The Approximate Solution of the Nonlinear Exact Equation of Deflection of an Elastic Beam with the Galerkin Method

Calculating the large deflection of a cantilever beam is one of the common problems in engineering. The differential equation of a beam under large deformation, or the typical elastica problem, is hard to approximate and solve with the known solutions and techniques in Cartesian coordinates. The exact solutions in elliptic functions are available, but not the explicit expressions in elementary functions in expectation. This paper attempts to solve the nonlinear differential equation of deflection of an elastic beam with the Galerkin method by successfully solving a series of nonlinear algebraic equations as a novel approach. The approximate solution based on the trigonometric function is assumed, and the coefficients of the trigonometric series solution are fitted with Chebyshev polynomials. The numerical results of solving the nonlinear algebraic equations show that the third-order approximate solution is highly consistent with the exact solution of the elliptic function. The effectiveness and advantages of the Galerkin method in solving nonlinear differential equations are further demonstrated.

Automated summary

This paper successfully solves the nonlinear differential equation of deflection of an elastic beam using the Galerkin method. It demonstrates the high consistency between the third-order approximate solution and the exact solution of the elliptic function. The effectiveness and advantages of the Galerkin method in solving nonlinear differential equations are further demonstrated.