Newmark-β method

The Newmark-β method is a widely used implicit time step method, 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 $\begin{array}{l}w\end{array}$ , the velocity $\begin{array}{l}\dot{w}\end{array}$ and the acceleration $\begin{array}{l}\ddot{w}\end{array}$ 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 $\begin{array}{l}t_i\end{array}$ are considered and, usually, a time increment $\begin{array}{l}\Delta t\end{array}$ is chosen. The values at the time step $\begin{array}{l}t_{i+1} = t_i + \Delta t\end{array}$ are determined from the values at time step $\begin{array}{l}t_i\end{array}$ and possibly $\begin{array}{l}t_{i-1}\end{array}$ , $\begin{array}{l}t_{i-2}\end{array}$ , $\begin{array}{l}\ldots\end{array}$ .

For implicit methods, the equation of motion for the yet unknown state is introduced into the time integration algorithm at time $\begin{array}{l}t_i + \Delta t\end{array}$ . Thereby, non-linear systems of equations can arise. An advantage of implicit methods is that they can be absolutely stable. This allows larger time steps as compared to explicit methods. This large group of methods is very widespread in structural dynamics as the methods are extremely effective and robust, especially in calculations with predominantly low-frequency structural responses. With an implicit procedure, the response values are obtained from both, the immediately preceding time step and the following time step.

Theory

The response of a single degree of freedom system can be obtained with:

$\begin{align} \dot{u}_{n+1} &= \dot{u}_n + \left( {1 - \gamma} \right) \Delta t \ddot{u}_n + {\gamma \Delta t \ddot{u}_{n+1}} \label{eq:numerische_untersuchung006}\\ % u_{n+1} &= u_n + \Delta t \dot{u}_n + \left( {\frac{1}{2} - \beta} \right) \Delta t^2 \ddot{u}_n + {\beta \,\Delta t^2 \ddot{u}_{n+1}} \label{eq:numerische_untersuchung007} \end{align}$

The parameters $\begin{array}{l}\beta\end{array}$ and $\begin{array}{l}\gamma\end{array}$ are used to control the method's stability and accuracy. Often, they are chosen as $\begin{array}{l}\beta=\frac{1}{4},\ \gamma=\frac{1}{2}\end{array}$ . This corresponds to a constant acceleration during the chosen time interval. Another approach is to consider linear acceleration during the time step. This corresponds to $\begin{array}{l}\beta=\frac{1}{6},\ \gamma=\frac{1}{2}\end{array}$ .

Choosing constant acceleration, the Newmark-β method is unconditionally stable (i.e. stable for an arbitrary time step $\begin{array}{l}\Delta t\end{array}$ ). This means, that the computed values lie in the range of the deformation of the system. In order to get reliable results, however, the time step has to be chosen such that the load characteristic and the characteristic of the free vibration can be represented accurately enough.

Constant acceleration

The acceleration and its integrates can now be represented as:

We now choose $\begin{array}{l}\beta=\frac{1}{4},\ \gamma=\frac{1}{2}\end{array}$ and from $\begin{array}{l}\ddot{u}(\tau),\ \dot{u}(\tau),\ u(\tau)\end{array}$ we obtain the relations between the consecutive time steps $\begin{array}{l}n\end{array}$ and $\begin{array}{l}n+1\end{array}$ :

$\begin{align} \label{eq:01} \ddot{u}_{avg} &= \frac{1}{2}(\ddot{u}_n +\ddot{u}_{n+1})\\ \label{eq:02} \dot{u}_{n+1} &= \dot{u}_n +\frac{\Delta t}{2}(\ddot{u}_n +\ddot{u}_{n+1})\\ \label{eq:03} u_{n+1}&=u_n +\dot{u}_n\Delta t+\frac{\Delta t^2}{4}(\ddot{u}_n +\ddot{u}_{n+1}) \end{align}$

These relations can be transferred into an explicit form:

(1)	$\begin{align} \dot{u}_{n+1} &= \frac{2}{\Delta t}(u_{n+1} - u_n) - \dot{u}_n \label{eq:t_n_1_01}\\ \ddot{u}_{n+1} &= \frac{4}{\Delta t^2} (u_{n+1} - u_n) - \frac{4}{\Delta t} \dot{u}_n - \ddot{u}_n \label{eq:t_n_1_02} \end{align}$

We now insert (1) into the equation of motion (Transient Analysis:1) at time $\begin{array}{l}t_{n+1}\end{array}$

$\begin{array}{l}\displaystyle m \ddot{u}_{n+1} + c \dot{u}_{n+1} + k u_{n+1} = f_{n+1} , \label{eq:numerische_untersuchung005}\end{array}$

and write all terms belonging to the current time step $\begin{array}{l}n\end{array}$ on the right hand side of the equation. We arrive at

$\begin{array}{l}\displaystyle \left(k + \frac{2 c}{\Delta t} + \frac{4 m}{\Delta t^2}\right) u_{n+1} = f_{n+1} + c \left( \frac{2 u_n}{\Delta t} + \dot{u}_n \right) + m \left( \frac{4}{\Delta t^2} u_n + \frac{4}{\Delta t} \dot{u}_n + \ddot{u}_n \right) .\end{array}$

This relation can now, given the exciting force $\begin{array}{l}f_{n+1}\end{array}$ , be solved for $\begin{array}{l}u_{n+1}\end{array}$ . From this, the velocity $\begin{array}{l}\dot{u}_{n+1}\end{array}$ and acceleration $\begin{array}{l}\ddot{u}_{n+1}\end{array}$ can be retrieved using (1).

Initial conditions

In the first time step, $\begin{array}{l}u_0\end{array}$ and $\begin{array}{l}\dot{u}_0\end{array}$ have to be given as initial conditions. The resulting acceleration can then be computed with

$\begin{array}{l}\displaystyle \label{eq:equili} \ddot{u}_0 = \frac{1}{m}\left( f_0 - c\dot{u}_0 - ku_0 \right).\end{array}$

Multi degree of freedom systems

Of course the method can be extended to multi degree of freedom systems. In this case, the scalars for stiffness, damping, and mass become matrices and the displacement, velocity, acceleration, and force become vectors. As the full dynamic stiffness matrix has to be inverted at each time step, the method can become computationally demanding for large systems and small time steps.

Seitenhierarchie

Newmark-β method

Theory

Constant acceleration

Initial conditions

Multi degree of freedom systems