R. N. Iyengar

Study of the Random Vibration of Nonlinear Systems by the Gaussian Closure Technique

A technique is developed to study random vibration of nonlinear systems. The method is based on the assumption that the joint probability density function of the response variables and input variables is Gaussian. It is shown that this method is more general than the statistical linearization technique in that it can handle non-Gaussian excitations and amplitude-limited responses. As an example a bilinear hysteretic system under white noise excitation is analyzed. The prediction of various response statistics by this technique is in good agreement with other available results.

Internal resonance and non-linear response of a cable under periodic excitation

The coupled non-linear equations of motion of a sagged cable in its first symmetric mode of in-plane and out-of-plane oscillations are solved by the method of multiple scales for its forced vibration response. The cases of both internal and external resonances are considered. A uniform lateral load is assumed to act along with an in-plane harmonic component, similar to a situation under vortex-induced oscillation. The quadratic nonlinearity terms in the equations of motion are shown to affect the cable behaviour significantly when the in-plane frequency is about twice the out-of-plane frequency of oscillation. A stability analysis is performed on the steady state solutions. The effect of cable sag on these solutionns and their stability is studied indirectly through the internal resonance parameter. The influence of the lateral load component, with and without internal resonance, on the stability regions is discussed.

Stochastic response and stability of the Duffing oscillator under narrowband excitation

The Duffing equation under narrowband Gaussian excitation is studied. Almost sure asymptotic stability analysis is used to discuss the realizability of multivalued steady states. The theoretical predictions are supported by numerical simulation results.

Dynamic response of a beam on elastic foundation of finite depth under a moving force

SummaryIn this paper, dynamic response of an infinitely long beam resting on a foundation of finite depth, under a moving force is studied. The effect of foundation inertia is included in the analysis by modelling the foundation as a series of closely spaced axially vibrating rods of finite depth, fixed at the bottom and connected to the beam at the top. Viscous damping in the beam and foundation is included in the analysis. Steady state response of the beam-foundation system is obtained. Detailed numerical results are presented to study the effect of various parameters such as foundation mass, velocity of the moving load, damping and axial force on the beam. It is shown that foundation inertia can considerably reduce the critical velocity and can also amplify the beam response.

Response of nonlinear systems to narrow-band excitation

The hardening cubic spring oscillator is studied under narrow-band gaussian excitation. Equivalent linearization leads to multiple steady states. The realizability of the solution is discussed through stochastic stability analysis. Theoretical results are supported by numerical simulation.

A new model for non-Gaussian random excitations

Abstract Very often one is called upon to model time series data which are clearly non-Gaussian, but which retain some aspects of a Gaussian process. In the present paper, a novel methodology which helps in modelling such data is presented. The method is essentially to express the process as a series with finite number of terms, wherein the first term is a Gaussian process with zero mean and unit standard deviation. Non-Gaussian higher order correction terms are added to this such that each succeeding term is orthogonal or uncorrelated with all the previous terms. The unknown coefficients in the series representation can be expressed in terms of the estimated moments of the data. Further the autocorrelation or PSD of the data can be exactly reproduced by the non-Gaussian model. The use of the proposed model is illustrated by considering the unevenness data of railway tracks. Application to response of systems under non-Gaussian excitation is also briefly discussed.

Rocking response of rectangular rigid blocks under random noise base excitations

The response of a rigid rectangular block resting on a rigid foundation and acted upon simultaneously by a horizontal and a vertical random white-noise excitation is considered. In the equation of motion, the energy dissipation is modeled through a viscous damping term. Under the assumption that the body does not topple, the steady-state joint probability density function of the rotation and the rotational velocity is obtained using the Fokker-Planck equation approach. Closed form solution is obtained for a specific combination of system parameters. A more general but approximate solution to the joint probability density function based on the method of equivalent non-linearization is also presented. Further, the problem of overturning of the block is approached in the framework of the diffusion methods for first passage failure studies. The overturning of the block is deemed incipient when the response trajectories in the phase plane cross the separatrix of the conservative unforced system. Expressions for the moments of first passage time are obtained via a series solution to the governing generalized Pontriagin-Vitt equations. Numerical results illustra- tive of the theoretical solutions are presented and their validity is examined through limited amount of digital simulations.

Free vibrations and parametric instability of a laterally loaded cable

In this paper a linear analysis is presented for the coupled vertical, longitudinal and lateral oscillations of a cable with a small sag-to-span ratio. Generally, the lateral oscillations are treated as uncoupled from the motion in the other directions. In the present study it is shown that existence of initial lateral displacement can induce coupling of all the oscillations in the three directions. The effect of a periodic component in the lateral load on the dynamic stability of the cable is also investigated.

A Nonlinear System Under Combined Periodic and Random Excitation

The anharmonic oscillator under combined sinusoidal and white noise excitation is studied using the Gaussian closure approximation. The mean response and the steady-state variance of the system is obtained by the WKBJ approximation and also by the Fokker-Planck equation. The multiple steadystate solutions are obtained and their stability analysis is presented. Numerical results are obtained for a particular set of system parameters. The theoretical results are compared with a digital simulation study to bring out the usefulness of the present approximate theory.

Entrainment in Van der Pol's oscillator in the presence of noise

The response of Van der Pal’s oscillator to a combination of harmonic and white noise excitations is considered. The harmonic excitation frequency is taken to be in the neighbourhood of the system limit cycle frequency. The effect of addition of noise on the entrainment behaviour is investigated using a combination of methods of stochastic averaging and equivalent nonlinearixation. Results based on the gaussian closure technique are also obtained and the theoretical solutions are compared with digital simulations.