Multiple-scale analysis

About: Multiple-scale analysis is a research topic. Over the lifetime, 1360 publications have been published within this topic receiving 27530 citations.

Nonlinear vibrations of a beam-spring-mass system

Abstract: The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.

On Direct Methods for Constructing Nonlinear Normal Modes of Continuous Systems

Abstract: A direct method based on the method of normal forms is proposed for constructing the nonlinear normal modes of continuous systems. The proposed method is compared with the method of multiple scales and the methods of Shaw and Pierre and King and Vakakis by applying them to three conservative systems with cubic nonlinearities: (a) a hinged-hinged beam resting on a nonlinear elastic foundation, (b) a model of a relief valve (linear elastic spring attached to a nonlinear spring with a mass), and (c) a simply supported linear beam with nonlinear torsional springs at both ends. In the absence of internal resonance, the constructed nonlinear modes with all four methods are the same. The method of multiple scales seems to be the simplest and the least computationally demanding. The methods of multiple scales and normal forms are applicable to problems with and without internal resonances, whereas the present forms of the methods of Shaw and Pierre and King and Vakakis are not applicable to problems with internal...

Identification of cubic nonlinearity in disbonded aluminum honeycomb panels using single degree-of-freedom models

Abstract: Prior work on a disbonded aluminum honeycomb panel showed evidence of a quadratic stiffness nonlinearity, as well as the presence of an unknown cubic nonlinearity. Approximations to higher order nonlinear single degree of freedom (SDOF) models were solved using the method of multiple scales. These approximations were then used to fit displacement data from a sinusoidal excitation test and determine the coefficients of the model as a function of damage size. Confirmation of the quadratic stiffness nonlinearity was achieved through examination of force restoration curves excited at one-half the primary resonance in conjunction with coefficient fitting of the test data to the model. The data were fit against the higher order models to determine whether the cubic nonlinearity could be stiffness or damping related. The coefficient fitting shows that the cubic nonlinearity is a stiffness nonlinearity. This confirmed what was seen in the force restoration curves when the system was excited at one-third the primary resonance. The ability to match the vibratory behavior of the damage to a SDOF model shows that the use of single frequency excitation at lower frequencies can isolate the nonlinear behavior of the damaged area and identify what damage mechanisms may be involved.