![]() Analytical solution of spatial elastic and its application to kinking problem. Stability of equilibrium of a helical rod under axial compression (in Chinese). Stability of a thin elastic helical rod with no circular cross section in relaxed state (in Chinese). The stability of elastic curved bars (in Chinese). The present work is useful to the research on the behavior of the post-buckling of the compressed helical spring. The reaction loads of the spring caused by the axial load subjected to the center point are also discussed, giving the boundary conditions for the solution to the equilibrium equations. Hence, the equations can be simplified to the functions of the twist angle and the arc length, which can be solved by a numerical method. By using a small deformation assumption, the variables of the deflection can be expanded into Taylor’s series, and the terms of high orders are ignored. The equilibrium equations are established by introducing two coordinate systems, the Frenet and the principal axis coordinate systems, to describe the spatial deformation of the center line and the torsion of the cross section of the spring, respectively. A three-dimensional (3D) helical spring model is considered in this paper. The number of unknowns that you will be able to solve for will again be the number of equations that you have.In most cases, the research on the buckling of a helical spring is based on the column, the spring is equivalent to the column, and the torsion around the axial line is ignored. Once you have your equilibrium equations, you can solve these formulas for unknowns. All moments will be about the \(z\) axis for two-dimensional problems, though moments can be about the \(x\), \(y\) and \(z\) axes for three-dimensional problems. To write out the moment equations, simply sum the moments exerted by each force (adding in pure moments shown in the diagram) about the given point and the given axis, and set that sum equal to zero. Remember that any force vector that travels through a given point will exert no moment about that point. Any point should work, but it is usually advantageous to choose a point that will decrease the number of unknowns in the equation. To do this you will need to choose a point to take the moments about. Next you will need to come up with the the moment equations. Your first equation will be the sum of the magnitudes of the components in the \(x\) direction being equal to zero, the second equation will be the sum of the magnitudes of the components in the \(y\) direction being equal to zero, and the third (if you have a 3D problem) will be the sum of the magnitudes in the \(z\) direction being equal to zero. Once you have chosen axes, you need to break down all of the force vectors into components along the \(x\), \(y\) and \(z\) directions (see the vectors page in Appendix 1 page for more details on this process). If you choose coordinate axes that line up with some of your force vectors you will simplify later analysis. These axes do need to be perpendicular to one another, but they do not necessarily have to be horizontal or vertical. ![]() Next you will need to choose the \(x\), \(y\), and \(z\) axes. In the free body diagram, provide values for any of the known magnitudes, directions, and points of application for the force vectors and provide variable names for any unknowns (either magnitudes, directions, or distances). ![]() ![]() This diagram should show all the force vectors acting on the body. \Īs with particles, the first step in finding the equilibrium equations is to draw a free body diagram of the body being analyzed. ![]()
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