode-1 response with respect to the harmonic load

Figure

Other than the Poincare map, the Lyapunov exponent provides an alternative indicator for chaos. The Lyapunov exponents for the Duffing system were computed using the algorithms developed for low-dimensional systems by Dieci [

These results point out that a large amplitude harmonic forcing can transform a stable Duffing system into an unstable system. In addition, chaos shows a sensitive dependence on the initial and loading conditions, as observed from the comparisons of Figures

The MWR method used in the present study, for obtaining the Duffing equation, eliminated the modal coupling resulting from a use of the standard Galerkin method. The modal decoupling avoids the computational difficulties associated with coupled modal coefficients in integral forms, such as those obtained by Sundara Raja Iyengar and Naqvi [

It has been discussed by Crandall et al. [

Our decoupling method would not be valid for the approximating function for the clamped laminate as that used by Sundara Raja Iyengar and Naqvi [

We note that our reduction procedure resulted in a decoupled modal form Duffing equation. This differs from the reduction for the normal forms of a nonlinear vibration system, as studied by Shaw and Pierre [

A decoupled modal form Duffing equation with constant coefficients is obtained for the nonlinear vibration of a thin symmetric isotropic laminate for both the rigidly clamped and loosely clamped boundary conditions. This equation results from a reduction of the governing equation of motion by using the method of weighted residuals, with a total conservation of energy in the sense of Galerkin’s averaging. The benefit, other than the modal decoupling, is an improved accuracy and computation efficiency. The decoupled modal form Duffing equation provides a direct formulation for the laminates’ modal response. Our analytical and computation studies identified the transitional phenomena of the Duffing equation from a stable system to an unstable system subject to a large load. The weak Lyapunov function justifies the weak attractors of the stable system as a path to chaos. The results also illustrate that chaos has a sensitive dependence on the loading and the initial conditions. By generalizing the Galerkin approach to the MWR, the present study found a particular procedure for modal decoupling of the Duffing equation for the dynamics of laminates with clamped boundary conditions.

Laminate rigidity matrix of the laminate

Inverse of A

Elements of

Coupling rigidity matrix of the laminate

Stiffness matrix of the laminate

Stiffness coefficients of the laminate

Complementary function coefficients of the Airy stress function

Flexural rigidity matrix of the laminate

Matrix components of

Residual of the equation of motion

Normalized load

Airy’s stress function coefficients

Airy's stress function

Inertia of the laminate

integrant functions.

Applied moment vector

In-plane force vector

In-plane force as a function of deflection

Laminate pressure load

Fime step in numerical integration

Forcing frequency index

Stiffness coefficient

Duffing equation forcing load

Stiffness coefficient

Stiffness in Duffing’s equation

Normalized stiffness in Duffing’s equation

Mid-plane deformation in

Deformation in

Mid-plane deformation in

Deformation in

Velocity

Initial velocity of

Time-dependent deflection function in modal form

deflection

Approximate deflection

Velocity of the approximate deflection

Acceleration of the approximate deflection

initial deflection of

Strain components

Modal coefficient in

Modal coefficient in

Damping ratio

The normalized damping ratio

Factor

Normalized time.

Threshold time

Mass density of the

Poisson ratio of the

Natural frequency

Forcing frequency

weighting function

The author would like to express her appreciation to Professor Stallybrass, Professor Dieci, and Professor Weiss in the School of Mathematics at Georgia Institute of Technology, for helpful discussions.