In Newmark’s Beta method there are two special cases:

  1. Average acceleration method : \(\alpha = 1/2 \\ \beta = 1/4\)

  2. Linear acceleration method:

\[\alpha = 1/2 \\ \beta = 1/6\]

These parameters define the variation of acceleration over a time step and determine the stability and accuracy characteristics of the method.

The general equation for SDOF problem is written as : \(m \ddot{u}_{t+1}+c \dot{u}_{i+1}+k {u}_{i+1}=p_{i+1}\) We can calculate the (i+1) terms using the i^th^ terms.

Steps involved in the calculation:

  1. Initial calculations: \(\ddot{u}_{0}=\frac{p_{0}-c \dot{u_{0}}-k{u_{0}}}{m_{0}}\)

    ​ Select $\Delta$ t

\[\begin{array}{l} a=\frac{1}{\beta \Delta t} m+\frac{\nu}{\beta} c ; \ \\ b=\frac{1}{2 \beta} m+\Delta t\left(\frac{\nu}{2 \beta}-1\right) c . \end{array}\]

  1. Calculate for each time step. \(\Delta \hat{\rho}_{i}=\Delta p_{i}+a \dot{u}_{i}+b \ddot{u}_{i}\) Determine for tangent stiffness k~i~ $$ \hat{k}{i}=k{i}+\frac{\nu}{\beta \Delta t} C+\frac{1}{\beta(\Delta t)^{2}} m
    \Delta \dot{u}{i} =\frac{\gamma}{\beta \Delta t} \Delta u{i}-\frac{\gamma}{\beta} \dot{u}{i}+\Delta t\left(1-\frac{\gamma}{2 \beta}\right) \ddot{u}{i} \

    \Delta \ddot{u}{i} =\frac{1}{\beta(\Delta t)^{2}} \Delta u{i}-\frac{1}{\beta \cdot \Delta t} \dot{u}{i}-\frac{1}{2 \beta} \ddot{u}{i} $$

  2. Repeat this cycle for the next steps.