Difference schemes for dynamics problems
Keywords:difference scheme, stability of difference scheme, modified central difference method, Newmark's method
To solve dynamics problems in the LIRA 10.12 software package, the difference scheme is used, known as modified central difference method. The goal of dynamic problems solving is to get a good approximation of actual dynamic response of a given structure. It's a matter of convergence conditions of the difference scheme used in numerical integration of motion equations.
The solution of the linear dynamic problem for all possible displacements satisfies the equations
, , – symmetric positive-definite bilinear functionals of possible work of the internal and inertial forces and motion resistance forces, they correspond to stiffness matrices, mass and damping matrices, – linear functional of possible work of the external forces.
After approximation in spatial variables (usually the finite element method is applied) we obtain the system of ordinary differential equations
, , – stiffness matrices, mass and damping matrices, – external forces.
Difference scheme of modified central difference method we obtain by replacing the values of functions and derivatives with corresponding difference relations. Difference relations, which are applied in modified central difference method, approximate accelerations , velocities and displacements with an error proportional to the square of the time step.
Approximation and stability are the convergence conditions of the difference scheme. To study the stability of difference schemes, we assume that and apply the obvious equations
Let us denote , apply the Cauchy inequality on the right-hand side and integrate the resulting inequality. Then
where are positive constants. The boundedness follows from the Gronwall's lemma. In addition to the continual version of study of the difference schemes stability, the discrete version is also considered.
Unconditional practical efficiency of the modified central difference method and the simplicity of its underlying principles allow it to be successfully applied to a wide range of dynamic problems.
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