Restoring Force Function Research Articles

Traveling wave solutions to beam equation with fast-increasing nonlinear restoring forces

On studying traveling waves on a nonlinearly suspended bridge, the following partial differential equation has been considered: uu+uxxxx+f(u)=0, where f(u)=u+−1. Here the bridge is seen as a vibrating beam supported by cables, which are treated as spring with a one-sided restoring force. The existence of a traveling wave solution to the above piece-wise linear equation has been proved by solving the equation explitly (McKenna & Walter in 1990). Recently the result has been extended to a group of equations with more general nonlinearities such as f(u)=u+−1+g(u) (Chen & McKenna, 1997). However, the restrictions on g(u) do not allow the resulting restoring force function to increase faster than the linear function u−1 for u>1. Since an interesting “multiton” behavior, that is, two traveling waves appear to emerge intact after interacting nonlinearly with each other, has been observed in numerical experiments for a fast-increasing nonlinearity f(u)=e−1−1, it hints that the conclusion of the existence of a traveling wave solution with fast-increasing nonlinearities shall be valid as well.

Energy and the nonsymmetric nonlinear spring

We give aderivation of the nonlinear spring equation mx - f(x) = 0 where f(x) represents the restoring force and m is the mass of a weight attached to the spring. This derivation shows that even powers may appear in the Maclaurin expansion of the restoring force function. We demonstrate how the ‘energy approach’ facilitates the investigation of the resultant motions and makes phase plane analysis easy by reducing the characterization of critical points to determining the sign of asecond-order partial derivative of the energy function. This circumvents the sometimes difficult eigenvalue determination of the linearization procedure commonly taught. This approach is no more difficult than the approach usually takes in textbooks for much simpler equations and it leads to interesting harmonic oscillator examples and problems suitable for undergraduate research.

Final Solution of Duffing Equation of Mixed Parity

For nonlinear free vibration of laminated anisotropic plates, the restoring force function in the equation of motion is found to be a cubic polynomial, which is in the form of a Duffing type equation or a combination of quadratic and cubic terms. For nonodd restoring force function, the behavior of oscillations is different for positive and negative amplitudes. Hence, it is necessary and useful for all practical purposes to examine the uniqueness of angular frequency from the equation of motion and its energy relation, using the harmonic balance with lower order harmonics

Nonlinear vibrations of cross-ply plates

Large amplitude free vibrations of simply supported rectangular cross-ply plates are examined. The restoring force function in the equation of motion is a combination of quadratic and cubic terms. The equation of motion is integrated using a Runge-Kutta numerical scheme of the MATLAB software. The results obtained are in good agreement with the existing results.

Analysis of nonlinear oscillators under stochastic excitation by the Fokker-Planck-Kolmogorov equation

The paper deals with the analysis of stochastic mechanical systems with one degree of freedom and proposes a simple procedure to obtain a representation of the dynamical response. In particular, approximate solution of the FPK equation is obtained for a system subjected to a stochastic force term. The resolving procedure is implemented with reference to a polynomial expansion of the restoring force function. Numerical tests are performed with reference to Duffing and van der Pol oscillators, showing good agreement with simulated response.

Reinvestigation Of Non-linear Vibrations Of Simply Supported Rectangular Cross-ply Plates

Large amplitude free vibrations of simply supported rectangular cross-ply plates are examined here. The restoring force function in the equation of motion is found to be a cubic polynomial, giving a Duffing type equation, or a combination of quadratic and cubic terms. The exact solution of the equation of motion is presented in a form that can be evaluated numerically to any desired degree of accuracy. The results obtained for isotropic as well as six-layered antisymmtric cross-ply rectangular plates are found to be in good agreement with the existing results.

NONLINEAR WAVE PROPAGATION ANALYSIS OF LIQUEFYING MULTI-LAYERED SURFACE GROUND BY USING REAL RESTORING FORCE

This paper describes an analytical theory and method to compute the nonlinear surface ground motion by using a real restoring force. A surface ground motion is mathematically expressed in a wave propagation equation or vibration equation. A restoring force function of soil is given in mathematical model. The most important point of the analytical method suggested in this paper is to use real restoring force for a shearing stress necessary for calculation. The authors don't use a mathematical model but a physical model to represent a restoring force characteristics of soil. The analytical apparatus is a micro-computer connected a dynamic triaxial soil testing machine. We have got the nonlinear response characteristics of surface ground motion in liquefaction process.