Download Planar kinematics of a rigid body: Review

ME 316
2008/2009 year
Lecture note 1
Lecture Note (1)
Kinematics of a rigid body
Table of contents:
1. Basic concepts
2. Classification of problems
3. The scope of problem solving using the equation in the GE226 course
1. Basic concepts
We begin with the review of the following concepts:
(1)
(2)
(3)
(4)
Rigid body,
Planar motion / mechanism vs. spatial motion and mechanism,
Types of planar motions, and
Relative motions.
1.1 Rigid body and rigid body versus particle
There are two definitions for Rigid Body:
Definition 1.1: A rigid body consists of a set of particles that have a fixed distance. The
relationship among these particles does not change with respect to time.
Definition 1.1 takes a view from the constitution of a physical object.
Definition 1.2: A body is considered as rigid body when there are at least two particles of
the body that have different motions or when there are at least two forces along the motion
direction that do not pass one point.
Definition 1.2 takes a view from purposes of motion or force analysis.
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A
Particle: A & B have the same motion
B
(a)
A
Rigid body: A and B may have
different motions
B
Figure 1.1
(b)
Motion
In Figure 1.1a, two particles on a body have the same motion (both the direction and
magnitude), thus the whole body is a particle instead of rigid body, though according to
Definition 1.1, the body is a rigid body (because the relationship among all the points of the
body does not change with respect to time).
In Figure 1.1b, when two particles on the body perform different motions, they should be
treated as a body not a particle.
The simplest example of a rigid body in practice is a disk that rotates about a fixed point
with a constant angular velocity (Figure 1.2).
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A
VA
ω
B
VB
Figure 1.2
Since the velocity of point A and the velocity of point B on the disk are different (note:
their directions are different, though they have the same magnitude), the whole disk cannot
be treated as a particle; it has to be treated as a rigid body.
Force
Figure 1.3
The distinction of the particle from the rigid body may also be attributed to the purpose of
force analysis. Figure 1.3 shows a sketch of a car. When we are interested in its motion,
which is a translation along the flat road, the entire car is viewed as a particle not a rigid
body. For example we can write the Newton’s second law for the entire car as F=ma. But
when we want to consider the support force – the force acting on the car through the wheels
from the flat road, the car is considered as a rigid body because the two forces may be
different at the two wheels, respectively.
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In summary, the notion of particle is the abstraction of the real world body or entity or
object. In particular, this notion allows us to view a real body as a point or particle so that
the analysis can be simplified. When a body is considered as a particle in motion analysis,
all the particles on the body must have the same motion. When a body is considered as a
particle in force analysis, all the forces must pass through one point on the body (see Figure
1.4).
Force
Force
Figure 1.4
1.2 Planar motion
Definition 1.3 (for a single body):
All particles on the body move in one plane or several planes that are, however, in parallel.
Definition 1.4 (for a group of bodies):
All bodies must move in one plane or several planes that are in parallel.
Examples:
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Example 1.1 (Figure 1.5): To determine whether the system is a planar system or spatial
system.
Figure 1.5
Analysis 1.1: All the moving bodies (the pulley and the rope) lie in one plane, i.e., the sheet
plane. Therefore this system is a planar system.
Example 1.2 (Figure 1.6): To determine whether the system is a planar system or not.
Analysis 1.2: Because the small gear and the large gear move in the planes which are not in
parallel; in fact they are in perpendicular to each other. So the motion system or mechanism
is not a planar motion system or planar mechanism. This system is a spatial mechanism or
motion system.
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Figure 1.6
Example 1.3 (Figure 1.7): To determine whether the system is a planar or spatial system.
Figure 1.7
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Analysis 1.3: In this mechanism, because Bar AB and the disk do not move in one plane or
planes which are in parallel to each other, the mechanism is certainly a spatial mechanism.
1.3 Type of Planar motions
In motion analysis, after we determine that a system under interest is a planar one, we need
to understand types of motions. This is because the governing principles for kinematics of
different types of motions are different. The simplest situation is well known to us, i.e.,
translation. When a body performs the translation motion, all the particles (or points) on the
body have the same motion. In fact, in the case of the translation motion, the whole body
can be viewed as a particle.
There are three types of planar motions:
Type 1: Translation (Figure 1.8)
1
1
1
1
1
(a)
1
1
1
(b)
Figure 1.8
In the case of translation motion, every line segment on the body remains parallel to its
original direction during motion. In Figure 1.8, although the gross path of the object
moving from the left to the right is a curve, the line denoted by 1-1 remains parallel for
these two positions, and thus, the motion is still translation in nature.
Example 1.4 (Figure 1.9) (Translation):
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In machine design practice, a typical system is shown in Figure 1.9, where two opposite
side links have the same lengths. The system is called the parallelogram mechanism. In this
system, link 2 performs a planar translation.
2
Ground or Frame
Figure 1.9
Type 2: Rotation (Figure 1.10).
When a body rotates about a fixed axis, all particles on the body move along circular paths
(see Figure 1.10)
In Figure 1.10, the body has infinite number of planes which cut the axis perpendicularly.
All the particles on these planes move along their corresponding circular paths,
respectively. It is noted that we do not require that all the particles move along one circular
path, but every particle must move along a circular path in the plane which cuts the axis
perpendicularly.
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axis
Particle moves along
a circular path in a
plane which cuts its
axis perpendicularly.
Figure 1.10
Type 3: General motion (Figure 1.11)
1
1
1
1
Position
Figure 1.11
The general planar motion is a superposition of the translation and the rotation.
Consequently, a line on a body will change both its orientation and its translation at two
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positions, and there will be a point on a body which changes its position; see Figure 1.10
(see Figure 1.11, line 1-1 at two positions).
One needs to notice that the sequence of the translation and rotation does not affect the final
result. In analysis, we usually consider that a body first translates to a position and then
rotates about the axis that passes through the center of mass of the body and is
perpendicular to the motion plane.
1.4 Relative Motion
Motion of any object will have a reference object. When we say a motion of the object, see
Figure 1.12, there is a reference object, that is, the earth. The earth is assumed to have
motionless. In this case we call the motion of the object “absolute motion”.
Definition 1.5: When the motion of an object is viewed by taking the earth as the reference
body, the motion of the object is called the absolute motion.
A reference object
An object
The earth
Figure 1.12
When the reference object is not the earth, we are talking about the relative motion between
object A and object B. In the case of planar motions, we usually consider two types of
relative motions: the relative rotation and relative translation.
The relationship between the relative rotation (angular velocity) and two absolute rotations
(angular velocities) of the two objects, respectively, is expressed by
 A / B   A  B
(1.1)
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where ωA and ωB: the absolute angular velocities of body A and body B, respectively, and
 A / B is the relative angular velocity between body A and body B – in particular, with the
subscript ‘a/B’, it means tha the rotation of body A is our interest, and body B is a
reference body for observing the rotation of body A.
There are several cases about the relative rotation that are commonly seen.
Case (i): Two bodies are joined by a sliding/slot; see Figure1.14. In this case, the relative
rotation between body A and body B is zero. That is to say; we have  A / B  0 ,  B / A  0 ,
and  A   B .
A
B
 A  B
Figure 1.14
Case (ii): Body A performs translation, while body B rotates, or vice versa (Figure 1.15).



In this case, we have  B  0 , so  A / B   A  B   A . It is noted that when we discuss the
relative motion, we do not care about whether the two bodies are in fact connected or not.
In other words, contrary to the intuition of most people, we still can talk about their relative
motion even two objects are not in a direct connection.
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A
B
V0
Figure 1.15
Let us consider back the connection pattern as shown in Figure 1.14 where two bodies A
and B perform a relative translation motion, as well as have the same rotation, because they
are connected through a slider. Their relative translation velocity can always be found by
picking up one point on body A (PA) and one point on body B (PB) – see Figure 1.16, and
then by the following equation:
T-axis
Figure 1.16
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V A / B (T )  VPA / PB(T )  VPA  VPB
(1.2)

Where V A / B (T ) :
the relative translation velocity of body A with respect to body B.

VPA / PB(T ) : the relative translation velocity of point PA (on body A) with respect to
point PB (on body B).
the absolute velocity of point PA.
V PB :
the absolute velocity of point PB.
V PA :
If the PA and PB are instantaneously coincident on the same location (see Figure 1.17
where B1 and B2 are coincident on the same location B), the distance between PA (B1 of
Figure 1.17) and PB (B2 of Figure 1.17) is zero; yet Equation (1.2) is still valid. That is to
say; contrary to the intuition of most people, the relative velocity between point B1 and B2
is not zero (see Figure 1.17).
  2
1
1
2
3
B (B1, B2, B3)
Figure 1.17 (B1: on body 1; B2: on body 2; B3: on body 3; body 3 and body 2 are
connected by a revolute joint or pivot; body 1 and body 2 are connected to able to translate
relative to each other.
Example 1.5 (Figure 1.18)
Figure 1.18 is an assembly of gears and link (i.e., arm). Arm AB rotates with an angular
velocity 2 rad/sec counterclockwise. The angular velocity of B is constant 0 rad/sec. Gear A
is free to rotate on its axle. Answer the following questions:
Question (a). What is the relative angular velocity between gear B and arm AB, in
particular taking arm AB as a reference object?
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Question (b). What is the angular velocity of gear A if we know that the relative angular
velocity of gear A with respect to gear B is 0.5 rad/sec clockwise?
Solution to (a).  B / AB   B   AB  0  2  2 (clockwise)
Solution to (b).  A   B   A / B  0  (0.5)  0.5 (clockwise)
Figure 1.18 (A: gear; B: gear; arm rotates with gear A; arm rotates with gear B;
gear A and arm both rotate with the ground).
1.2 Classification of problems of kinematics
1.2.1 Kinematic pairs
In kinematic design and analysis, we do not care about the mass, mass distribution and
shape of a body. In Figure 1.19, the body with a shape is abstracted to the link. For the link,
we only care about the distance between A and B, i.e. L (Figure 1.19).
B
B
A
A
Body
Link
Figure 1.19
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Bodies are connected through joints. Though the joint has geometry, we only care about the
relative motions that are permitted, by the joint, between two bodies. Figure 1.20 shows
types of joints, their permitted relative motions between two bodies, and their graphical
representations.
Types of joints
Graphical Representation
Revolute joint (or pair)
Sliding joint (or pair)
Fixed joint (or pair)
Point contact joint (or pair)
Relative motion
Relative rotation
Relative translation
Fixed (no relative
motion)
Both relative rotation
and translation
Low pair: revolute and sliding pair; High pair: point contact pair
Figure 1.20
1.2.2 Classification of kinematic mechanisms (see Figure 1.21)
Type 1: All joints are a revolute pair.
Type 2: There is a sliding pair but its sliding axis is fixed with respect to the ground.
Type 3: There is a high pair with the bar body fixed with respect to the ground.
Type 4: There is a high pair with both the bar and cam body in motion.
Type 5: There is a sliding pair with its sliding axis having a rotational motion.
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Figure 1.21
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1.3 The General Equation in GE 226 and its Application to the Type of Problems
The basic equation in GE 226 is the one below (see Figure 1.22):
VB  VA   AB  r B / A
(1.3)
Where VA : velocity at A; VB : velocity at B; r B / A : distance vector from A to B;  AB :
angular velocity of the distance line AB.
Equation (1.3) can only solve problems of Type 1, Type 2, and Type 3.
B
A
The distance between A and B
will not change with respect to
time.
Figure 1.22
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