Strand7:  Web notes:  Mechanisms

Using Strand7 to analyse mechanisms
The nonlinear geometric solver in Strand7 is capable of analysing the behaviour of mechanisms, similarly to some kinematics packages on the market. The main difficulty we need to overcome when attempting this kind of analysis with finite elements is that a pure mechanism will generate a singular stiffness matrix. There are three ways to deal with this:

  1. Add a small amount of stiffness somewhere;
  2. Use constraint equations to eliminate the singularity altogether; or
  3. Use the nonlinear transient dynamic solver and solve the problem in the time domain.
The first option works provided we can determine the correct amount of stiffness. Basically if you add too much stiffness, you no longer have a mechanism, and if you add too little stiffness, you may still have a "weak" singularity. You can also generate an ill conditioned matrix using this approach if you have a relatively high stiffness attached to a low stiffness.

The second option works provided you can specify a general enforced displacement that can vary with load increment. This provides a convenient way of defining the path for the mechanism.

For some applications, the finite element approach is superior to the traditional kinematic analysis method because with FEA you can model the true stiffness of the linkages, without assuming that they are rigid.

Two simple examples are shown below:
  1. The movement of a racing car double wishbone suspension;
  2. A crane boom being lowered.
The method is sufficiently general that it may be used in 2D as well as 3D, and is suitable for analysing a wide range of structures such as scissor-lift platforms, robotic arms, deployable structures, etc.

EXAMPLE 1: Four-link suspension including spring
Double Wishbone Suspension

The above picture shows a 2D representation of a double wishbone suspension system. This is typical of the setup found on most open-wheeler racing cars and also on some roadgoing sports cars. With this system, the linkage geometry (i.e. lengths, locations of pick-up points, etc.) determines the behaviour of the wheel in "bump" (one wheel moving up and down), "droop" (car body moving up and down) or "roll" (car body rolling due to lateral forces such as going around a corner). The stiffness of the spring determines how much the suspension moves relative to the body.

Using the Strand7 geometric nonlinear solver, we can determine not only the path the wheel will take under a given loading condition, but also how much it will move and the forces in all the members. It would also be possible to model the wheel and tyre, taking into account the friction between the tyre and road and the deformation of the tyre. If you model the system using the nonlinear transient dynamic solver, you can also take into account rates of change and the effect of a damper.

EXAMPLE 2 : Crane boom being lowered

Crane Boom

The above picture shows a crane boom being lowered. The Strand7 nonlinear solver was used to determine the force F required to balance the weight W, over the range of angles considered. The graph below shows the required force as a function of boom angle.

Force vs Displacement for boom

To solve this model, we apply a prescribed displacement at point A on the model, of the form DY(A)=constant. We use the geometric nonlinear solver to increment the displacement and we recover the force F.