Abstract
The aim of this paper is to investigate the free vibration of functional-graded (FG) cylindrical helical springs. Model differential equations of homogeneous helical springs are extended to the vibration of FG helical springs. The equations are discretized using finite difference method for space. The time dependent equations are solved using a GMRES method. The initial axial and rotational displacements are applied at the free end of the spring manually and then released. The validated numerical model is then adopted to establish the effects of the FG material index on the model natural frequencies obtained by FFT analysis. According to the results, in both homogeneous and FG helical springs, the amplitudes of axial and rotational displacements increase as they approach the free end of the spring. The numerical results indicate that the FG material index strongly affects the dynamic behavior of the cylindrical helical springs. The amplitudes of the oscillations are damped efficiently and by increasing the material gradient index.
Highlights
Helical springs are widely used in engineering applications
The aim of this paper is to investigate the free vibration of functional-graded (FG) cylindrical helical springs
Michalczyk et al [14] presented a simple formula for calculating the natural frequency of transverse vibrations of axial steel helical springs using a modified equivalent Timoshenko beam theory
Summary
Helical springs are widely used in engineering applications. The numerical modeling of the helical springs is of particular importance, due to their nonlinear and complex behavior. Lee et al [3] investigated the governing equations of helical cylindrical springs based on the Timoshenko beam theory in a curvilinear coordinate system. Kacar et al [12] utilized the stiffness matrices to obtain natural frequencies and buckling loads of non-uniform helical springs made of composite materials, based on first-order shear deformation theory. Michalczyk et al [14] presented a simple formula for calculating the natural frequency of transverse vibrations of axial steel helical springs using a modified equivalent Timoshenko beam theory. They have determined a critical axial force, when the first natural frequency becomes zero
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