ME 316 2005/2006 yearLecture note 1
Lecture Note (1)
Planar kinematics of a rigid body: Review
We begin with the review of the following concepts:
(1)rigid body,
(2)planar motion / mechanism vs. spatial motion and mechanism,
(3)types of planar motions,
(4)relative motions, and
(5)basic equations we have learned in GE 226 for solving planar kinematics problems.
After that, we classify the problems we were able to solve with the knowledge of GE 226 and further explain those unresolved planar kinematics problems that will be solved in this course.
Rigid body
There are two definitions for the Rigid Body:
Definition 1.1. A rigid body consists of a set of particles that have a fixed distance without change under forces.
This definition appeared in the course “General Engineering 124, Engineering mechanics – statics”.This definition stems from the constitution of a physical object. When we go to certain types of analysis, a rigid body based on this physical definition may be treated as a particle; see the following definition.
Definition 1.2. A body is considered as a rigid body when the motion of any particles that constitute the body is the same.
Figure 1.1 explains this definition further. In the case of Figure 1.1a, two particles on a body have the same motion (both the direction and the magnitude), thus we can treat the whole body as a particle. Here the shape of the body does not affect our view.
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).
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 body, in particular a rigid body (because the distance between A and B is not changed).
The distinction of the particle from the rigid body may also be attributed to the 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 can be viewed as a particle, not a rigid body. We can write the Newton’s second law for the entire car as F=ma. But when we need 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.
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). So a rigid body has the following features depending on a different view of point of motion analysis or force analysis.
- From motion viewpoint, all the points on the body have the same motion.
- From the force viewpoint, all the forces pass through one point, see Figure 1.3.
Planar motion
Definition 1.3 (for a single body).
All particles on the body move in one plane or several planes that are 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:
Example 1.1 (Figure 1.4). To determine whether the system is a planar system or spatial system.
Figure 1.4
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 1.5). To determine whether the system is a planar system or not.
Analysis 1.2. Because the smaller gear and the larger 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.
Figure 1.5
Example 1.3 (Figure 1.6). To determine whether the system is a planar or spatial system.
Figure 1.6
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.
Type of Planar motions
In motion analysis, after we have determined that a system under interest is a planar one, we need to understand types of motions. This is because the governing principle for kinematics of different types of motions is 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.
There are three types of planar motions:
Type 1. Translation (Figure 1.7)
(a)(b)
Figure 1.7
In the case of translation motion, every line segment on the body remains parallel to its original direction during the motion. In Figure 1.7, although the gross path of the object moves 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 the translation.
Example 1.4 (Figure 1.8). (Translation)
In machine design practice, a typical system is shown in Figure 1.8, 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, because line 1-1 keeps its orientation unchanged (1’-1’) when the machine moves to a new position.
Type 2. Rotation (Figure 1.9).
When a body rotates about a fixed axis, all particles on the body move along circular paths. (see Figure 1.9)
In Figure 1.9, 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.
Figure 1.9
Type 3. General motion (Figure 1.10)
Figure 1.10
The general planar motion is a superposition of the translation and the rotation. Consequently, a line on a body will changeboth its orientation and its translation at two positions, and there will be a point on a body which changes its position, see Figure 1.10. (see Figure 1.10, 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 massof the body and is perpendicular to the motion plane.
Relative Motion
Motion of any object will have a reference object. When we say a motion of the object, see Figure 1.11, there is a reference object, that is, the earth. The earth is assumed to be at rest. In this case we call the motion of the object “absolute motion”.
When the reference object is not the earth, then 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 the relative translation; see Figure 1.12.
The relation between the relative rotation (angular velocity) and two absolute rotations (angular velocities) is expressed by
(1.1)
In the above equation, the rotation of body A is our interest, and body B is a reference body for observing the rotation of body A.
Figure 1.12.There are several cases about the relative rotation that are commonly seen.
Case (i). Two bodies are joined by a sliding/slot; see Figure1.13 In this case, the relative rotation between body A and body B is zero. That is to say, we have , , .
Figure 1.13
Case (ii). Body A performs translation, while body B rotates, or vice versa (Figure 1.14). In this case, we have, so . It is noted that when we discuss the relative motion, we do not restrict to the situation where two bodies are in fact connected or not.
Figure 1.14
Let us consider the connection pattern as shown in Figure 1.15 where two bodies A and B perform a relative motion, and as well as they have the same rotation as they are connected through a slider. Let us set the slider direction as the T-axis, which is also the direction of relative translation between bodies A and B. Further, we select two points on object A and object B along the relative translation axis, respectively; see Figure 1.15. Let PA and PB denote these points on object A and object B, respectively. We then have the following equation:
(1.2)
Where : the relative translation velocity of body A with respect to body B.
: the relative translation velocity of point PA (on body A) with respect to
point PB (on body B).
: the angular velocity of translation axis.
: the displacement vector from PA to PB.
Figure 1.15
If the assembly of A and B in Figure 1.15 do not have a rotation, Equation (1.2) becomes
(1.3)
If the PA and PB are instantaneously coincident (on the same location), the distance between PA and PB is zero. This implies that in Equation (1.2), the magnitude of vector is zero. We will also have Equation (1.3).
Example 1.5 (Figure 1.16)
Figure 1.16 is an assembly of gears and link (i.e., arm). ArmAB rotates with an angular velocity 2rad/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?
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). (clockwise)
Solution to (b). (clockwise)
Figure 1.16
Example 1.6 (Figure 1.17).
Figure 1.17
There is a pivot at point A between bar AB and the flat road. Vo is constant. Θ is known. Find: the angular velocity of Bar AB, ω(θ).
Solution:
The strategy to find solutions is very important. In general, the solving process is a process to find unknown variables from known variables. It can happen that the known variable appears as one body (say A) but the unknown variable appears on another body (say B). To relate the unknown variable withthe known variable in an equation, we need to apply the relative motive theory, as this theory has equations that relate motion variables across two bodies; see equations (1.1)-(1.3).
Analysis:In this problem, the known variable is on the block (i.e., Vo) and the unknown variable is on bar AB (i.e., ωAB). Therefore we need to consider applying equations (1.1) to (1.3), anyway, to make these two variables related to each other.
It is further noted that Bar AB and the block have both the relative rotation and the relative translation; i.e., the relative motion between these two bodies is the general motion (translation plus rotation).
We choose a contact point at B where there two points corresponding two bodies, bar AB and the block, respectively. We denoted these two points as B1 and B2 (B1 on bar AB and B2 on the block), respectively. Then we can write the following equation, using equation (1.2) and notice =0, which describes the relative translation between points B1 and B2.
(1.4)
The following table shows the analysis of the known and unknown variables.
Direction:x x x
Magnitude:? ? x
x: means known.
?: means unknown
The direction of : along with the line AB.
The direction of : perpendicular with the line AB.
From this analysis, we understand that we have two unknown variables (see the question marks in the above analysis) while we have one vector equation (i.e. two scalar equations). So the number of equations meets the number of unknowns, and the problem can be well solved. The detailed solution is given below. First, we write out all the items in the above equations in terms of the complex number:
From above equations, we obtain:
(1.5)
Two equations can be obtained from the above complex number equation.
(1.6a)
(1.6b)
Further,
(1.7a)
(1.7b)
From equations (1.7a), (1.7b), we have two unknowns: and. So we can work out them as follows:
(1.8a)
(1.8b)
(1.9)
Substituting (1.9) into (1.8a), we obtain:
(1.10)
It is noted that the relative translation between B1 and B2, especially B1 with respect to B2 is along the direction as shown in Figure 1.18.
Figure 1.18
From the above analysis, it can be summarized that the establishment of a relationship between unknown variables and known variables has the following cases.
Case 1. Two variables are on the same body (see Figure 1.19).
See two points A and B on body 1. Suppose we know the velocity of point A and the angular velocity of body 1, then we have
(1.11)
Case 2. When two bodies are pivoted at a location; see Figure 1.19 where body 1 and body 2 are pivoted at location denoted by B, we identify B1 (on body 1) and B2 (on body 2); both are at location B. In this case, we have
Suppose the velocity of point A is known. Through equation (1.11), we know VB(VB1, Vb2). Since B1 is on body 1, we actually transfer the motion (known) from body 2 to body 1. After the motion of B1 on body 1 is known, we can find motion of any other particle on body 1 using equation (1/12) in a similar manner that is applied to body 2.
Case 3. When two bodies are jointed with a translating guilder (see Figure 1.20), their angular velocities are the same. It is further noted that at the location denoted by A there are three particles or elements on three bodies, respectively, i.e., A1 (on body 1), A2 (on body 2), and A3 (on body 3). There are the following relationships existed among the velocities of these points:
Figure 1.20
because body 2 and body 3 are pivoted.
, while
where we want to emphasize that stands for the relative velocity of point A1 with respect to point A2. Its direction is along the sliding guider.
The type of problem discussed in GE226
The basic equation in GE 226 is the one below:
(1.12)
With this equation, GE 226 can only solve the problems with the following features.
(1)In a mechanical system all bodies are connected by a so-called pivot.
(2)If there is a slider, then the guider for the slider is fixed.
For the problem shown in Figure 1.17, use of equation (1.12) without any other concept introduced cannot solve such a kind of problem. The main reason is that equation (1.12) only establishes the relation for two points on the same body. Therefore, using equation (1.12), we can only establish a velocity relation between A and B1(Figure 1.18) i.e.,
In equation (1.13), we know =0 and we know, but we do not know and. We have in fact three unknown variables in equation (1.13), namely, the magnitude of (i.e. VB1), the direction of and the magnitude of . Often we may mistakenly think the direction of is horizontal. In this case, we may write:
We extend the above equation, leading to
From equation (1.15), we get =0, but from equation (1.14), we get
implying that we have a contradictory result.
There may be an attempt to not distinguish B1, B2 at B; as such, one can often see the following equation in solutions.
VB is taken as (the velocity of the block; see Figure 1.17). Such a treatment is wrong as well because the distance between A and B changes with respect to time, while in order to use equation (1.12), we should examine whether the distance between A and B is fixed. Only two points on the same body can we say that the distance between these two points is fixed.
However, we can use the concept, mathematically expressed in equations (1.1) to (1.3), to solve this kind of problem, as shown above. A more general method will be described in Lecture Note 2.
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