Numerical Integration of FGM Beam Vibration Equations Using a Discrete Variational Approach with Algebraic Constraints

This paper develops a constrained differential-quadrature/discrete-variational framework for vibration analysis of functionally graded beams. Spatial derivatives are discretized on a Chebyshev–Lobatto grid, while time integration is constructed from Lagrange interpolation and Gauss–Legendre quadrature. Boundary conditions are retained as algebraic constraints and enforced with Lagrange multipliers, leading to the semi-discrete operator K=k1I+kf2A(4)−k2A(2). For the default two-node formulation, the DVM keeps position-level constraint residuals near machine precision and suppresses cumulative position- and velocity-level drift relative to the RK4 comparison, while acceleration-level residuals remain of comparable order. The DVM response remains bounded in simulations up to 10,000 s. An empirical time-step scan is stable through h=0.30 and unstable at h=0.35 for the tested problem. Temporal convergence against an exact-in-time solution of the same constrained semi-discrete system is essentially second order, and pairwise self-convergence gives the same result. Spatial refinement from 5 to 11 Chebyshev–Lobatto nodes reduces the L∞ displacement error from 4.01×10−3 to 8.26×10−10, consistent with spectral-type DQM convergence. Additional sensitivity studies confirm that the principal conclusions are robust to algebraic-solver initialization, stopping tolerance, and several boundary-condition choices within the tested parameter range.