Papers by Keyword: Method of Multiple Scales

Abstract: In this study, linear vibrations of axially moving beam simply supported between the guides are examined and natural frequencies are calculated numerically. The vibrations of axially tensioned Euler-Bernoulli beam are investigated under clamped-clamped end conditions. Governing differential equations of motion are derived using Hamilton’s Principle for two regions of the beam. The boundaries at the outer ends of the beam are assumed immovable. Non-dimensional equations of motion are derived and the solutions of the linear problem are obtained. Assuming a weakly non-linear system, linear equations are obtained using the Method of Multiple Scales. First seven natural frequencies are calculated numerically based on the flexural rigidity, axial velocity and locations of the intermediate support.

Abstract: Honeycomb sandwich plate have been widly applied in industry design in recent years. In this paper, we study the cubic hypernormal form (the simplest normal form and the unique normal form) for honeycomb sandwich plate dynamics model with the help of Maple symbolic computation. Firstly, we get the average equation of four dimensional cartesian form by using the method of multiple scales perturbation analysis. Based on the method combined new grading function with multiple Lie brackets, we obtain the hypernormal form of cubic truncated. The results will further enrich the research for dynamics of honeycomb sandwich plate model, and is also the basis for higher order normal form research.

Abstract: Based on the electro-magneto-elastic theory and the theory of the bending vibration of the electric beam, nonlinear vibration equation of current-carrying beam in thermal-magneto-elasticity field is studied. The Lorentz force and thermal force on the beam are derived. According to the method of multiple scales for nonlinear vibrations the approximation solution of 1/3 subharmonic resonance of the system is obtained. Numerical analysis results show that the amplitude changed with the system parameters.

Abstract: Transverse free dynamics of a beam-like nanostructure with axial load is investigated. The effects of a small size at nano-scale unavailable in classical mechanics are presented. Explicit solutions for natural frequency, vibration mode and transverse displacement are obtained by separation of variables and multiple scales analysis. Results by two methods are in close agreement.

Abstract: An analysis of linear and nonlinear dynamics of rotors is presented taking into account the shear effects. The nonlinearity arises due to the consideration of large deformations in bending. The rotor system studied is composed of a rigid disk and a circular shaft. In order to study the combined effect of rotary inertia and shear effects the shaft is modeled as a Timoshenko beam of circular cross section. A mathematical model is developed consisting of 4th order coupled nonlinear differential equations of motion. Method of multiple scales is used to solve these nonlinear equations. Linear and nonlinear dynamic behavior is studied numerically for different values of slenderness ratio r. Resonant curves are plotted for the nonlinear analysis. Due to nonlinearity these curves are of hard spring type. This spring hardening effect is more visible for lower values of r. Also the nonlinear response amplitude is higher when shear deformations are taken into account.

Abstract: Based on the Maxwell equation and Kirchhoff assumption of thin plate, nonlinear magneto-elastic vibration equation, electrodynamics equation and electromagnetic force expressions of current-conducting thin plate were deduced. Furthermore, nonlinear super-harmonic resonance of thin beam-plate under lateral mechanical motive load in longitudinal magnetic field was studied. Considering the thin plate simply supported on two opposite sides, the magneto-elastic coupled vibration differential equations about function of displacement of vibration and electric field intensity were obtained by the method of Galerkin. Then, the amplitude-frequency response equation under super-harmonic resonance was derived by using method of Multiple scales. Correspondingly the stability of stable solution was analyzed. Through the numerical calculation, characteristic curves of amplitude changing with detuning parameter, the excitation amplitude and the magnetic intensity. At last, the influence of electric-magnetic and mechanic parameter on resonance phenomenon and stability of solution was analyzed.

Abstract: On base of the electro-magneto-elastic theory and the theory of the bending vibration of the electric beam, nonlinear vibration equation of current-carrying beam subjected to thermal-magneto-elasticity field is studied. The Lorentz force and thermal force on the beam are derived. According to the method of multiple scales for nonlinear vibrations the approximation solution of the primary resonance of the system is obtained. Numerical analysis results show that the amplitude changed with the system parameters. With the decrease of magnetic intensity, the amplitude increases rapidly. The response curve occurs bending phenomenon and soft features is increased gradually. Increasing current, the amplitudes increase. With the decrease of temperature, the peak of response curves decrease. With the increase of temperature, natural frequency decreased. It is useful in practical engineering.

Abstract: An analytical technique, namely the method of multiple scales, is applied to solve the differential equations of free oscillations with even nonlinearities in a mass-spring system. Unlike other perturbation methods, the method of multiple scales is effective in determining the transient response as well as determining the approximation to the frequency of the nonlinear system. In this paper, the periodic solutions of the even nonlinear differential equations have been obtained by using the method of multiple scales. Compared with the numerical examples, the approximate solutions are in good agreement with exact solutions. The numerical and analytical solutions have clearly shown that there exists the so-called drift phenomenon in the free oscillations of systems with even nonlinearities without any external excitation.