Introduction

Dynamic analysis of structures is governed by the equations of motion derived from continuum mechanics and finite element discretization. In practical engineering applications, analytical solutions are rarely available, especially for complex three-dimensional systems subjected to arbitrary time-dependent loading. Consequently, numerical time integration methods are required.

One of the most widely used approaches in computational structural dynamics is the \textbf{Newmark-$\beta$ method}, introduced by Nathan M. Newmark in 1959. The method belongs to the family of direct integration schemes and is extensively used in finite element software for transient dynamic analysis.

The Newmark method computes the structural response step-by-step in time by approximating the evolution of displacements, velocities, and accelerations within each time increment.

The method is especially important because:
• it is suitable for large finite element systems,
• it can be unconditionally stable,
• it naturally incorporates damping effects,
• it is applicable to nonlinear and linear dynamics,
• it is computationally efficient and robust.

Need for Time Integration

The equation of motion is a system of second-order ordinary differential equations:

\begin{equation}
\mathbf{M}\ddot{\q}
+
\mathbf{C}\dot{\q}
+
\mathbf{K}\q
=
\mathbf{F}(t)
\end{equation}

For realistic structures and arbitrary loading functions, closed-form analytical solutions generally do not exist.

Therefore, the response must be computed numerically over discrete time intervals:

\begin{equation}
t_n \rightarrow t_{n+1}=t_n+\Delta t
\end{equation}

where $\Delta t$ is the time step.

The objective is to determine:
• displacement $\q_{n+1}$,
• velocity $\dot{\q}_{n+1}$,
• acceleration $\ddot{\q}_{n+1}$,

from the known state at time $t_n$.

Basic Idea

The Newmark method approximates the evolution of displacement and velocity within a time step using assumed acceleration interpolation.

Two parameters control the method:$\beta$, $\gamma$. The displacement at time $t_{n+1}$ is approximated as:

$$\q_{n+1} = \q_n + \Delta t \dot{\q}_n + \Delta t^2 \left[ \left( \frac{1}{2}-\beta \right) \ddot{\q}_n + \beta \ddot{\q}_{n+1} \right]$$

Similarly, the velocity is approximated by:

$$\dot{\q}_{n+1} = \dot{\q}_n + \Delta t \left[ (1-\gamma)\ddot{\q}_n + \gamma \ddot{\q}_{n+1} \right]$$

These equations contain the unknown acceleration at the new time step.The key feature of the implicit Newmark method is that the acceleration at time $t_{n+1}$ is not known in advance. Instead, it is determined by enforcing dynamic equilibrium at the new time step:

$$\mathbf{M}\ddot{\q}_{n+1} + \mathbf{C}\dot{\q}_{n+1} + \mathbf{K}\q_{n+1} = \mathbf{F}_{n+1}$$

Substituting the Newmark approximations into the equilibrium equation yields a system of algebraic equations for the unknown displacement vector $\q_{n+1}$.

After rearrangement, the system can be written as:

\begin{equation}
\mathbf{K}_{\mathrm{eff}}
\q_{n+1}
=
\mathbf{F}_{\mathrm{eff}}
\end{equation}

where the effective stiffness matrix is:

\begin{equation}
\mathbf{K}_{\mathrm{eff}}
=
\mathbf{K}
+
\frac{\gamma}{\beta \Delta t}\mathbf{C}
+
\frac{1}{\beta \Delta t^2}\mathbf{M}
\end{equation}

The effective force vector contains contributions from the previous-step displacement, velocity, and acceleration.

Once $\q_{n+1}$ is obtained, the acceleration and velocity are updated.

The most commonly used parameters are:

\begin{equation}
\beta = \frac{1}{4},
\qquad
\gamma = \frac{1}{2}
\end{equation}

Single-Degree-of-Freedom Example

Consider a damped spring-mass system with one degree of freedom.

The equation of motion is:

\begin{equation}
m\ddot{u}
+
c\dot{u}
+
ku
=
f(t)
\end{equation}

where:

• $m$ is the mass,
• $c$ is the damping coefficient,
• $k$ is the stiffness,
• $u$ is displacement,
• $f(t)$ is the external force.

The displacement approximation becomes:

\begin{equation}
u_{n+1}
=
u_n
+
\Delta t \dot{u}_n
+
\Delta t^2
\left[
\left(
\frac12-\beta
\right)\ddot{u}_n
+
\beta \ddot{u}_{n+1}
\right]
\end{equation}

The velocity approximation is:

\begin{equation}
\dot{u}_{n+1}
=
\dot{u}_n
+
\Delta t
\left[
(1-\gamma)\ddot{u}_n
+
\gamma \ddot{u}_{n+1}
\right]
\end{equation}

The objective is to isolate $\ddot{u}_{n+1}$.

First, move all known terms to the left-hand side:

\begin{equation}
u_{n+1}-
u_n-
\Delta t \dot{u}_n
=
\Delta t^2
\left[
\left(
\frac12-\beta
\right)\ddot{u}_n+
\beta \ddot{u}_{n+1}
\right]
\end{equation}

Divide both sides by $\Delta t^2$:

$$\frac{u_{n+1}-u_n-\Delta t \dot{u}_n}{\Delta t^2} =\left(\frac12-\beta\right)\ddot{u}_n+\beta \ddot{u}_{n+1}$$

Now isolate $\ddot{u}_{n+1}$:

\begin{equation}
\beta \ddot{u}_{n+1}=\frac{u_{n+1}-u_n-\Delta t \dot{u}_n}{\Delta t^2}-
\left(\frac12-\beta\right)\ddot{u}_n\end{equation}

Finally:

\begin{equation}
\ddot{u}_{n+1}=
\frac{1}{\beta \Delta t^2}\left(u_{n+1}-u_n-\Delta t \dot{u}_n\right)-\left(\frac{1}{2\beta}-1
\right)\ddot{u}_n
\end{equation}

Thus, the acceleration is expressed entirely in terms of the unknown displacement $u_{n+1}$ and known quantities from the previous time step. Next, we substitute this acceleration expression into the Newmark velocity equation:

$$\dot{u}_{n+1}=\dot{u}_n+\Delta t\left[(1-\gamma)\ddot{u}_n+\gamma \ddot{u}_{n+1}\right]$$

Substituting the previous formula for $\ddot{u}_{n+1}$ gives:

$$\dot{u}_{n+1}= \dot{u}_n+\Delta t (1-\gamma)\ddot{u}_n+ \gamma \Delta t \left[ \frac{1}{\beta \Delta t^2} \left( u_{n+1}-u_n-\Delta t \dot{u}_n\right)-\left( \frac{1}{2\beta}-1 \right)\ddot{u}_n \right]$$

Rearranging terms:

$$\dot{u}_{n+1}=\frac{\gamma}{\beta \Delta t}u_{n+1}+\Bigg[\dot{u}_n- \frac{\gamma}{\beta}\dot{u}_n\Bigg]+ \Bigg[\Delta t(1-\gamma)-\gamma \Delta t\left(\frac{1}{2\beta}-1\right)\Bigg]\ddot{u}_n- \frac{\gamma}{\beta \Delta t}u_n$$

At this point, both $\ddot{u}_{n+1}$ and $\dot{u}_{n+1}$ are written as functions of the unknown displacement $u_{n+1}$.

We now substitute these expressions into the equilibrium equation evaluated at time $t_{n+1}$:

\begin{equation}
m\ddot{u}_{n+1}
+
c\dot{u}_{n+1}
+
ku_{n+1}
=
f_{n+1}
\end{equation}

Substituting the acceleration expression:

$$m\ddot{u}_{n+1}=m\Bigg[\frac{1}{\beta \Delta t^2}u_{n+1}-\frac{1}{\beta \Delta t^2}u_n-\frac{1}{\beta \Delta t}\dot{u}_n-\left(\frac{1}{2\beta}-1\right)\ddot{u}_n\Bigg]$$

Similarly, the velocity term contributes:

$$c\dot{u}_{n+1}=c\left[\frac{\gamma}{\beta \Delta t}u_{n+1}+(\text{known terms})\right]$$

Finally, the stiffness contribution is simply:

$$ku_{n+1}$$

Now collect all terms multiplying the unknown displacement $u_{n+1}$:

\begin{equation}
\left(
k
+
\frac{\gamma}{\beta \Delta t}c
+
\frac{1}{\beta \Delta t^2}m
\right)
u_{n+1}
=
f_{\mathrm{eff}}
\end{equation}

The quantity inside parentheses is called the effective stiffness:

$$k_{\mathrm{eff}}=k+\frac{\gamma}{\beta \Delta t}c+\frac{1}{\beta \Delta t^2}m$$

The right-hand side $f_{\mathrm{eff}}$ contains all known quantities from the previous time step.

Therefore, instead of solving directly for acceleration, the Newmark method transforms the dynamic equilibrium equation into an algebraic equation for the unknown displacement $u_{n+1}$.

Once $u_{n+1}$ is obtained, the acceleration and velocity can be updated explicitly using the previously derived formulas.