Newmark Numerical Integration Method for Solving Dynamic Equation Systems

The Newmark numerical integration method is a robust technique for solving systems of dynamic equations, widely applied in engineering and physics due to its computational stability and versatility. This method discretizes the time domain into incremental steps and computes solutions recursively using update formulas that incorporate previous states—displacements, velocities, and accelerations. Key implementation steps include: initializing system matrices (mass, damping, and stiffness), selecting Newmark parameters (γ and β) to control numerical stability, and iterating over time steps to solve for new displacements and velocities. A typical code implementation involves updating acceleration using the equation of motion, followed by velocity and displacement integrations via the Newmark formulas. Its unconditional stability allows larger time steps without sacrificing accuracy, making it efficient for long-duration simulations. Researchers and practitioners are encouraged to master this method to leverage its full potential in structural dynamics, seismic analysis, and real-time simulations.