Lee, J. and Thompson, D.J.
Dynamic stiffness formulation, free vibration and wave motion of helical springs.
Journal of Sound and Vibration, 239, (2), . (doi:10.1006/jsvi.2000.3169).
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A coil spring can only be treated as a simple massless force element at low frequencies, the effects of internal resonances leading to significant dynamic stiffening. For an automotive suspension spring this occurs at frequencies as low as about 40 Hz. This paper presents an efficient method for calculating the dynamic stiffness of a helical coil spring. The partial differential equations of motion are used to derive the relation between wavenumber and frequency along with the associated wave shapes. By expressing the response in terms of these waves, the dynamic stiffness matrix is assembled. Natural frequencies are obtained from the reduced stiffness matrix, allowing for different boundary conditions, making use of the Wittrick–Williams algorithm. The results of the dynamic stiffness method are compared with those of the transfer matrix method and the finite element method. The nature of the wave propagation is also investigated. Although at low frequencies four wave types propagate, above a particular frequency only two propagating waves remain. These are composite waves which are excited by both axial and transverse motion. For lower values of helix angle an intermediate frequency range exists where six propagating waves can occur.
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