On the linearized disturbance dynamic equations for buckling and free vibration of cylindrical helical coil springs under combined compression and torsion

In this study a set of twelve linearized disturbance dynamic equations in canonical form is derived systematically and in a comprehensive manner based on the first order shear deformation theory to study the buckling and vibration analysis of helical coil springs made of isotropic linear materials. Those complete equations comprise the axial and shear deformation effects together with rotatory inertia effects. The special case of these equations corresponds also to the equations for straight and circular rods. The main differences among the existing formulations based on the same approach are discussed briefly. The resulting equations are used for numerical buckling and free vibration analyses to show its soundness.