Finite amplitude oscillations of a hyperelastic spherical cavity

We present numerical results for the finite oscillations of a hyperelastic spherical cavity by employing the governing equations for finite amplitude oscillations of hyperelastic spherical shells and simplifying it for a spherical cavity in an infinite medium and then applying a fourth-order Runge-Kutta numerical technique to the resulting non-linear first-order differential equation.The results are plotted for Mooney-Rivlin type materials for free and forced oscillations under Heaviside type step loading. The results for Neo-Hookean materials are also discussed. Dependence of the amplitudes and frequencies of oscillations on different parameters of the problem is also discussed in length.     

Large amplitude non-linear oscillations of a general conservative system

This paper presents new, approximate analytical solutions to large-amplitude oscillations of a general, inclusive of odd and non-odd non-linearity, conservative single-degree-of-freedom system. Based on the original general non-linear oscillating system, two new systems with odd non-linearity are to be addressed. Building on the approximate analytical solutions of odd non-linear systems developed by the authors earlier, we construct the new approximate analytical solutions to the original general non-linear system by combinatory piecing of the approximate solutions corresponding to, respectively, the two new systems introduced. These approximate solutions are valid for small as well as large amplitudes of oscillation for which the perturbation method either provides inaccurate solutions or is inapplicable. Two examples with excellent approximate analytical solutions are presented to illustrate the great accuracy and simplicity of the new formulation.     

Coupled shear-torsional motion of a rubber support system

The finite amplitude, coupled shear-torsional motion of a circular disk supported between identical rubber spring cylinders is studied. The material of the springs is assumed to be an incompressible elastic material. The oscillatory motion oscillatory of the disk is studied for two different cases. In the first case, the material of the spring is assumed to be an incompressible elastic material whose response functions are constants. Typical examples include the Mooney-Rivlin model. The motion of the disk in this case is governed by two independent equations whose closed form solutions are noted. For the second case, the material of the spring is assumed to be an incompressible quadratic material. The motion oscillatory of the disk in this case is governed by two coupled nonlinear differential equations. The stability properties of small shearing oscillation superimposed on finite torsion and small torsional oscillation superimposed on finite shearing are studied.     

Large amplitude oscillations of a hollow spherical dielectric

Based on the Truesdell theorem on quasi-equilibrated motions and an electroelastostatics theory of elastic dielectrics due to Toupin and Eringen, the problem of large amplitude radial oscillations of hollow spherical dielectrics is analysed, assuming a semi-general strain energy density function for the hyperelastic, incompressible dielectric material under consideration. Criteria of existence of oscillatory motions are established. The expressions for the periods of free oscillations initiated under large initial deformation as well as forced oscillations due to a sudden Heaviside impact are given in integral form.     

Finite amplitude vibrations of a Mooney-Rivlin oscillator

The solution for the finite amplitude, uniaxial motion of a Mooney-Rivlin oscillator on suspension, compression, or horizontal supports, and for arbitrary initial data is presented. The problem is unusual. Depending upon the initial data, the type of support, the amount of static stretch, and the value of a Mooney-Rivlin parameter, the solution may have one of three distinct possible periodic forms. The three cases are solved exactly, and simple bounds on the period of the finite motion are given. Some special situations are illustrated, both analytically and graphically. The effect of the amount of static stretch on the period of superimposed small amplitude oscillations also is described. The appropriate results are compared with those obtained previously for the neo-Hookean model.Dedicated to Clifford Truesdell, in admiration and gratitude, on the occasion of the twenty-fifth anniversary of the Society for Natural Philosophy.     

Finite amplitude vibrations of a circular plate

The problem of finite-amplitude, axisymmetric free and forced vibration of a circular plate is examined with various boundary conditions. The non-linear boundary-value problem is converted into the corresponding eigenvalue problem by elimination of the time variable. Then by a Newton-Raphson iteration scheme, and the concept of analytical continuation, the solution to the non-linear eigenvalue problem for the vibrations is obtained in a discrete form. It is seen that the removal of radial restraint causes drastic changes in the plate responses and the patterns of membrane stresses. Comparison with solutions based on the Berger assumption reveals the unsuitability of the assumption when the plate is not radially restrained.     

On the large amplitude oscillations of a thin elastic beam

This paper is concerned with the finite amplitude, free, planar oscillations of a thin elastic beam. By assuming the motion to be inextensional but at the same time recognizing the existence of a resultant normal force acting on each cross-section of the beam a system of governing equations is derived which is manageable but still meaningful. For the case of the simply-supported beam a finite difference, Galerkin, and (regular) perturbation solutions are explicitly obtained. The results are compared and discussed. In the course of obtaining the various solutions it is found that an additional simplification in the form of the governing equations is possible. This simplification turns out to be quite important from a general point of view of obtaining approximate analytical solutions.     

Large amplitude oscillations of thick hyperelastic cylindrical shells

We present numerical solutions to the problem of large amplitude oscillations of a thick-walled hyperelastic cylindrical shell employing the general theory of finite dynamic deformations of elastic bodies. The material of the shell is considered incompressible and of Mooney-Rivlin type rubbers.

We apply a fourth-order Runge-Kutta numerical technique to the governing equation which was originally derived by J.K. Knowles in 1960.

We consider the free as well as forced oscillations due to a Heaviside step load and display graphs for the variations of amplitude against time and frequencies for different thicknesses and material constants. Discussions are presented on the significances of the results obtained.     

Finite amplitude one-dimensional wave propagation in a thermoelastic half-space

The problem of finite wave propagation in a nonlinearly thermoelastic half-space is considered. The surface of the half-space is subjected to a time-dependent thermal and normal mechanical loading. The solution is obtained by a numerical procedure, which is shown to furnish accurate results, and linear dynamic thermoelastic problems are obtained as special cases. The accuracy of the results is checked by comparison with some known analytical solutions which can be obtained in some special cases of both the linear and the nonlinear problems. In those cases where the solution contains shocks, it is shown that the numerical results satisfy the necessary jumps conditions which need to hold across such discontinuities.     

Modelling of ER squeeze films at low amplitude oscillations

An experimental investigation of electrorheological squeeze film dynamics is presented for constant applied voltage and low strain amplitude. Both broadband random and sinusoidal motion are examined to explain complex film dynamics. Spectral results indicate a primarily elastic response with slip at the plate boundaries. By examining the evolution of an effective shear modulus over time, sinusoidal results show that slip at the boundaries is due to a solvent layer which may be modelled as a separate variable thickness squeeze film.     

Large amplitude radial oscillations of layered thick-walled cylindrical shells

Finite breathing motions of multi-layered, long, circular cylindrical shells of arbitrary wall thickness are investigated on the basis of the theory of large elastic deformations. The materials of the layers are assumed to be isotropic, elastic, homogeneous and incompressible. The governing non-linear ordinary differential equation is solved partially to give the frequencies of oscillations in an integral form. An approximate solution technique based on Galerkin's orthogonalization process is also formulated to give complete solutions. A tube consisting of two layers of neo-Hookean materials is solved both by exact and approximate methods. An excellent agreement is observed between the two sets of results.     

A damped simple pendulum of constant amplitude

A simple pendulum acted on by gravity and subjected to a resistance proportional to the velocity of the bob is considered. If the length of the string and the mass of the bob are held constant, the amplitude of the bob decreases gradually because of the damping. We want to keep the maximum swing of the bob constant for all time; this we achieve by varying the length of the string, the mass of the bob or both. The key to the solution of our problem is a second-order nonlinear differential equation having arbitrary nonlinearity and an arbitrary coefficient function, for which we give the exact integral. We also give an application of this differential equation to a boundary-value problem for a nonlinear generalization of a hypergeometric equation.     

Drop oscillations under simple shear in a highly viscoelastic matrix

Recent two-dimensional numerical simulations and experiments have shown that, when a drop undergoes shear in a viscoelastic matrix liquid, the deformation can undergo an overshoot. I implement a volume-of-fluid algorithm with a paraboloid reconstruction of the interface for the calculation of the surface tension force for three-dimensional direct numerical simulations for a Newtonian drop in an Oldroyd-B liquid near criticalities. Weissenberg numbers up to 1 at viscosity ratio 1 and retardation parameter 0.5 are examined. Critical capillary numbers rise with the Weissenberg number. Just below criticality, drop deformation begins to undergo an overshoot when the Weissenberg number is sufficiently high. The overshoot becomes more pronounced, and at higher matrix Weissenberg numbers, such as 0.8, drop deformation undergoes novel oscillations before settling to a stationary shape. Breakup simulations are also described.     

Finite amplitude spherically symmetric wave propagation in a prestressed hyperelastic shell

Spherically symmetric finite amplitude wave propagation in a prestressed compressible hyperelastic spherical shell is considered. The prestress results from quasi-static application of internal pressure and a numerical solution for this elastostatic problem is obtained first. Dynamic change of the internal pressure results in the propagation of a spherically symmetric wave. A Godunov type finite difference scheme is proposed for the solution of the wave propagation problem and numerical results, which are valid until the first reflection, are presented for a particular isotropic strain energy function and for the special cases of sudden removal and sudden increase of the internal pressure.     

On forced oscillations of a simple model for a novel wave energy converter

The dynamics of a simple model for an ocean wave energy converter is discussed. The model for the converter is a hybrid system consisting of a pair of harmonically excited mass–spring–dashpot systems and a set of four state-dependent switching rules. Of particular interest is the response of the model to a wide spectrum of harmonic excitations. Partially because of the piecewise-smooth dynamics of the system, the response is far more interesting than the linear components of the model would suggest. As expected with hybrid systems of this type, it is difficult to establish analytical results, and hence, with the assistance of an extensive series of numerical integrations, an atlas of qualitative results on the limit cycles and other forms of bounded oscillations exhibited by the system is presented. In addition, the presence of unstable limit cycles, the stabilization of the unforced system using low-frequency excitation, the peculiar nature of the response of the system to high-frequency excitation, and the implications of these results on the energy harvesting capabilities of the wave energy converter are discussed.