3.4 SOLVING FOR POSITIONS

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We can solve kinematics problems using a variety of techniques. When the appropriate technique is selected at the right time, it will simplify calculations.

Some of the basic techniques are,

- Trigonometry

- Complex numbers

- Numerical

- Simulation

There are also some specialized techniques for analysis of mechanisms,

Chase Solutions - a closed form approach for planar mechanisms that can be used for numerical calculations.

3.4.1 Trigonometry

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Basically we generate a set of equations that define the shape of the mechanism. To do this we will need to call upon the methods of trigonometry. (Note: recall the sine and cosine laws, as well as the basic trig identities)

Consider the example from the textbook. The offset crank-piston can be solved easily with careful observation,

Next, consider the (crank-rocker) four bar linkage below,

3.4.2 Complex Numbers

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Complex numbers can be very useful when dealing with planar mechanisms.

Recall the basic forms for representing complex numbers,

When adding or subtracting complex vectors we can often use a calculator with complex or polar notations. This will save time and reduce errors.

Consider the example below,

3.4.3 Numerical Solutions

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At times it will be very difficult, or unreasonable to isolate a single variable in an equation.

When this occurs we may take the resulting equation, and set it equal to zero by moving all terms to one side. Then, by trial and error, we can make guesses that will lead to a value equal to zero. One popular way for making these guesses is the Newton-Raphson technique.

Consider the example below,

3.4.4 Simulations

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We can use packages such as Working Model.

A typical method for problem solution might be,

1. lay out the basic components in the mechanism (not necessarily in the final configuration)

2. Apply a driving function to the input joint (e.g., the input angle on the crank)

3. Run the simulation, and generate data/graphs.

4. If motion is satisfactory, check for forces, velocities, accelerations, etc.

Quite often it is possible to write a script that will run a simulation, adjust the mechanism, run the simulation again, and eventually by trial and error an ideal mechanism can be found.

The simulation packages typically contain an integration technique (e.g., Runge-Kutta) that will deal with the dynamics of the problem. The basic process is,

1. Set the initial conditions of the problem, based on the user setup. A time step will also be chosen for the period of integration.

2. Determine the forces on each body in the system. In effect a free body diagram like analysis is done based on applied forces, contact forces, friction, gravity, etc.

3. For each body in the system, integrate the dynamic effects to get a new position, velocity and acceleration.

4. Look at the amount of change that occurred. If too large, reduce the time step. and repeat step 3.

5. Update the model of the system. If the system is no longer changing, stop the simulation.

6. Update the display, and go back to step 2.