Newmark-Beta Method for Structural Dynamic Response Analysis

The MATLAB implementation process for solving structural dynamic responses using the Newmark-Beta method involves the following key steps: First, initialize structural parameters including mass matrix, stiffness matrix, and damping coefficients through input functions or direct matrix assignment. Second, select appropriate integration parameters based on computation time steps and natural frequencies, typically employing the linear acceleration integration scheme (with default β=1/6, γ=1/2) to balance numerical accuracy and computational efficiency. The core algorithm implements the Newmark time integration formulas: displacement update using x(t+Δt)=x(t)+Δt·v(t)+Δt²[(0.5-β)a(t)+βa(t+Δt)] and velocity update v(t+Δt)=v(t)+Δt[(1-γ)a(t)+γa(t+Δt)]. Third, develop and debug the numerical solution procedure incorporating effective stiffness matrix formulation K_eff=K+γ/(βΔt)C+1/(βΔt²)M and iterative solution of acceleration vectors. The implementation should include convergence checks and stability verification through eigenvalue analysis. Finally, perform result validation by comparing with analytical solutions or benchmark problems to ensure the structural dynamic responses meet design specifications through response spectrum analysis and peak response evaluation.