21. Introduction to Kinematics of Mechanisms

• Consider a pair of adjustable vice grips.

•Some definitions,

Machine: a collection of components that will do work.

Mechanism: a collection of components to transform motion

Kinematics: consider positions/velocities/accelerations in mechanical systems

Structure: a collection of components to make larger static structures

Statics: estimate forces in mechanisms that are in equilibrium

Dynamics: determine motion that results when forces are out of balance

Link: rigid body between joints

Binary Link: has two joints only

Ternary Link: has three joints

Quaternary Link: has four joints

Pair or Joint: a connection between two links

Driver / Follower: the driver link will be driving the follower

Kinematic Chain: a sequence of links making up a mechanism

Open Loop: a snake like set of connected links

Closed Loop: a kinematic chain has one or more links that go back in the chain

Frame: a grounded or fixed link in a mechanism

Spatial: in 3 dimensions

Relative/Absolute: a position, velocity, etc. is measured based on a fixed (absolute) or moving (relative) point.

• A Degree Of Freedom (DOF) is an independently controllable variable. As an example, a machine that has two degrees of freedom might need two motors to control it.

• Lower Pairs,: constrained position/orientation of both sides of the joints are identical

Turning / Revolute: basically a pin joint (R)

Prismatic: a slider (P)

Screw/Helix: a nut and screw pair (H)

Cylindric: a shaft in a collar (C)

Globular/Spherical: a ball joint (S)

• Higher pairs include,: typically other equations are needed to constrain the joints, such as gear ratios (if the joint has more than a single degree of freedom)

flat/planar: constrained to move over a plane

belt on pulley

meshing gears

sliding wheel on a surface


• The definition of higher/lower pairs given in Shigley [1995] is, “the lower pairs, such as the pin joint, have surface contact between the pair elements, while higher pairs, such at the connection between a cam and its follower, have line or point contact between the surface elements.” They go on to point out that the definition is not exact, which is somewhat disappointing.

• A better definition of a higher pair is: A higher pair is not a lower pair, where a lower pair permits the following relative motions between links; circular, linear, helical, cylindrical, spherical, planar.

• If a link has one joint, it is a unary link. A link with two joints is binary, with three it is ternary, with four it is quaternary, etc.

• Planar linkages use lower order pairs, and are constrained to a single plane of motion.

21.1 Some Popular Mechanisms

• Some basic mechanism types are listed below, and split into some suggested categories

21.1.1 Locking/Engaging


Indexing Mechanisms: e.g. the Geneva mechanism


Reversing Mechanisms: A mechanism that can disengage a transmission, and reverse direction of transmission

21.1.2 Motion Transmission/Transformation

21.1.3 Four Bar Linkages

Swinging or Rocking Mechanism: produce cyclic motions


Stop/Pause/Hesitation: a motion is produced that appears to come to a stop for a short period of time.


Curve Generators: mechanisms set up to follow complex paths: typically four bar linkages.

Straight Line: mechanisms are set up to generate straight line motions


21.1.4 Reciprocating

Reciprocating Mechanisms: converts a rotational motion to a linear motion



21.1.5 Six Bar Linkages

• These allow more complex motion, especially when ternary links are used.

• In Watt linkages there are two ternary links touching,


• In Stephenson linkages the ternary links don’t touch,


21.2 Skeleton Diagrams for Mechanisms

• Many of the mechanisms drawn so far have been in a ‘stick’ or ‘skeleton’ form.

• We will use this representation to do abstract work.

• Convert the mechanism below to a skeletal form,


21.3 DOF and the Kutzbach/Gruebler Criterion

• Planar mechanisms can be made up of a number of joints and simple links.

• We can calculate a number that represents the mobility of a mechanism. If the value of ‘m’ is zero, then the mechanism will be rigid. If ‘m’ is less than zero, then the mechanism is rigid and over-constrained. When larger than zero, there are ‘m’ d.o.f. (degrees of freedom)


• Some examples of joints are

lower pairs (1 dof)

pin joint


roller with no sliding

higher pair (2 dof)

roller with sliding

even higher pairs (3 dof)


• Consider the following examples with simple links,




• Next consider a more complex system. In this case the slider acts as another link.


• In trusses we have a number of links meet at a joint. In this case we assume that one of the links provides the pin to the joint: each other link adds a new joint acting on that pin.



• If we have higher pairs, then we must include these,


21.4 Kinematic Geometric Inversion

• Although relative motions between components remain the same, we can pick different links to be the stationary frame.

• As different links are chosen to be grounded the other links will move, and this can lead to radically different effects.

• Consider the example below,


• For a geometric inversion we reconnect links so that they are different from their original configuration.

21.5 Grashof’s Law

• If we want a mechanisms that will undergo continuous motion we must satisfy the Grashof criterion. In particular the shortest link, with length ‘s’ will be in constant relative motion if the Grashof Criterion is satisfied. (In other words we can rotate the smallest link continuously)


•. Consider the equation, and the four basic kinematic inversions below. Keep in mind that the crank will be the shortest link, with length ‘s, and in all four cases will rotate continuously.




• Consider the example problem below,


21.6 Mechanical Advantage

• As a mechanism moves over a range of motion its geometry changes. If we are using a mechanisms to transmit torque, or force then we must consider the ratio between the input and output force in various positions.

• Transmission angle is the angle between the coupling member and the output member in a mechanism. As this angle approaches ±90°, the mechanical advantage of the mechanism typically increases.

• Toggle positions occur when the input crank has near infinite mechanical advantage. Note: this also applies that the follower has no mechanical advantage on the crank.

• Consider the example below, [prob. 1-3 from Shigley & Uicker],



21.7 Problems

Problem 21.1 Draw the kinematic inversions of the linkage below. Does it satisfy the Grashof Criterion? What is the mobility of the mechanism? Determine the maximum transmission angles, and the toggle angles if CD is the crank. What is the advance-to-return time ratio?

Problem 21.2 Draw the kinematic inversions of the linkage below. Does it satisfy the Grashof Criterion? What is the mobility of the mechanism?

Problem 21.3 The human arm is an open loop kinematic chain. Assuming the shoulder is the ground, what is the mobility of the human arm, including all joints down to the finger tips?

21.8 References

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

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