22. Position and Displacement of Points and Mechanisms

• The calculation of the position of mechanisms is an essential first step to modeling how they work.

• To do the calculations a number of basic mathematical tools are used.

22.1 Mathematical Tools

• Recall the basics,

coordinate systems


Left/Right Hand Rules






unit vectors

dot products and projected values

cross products

triple product

we can define locations of points using,

direction cosines

polar location

Cartesian values

projected coordinates

cylindrical coordinates.

spherical coordinates

parametric values


Complex math, including exponential form

basic graphing

rotation and translation






22.2 Defining Positions and Displacements

• A point can be defined using a variety of methods. For our purposes, Cartesian coordinates will be predominantly used.

• We will largely rely upon vectors to define positions.


• We must pay attention to the fact that the vector has a direction and magnitude. The direction is relative to the orientation of the coordinates (which YOU will choose arbitrarily). In more advanced problems we will use multiple coordinate axes.

• We can add vectors easily if they are relative to the same reference coordinates,


• So far we have reviewed vectors that are all relative to the same coordinate axes.

• Now, consider what would happened if we had multiple coordinate axes.


Problem 22.1 If the apparent position of point P is (-1,-2,4), what is the absolute position of P? What is the absolute position difference between O2 and P?

22.3 Closed Loop Mechanisms

• If we have a kinematic chain that forms a closed loop, the sum of the vectors for the links must total zero.



• 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,


Complex numbers



• 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.

22.4.1 Trigonometry

• 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,


Problem 22.2 Given the crank angle (at A) find the rocker angle (at D). Given the rocker angle, find the crank angle. Assume, and add, and all required dimensions.

22.4.2 Complex Numbers

• 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.

Problem 22.3 The crank slider mechanism is in the position where BC is vertical.a) If the crank angle is 25°, find the position of C using Complex Polar Notation, b) repeat using complex numbers only. c) How can the solution be checked for accuracy?

22.4.3 Numerical Solutions

• 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.

Problem 22.4 Find the final resting position for the disk on the curved surface.

22.4.4 Simulations

• 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.

22.5 Graphing Positions

• We can graph the positions of point on mechanisms using the techniques covered so far.

• Note that we have been plotting joint positions so far, but in actual practice we will be using other point on the links.

Problem 22.5 Trace the path for point E, as the crank is rotated about joint A. Try this using calculations, mathcad, working model, and a computer program.

22.6 Displacement, Translation, and Rotation

• As a mechanism is moved, points on the mechanism will also move through space. The problem is that they will continue to be the same points, but we need to indicate the difference in position (displacement) for calculation purposes.

• We will indicate position using the traditional delta symbol,


• We can also consider relative positions between points on rigid bodies,


• Translation of a rigid body indicates that all points are traveling in the same direction at the same velocity. In this case the displacements of both points will be equal for pure translation.


• Rotation occurs when the displacements of two points on a body are not equal. If one of the displacements is zero, then it is the center of rotation.

• Apparent displacement considers that the coordinate axes are also moving. In this case the displacement and apparent position notations are combined.


Problem 22.6 Find the motion of the point B using complex polar notation, and then matrices.

22.7 Problems

Problem 22.7 Crank CB is 6” in length. Find the location of point D as a function of the crank angle at B using either complex numbers or complex polar notation.

Problem 22.8 The test pilot of the aircraft below is doing loops to evaluate the cockpit cup holders. Part way through one loop he notices (apparently) another plane one mile straight ahead from his position. If he is flying upside down at the location shown below, what is the absolute position of the other plane relative to the control tower. Note: use proper notation to solve this problem.

22.8 References

22.1 Erdman, A.G. and Sandor, G.N., Mechanism Design Analysis and Synthesis, Vol. 1, 3rd Edition, Prentice Hall, 1997.

22.2 Shigley, J.E., Uicker, J.J., “Theory of Machines and Mechanisms, Second Edition, McGraw-Hill, 1995.