Forced Vibrations - Programming the Finite Element Method - page 481

val applied nodal load weightings
weights weighting coefficients
x0 “old” displacements

“new” displacements

x1

In this program whose structure chart is given as Figure 11.8, the same problem as
that previously analysed is solved again using a direct time-integration procedure. In this
example the tip loading takes the form of a continuous cosine function defined in a FUNC-
TION subprogram called load placed at the end of the main program. The specific method
is described in Section 3.13.2 where it was shown to be the same implicit technique as was
used for first order problems in Program 8.1, where it is often called the “Crank-Nicolson”
approach ( θ = 0 . 5). In second-order problems, it is also known as the “Newmark β = 1 / 4”

method in which form it was used in Program 11.1.
To step from one time instant to the next, a set of simultaneous equations has to be
solved. Since the differential equations are often linearised, this is not as great a numerical
task as might be supposed because the equation coefficients are constant, and need be
factorised only once before the time-stepping procedure commences (see equation (3.139)).
Velocities and accelerations are computed by ancillary equations (3.140) and (3.141).

Read data
Allocate arrays
Find problem size
Null global stiffness and mass matrices

For all elements

Find element geometry
and steering vector

For all Gauss points

Compute element stiffness and
mass contributions
to km and mm

Assemble km into kv
Assemble mm into mv

Set the initial conditions
Reduce left-hand side

For all the time steps

Update time
Set the forcing function
Assemble the new right-hand side
Complete the equation solution.
Compute new velocities and accelerations
Print results

Figure 11.8 Structure chart for implicit algorithms used in Programs 11.3 and 11.4