Solving Beam Vibration Problem Under Moving Load Using Newmark Method

This study addresses the beam vibration problem under moving load using the Newmark method. The Newmark method represents a widely adopted numerical approach for solving dynamic problems in structural mechanics, particularly effective for analyzing dynamic responses of engineering structures. From an implementation perspective, this method requires discretizing time into small increments and solving the system's state equations at each time step through iterative calculations. When applying this methodology to beam vibration under moving load, several critical factors must be considered: initial conditions of the beam, time-varying position and magnitude of the moving load, and material properties of the beam structure. The implementation typically involves formulating the equation of motion Mü + Ců + Ku = F(t), where the force vector F(t) needs updating at each time step to reflect the moving load's current position. The Newmark-beta parameters (γ and β) control numerical stability and accuracy, with common values being γ=0.5 and β=0.25 for unconditional stability. The computational algorithm generally follows these steps: initial configuration setup, stiffness/mass/damping matrix assembly, time loop implementation with force vector updating, and Newmark integration within each time step. Key functions would include position tracking for the moving load, matrix operations for system equations, and convergence checks for numerical stability. Beyond the basic implementation, this methodology offers significant expansion potential. Researchers can extend the approach to analyze other structural types like plates and shells, or conduct comparative studies with alternative numerical methods (e.g., Wilson-θ, Runge-Kutta) to identify more accurate and efficient solutions. The problem possesses substantial research value and practical application prospects in structural dynamics and transportation engineering.