Kinematics and Dynamics of Machines
Instructor: S. Farhadi
1.1. The Subject of Kinematics and Dynamics of
•The subject is a continuation of statics and dynamics
•The objective of kinematicsis to develop various means of
transforming motion to achieve a specific kind needed in
•The objective of dynamicsis analysis of the behavior of a
given machine or mechanism when subjected to dynamic
1.2. Kinematics and Dynamics as Part of the
•The role of kinematics is to ensure the functionality of the
•The role of dynamics is to verify the acceptability of induced
forces in parts
1.3. Is It a Machine, a Mechanism, or a Structure?
•The term machineis usually applied to a complete product. A
car is a machine, as is a tractor, a combine, an earthmoving
•Each machine may have some devices performing specific
functions, like a windshield wiper in a car, which are called
•A structuredoes not have moving parts; its function is purely
structural, i.e., to maintain its form and shape under given
external loads, like a bridge, a building, or an antenna mast.
1.4. Examples of Mechanisms; Terminology
Skeleton representation of the
Windshield wiper mechanism
•Crank: A links which is able to make a complete revolution
and may be driven by a motor
•Rocker: A links which is not able to make a complete
•Coupler: A link which connects driver and follower
•Driver: the input link
•Follower: the output link
•Frame (base link):The fixed link
•Skeleton: A representation of the mechanism (replacing the
members with some links which connect essential points)
•Kinematic chain: an interconnected system of links in which
not a single link is fixed. Such a chain becomes a mechanism
when one of the links in the chain is fixed.
•Planar mechanism: a mechanism in which all points move in
•Binary links: links with two connections to other links
•Ternary link: links with three connections to other links.
1.5. Mobility of Mechanisms
•The mobilityof a mechanism is its number of degrees of freedom. This
translates into a number of independent input motions leading toa single
•A single unconstrained link has three DOF in planar motion.
•If the two links are welded together, they form a single link having three
•A revolute joint in place of welding allows a motion of one linkrelative to
another. Thus, the two links connected by a revolute joint have four DOF.
•A revolute joint eliminates two DOF.
•If the constraining condition allows only one DOF between the
two links, the corresponding joint is called a lower-pair joint.
•If the constraint allows two DOF between the two links, the
corresponding joint is called a high-pair joint.
•A low-pair joint reduces the mobility of a mechanism by two
•A high-pair joint reduces the mobility of a mechanism by one
Kutzbach’scriterion of mobility
m = 3(n –1) –2j1 –j2
n: the number of links,
j1 : the number of low-pair joints
j2 : the number of high-pair joints.
Note that1 is subtracted from n in the above equation to take
into account that the mobility of the frameis zero.
Example: Mobility of various configurations of connected links:
(a)n = 3, j1 = 3, j2 = 0, m = 0;
(b)n = 4, j1 = 4, j2 = 0, m = 1;
(c)n = 4, j1 = 4, j2 = 0, m = 1;
(d)n = 5, j1 = 5, j2 = 0, m = 2.
Effect of additional links on mobility
(a) m = 1,
(b) m = 0,
(c) m = -1.
When a structure has negative mobility, it is called an over-constrainedstructure.
•Kutzbach’sformula for mechanism mobility does not take into
account the specific geometry of the mechanism, only the
connectivity of links and the type of connections (constraints).
•Kutzbach’scriterion can be violated due to the non-
uniqueness of geometry for a given connectivity of links.
•In compound mechanisms, there are links with more than two
joints. Kutzbach’scriterion is applicable to such mechanisms
provided that a proper account of links and joints is made.
An example of a compound mechanism with coaxial joints at B.
1.6. Kinematic Inversion
•The process of choosing different links in the chain as frames
is known as kinematic inversion. In this way, for an n-link
chain n different mechanisms can be obtained.
Four inversions of the slider-crank chain: (a) an internal combustion engine, (b) rotary
engine used in early aircraft, quick-return mechanism, (c) steam engine, crank-shaper
mechanism, (d) farm hand pump.
1.7. Grashof’s Law for a Four-Bar Linkage
•Consider a four-bar linkage as presented. In this figure, s
identifies the smallest link, l is the longest link, and p, q are
two other links. Grashof’s law, states that if the sum of the
shortest and longest links is not greater than the sum of the
remaining two links, at least one of the links will be revolving.
•Grashof’s law (condition) is expressed in the form:
s + l =<p + q
Inversions of the four-bar linkage: (a) and (b) crank-rocker
mechanisms, (c) double-crank mechanism, (d) double-rocker
Rotational Speed Ratio
Constant Rotational Speed Ratio
•In order to have a constant rotational speed ratio, the
transmitting line should intersect the center-points line in a
•This condition is valid for a wheel-belt mechanism, but is not
valid for a four-bar mechanism
In order to have a pure rolling contact, the tangential
components of velocity for the contacting points have to be
equal and unidirectional. That happens solely when the
contact point lies on the centers point line.