Elliptic Function Solution To Large-deflection Problems Of Cantilever Beams

In modern mechanisms, compliant mechanism exhibits many advantages over its rigid-body counterpart, such as increased precision, simpler structure, and reduced cost of manufacturing. Therefore, it was widely used in Micro-Electro-Mechanism System(MEMS),Precise Control, and other High-End Application. Most compliant mechanisms can be simplified as flexible beams which is connected by each other. The performance of compliant mechanism is also depend on the deflection of the flexible beams. So the operation process of compliant mechanism, is usually the complex nonlinear deflection process of the flexible beams. Therefore, the exact solution of large deflection problems of beam element is the key to compliant mechanism design and analysis. What’s more, In all of the beam elements, cantilever beam’s large deflection problem is relatively simple, but is also the base of the analysis of large deflection problem for other beams. So the exact solution of the cantilever beam large deflection problems has major significance for the design of compliant mechanism.In this paper, we combine Bernoulli-Euler beam theory with Jacobian Elliptic Functions theory, to analyze the large deflection problem of cantilever beam. Firstly, establish the physical and mathematical model of the cantilever beam’s deflection problem based on the Euler-Bernoulli equation, and simplified them into two boundary value differential equations. Then introduced a variable to classify the load situation, and using different Jacobi elliptic functions to obtain the corresponding analytical solutions, based on different load situations. Finally, designed the calculation programs based on the derivation before. And compared the results calculated by the elliptic function solution with the results calculated by the finite element algorithm, to validate the analytical solution.In the end of the paper, analyzed the characteristics of the deformed beams, based on the elliptic function solutions, and analyzed the large deflection problem of cantilever beams from the view point of energy, obtained the elliptic function expressions for strain energy. And finally give a simple example to illustrate the usage of elliptic function solutions.