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Dynamic Analysis Methods

This section describes analysis methods for dynamic problems using direct time integration. The formulations of the implicit and explicit methods are presented below.

Discretization of the Equation of Motion (Common Framework)

TBD (to be completed in the next phase).

Implicit Method (Newmark-β Method)

For dynamic problems, a direct time integration method is applied to solve the equation of motion shown below.

\[\begin{equation} M( t + \Delta t ) \ddot{U} (t + \Delta t) + C( t + \Delta t ) \dot{U}(t + \Delta t) + Q( t + \Delta t ) = F( t + \Delta t ) \label{eq:2.5.1} \end{equation}\]

Here, \(M\) is the mass matrix, \(C\) is the damping matrix, \(Q\) is the internal-force vector, and \(F\) is the external-force vector. The mass matrix is assumed to remain constant regardless of deformation, even in nonlinear analysis.

The changes in displacement, velocity, and acceleration over the time increment \(\Delta t\) are approximated using the Newmark-\(\beta\) method as shown in Eq. \(\eqref{eq:2.5.2}\) and Eq. \(\eqref{eq:2.5.3}\).

\[\begin{equation} \dot{U}(t + \Delta t) = \frac{\gamma}{\beta \Delta t} \Delta U( t + \Delta t ) - \frac{\gamma - \beta}{\beta} \dot{U}( t ) - \Delta t \frac{\gamma - 2\beta}{2\beta} \ddot{U}(t) \label{eq:2.5.2} \end{equation}\]
\[\begin{equation} \ddot{U}(t + \Delta t) = \frac{1}{\beta \Delta t^2}\Delta U(t + \Delta t) - \frac{1}{\beta \Delta t} \dot{U}(t) - \frac{1 - 2\beta}{2\beta} \ddot {U}(t) \label{eq:2.5.3} \end{equation}\]

Here, \(\gamma\) and \(\beta\) are parameters of the Newmark-\(\beta\) method.

As is well known, the following values of \(\gamma\) and \(\beta\) correspond to the linear acceleration method and the trapezoidal rule, respectively.

\(\gamma = \displaystyle \frac{1}{2}\), \(\beta = \displaystyle \frac{1}{6}\) (linear acceleration method)

\(\gamma = \displaystyle \frac{1}{2}\), \(\beta = \displaystyle \frac{1}{4}\) (trapezoidal rule)

Substituting Eq. \(\eqref{eq:2.5.2}\) and Eq. \(\eqref{eq:2.5.3}\) into Eq. \(\eqref{eq:2.5.1}\) gives the following equation.

\[\begin{align} \nonumber \left( \frac{1}{\beta \Delta t^2} \mathbf{M} + \frac{\gamma}{\beta \Delta t} C + K \right) \Delta U ( t + \Delta t ) &= F ( t + \Delta t ) - Q ( t + \Delta t ) \\\ \nonumber &+ \frac{1}{\beta \Delta t} M \dot{U} ( t ) + \frac{1 - 2\beta}{2\beta} M \ddot{U} ( t ) \\\ &+ \frac{\gamma - \beta}{\beta} C \dot{U} (t) + \Delta t \frac{\gamma - 2\beta}{2 \beta} C \ddot{U}(t) \label{eq:2.5.4} \end{align}\]

In particular, for a linear problem, \(K_L\) is the linear stiffness matrix and \(Q ( t + \Delta t ) = K_L U (t + \Delta t)\). Substituting this relation into the equation above gives the following equation.

\[\begin{align} \nonumber M \left\lbrace -\frac{1}{\beta \Delta t^2} U(t) -\frac{1}{\beta \Delta t}\dot U(t) - \frac{2\beta}{1-2\beta} \ddot U(t) \right\rbrace &+ C\left\lbrace - \frac{\gamma}{\beta \Delta t} U(t) + \left(1 - \frac{\gamma}{\beta}\right) \dot U(t) + \Delta{t}\frac{ 2\beta-\gamma}{2\beta}\ddot U(t)\right\rbrace \\\ & + \frac{1}{\beta \Delta{t}^2} M + \frac{\gamma}{\beta \Delta{t}} C + K_L U(t+\Delta{t}) = F(t+\Delta{t}) \label{eq:2.5.5} \end{align}\]

At locations where acceleration is prescribed as a geometric boundary condition, the displacement is obtained from Eq. \(\eqref{eq:2.5.2}\) as follows.

\[\begin{equation} u_{is} (t+\Delta{t}) = u_{is} (t) + \Delta t \dot{u}(t) + \Delta t^2 \left(\frac{1}{2} -\beta \right) {\ddot{u}}_{is} (t + \Delta t) \label{eq:2.5.6} \end{equation}\]

Similarly, at locations where velocity is prescribed, the displacement is obtained from Eq. \(\eqref{eq:2.5.6}\) as follows.

\[\begin{equation} u_{is}(t+\Delta{t})= u_{is}(t)+\Delta t \frac{ \gamma - \beta}{ \gamma}\dot{u_{is}}(t) +(\Delta{t})^2 \frac{ \gamma - 2\beta}{ 2\gamma} \ddot{u_{is}}(t) +\Delta t \frac{\beta}{ \gamma}\dot{u_{is}}(t+\Delta{t}) \label{eq:2.5.7} \end{equation}\]

Here, \(u_{is}(t+\Delta{t})\) is the nodal displacement at time \(t+\Delta{t}\), and \(\dot{u_{is}}(t+\Delta{t})\) is the nodal velocity 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 number, and \(s\) is the node number. The mass and damping terms are handled as follows.

Treatment of the Mass Term

In principle, the mass matrix is treated as a lumped mass matrix.

Treatment of the Damping Term

The damping term is treated as Rayleigh damping expressed by Eq. \(\eqref{eq:2.5.8}\).

\[\begin{equation} C = R_m M + R_k K_L \label{eq:2.5.8} \end{equation}\]

Here, \(R_m\) and \(R_k\) are the Rayleigh damping parameters.

Explicit Method (Central Difference Method)

The explicit method is based on the equation of motion at time t shown below.

\[\begin{equation} M \ddot{U}(t) + C (t) \dot{U}(t) + Q(t) = F(t) \label{eq:2.5.9} \end{equation}\]

Expressing the displacements at times \(t + \Delta t\) and \(t - \Delta t\) by Taylor expansions about time \(t\) and retaining terms through second order in \(\Delta t\) gives the following equations.

\[\begin{equation} U(t+\Delta{t}) = U(t)+\dot{U}(t)(\Delta{t}) +\frac{1}{2!}\ddot{U}(\Delta{t})^2 \label{eq:2.5.10} \end{equation}\]
\[\begin{equation} U(t-\Delta{t})=U(t)-\dot{U}(t)(\Delta{t}) +\frac{1}{2!}\ddot{U}(\Delta{t})^2 \label{eq:2.5.11} \end{equation}\]

Taking the difference and sum of Eq. \(\eqref{eq:2.5.3}\) and Eq. \(\eqref{eq:2.5.4}\) gives the following equations.

\[\begin{equation} \dot{U}(t)=\frac{1}{2\Delta{t}} (U(t+\Delta{t})-U(t-\Delta{t})) \label{eq:2.5.12} \end{equation}\]
\[\begin{equation} \ddot{U}= \frac{1}{(2\Delta{t})^2} (U(t+\Delta{t})-2U(t)+U(t-\Delta{t})) \label{eq:2.5.13} \end{equation}\]

Substituting Eq. \(\eqref{eq:2.5.12}\) and Eq. \(\eqref{eq:2.5.13}\) into Eq. \(\eqref{eq:2.5.9}\) gives the following equation.

\[\begin{equation} \left( \frac{1}{\Delta t^{2}} M + \frac{1}{2\Delta t} C \right) U ( t + \Delta t ) \\\ = F(t) - Q(t) - \frac{1}{\Delta t^{2}} 2 U(t) - U( t - \Delta t) - \frac{1}{2\Delta t} C U(t - \Delta t) \label{eq:2.5.14} \end{equation}\]

In particular, for a linear problem, \(Q(t) = K_L U(t)\), and the equation above becomes

\[\begin{equation} \left( \frac{1}{\Delta t^{2}} M + \frac{1}{2\Delta t} C \right) U( t + \Delta t ) \\\ = F(t) - K_L U(t) - \frac{1}{\Delta t^{2}} M U(t) - U(t - \Delta t) - \frac{1}{2\Delta t} C U (t - \Delta t) \label{eq:2.5.15} \end{equation}\]

If the mass matrix \(M\) is taken as a lumped mass matrix and the damping matrix as a proportional damping matrix \(C = R_m M\), Eq. \(\eqref{eq:2.5.15}\) requires no solution of simultaneous equations.

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

\[\begin{equation} U( t + \Delta t ) \\\ = \frac{1}{( \frac{1}{\Delta t^{2}} M + \frac{1}{2\Delta t} C )} \{ F(t) - Q(t) - \frac{1}{\Delta t^{2}} M U(t) - U(t - \Delta t) - \frac{1}{2\Delta t} C U(t - \Delta t) \} \label{eq:2.5.17} \end{equation}\]