ABSTRACT
This chapter shows that the coupled partial differential equations describing vibrations of a helical spring using separation of variables. Springs are widely used in most engineering applications and biomechanics. In fact, the spring is a nonlinear mechanical element, and an extension or compression is always accompanied by torsion and vice versa. J. H. Michell in 1890 showed the existence of independent radial and axial waves in spring oscillations. The free vibration of helical springs and rods with fixed-free boundary conditions are complex. Separation of variables is a powerful method of solving boundary value problems for helical springs and rods. A numerically investigation of the two vibrating modes for the helical spring and rod show that their mode shapes are trapezoidal. This means that the motion of the particles comprising the spring or rod during each mode is not continuous, but is frozen for a certain duration of time at the maximum amplitude of the oscillations.
