MATLAB Implementation of the Pseudo Excitation Method for Random Vibration Analysis

The pseudo excitation method is an efficient frequency domain analysis technique particularly suitable for solving random vibration problems. This method transforms complex random vibration analysis into deterministic response calculations by constructing pseudo excitations, significantly improving computational efficiency. For beginners, mastering this approach provides deep understanding of random vibration fundamentals and enables quick application to practical engineering problems.

For 4-degree-of-freedom (4-DOF) random vibration problems, implementation typically involves: First, establishing the system's equations of motion, usually represented in mass, damping, and stiffness matrix forms (coded as M, C, K matrices in MATLAB). Next, transforming time-domain problems to frequency domain via Fourier transform to construct pseudo excitations. The pseudo excitation is a complex harmonic excitation whose amplitude spectral density matches the actual random excitation, implemented using MATLAB's fft function or analytical spectral formulations.

Subsequently, calculating the system's frequency response function (FRH) to the pseudo excitation involves solving complex frequency responses. For 4-DOF systems, the FRH is a 4x4 matrix where each element represents transfer characteristics between different degrees of freedom, computed using MATLAB's matrix inversion operations like inv(-omega^2*M + 1i*omega*C + K). The virtual response is obtained by multiplying the FRH matrix with the pseudo excitation vector using MATLAB's matrix multiplication operator (*).

Finally, auto-power spectral density and cross-power spectral density of the virtual response are obtained through simple complex number operations (using conj() for complex conjugation and element-wise multiplication .* in MATLAB), extracting the system's random vibration response characteristics. This approach avoids extensive numerical integration required in traditional time-domain analysis, significantly improving computational efficiency.

For beginners, key understanding involves mastering frequency domain analysis concepts and the methodology of transforming random problems into deterministic ones. Additionally, familiarity with MATLAB's matrix operations and complex number handling greatly simplifies code implementation. Through 4-DOF system practice, the method can be progressively extended to more complex multi-degree-of-freedom systems or continuum systems.