In this study, all the translational and rotational dynamic motions of a helical spring described with heavily coupled partial differential equations are analyzed by using an extended finite difference-type numerical scheme.
The proposed numerical scheme is a single-step, second-order accurate and implicit finite difference method. The stability, consistency and convergence are examined analytically with the second-order hyperbolic partial differential equation. Since a transformed difference equation is constructed using the displacement and known partial differential terms, the proposed numerical scheme can directly satisfy the natural boundary conditions and also the interpretation of the physical meanings of partial differential values is very easy. The proposed numerical scheme is easily applicable to high order partial differential equations, also coding and remeshing are very simple. From the numerical examples, it can be said that the proposed method is more accurate and effective than the already well-known finite difference methods.
The moving load and moving boundary problems, space and time dependent systems, are prepared to demonstrate the validity and capability of the proposed numerical scheme.
The moving load and moving boundary problems with variable moving velocities are effectively solved by using the proposed numerical scheme. The equations of motion for a spatially curved and twisted beam are derived and the forced transient responses of a helical spring having all translational and rotational motions are obtained by using the proposed numerical scheme. From these dynamic responses of a helical spring, it can be said that the coupling effects play an important role in the dynamic behavior of a helical spring. The static and dynamic analysis of a helical spring are verified by comparing with the result from ANSYS FEM package.
Also the optimum vertical force to minimize the unwanted radial motion (side sway motions) of a helical spring is designed by optimizing the equations. The designed optimum vertical force point shows that the acting position deviated from the center of the helix in vertical direction, and the deviated optimum excitation force generates moments which cut off the effect of the radial motion. The radial motion is drastically reduced by the optimum vertical force and this result is very useful to the surface design of forcing plate facing to a helical spring. Since usual design optimization problems require heavy computational work, the proposed numerical scheme is very efficient in the computation efforts.
Finally, a practical example of a helical spring, DOHC type cam-valve train with distributed parameter model of a valve spring, is illustrated. The valve response, contact force and valve spring force are accurately predicted and the guideline on ramp design using continuous impact force model is proposed.