Formulation of Statistical Linearization for M-D-O-F Systems Subject to Combined Periodic and Stochastic Excitations

Abstract

A formulation of statistical linearization for multi-degree-of-freedom (M-D-O-F) systems subject to combined mono-frequency periodic and stochastic excitations is presented. The proposed technique is based on coupling the statistical linearization and the harmonic balance concepts. The steady-state system response is expressed as the sum of a periodic (deterministic) component and of a zero-mean stochastic component. Next, the equation of motion leads to a nonlinear vector stochastic ordinary differential equation (ODE) for the zero-mean component of the response. The nonlinear term contains both the zero-mean component and the periodic component, and they are further equivalent to linear elements. Furthermore, due to the presence of the periodic component, these linear elements are approximated by averaging over one period of the excitation. This procedure leads to an equivalent system whose elements depend both on the statistical moments of the zero-mean stochastic component and on the amplitudes of the periodic component of the response. Next, input–output random vibration analysis leads to a set of nonlinear equations involving the preceded amplitudes and statistical moments. This set of equations is supplemented by another set of equations derived by ensuring, in a harmonic balance sense, that the equation of motion of the M-D-O-F system is satisfied after ensemble averaging. Numerical examples of a 2-D-O-F nonlinear system are considered to demonstrate the reliability of the proposed technique by juxtaposing the semi-analytical results with pertinent Monte Carlo simulation data.

A time and ensemble equivalent linearization method for nonlinear systems under combined harmonic and random excitation

“…Firstly, a single degree of freedom Duffing oscillator is assessed, where the method is shown to be capable of reproducing the results of Monte Carlo Time Integration with several orders of magnitude reduction in computational cost. It is also demonstrated that performance is comparable to an alternative analysis approach recently proposed by Spanos et al 11,39 Secondly, a single degree of freedom system with end-stops providing a discontinuous nonlinearity is examined, with the proposed approach again shown to compare well with Monte Carlo time integration. Finally, the method is applied to a multi degree of freedom cantilever beam with a localized smooth nonlinearity at the tip of the beam, for which experimental results were obtained from a laboratory test rig.…”

“…Therefore, while the ''tangent matrix'' simplification may be useful algebraically, for the linearization performed here it does not describe the complete solution as it does for zero mean stationary problems. Finally, it is also necessary to point out how the derivation presented above differs from that proposed by Spanos et al, 11,39 which for simplicity is termed the ''Spanos Method'' in this paper. The linearization presented in equation ( 2) makes no distinction between the random and deterministic response components; this is only introduced in equation ( 14) after the linearization matrices are derived.…”

Stochastic Nonlinear Response of Structural Systems Endowed With Singular Matrices Subject to Combined Periodic and Stochastic Excitations

“…x 1 = 0.0704 and σ 2 x 3 = 0.0617, σ 2 x 3 = 0.1098. Finally, applying the standard solution framework proposed in [15] for deriving the system response variance yields σ 2 q 1 = 0.0705, σ 2 q 1 = 0.0704 and σ 2 q 2 = 0.0617, σ 2 q 2 = 0.1098. Clearly, comparing the results above, it is seen that the herein proposed framework is in total agreement with the standard approach in [15].…”

“…Combining the two methods results in the formulation of a coupled system of algebraic equations, which is solved iteratively by invoking the generalized matrix inverse theory [14]. Overall, the proposed technique consists a generalization of the framework developed in [15] to account for systems with singular matrices. Its validity is demonstrated by pertinent numerical examples.…”

Non-Linear Response of Cable-Mass-Spring System in High-Rise Buildings under Stochastic Seismic Excitation

“…It has also been combined with other advanced methods and techniques to conduct analysis in various domains of stochastic dynamics and more. Some examples can be found, for example, in [23][24][25][26][27][28].…”