step2¶

Purpose:

Compute the dynamic solution to a set of second order differential equations.

Syntax:

[a, da, d2a] = step2(K, C, M, f, a0, da0, bc, ip)
[a, da, d2a] = step2(K, C, M, f, a0, da0, bc, ip, times)
[a, da, d2a, ahist, dahist, d2ahist] = step2(K, C, M, f, a0, da0, bc, ip, times, dofs)

Description:

step2 computes at equal time steps the solution to a set of second order differential equations of the form:

\[\begin{split}\mathbf{M} \ddot{\mathbf{a}} + \mathbf{C} \dot{\mathbf{a}} + \mathbf{K} \mathbf{a} = \mathbf{f}(x, t), \\ \mathbf{a}(0) = \mathbf{a}_0, \\ \dot{\mathbf{a}}(0) = \mathbf{v}_0.\end{split}\]

In structural mechanics problems, K, C and M represent the \(n \times n\) stiffness, damping and mass matrices, respectively. a is the displacement, da ( = \(\dot{\mathbf{a}}\) ) is the velocity and d2a ( = \(\ddot{\mathbf{a}}\) ) is the acceleration.

The matrix f contains the time-dependent load vectors. If no external loads are active, use [] for f. The matrix f is organized as:

\[\begin{split}f = \begin{bmatrix} \text{time history of the load at } dof_1 \\ \text{time history of the load at } dof_2 \\ \vdots \\ \text{time history of the load at } dof_n \end{bmatrix}\end{split}\]

The dimension of f is:

(number of degrees-of-freedom) × (number of timesteps + 1)

The initial conditions are given by the vectors a0 and da0, containing initial displacements and initial velocities.

The matrix bc contains the time-dependent prescribed displacement. If no displacements are prescribed, use [] for bc. The matrix bc is organized as:

\[\begin{split}bc = \begin{bmatrix} dof_1 & \text{time history of the displacement} \\ dof_2 & \text{time history of the displacement} \\ \vdots & \vdots \\ dof_{m_2} & \text{time history of the displacement} \end{bmatrix}\end{split}\]

The dimension of bc is:

(number of dofs with prescribed displacement) × (number of timesteps + 2)

The time integration procedure is governed by the parameters given in the vector ip defined as:

\[ip = [dt, T, \alpha, \delta]\]

where dt specifies the time increment, T the total time, and alpha and delta are time integration constants for the Newmark family of methods.

Frequently used values:

\(\alpha\)	\(\delta\)	Method
\(\frac{1}{4}\)	\(\frac{1}{2}\)	Average acceleration (trapezoidal) rule
\(\frac{1}{6}\)	\(\frac{1}{2}\)	Linear acceleration
0	\(\frac{1}{2}\)	Central difference

The computed values of \(\mathbf{a}\), \(\dot{\mathbf{a}}\) and \(\ddot{\mathbf{a}}\) are stored in a, da and d2a, respectively. The first column contains the initial values and the following columns contain the values for each time step.

The dimension of a, da and d2a is:

(number of degrees-of-freedom) × (number of time steps + 1)

If the values are to be stored only for specific times, the parameter times specifies at which times the solution will be stored. The values are stored in a, da and d2a, one column for each requested time according to times. The dimension is then:

(number of degrees-of-freedom) × (number of requested times + 1)

If the history is to be stored in ahist, dahist and d2ahist for some degrees of freedom, the parameter dofs specifies for which degrees of freedom the history is to be stored. The computed time histories are stored in ahist, dahist and d2ahist, one row for each requested degree of freedom according to dofs. The dimension is:

(number of specified degrees of freedom) × (number of timesteps + 1)

In most cases only a few degrees-of-freedom are affected by the exterior load, and hence the matrix contains only few non-zero entries. In such cases it is possible to save space by defining f as sparse (a MATLAB built-in function).