Calculating Natural Frequencies and Mode Shapes of Springs with Consideration of the Rotational Inertia and Shear

To calculate natural frequencies and mode shapes of «long» cylindrical helical springs, a finite element in the form of a single coil is proposed, in which nodes are located on the axis of the spring. The stiffness matrix of the finite element is obtained by numerical integration of a system of differential equations for a spatial rod, therefore it can be regarded as numerically exact. The shape functions corresponding to the identity nodal displacements of the finite element are a by-product of the stiffness matrix calculation. The shape functions are also numerically exact as they are derived from a system of differential equations. The shape functions are used to build the mass matrix using the standard finite element method. When compiling the stiffness matrix, transverse shear and extension of the helical rod axis are taken into account, while the rotational inertia is considered when constructing the mass matrix. The main advantage of the developed finite element is that it is «soft» compared to a conventional beam finite element, whose tensile and bending stiffness values differ by orders of magnitude. In addition, the departure from partitioning the coil into rectilinear finite elements makes it possible to reduce the dimensionality of the problem by dozens of times. The comparison of the calculation and the experimental results has shown high accuracy of the proposed finite element.

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