The central difference method is an example for explicit time integration, which can for example be used for Transient Analysis.
Time step procedures enable the numerical calculation of vibration problems. They are applicable in a wide range of problems, but are usually applied, when analytic methods become infeasible, e.g. in case of general aperiodic loads. The basis is the concept to determine the temporal progression of the displacement w, the velocity \dot{w} and the acceleration \ddot{w} step-wise. The time domain is discretized into finite time intervals and the equations of motion are then only fulfilled in an approximate manner. Thus, only discrete points in time t_i are considered and, usually, a time increment \Delta t is chosen. The values at the time step t_{i+1} = t_i + \Delta t are determined from the values at time step t_i and possibly t_{i-1}, t_{i-2}, \ldots.
For an explicit method, the equilibrium of a system is formulated at the beginning of the respective time increment (time t_i) with the known quantities of motion. The motion quantities at time t_{i+1} = t_i + \Delta t can be determined directly by solving a system of equations. Explicit methods are easy to apply and require only a few arithmetic operations per time step. However, explicit methods are only conditionally numerically stable, i.e. for too long time steps the solution can grow exponentially and thus become useless. They must therefore be the adjusted to the highest natural frequency that occurs in the system. Therefore, the step size \Delta t of the method has to be chosen sufficiently small.
Theory
We use the central differencing scheme to numerically derive velocity and acceleration from the displacement:
| (1) | \begin{align} \label{eq:cds01} \dot{u}_n&=\frac{1}{2\Delta t}\left(-u_{n-1} + u_{n+1}\right)\\ \label{eq:cds02} \ddot{u}_n&=\frac{1}{\Delta t^2}\left(u_{n-1} -2u_n + u_{n+1}\right). \end{align} |
By inserting into the equation of motion at time t_n
| m \ddot{u}_{n} + c \dot{u}_{n} + k u_{n} = f_{n} |
and bringing all terms regarding time steps n and n-1 to the right hand side, we arrive at
| \left( \frac{1}{\Delta t^{2}} m + \frac{1}{2 \Delta t} c \right) u_{n+1} = f - \left( k - \frac{2}{\Delta t^{2}} m \right) u_n - \left( \frac{1}{\Delta t^{2}} m - \frac{1}{2 \Delta t} c \right) u_{n-1} . |
This can now be solved for u_{n+1} and (1) yield the velocity and acceleration for the new time step.
Initial conditions
Because not only the current but also the previous time step is required for the central difference method, we need to compute u_{-1} in order to start the time stepping method at n=0. It is calculated as
| u_{-1} = u_0 - \Delta t \dot{u}_0 + \frac{\Delta t^2}{2}\ddot{u}_0, |
where the initial conditions u_0,\ \dot{u}_0 have to be specified and the initial acceleration \ddot{u}_0 can be calculated from the equilibrium relation
| \ddot{u}_0 = \frac{1}{m}\left( f_0 - c\dot{u}_0 - ku_0 \right). |