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DE2-EA 2.1: M4DE Dr Connor Myant

2. Planar Kinematics

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Lecture resources may be found on Blackboard and at http://connormyant.com

http://connormyant.com/

Dr Connor Myant DE2-EA2.1 M4DE 2

Contents Design in Motion ..................................................................................................................................... 3

Rigid Bodies and Types of motion ....................................................................................................... 4

Rotation around a fixed axis: .............................................................................................................. 4

General Motion: Velocity .................................................................................................................... 6

Relative Velocities .............................................................................................................. 6

The Angular Velocity Vector .............................................................................................. 7

Instantaneous Centres ....................................................................................................... 8

Reminder: Cross Product ...................................................................................................................... 10


Design in Motion

This chapter is concerned with the principles of Kinematics; a branch of mechanics concerned with

the motion of objects and systems without reference to the masses of those objects or the forces

which may have caused the motion.

Kinematics is closely linked with geometry; and begins with a description of the geometry of the

system, the initial conditions of position, velocity and or acceleration of various points within the

system. Employing geometrical arguments, it is then possible to determine the position, velocity and

acceleration of any part of the system.

We use kinematics in various engineering displaces to determine the motion of a single body or a

collection of bodies; in mechanical engineering, robotics, biomechanics. As design engineers, we

may use kinematic analysis to describe the range of movement of a given mechanism, or the desired

range of a mechanism.

Design Engineering Example: As design engineers is it vital to have an in-depth understanding of an object’s motion, and function, so we can find the optimal design solution. Consider a bicycle. The gear that is engaged determines the ratio of the angular velocity of the pedals and sprocket to that of the bicycle’s rear wheel. The ratio of the sprocket’s radius to that of the gear equals the ratio of the wheel’s angular velocity to that of the sprocket.


Rigid Bodies and Types of motion

Before we can investigate the motion of objects, we must define the object. In our case we will only

consider rigid bodies; that is an idealised model of an object that does not deform or change shape

during motion or use; i.e. the distance between every pair of points of a rigid body remains constant.

In many cases, objects do deform, but the deformation is so small we can ignore it (negligible).

Describing the motion of rigid bodies requires a reference frame (coordinate system) relative to

which the motions of the points of the rigid body and its angular motion are measured. In most cases

we will employ a reference system that is fixed (to the earth). However, in a later chapter we will

introduce the idea of a moving reference system.

There are three main types of motion that we will study;

1. Translation: If a rigid body in motion relative to a given reference frame does not rotate, it is

said to be in translation.

2. Rotation about a fixed axis: After translation, the simplest type of rigid body motion is rotation

about an axis which is fixed relative to a given reference frame. Each point of the rigid body on

the axis is stationary, and each point not on the axis moves in a circular path about the axis as

the rigid body rotates.

3. Planar motion: Consider a plane that is fixed relative to a given reference frame and a rigid body

intersected by the plane. If the rigid body undergoes a motion in which the points intersected

by the plane remain in the plane, the body is said to be in two-dimensional, or planar, motion.

The fixed plane is referred to as the plane of motion.

Rotation around a fixed axis:

Figure 2.1 shows a rigid body rotating about a fixed axis with two lines perpendicular to the axis.

The angle, 𝜃, between the reference line and the body-fixed line describes the position, or

orientation, of the rigid body about the fixed axis. The rigid body’s angular velocity, 𝜔 (rate of

rotation), and its angular acceleration, 𝛼, are;

𝜔 =𝑑𝜃

𝑑𝑡, 𝛼 =

𝑑2𝜃

𝑑𝑡2 =𝑑𝜔

𝑑𝑡 (2.1)


Figure 2.1. Specifying the orientation of an object rotating about a fixed axis

Each point of the object not on the fixed axis moves in a circular path about the axis. Using our

knowledge of the motion of a point in a circular path, we can relate the velocity and acceleration of

a point to the object’s angular velocity and angular acceleration. Let’s view the object in Figure 2.1

down the axis of rotation, shown in Figure 2.2. The velocity of a point at a distance, 𝑟, from the fixed

axis is tangent to the point’s circular path (LHS) and is given in terms of the angular velocity of the

object by;

𝑣 = 𝑟𝜔 (2.2)

The point has components of acceleration tangential and normal to its circular path (RHS). In terms

of the angular acceleration of the object, the components of acceleration are;

𝑎𝑡 = 𝑟𝛼, 𝑎𝑛 =𝑣2

𝑟= 𝑟𝜔2 (2.3)

Using these equations we can analyse problems involving objects rotating about a fixed axis.

Figure 2.2. (a) Velocity and (b) acceleration of a point of a rigid body rotating about a fixed axis


General Motion: Velocity

In translation every point of a rigid body undergoes the same motion. Under rotation each point of

a rigid body experiences circular motion (about the fixed axis). In order to study more complex

motions that involve both translation and rotation, we must develop equations that relate the

relative motions of points of a rigid body to its angular motion.

Relative Velocities

Figure 2.3(A) shows a rigid body from a perspective perpendicular to the plane of its motion. Points

𝐴 and 𝐵 are points of the rigid body contained in the latter plane, and 𝑂 is the origin of a given

reference frame. The position of 𝐴 relative to 𝐵, 𝑟𝐴/𝐵, is related to the position of 𝐴 and 𝐵 relative

to 𝑂 by;

𝑟𝐴 = 𝑟𝐵 + 𝑟𝐴/𝐵 (2.4)

Taking the derivative with respect to time we get;

𝑣𝐴 = 𝑣𝐵 + 𝑣𝐴/𝐵 (2.5)

Where 𝑣𝐴 and 𝑣𝐵 are the velocities of 𝐴 and 𝐵 relative to the given reference frame and 𝑣𝐴/𝐵 is the

velocity of 𝐴 relative to 𝐵.

We can show that 𝑣𝐴/𝐵 is related in a simple way to the rigid body’s angular velocity. Since 𝐴 and

𝐵 are points of the rigid body, the distance between them, 𝑟𝐴/𝐵 = |𝑟𝐴/𝐵| is constant. That means

that, relative to 𝐵, 𝐴 moves in a circular path as the rigid body rotates. That velocity of 𝐴 relative

𝐵 is therefore tangent to the circular path and equal to the product of 𝑟𝐴/𝐵 and the angular velocity

𝜔 of the rigid body (Figure 2.3(B)). From Equation (2.5), the velocity of 𝐴 is the sum of the velocity

of 𝐵 and the velocity of 𝐴 relative to B (Figure 2.3(C)). This result can be used to relate velocities of

points of a rigid body in planar motion when the angular velocity of the body is known.

Design Engineering Example: Consider a piston system to observe all three types of motion. The piston will be enclosed in a housing, and restricted to linear translation. A crankshaft will engage with the connecting push rod to cause the piston to move up and down in a reciprocal fashion. The crankshaft is undergoing rotational motion, whilst the connecting rod is experiencing planar motion.

The piston’s linear velocity and acceleration will be determined by the angular velocity of the crank and the dimensions of the connecting push rod.


Figure 2.3. A) rigid body in planner motion, B) The velocity of A relative to B, C) The velocity of A is

the sum of its velocity relative to B and the velocity of B

The Angular Velocity Vector

We can express the rate of angular rotation of a rigid body as a vector. This is termed the angular

velocity vector, and given as �⃑⃑� = 𝜔𝑘. Let’s consider points 𝐴 and 𝐵 of a rigid body with angular

velocity �⃑⃑� (Figure 2.4(A)). We can show that the velocity of 𝐴 relative to 𝐵 is;

𝑣𝐴/𝐵 =𝑑𝑟𝐴/𝐵

𝑑𝑡= �⃑⃑� × 𝑟𝐴/𝐵 (2.6)

Relative to 𝐵, point 𝐴 is moving in a circular path of radius |𝑟𝐴/𝐵| sin𝛽, where 𝛽 is the angle between

the vectors 𝑟𝐴/𝐵 and �⃑⃑� (Figure 2.4(B)). The magnitude of the velocity of 𝐴 relative to 𝐵 is equal to

the product of the radius of the circular path and the angular velocity of the rigid body;

|𝑣𝐴/𝐵| = (|𝑟𝐴/𝐵| sin𝛽)|�⃑⃑� |

The right hand side of this equation is the magnitude of the cross product of 𝑟𝐴/𝐵 and �⃑⃑� . In addition,

𝑣𝐴/𝐵is perpendicular both to �⃑⃑� and 𝑟𝐴/𝐵.

By combining Equations. (2.5) and (2.6) we obtain an equation for the relation between the

velocities of two points of a rigid body in terms of its angular velocity;

𝑣𝐴 = 𝑣𝐵 + �⃑⃑� × 𝑟𝐴/𝐵 (2.7)

If the angular velocity vector and the velocity of one point of a rigid body are known, Eq. (2.7) can

be employed to determine the velocity of any point of the rigid body.


Figure 2.4. a) Point A and B of a rotating rigid body, b) A is moving in circular path relative to B

Instantaneous Centres

A point on a rigid body whose velocity is zero at a given instant is called an instantaneous centre.

Meaning the point may have zero velocity only at the instant under consideration. Although we also

refer to a fixed point, such as a point on a fixed axis about which a rigid body rotates, as an

instantaneous centre.

When we know the location of an instantaneous centre of a rigid body in two-dimensional motion

and we know its angular velocity, the velocities of other points are easy to determine. The

instantaneous centre of a rigid body in planar motion can often be found by a simple procedure.

Suppose that the directions of the motion of two points 𝐴 and 𝐵 are known and are not parallel

(Figure 2.5). If we draw lines through 𝐴 and 𝐵 perpendicular to their directions of motion, then the

point 𝐶 where the lines intersect is the instantaneous centre.

Design Engineering Example: A cam is a rotating shape whose angular motion is converted into output motion by a cam follower which rides on the cam surface. As the cam turns, driven by the circular motion, the cam follower traces the surface of the cam transmitting its motion to the required mechanism. Cams are employed in numerous applications and products; from engines and pumps to door locks and toys.

To determine the motion of the follower we need to calculate the motion of the cam at any point; we normally do this relative to its centre of rotation, or the cam shaft.


Figure 2.5. Locating the instantaneous center in planar motion.

Design Engineering Example: Instantaneous centres is a useful method for studying linkage systems. Linkages can be used to change direction, alter speed and change the timing of moving parts. You can find simple linkages on bathroom mirror stands, moving platforms and clamps.

The linkage system on a Digger Arm is a good example of a linkage system being employed to gain a mechanical advantage.


Reminder: Cross Product

The cross product of two vectors 𝑎 and 𝑏 is defined only in three-dimensional space and is denoted by 𝑎 × 𝑏.

Figure 2.7. The cross-product in respect to a right-handed coordinate system.

The cross product 𝑎 × 𝑏 is defined as a vector 𝑐 that is perpendicular to both 𝑎 and 𝑏, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.

Figure 2.8 Finding the direction of the cross product by the right-hand rule.

The cross product is defined by the formula;

𝑎 × 𝑏 = ‖𝑎‖‖𝑏‖ sin(𝜃)𝑛

where 𝜃 is the angle between 𝑎 and 𝑏 in the plane containing them (hence, it is between 0° and 180°), 𝑎 and 𝑏 are the magnitudes of vectors 𝑎 and 𝑏, and 𝑛 is a unit vector perpendicular to the plane containing 𝑎 and 𝑏 in the direction given by the right-hand rule (illustrated). If the vectors 𝑎 and 𝑏 are parallel (i.e., the angle 𝜃 between them is either 0° or 180°), by the above formula, the cross product of 𝑎 and 𝑏 is the zero vector 0.


For our purposes we will most likely use a matrix notation for the cross product. Lets start by expressing a cross product as:

𝑢 × 𝑣 = |𝑖 𝑗 𝑘𝑢1 𝑢2 𝑢3

𝑣1 𝑣2 𝑣3

|

This determinant can be computed using Sarrus' rule or cofactor expansion. Using Sarrus' rule, it expands to;

𝑢 × 𝑣 = (𝑢2𝑣3𝑖 + 𝑢3𝑣1𝑗 + 𝑢1𝑣2𝑘) − (𝑢3𝑣2𝑖 + 𝑢1𝑣3𝑗 + 𝑢2𝑣1𝑘)

𝑢 × 𝑣 = (𝑢2𝑣3 − 𝑢3𝑣2)𝑖 + (𝑢3𝑣1 − 𝑢1𝑣3)𝑗 + (𝑢1𝑣2 − 𝑢2𝑣1)𝑘

Using cofactor expansion along the first row instead, it expands to

𝑢 × 𝑣 = |𝑢2 𝑢3

𝑣2 𝑣3| 𝑖 − |

𝑢1 𝑢3

𝑣1 𝑣3| 𝑗 + |

𝑢1 𝑢2

𝑣1 𝑣2| 𝑘

which gives the components of the resulting vector directly.

Figure 2.9. Use of Sarrus' rule to find the cross product of u and v

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2. Planar Kinematics - WordPress.com · 2.10.2017 · 2. Planar Kinematics ... intersected by the plane. If the rigid body undergoes a motion in which the points intersected by the