Chun-Sheng Chen

A further study on nonlinear vibration of initially stressed plates

Abstract Numerical solutions of an arbitrary initially stressed plate based on various displacement fields under nonlinear vibration are presented. Nonlinear partial differential equations of plate vibration are formulated from Lo’s high order transverse shear and transverse normal deformation plate theory. The higher order terms for Lo’s displacement field are neglected to obtain various simpler forms of equations such as the first order theory and other higher order theories for isotropic plate. These nonlinear partial equations of different forms are then transformed by the Galerkin method to ordinary nonlinear differential equations. The Runge–Kutta method is used to obtain the ratio of nonlinear frequency to linear frequencies of vibration. By using these equations, the nonlinear vibration of simply supported initially stress plates with various plate theories are investigated. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, Poisson ratio, initial stress and plate theory. The study concludes that the surprising discrepancies exist among the various plate theories, which indicate the transverse normal strain, nonuniform shear stress and initial stress have great effect on the vibration behavior of plate under nonlinear vibration.

DYNAMIC STABILITY CHARACTERISTICS OF FUNCTIONALLY GRADED PLATES UNDER ARBITRARY PERIODIC LOADS

The dynamic instability of functionally graded material (FGM) plates under an arbitrary periodic load is studied. The properties of the functionally graded plates (FGPs) are assumed to vary continuously across the plate thickness according to a simple power law. With the derived Mathieu equations, the dynamic instability regions of the FGPs are determined by using the Bolotins method. The in-plane periodic load is taken to be a combination of periodic axial and bending stress in the example problems. The influences of the volume fraction index, layer thickness ratio, static and dynamic load on the dynamic instability of ceramic-FGM-metal plates are discussed. The results reveal that the excitation frequency, instability region and dynamic instability index of these plates are significantly affected by the static load, dynamic load, volume fraction index and layer thickness.

The Investigation on the Vibration and Stability of Functionally Graded Plates

An analytical formulation is derived to investigate the vibration frequencies and buckling coefficients of a hybrid functionally graded plate (FGP) with an arbitrary initial stress. The governing equations are derived using the average stress method. The ceramic—functionally graded material (FGM)—metal plates are studied in examples. The properties of FGM are assumed to vary continuously from one free surface to another, according to a simple power law of the constituent volume fractions. The initial stress is a combination of a uniaxial extensional stress and a pure bending stress. The effects of volume fraction index, initial stress and other factors on the natural frequencies and buckling loads are studied.

Stability of parametric vibrations of laminated composite plates

The dynamic stability of laminated composite plates subjected to arbitrary periodic loads is studied based on the first-order shear deformation plate theory. The in-plane load is taken to be a combination of periodic biaxial and bending stress. A set of second-order ordinary differential equations with periodic coefficients of Mathieu-Hill type is formed to determine the regions of dynamic instability based on Bolotins method. Numerical results reveal that the dynamic instability is significantly affected by the modulus ratio, number of layer, static and dynamic load parameters. The effects of various important parameters on the instability region and dynamic instability index are investigated.

Investigations on the Weldline Tensile Strength of Thin-wall Injection Molded Parts

It is known that the weldline reduces the mechanical strength of the conventional injection molded parts. Systematic studies on the weldline strength of thin-wall molded parts were not yet reported. This study investigates the effect of processing conditions including melt temperature, mold temperature, injection speed and packing pressure on the weldline strength of thin-wall parts. Tensile test specimen of 2.5, 1 and 0.8 mm thick were injection molded under specified conditions. Both single gate and double gates were used to form parts with and without weldlines. Part tensile strengths were measured experimentally. From the experimental results, it was found that higher melt temperature and mold temperature as well as faster injection speed will increase weldline strength whereas higher packing pressure would decrease weldline strength. Melt temperature and mold temperature are two parameters that affect weldline strength most significantly within the current molding window. Higher melt and mold temperatures not only lower the residual stress but also help the diffusion of molecular chains leading to a higher degree of bonding at the weldline interface. On the other hand, high packing pressure leads to higher residual stress formation and reduces the molecular bonding rate. Part thickness also exhibits significant effect on weldline strength reduction. Thinner parts resulting in higher percentage of frozen layer in filling process thus limit the bonding rate at weldline interface. From the regression analysis, the KIM model with corrected terms in the fitting coefficient was found to correlate process conditions and weldline strength reduction quite well.

Imperfection sensitivity in the nonlinear vibration oscillations of initially stressed plates

Abstract In this paper, nonlinear partial differential equations for the vibrating motion of an initially stressed imperfect isotropic plate are presented. The derived equations include the effects for imperfections, transverse shear deformation, rotary inertia and non-uniform initial stresses. Using these derived governing equations, the nonlinear vibration of initially stressed plates with initial imperfection is studied. Present approach employs perturbation technique, Galerkin method and Runge–Kutta method. The perturbation technique was used to derive the nonlinear governing equations. The motion of imperfect plates is obtained by performing the Galerkin method and then solved by Runge–Kutta method. Numerical solutions are presented that relate to the performances of perfect and imperfect plates. A combination of pure bending stress plus an extensional stress is taken to be the initial stress in the plane of the plate. It is found that the existence of initial stresses and geometric imperfection may result in a drastic change on the nonlinear vibration behavior. The effects of various parameters on the nonlinear free vibrations of imperfect plates are presented.

Nonlinear Vibration of Orthotropic Plates with Initial Stresses on a Two-parameter Elastic Foundation

Nonlinear partial differential equations for vibration of an orthotropic thick plate on a two-parameter (Pasternak type) elastic foundation subjected to a nonuniform initial stress are derived. Both rotary inertia and transverse stress are considered in the derivation. The Galerkin method is used to transform the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge–Kutta method is used to obtain the ratio of nonlinear to linear frequency. Numerical examples are presented for the combination of pure bending stresses and extensional stresses are taken to be the initial stresses in the plane of the plate. These equations are then applied to a simply supported orthotropic plate on a Pasternak foundation to solve its motion of nonlinear vibration. The effects of the material properties, initial stress, amplitude of vibration, and foundation stiffness on frequency ratio are discussed.

Effects of Process Parameters on Replication Accuracy of Microinjection Molded Cyclic Olefins Copolymers Parts

In this study, the effects of various processing parameters of microinjection molding on the replication accuracy of the micro featured fluidic platform used for DNA/RNA tests are investigated. LIGA-like processes were utilized to prepare a silicon-based SU-8 photoresist, followed by electroforming to make a Ni–Co-based stamp. A cyclic olefin copolymer (COC) was used as the injection molding material. The molding parameters associated with the replication accuracy of micro channel parts were investigated. It was found that for microinjection molded devices, the replication accuracies of the imprint width and depth increase with increasing of mold temperature, melt temperature, injection velocity, and packing pressure.

Nonlinear vibration of initially stressed hybrid composite plates on elastic foundations

Nonlinear vibration equations of motion based on the Mindlin plate theory are derived for a hybrid composite plate with an initial stress on elastic foundations. Using the governing equations derived, the nonlinear vibration behavior of an initially stressed hybrid composite plate on Pasternak and Winkler elastic foundations is studied. The initial stress is taken to be a combination of a pure bending stress and a tensile stress in the plane of the plate. The Galerkin method is employed to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations, and the Runge–Kutta method is used to obtain the nonlinear frequencies. The linear frequency can be calculated by neglecting the nonlinear terms in the ordinary nonlinear differential equations. The results obtained reveal that the foundation stiffness and initial stresses lead to a drastic change in the nonlinear vibration behavior of the plate. The effects of various parameters on the nonlinear vibration of hybrid composite plates are investigated and discussed.

Effects of Locally Distributed Kelvin–Voigt Damping on Parametric Instability of Timoshenko Beams

The effect of partially distributed internal damping of the Kelvin–Voigt type on the parametric instability of a Timoshenko beam subjected to periodic axial loads is studied. To model the dynamic behavior of the beam, a coupled set of second-order linear ordinary differential equations with periodic coefficients is established by the finite element method. A quadratic eigenvalue equation is derived for a parametrically excited damped system to determine the instability regions of the beam of concern based on Bolotins method. The effects of internal damping, size and location of the damped segment, ratio of thickness to length and static load factor on the parametric instability of the beam are studied, along with the stabilizing effect of the Kelvin–Voigt damping on the primary parametric resonance presented. The results reveal that the beam with a larger damped segment positioned near the fixed end is dynamically more stable.