1 A numerical model for investigation of dynamic behavior and free vibration of functionally graded cylindrical helical springs https://jourcc.com/index.php/jourcc/article/view/jcc335 10.52547/jcc.3.3.5 20.1001.1.26765837.2021.3.8.5.2 1 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. 176 181 Zohreh Ebrahimi Payame Noor University Iran zh.ebr73@gmail.com Masoud Abasi Atibeh Payame Noor University Iran info@jourcc.com FG material Helical spring Axial and rotational displacement Gradient index Article5.pdf [1] W. Jiang, W. Jones, T. Wang, and K. Wu, Free vibration of helical springs, Journal of Applied Mechanics, 58(1) (1991) 222-228. ## [2] J. Banerjee and F. Williams, An exact dynamic stiffness matrix for coupled ex-tensional-torsional vibration of structural members, Computers and structures, 50(2) (1994) 161-166. ## [3] J. Lee and D. 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