Numerical Methods in Structural Dynamics

Numerical computation methods serve as essential tools for investigating dynamic responses in structural dynamics analysis. This article introduces several common numerical integration techniques and their implementation approaches in MATLAB, enabling engineers to efficiently perform dynamic simulations.

The Houbolt method is an implicit integration scheme particularly suitable for systems with high stiffness. Its core algorithm approximates acceleration using current and previous displacement values to establish recursive formulas. While computationally intensive, this method offers excellent numerical stability and is ideal for long-term dynamic response analysis. In MATLAB implementation, this typically involves solving matrix equations at each time step using backslash operators or iterative solvers.

Duhamel integration operates based on the superposition principle of linear systems, computing responses through convolution operations. The methodology decomposes excitation signals into impulse sequences, calculates individual impulse responses, and superimposes them. This approach excels in frequency-domain analysis and applies to linear systems under arbitrary excitation forms. MATLAB implementation often utilizes convolution functions or Fourier transform techniques for efficient computation.

The Wilson-θ method enhances the Newmark-β approach by introducing parameter θ (typically 1.4) to achieve unconditional stability. This technique employs linear acceleration assumptions within each time step and solves equilibrium equations iteratively, making it particularly effective for nonlinear dynamic problems. Implementation in MATLAB requires careful handling of iteration loops and convergence checks during the solving process.

For MATLAB implementation, developers can create unified function interfaces with algorithm selection through input parameters. Key implementation steps include: establishing system mass/damping/stiffness matrices, setting time increments, initializing response arrays, and performing loop calculations according to each method's recursive formulas. Post-processing modules should incorporate response history plotting, spectral analysis, and other diagnostic functionalities using MATLAB's visualization and signal processing tools.

Each method demonstrates unique advantages: Houbolt method suits long-term simulations, Duhamel integration excels in frequency-domain analysis, while Wilson-θ performs best for nonlinear problems. Practical applications require selecting appropriate methods based on system characteristics and computational requirements, with validation through comparative analysis of algorithm accuracy and efficiency.