# Three-Dimensional Load-Deformation Relationships of Arbitrarily Loaded Coiled Springs Doctoral thesis, 1995

This paper reports the development of a computer based tool that can be used to predict the three-dimensional load-deformation relationships of arbitrarily loaded coiled springs. The non-linear deformation of the spring is calculated by dividing the spring into small elements for which the load-deformation relationships can be considered to be linear. The load and deformation of each element are calculated recursively from one end to the other, and the total deformation of the spring is obtained as the sum of the contributions of all the elements. A linear, locally constant stiffness matrix is then established at the pre-loaded equilibrium point. This matrix is obtained by adding small increments to the initial load, calculating the corresponding changes in deformation and solving the linear system of equations in load increments and changes of deformation. An extended model for the linear stiffness of a short section of a coiled spring is developed in order to improve the accuracy of the calculations. This model calculates the deformation of the curved spring wire and includes the effects of pitch, curvature of the wire and distortion due to normal and transverse forces in the wire. The model shows some additional three-dimensional properties of the spring, e.g., rotational asymmetry. A three-dimensional static measurement of the load and deformation of a laterally loaded spring is performed and shows good agreement with the calculation of the non-linear deformation. Further, the natural frequencies of a mechanical system consisting of a rigid body and a laterally loaded spring are measured and compared with calculated results. This comparison verifies the merits of the linearized stiffness matrix. Both the static and dynamic experiments show that the accuracy of the calculations is improved when the extended stiffness model is used.

extended stiffness model

helical spring

non-linear deformation

stiffness matrix

coiled spring

arbitrary load

large deformation

three-dimensional

## Author

#### Lars Lindkvist

Department of Machine and Vehicle Design

#### Subject Categories

Mechanical Engineering

#### ISBN

91-7197-089-4

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 1081

10/7/2017