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Department of Mechanical, Materials and Manufacturing Engineering

Dynamics MM2DYN

Lecture 3: Kinematics of Planar Mechanisms

3.1 Introduction

3.2 Space Diagrams

3.3 Relative Motion

3.4 Relative Position

3.5 Relative Velocity

3.6 Relative Acceleration

3.7 Application to Planar Mechanisms

3.1 Introduction

Mechanisms are widely seen in engineering when a body must follow some specific movement.

Examples included production line machinery, car suspension, or a robotic arm. In fact, any

application that connects a number of rigid body objects together could be considered a

mechanism. Remember that the process of connecting the objects together constrains some of

the degrees of freedom of the system. These constraints allow us to analyse the motion of the

mechanism using the results found in the rigid body kinematics lecture.

Industrial engineers who design mechanisms in a professional capacity will use engineering

software to perform the kinematic analysis and simulate the motion. However, the engineer will

often carry out rough calculations before committing to the design in CAD to ensure the design

is roughly correct.

The aim here is to present the basic principles involved in analysing the kinematics of

mechanisms in order to develop some engineering insight, so that you can take away some

experience that will help in designing mechanisms in the future.

The equations and concepts presented here build on the kinematics of rigid bodies (section 2)

and the kinematics of particles (section 1.3 and EM1 notes on WebCT).

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3.2 Space Diagrams

A space diagram is a sketch of the mechanism at the particular instant in time that the analysis

is performed. It shows the geometry and relative motion of bodies in the mechanism. Below is

an example of a space diagram for a mechanism:

3.3 Relative Motion

At the end of the previous lecture, we noted that the motion of some point on a connected body

could be found more easily by considering the relative motionof each connected body. So for

the example simple robotic arm with two pin joints and a sliding joint:

The absolute motion at C can be found in terms of the relative motion of each section back to

the absolute position O. So:

_ = _ + _ + _

_ = _ + _ + _

_ = _ + _ + _

O

AB

C

O

A

B

CD

DC

BO

A

Real Mechanism Space Diagram

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It is necessary to establish a formal notation to describe the relative motion of one point relative

to another (the motion of A relative to B is not the same as the motion of B relative to A). For

the rest of this module, the subscript _refers to at point A relative to point B. This matches

the notation in the Introduction to Mechanical Engineering: Part 1 text book. So for example:

_ is the position vector of point C relative to point B

_ is the velocity vector of point C relative to point O

_ is the acceleration vector of point B relative to point A

3.4 Relative Position

Finding the position of one point relative to another can be achieved by analysing the geometry

of the space diagram. The vector position of point B relative to point A (_) can be shown

pictorially as:

Note that the vector position of B relative to A is opposite to the vector position of A relative to B,

i.e. _ = _ . As normal, the magnitude (or length) of the position vector of B relative to A

will be denoted by _ .

So now the position of B can be found from the position of A (relative to O) plus the position of B

relative to A:

_ = _ + _

A

B

_

_

_

O

A

B

_

_

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Note that we could also find the position of A if the position of B and their relative position is

known:

_ = _ _

= _ + _

3.5 Relative Velocity

The velocity at any point P is equal to the velocity at point A plus the additional velocity due to

the movement of P relative to A:

= + _

This is true for any two moving points the relative velocity _ is the velocity that P would

appearto have if you imagine standing at point A (so poin t A doesnt appear to be moving).

Relative velocity within a rigid body revisited

In section 2.5.3 we have shown that for any point in one rigid body moving with planar motion

that the velocity at point P is the resultant of the linear velocity at A () and the rotation of the

body ( ). I.e. in the case of a point within one rigid body, the relative motion is just due to the

rotation.

Similarly to relative position, the velocity at A could be found in terms of the velocity at P too -

= v . This makes sense as if you imagine standing at point P looking back at point A,

the clockwise rotation of the body would create a relative velocity term in the opposite direction

at A.

A

P

= + v

Relative velocity within one rigid body

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When rigid body objects are connected together in a mechanism, we need to look at the

connection constraints to establish the relative velocity terms. Unlike any two points within a

single rigid object, the distance between two connections in a mechanism may change (e.g. the

length from A to B changes in the mechanism below because of the sliding connection).

Working from the polar particle kinematics (see section 1.3), we can say that the velocity of B

relative to A has a radial term and an angular term:

In the example above, point A isnt moving, so the velocity at B is the same as the relative

velocity_.

In more general terms, if point A wasnt grounded, but free to move with some velocity, then

the total velocity at some point P would be = + _ :

A

P_

_

_

__

A

B_

__

_

_

_

O

A

B

CD

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3.6 Relative Acceleration

The acceleration at any point P is equal to the acceleration at point A plus the additional

acceleration due to the movement of P relative to A:

= + _

This is true for any two moving points the relative acceleration _ is the acceleration that P

would appearto have if you imagine standing at point A.

When rigid body objects are connected together in a mechanism, we need to look at the

connection constraints to establish the relative velocity terms. Unlike any two points within a

single rigid object, the distance between two connections in a mechanism may change (e.g. the

length from A to B changes in the mechanism below because of the sliding connection).

Working from the polar particle kinematics (see section 1.3), we can say that the acceleration of

B relative to A has a radial term and an angular term:

In the example above, point A isnt moving, so the acceleration at B is the same as the relative

acceleration_.

In more general terms, if point A grounded, but free to move with some acceleration , then the

total acceleration at some point P would be = + _ :

A

B _ __2

__ + 2_

_

_

_

_

_

O

A

B

CD

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