# Dynamic Analysis Method

## Dynamic Analysis Method

In this section, the dynamic problem analysis method with a direct time integration method applied is described. As presented below, with this development code, it is possible to perform time history response analysis by an implicit or explicit method.

### Formulation of the implicit method

Focusing on dynamic problems, the direct time integration method was applied to solve the following equation motion indicated:

where $M$ and the mass matrix, $C$ is the damping matrix, and $Q$ is the internal stress vector, and $F$ is the external force vector. This software does not consider the changes in mass; thus, the mass matrix is non-linear and constant regardless of deformation.

The displacement within time increment $\Delta t$, and the change in speed and acceleration are approximated with the Newmark-$\beta$ method, as expressed in Eq.$\eqref{eq:2.5.2}$ and Eq.$\eqref{eq:2.5.3}$:

where $\gamma$ and $\beta$ are the parameters of the Newmark-$\beta$ method.

if $\gamma$ and $\beta$ have the following values, it coincides with the linear acceleration method or the trapezoidal rule.

As it is already known, when $\gamma$ and $\beta$ are substituted into the following values, it will match the linear acceleration method, or trapezoid rule.

• $\gamma=\displaystyle \frac{1}{2}$, $\beta=\displaystyle \frac{1}{6}$ (Linear acceleration method)
• $\gamma=\displaystyle \frac{1}{2}$, $\displaystyle \beta=\frac{1}{4}$ (Trapezoid rule)

If Eq.$\eqref{eq:2.5.2}$ and Eq.$\eqref{eq:2.5.3}$ are substituted into Eq.$\eqref{eq:2.5.1}$, the following equation is obtaind:

$K_L$ is linear stiffness matrix for linear problem; thus, $Q ( t + \Delta t ) = K_L U (t + \Delta t)$. If this equation is substituted into the equation above, the following equation is obtained:

In the portion, where the acceleration is specified as a geometric boundary condition, the displacement of the following equation is obtained from Eq.$\eqref{eq:2.5.2}$.

Similarly, if the speed is specified, the displacement of the following equation is obtained from Eq.$\eqref{eq:2.5.6}$:

Where

• $u_{is}(t+\Delta t)$ is the nodal displacement at $t + \Delta t$,
• $\dot{u}_{is}(t+\Delta t)$ is the nodal speed at time $t + \Delta t$
• $\ddot{u}_{is}(t+\Delta t)$ is the nodal acceleration at time $t + \Delta t$
• $i$ is the nodal degree-of-freedom (DOF)
• $s$ is the Node number.

Furthermore, the mass and damping terms were handled as follows:

#### (1) Handling of mass term

The mass matrices are handled as concentrated mass matrices.

#### (2) Handling of damping terms

The damping terms are handled as Rayleigh damping expressed in Eq.$\eqref{eq:2.5.8}$.

where $R_m$ and $R_k$ are parameters of Rayleigh damping.

### Formulation of Explicit Method

The explicit method is based on the motion equation at time $t$ expressed in the following equation:

where the displacement at time $t + \Delta t$ and that at time $t - \Delta t$ are expressed by the Taylor expansion at time $t$. If it is expanded up to the secondary term with $\Delta t$, it becomes

From the difference and sum of Eq.$\eqref{eq:2.5.3}$ and Eq.$\eqref{eq:2.5.4}$, the following equation is obtained:

If Eq.$\eqref{eq:2.5.12}$ and Eq.$\eqref{eq:2.5.13}$ are substituted into Eq.$\eqref{eq:2.5.9}$, the following equation is obtained:

For linear problems, specifically, $Q(t)=K_L U(t)$, the above equation becomes

In this case, if mass matrix $M$ is set as a concentrated mass matrix, and the damping matrix as the proportional damping matrix $C=R_m M$, Eq.$\eqref{eq:2.5.15}$ eliminates the requirement of solving operations for simultaneous equations.

Therefore, from Eq.$\eqref{eq:2.5.15}$, $U(t+\Delta t)$ can be determined by the following equation: