Free vibration analysis of non-cylindrical helical springs by the pseudospectral method

The pseudospectral method is applied to the free vibration analysis of non-cylindrical helical springs. The entire domain is considered as a single element and the displacements and the rotations are approximated by the sums of Chebyshev polynomials. The internal forces and moments are substituted to give six equations of motion, which are collocated to yield the system of algebraic equations. The boundary condition is considered as the constraints, and the set of equations is condensed so that the number of degrees of freedom of the problem matches the number of the expansion coefficients. Numerical examples are provided for clamped–clamped, free–free, clamped–free and hinged–hinged boundary conditions.