Transverse vibrations of tightened bolts: Simplified modeling of tension-dependent boundary stiffness and damping

An axially pre-stressed Bernoulli–Euler beam with end linear springs and dampers is derived to model tightened bolts. The boundary stiffness and damping are analytically expressed as functions of the tensile axial force. The theoretical frequencies and damping ratios resulting from the beam model compare agreeably with the experimental data obtained from two types of bolts. An attempt to explain the apparently linear behavior of the boundary conditions is given, based on a model comprised of coupled mass-spring chains, which is reduced to a one-degree-freedom mass-spring system whose effective restoring force is sought depending on the number of mass-spring systems. The frequency response of the coupled chains and the shape of the restoring force of the reduced system indicates a nearly linear behavior of the coupled chains as the number of chains increases. The results of this work could potentially be used in a vibration based technique to estimate bolt tension from measured vibration.