Nonlinear Oscillations Analysis of the Elevator Cable in a Drum Drive Elevator System

This paper aims to investigate nonlinear oscillations of an elevator cable in a drum drive. The governing equation of motion of the objective system is developed by virtue of Lagrangian's method. A complicated term is broached in the governing equation of the motion of the system owing to existence of multiplication of a quadratic function of velocity with a sinusoidal function of displacement in the kinetic energy of the system. The obtained equation is an example of a well-known category of nonlinear oscillators, namely, non-natural systems. Due to the complex terms in the governing equation, perturbation methods cannot directly extract any closed form expressions for the natural frequency. Unavoidably, different non-perturbative approaches are employed to solve the problem and to elicit a closed-form expression for the natural frequency. Energy balance method, modified energy balance method and variational approach are utilized for frequency analyzing of the system. Frequency-amplitude relationships are analytically obtained for nonlinear vibration of the elevator's drum. In order to examine accuracy of the obtained results, exact solutions are numerically obtained and then compared with those obtained from approximate closed-form solutions for several cases. In a parametric study for different nonlinear parameters, variation of the natural frequencies against the initial amplitude is investigated. Accuracy of the three different approaches is then discussed for both small and large amplitudes of the oscillations.     

Internal-external resonance of beams on non-linear viscoelastic foundation traversed by moving load

Vibration of a finite Euler–Bernoulli beam, supported by non-linear viscoelastic foundation traversed by a moving load, is studied and the Galerkin method is used to discretize the non-linear partial differential equation of motion. Subsequently, the solution is obtained for different harmonics using the Multiple Scales Method (MSM) as one of the perturbation techniques. Free vibration of a beam on non-linear foundation is investigated and the effects of damping and non-linear stiffness of the foundation on the responses are examined. Internal-external resonance condition is then stated and the frequency responses of different harmonics are obtained by MSM. Different conditions of the external resonance are studied and a parametric study is carried out for each case. The effects of damping and non-linear stiffness of the foundation as well as the magnitude of the moving load on the frequency responses are investigated. Finally, a thorough local stability analysis is performed on the system.     

Approximate solution for acoustic scattering by a nonlinear composite beam

Nonlinear Dynamics - Dynamic behavior of a nonlinear composite beam under the action of acoustic incident waves is analyzed in this paper. Frequency responses are obtained in primary resonance...     

Existence of Periodic Solutions for the Generalized Form of Mathieu Equation

The generalized form of the well-known Mathieu differential equation, which consists of two driving force terms, including the quadratic and cubic nonlinearities, has been analyzed in this paper. The two-dimensional Lindstedt–Poincarés perturbation technique has been considered in order to obtain the analytical solutions. The transition curves in some special cases have been presented. It is shown that the periodic solution does indeed exist and in general they are dependent on the initial conditions. Results of this analytical approach were compared with those obtained from the numerical methods and it is found that they are in a good agreement.     

Application of a Bayesian algorithm for the Statistical Energy model updating of a railway coach

The classical statistical energy analysis (SEA) theory is a common approach for vibroacoustic analysis of coupled complex structures, being efficient to predict high-frequency noise and vibration of engineering systems. There are however some limitations in applying the conventional SEA. The presence of possible strong coupling between subsystems and the lack of diffuseness result in a significant uncertainty. This is the main motivation for the present study, where a procedure to update SEA models is proposed. The proposed procedure is the combination of the classical SEA method and a Bayesian technique. Due to reasons such as finding a limited number of important parameters, using a limited search range, avoiding matrix inversion and taking the effect of noise into account, the proposed strategy can be considered as a proper alternative to the experimental SEA approach. To investigate the performance of the proposed strategy, the SEA model updating of a railway passenger coach is carried out. First, a sensitivity analysis is carried out to select the most sensitive parameters of the SEA model. For the selected parameters of the model, prior probability density functions are then taken into account based on published data on comparison between experimental and theoretical results, so that the variance of the theory is estimated. The Monte Carlo Metropolis Hastings algorithm is employed to estimate the modified values of the parameters. It is shown that the algorithm can be efficiently used to update the SEA models with a high number of unknown parameters.     

Primary and secondary resonance analyses of clamped–clamped micro-beams

Nonlinear harmonic vibration of a micro-electro-mechanical beam is investigated, and the micro-actuator, which is considered in this study, is a special kind of electrostatic symmetric actuators. A fully clamped micro-beam with a uniform thickness is modeled as an electrostatic micro-actuator with two symmetric potential walls. The nonlinear forced vibration of the micro-beam is analyzed, and the non-dimensional governing equation of motion, using the Galerkin method, is developed. Higher-order nonlinear terms in the equation of motion are taken into account for the first time, and the perturbation method is utilized regarding these terms and hence, all the resonant cases have been considered. The multiple scales method is employed to solve the nonlinear equations, and therefore, the problem does not deal with the large deformations. The primary and secondary resonance conditions are determined, and the corresponding secular terms in each case are recognized. Harmonic responses are obtained for different cases of resonance, and eventually, the stable and unstable portions of the responses are identified. A parametric sensitivity study is carried out to examine the effects of different parameters on the amplitude–frequency characteristic equations.     

Parametrically Excited Vibration of a Timoshenko Beam on Random Viscoelastic Foundation jected to a Harmonic Moving Load

The vibration response of a Timoshenko beam supported by a viscoelastic foundation with randomly distributed parameters along the beam length and jected to a harmonic moving load, is studied. By means of the first-order two-dimensional regular perturbation method and employing appropriate Green's functions, the dynamic response of the beam consisting of the mean and variance of the deflection and of the bending moment are obtained analytically in integral forms. Results of a field measurement for a test track are utilized to model the uncertainty of the foundation parameters. A frequency analysis is carried out and the effect of the load speed on the response is studied. It is found that the covariance functions of the stiffness and the loss factor both have the shape of an exponential function multiplied by a cosine function. Furthermore, it is shown that in each frequency response there is a peak value for the frequency, which changes inversely with the load speed. It is also found that the peak value of the mean and also standard deviation of the deflection and bending moment can be a decreasing or increasing function of the load speed depending on its frequency. An erratum to this article is available at .     

Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads

The vibration of an Euler-Bernoulli beam, resting on a nonlinear Kelvin-Voight viscoelastic foundation, traversed by a moving load is studied in the frequency domain. The objective is to obtain the frequency responses of the beam and the effects of different parameters on the system response. The parameters include the magnitude and speed of the moving load and the foundation nonlinearity and its damping coefficient. The solution is obtained by using the Galerkin method in conjunction with the multiple scales method (MSM). The governing nonlinear partial differential equations of motion are discretized into sets of nonlinear ordinary differential equations. Subsequently, the solution is calculated for different harmonics by using the MSM as one of the powerful perturbation techniques. The steady-state responses of the main harmonic as well as its two super-harmonics are then obtained. As a case study, a conventional railway track is dynamically simulated and the jump phenomenon in the response is observed for three harmonics. Moreover, a thorough stability analysis of the system is carried out.