# Solution of the Equation of Motion

• Sergio Oller
Part of the Lecture Notes on Numerical Methods in Engineering and Sciences book series (LNNMES)

## Abstract

This chapter deals with the solution of the equation of motion in its semi-discrete form in the time domain (see equilibrium equation, section 2.5). Below is the assembly of equation 2.72 (or 2.73 if the equilibrium is achieved in the reference configuration) which defines the equilibrium in the solid at time t + Δt,
$$\underset{\varOmega^e}{\mathrm{A}}{}^i\left[{\displaystyle \underset{{\mathrm{V}}^e}{\int }{\sigma}_{ij}{\nabla}_i^S{N}_{jk} dV}\right]_{\varOmega^e}^{t+\varDelta t}=\underset{\varOmega^e}{\mathrm{A}}{}^i\left[{\displaystyle \underset{S^e}{\oint }{t}_i{N}_{ik}\ dS}+{\displaystyle \underset{V^e}{\int}\uprho\ {b}_i{N}_{ik} dV}\right]_{\varOmega^e}^{t+\varDelta t}-\underset{\varOmega^e}{\mathrm{A}}{}^i\left[{\displaystyle \underset{V^e}{\int}\uprho\;{N}_{ki}\;{N}_{ij} dV}\right]_{\varOmega^e}^{t+\varDelta t}\;{\left.{\ddot{U}}_j\right|}_{\varOmega^e}^{t+\varDelta t}$$