Abstract
Volterra series is a powerful mathematical tool for nonlinear system analysis, and there is a wide range of nonlinear engineering systems and structures that can be represented by a Volterra series model. In the present study, the random vibration of nonlinear systems is investigated using Volterra series. Analytical expressions were derived for the calculation of the output power spectral density (PSD) and input-output cross-PSD for nonlinear systems subjected to Gaussian excitation. Based on these expressions, it was revealed that both the output PSD and the input-output cross-PSD can be expressed as polynomial functions of the nonlinear characteristic parameters or the input intensity. Numerical studies were carried out to verify the theoretical analysis result and to demonstrate the effectiveness of the derived relationship. The results reached in this study are of significance to the analysis and design of the nonlinear engineering systems and structures which can be represented by a Volterra series model.
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Abbreviations
- m, c, k :
-
Mass, damping and stiffness coefficients respectively
- k 2, k 3 :
-
Quadratic and cubic nonlinear stiffness parameters respectively
- x, y :
-
System input and output respectively
- X(·), Y(·):
-
Fourier spectra of system input and output respectively
- S yy (·), S xy (·):
-
Power spectral density (PSD) and cross PSD of system output respectively
- H(·):
-
Frequency response function (FRF)
- h(t):
-
Impulse response function
- y n (t):
-
The n-th order Volterra output
- h n (τ1, τ2, …, τ n ):
-
The n-th order Volterra kernel
- E{·}:
-
Expected value operator
- H n (Ω 1, Ω 2, …, Ω n ):
-
Then-th order generalized frequency response function
- \(\mathcal{F}\left[ \cdot \right]\) :
-
Fourier transform operator
- δ(·):
-
Dirac delta function
- ⊗:
-
Kronecker product
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The project was supported by the National Science Fund for Distinguished Young Scholars (11125209), the National Natural Science Foundation of China (10902068, 51121063 and 10702039), the Shanghai Pujiang Program (10PJ1406000) and the Opening Project of State Key Laboratory of Mechanical System and Vibration (MSV201103).
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Dong, XJ., Peng, ZK., Zhang, WM. et al. Parametric characteristic of the random vibration response of nonlinear systems. Acta Mech Sin 29, 267–283 (2013). https://doi.org/10.1007/s10409-013-0019-0
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DOI: https://doi.org/10.1007/s10409-013-0019-0