2.1 Introduction. 2.2 Degree of Freedom DOF of a rigid body

Size: px

Start display at page:

Download "2.1 Introduction. 2.2 Degree of Freedom DOF of a rigid body"

Posy Parker
6 years ago
Views:

Chapter 2 Kinematics 2.1 Introduction 2.2 Degree of Freedom 2.2.1 DOF of a rigid body In order to control and guide the mechanisms to move as we desired, we need to set proper constraints.

In mechanics, it means how many independent motions a mechanism can possibly achieve.

In other words, a point can freely move at and Y directions in a plane and, Y and Z in a space.

In planar motion, there is only one direction it can rotate. That is Z direction, or the direction that is perpendicular to the plane.

When we put the rigid body in space, it will not only be able to move out of the plane, i.e., gain the translation at the Z direction, it will be able to rotate out from the plane, or possible to rotate about and Y axes.

1 Chapter 2 Kinematics 2.1 Introduction 2.2 Degree of Freedom DOF of a rigid body In order to control and guide the mechanisms to move as we desired, we need to set proper constraints. In order to set proper constraints, we need to study degree of freedom (DOF). In mechanics, it means how many independent motions a mechanism can possibly achieve. For a particle point, its DOF is 2 in 2D plane or 3 in 3D space without any constraint. In other words, a point can freely move at and Y directions in a plane and, Y and Z in a space. For a rigid body, since it comes with dimension (size and shape), it can rotate about certain axes. In planar motion, there is only one direction it can rotate. That is Z direction, or the direction that is perpendicular to the plane. Therefore, an unconstrained rigid body can have 3DOF in planar motion. When we put the rigid body in space, it will not only be able to move out of the plane, i.e., gain the translation at the Z direction, it will be able to rotate out from the plane, or possible to rotate about and Y axes. Please note that since we did not specify any coordinate system at this point, these, Y and Z axes are not necessary perpendicular to each other. As long as they are independent at the local point where the rigid body is residing, we can always observe 3 or 6 independent motions of a rigid body depending whether it is 2D or 3D. Z Y Z R 2D Plane O R Ry Y 3D 2-1

2 Figure 2-1: (Left) An unrestricted planar body can move at and Y directions and rotate at Z direction in a plane. The degree of freedom is 3. (Right) An unrestricted special body can move and rotate at, Y and Z directions. The degree of freedom is 6. You can get a more intuitive observation of the degree of freedom from this interactive animation (add a link here). Select among 2D & 3D, rotation, translation, or both. For a rigid body in planar motion, since there are three possible independent motions, we need to use three independent variables to identify them. We then define the two linear translations and the one rotation as, where we implied a Cartesian coordinate system here. As the motions are simply time derivative of the positions, their corresponding position variables are. However, we note that there is some confusion with the coordinate variables. Let us look at the bicycle wheel of Figure 2-2. For the rotation, when the wheel is turning, every single point will be turning at the same angular velocity. So we do not need to differentiate any specific point. However, that is not the case for the linear velocities. As we know from the Dynamics, when a wheel is rotating about its axis, even at a constant angular velocity, the velocities of any two points in the rigid body will be different. For example, two points A and C on the rim will have same speeds or magnitudes since they have same radii about the rotating center O. Meanwhile, points A and B will have same direction since they are along the same radial line. However, no two points will have same magnitudes and same directions. Therefore, we need to specify which point we are representing using subscripts on the linear velocity. Generally, we choose either the center of mass or the rotating center in mechanics, say point O. Then we can describe the motion of the rigid body as where a subscript is used to denote the point. For any other point, we can easily obtain its linear velocity as long as we can find its relative position. 2-2

3 C V C B A O V B V A Figure 2-2: A rotating wheel. Likewise, if we want to specify the pose (position and orientation) of a rigid body, we just need to specify the coordinate of one point and a direction. The direction can be obtained by measuring the angular displacement of a line that is fixed to the rigid body or simply by getting the coordinates of another point. Unsurprisingly, these two methods are essentially equivalent with some help of mathematics. For example, if we know the coordinates of two points and. Then we can choose any one as our base point, say A, and calculate the angle as: (2-1) However, there is a small problem with this equation. For example, look at Figure 2-3, point A and point B are in opposite quadrant, but their tangent values of the corresponding angles are the same. That is (2-2) 2-3

Y A( A, Y A ) β α π α O B(- B, -Y B ) Figure 2-3: Points A and B will have same tangent values. Therefore, the direct inverse tangent will not yield a unique answer between 0 and. Further, when, Eq.

For example, in Matlab or Octave, the is θ atan y x. So does Java and C++.

You can blame the engineers who designed the software for the inconsistency, but make sure you read the manual and type the argument in the right order. 2.

4 Y A( A, Y A ) β α π α O B(- B, -Y B ) Figure 2-3: Points A and B will have same tangent values. Therefore, the direct inverse tangent will not yield a unique answer between 0 and. Further, when, Eq. (2-1) will not yield any result. Therefore, we will introduce the four-quadrant inverse tangent,, such as { This function exists in most math software or programing language. For example, in Matlab or Octave, the is θ atan y x. So does Java and C++. However, if you use a calculator or a spreadsheet (such as MS Excel) to do the calculation, the function is more likely to be θ atan x y. You can blame the engineers who designed the software for the inconsistency, but make sure you read the manual and type the argument in the right order DOF of linked rigid bodies For a free floating rigid body with planar motion, its degree of freedom is 3. If we have multiple, say N, independent free floating rigid bodies, the total DOF will be 3N since the movements of them are completely independent from each other. This scenario might be true in the computer arcade games. However, it is rarely true in the engineering. Indeed, we can consider these independent rigid bodies as the parts of a mechanism. One purpose of mechanical synthesis or mechanical design is to find ways to connect these N free floating machine parts in a proper way so we will strip certain DOFs away with constraints (joints) and force the parts to move with right motions at right timings. 2-4

5 Ground Let's start from a fundamental yet often ignored case. That is, we fix one rigid body to the ground. In machinery, it is the case of bolting the engine to the chassis of the automotive or fixing the base of a robotic welder to the shop floor. In both cases, what we are trying to do is to find a ground, or reference, where the user or observer can easily conduct analysis or where the power source is initiated. The ground is not necessarily the earth ground, though it is often the case. Depending what we are analysis, it can be the flying space shuttle when we are studying the robotic arm or the casing when we are studying a mechanical watch. Mathematically, the ground is the inertial coordinate system that we will take as the reference. Figure 2-4: (A) and (B): Two common illustrations of the ground. (C) The coordinate system is fixed to the ground. (D) The joint is fixed to the ground. When a link is fixed to the ground, it will not move. Therefore, all of its DOFs will be lost. Or the link itself becomes part of the ground. This link is often called ground link. In many cases, since the ground link is not differentiated from ground, we will just skip it in the drawings and use a ground symbol to denote it. On the other hand, just like the earth is a continuous body, the ground is by default connected and hence one single rigid body. Likewise, a machine or a mechanism can be bolted or linked to the casing or base at multiple points, but the casing or base is one big ground. Although we may not link the ground points in the schematic drawings, they are implied to be rigidly connected and are simply the points in an imaginary rigid body. Joints (A) Y (B) (C) (D) In SolidWorks, the first part to put into the assembly drawing canvas is set as fixed by default. It means that the part is fixed to the ground. You can right click it and make it float, which means retrieving all the 6DOF back to the part. 2-5

A flash figure of two free floating links A pin joint 2 links with a pin joint L 2 is independent from L 1.

6 Pin Joint One of the most common joints between two links is a pin joint. A pin joint can be a pin, a bolt, or a shaft. Physically, two links will share a common axis to rotate about. A flash figure of two free floating links A pin joint 2 links with a pin joint L 2 is independent from L 1. Total #DOF=2N=6 When L 2 is pinned to L 1, it can still rotate about the pin joint, but it loses the freedom of moving and Y direction without dragging L 1 or break the joint. Hence, 2 degrees of freedom is lost due to the pin joint. The total #DOF=#DOF for 2 free bodies -#DOF eliminated by the joint =. 2-6

Figure 2-5: 3 links and 2 joints here where the 2 joints are different. Figure 2-6: 3 links and 2 joints here where the 2 joints are same and on a single shaft.

Each added link will add one DOF by adding 3DOF for an added rigid body and simultaneously removing 2DOF by attaching the new link with a pin joint.

7 Figure 2-5: 3 links and 2 joints here where the 2 joints are different. Figure 2-6: 3 links and 2 joints here where the 2 joints are same and on a single shaft. In the design of articulate robotic arm, many links are connected by pin joints. Each added link will add one DOF by adding 3DOF for an added rigid body and simultaneously removing 2DOF by attaching the new link with a pin joint. For the open link design in the articulated robotic arm, in order to control and guide the motion of this added arm, we need to introduce a power and control source, generally a micro-processor controlled servo. However, for the most machines, we don't have this luxury. We often obtain the controllability by using closed loop designs, which we will discuss in detail later in the course. Figure 2-7: Open loop links. (Obtain permission or draw with SolidWorks.) 2-7

Slider Joint When an object is sliding along the surface

loses the ability to rotate relative to the body it is

Like a pin joint, the slider will also remove 2DOF from

Both the pin joint and the slider joint mentioned above

We call them full joints and denote them as.

the one shown in Figure Figure 2-9: 2DOF slider joints.

8 Slider Joint When an object is sliding along the surface of the other one, like the train running on the track, it loses the ability to rotate relative to the body it is sliding on. It also cannot move away from the track. Like a pin joint, the slider will also remove 2DOF from the second object. Figure 2-8: Examples of slider joints. Both the pin joint and the slider joint mentioned above will remove 2 DOF when they are used to connect two joints. We call them full joints and denote them as. Half Joints There is another kind of slider joints like the one shown in Figure Figure 2-9: 2DOF slider joints. Figure 2-10: Comparing two folding chair designs with half slider joints and pin joints. 2-8

9 Sometime, we can transform a half joint to two full joints. For example, the slider joint in Figure [fig:a-2dof-slider]a can be considered as equivalent to the pin and slider in Figure [fig:a-2dof-slider]b. It is also a common practice in engineering to do so. Figure 2-11: A half joint can be replaced by one link and two full joints. For mechanisms with both full joints and half joints, we can calculate the overall DOF of a planar mechanism as where N is the total number of links including the ground link, and is the number of half joints. is the number of full joints, Example 5. A three bar link as shown in Figure. There are 3 links counting the ground link and 3 pin joints, which are full joints. The total DOF is. Y A O B Figure 2-12: A 3-bar linkage has 0DOF. It means that although we use free spinning joints to connect three links, the overall mechanism will behave like a rigid body. Although it is not very desirable to achieve motion, 2-9

it is widely used to achieve structure. In civil engineering, this triangular structure is called truss and are widely used in building high rises and bridges.

Figure 2-13: Truss is a typical construction structure. Example 6. A four bar link as shown in Figure 2-14. There are four links including the ground link and 4 pin joints.

That is, as long as we can control the configuration and motion of one point, generally the rotational angle or angular velocity of a link about one joint, the

10 it is widely used to achieve structure. In civil engineering, this triangular structure is called truss and are widely used in building high rises and bridges. In mechanical design, this structure is also broadly adopted to save space, material and cost and to make assembly easier. Figure 2-13: Truss is a typical construction structure. Example 6. A four bar link as shown in Figure There are four links including the ground link and 4 pin joints. The total DOF is Y A B O C Figure 2-14: A four bar linkage has 1DOF. The 1DOF of a four bar link means the one to one correspondence between input and output. That is, as long as we can control the configuration and motion of one point, generally the rotational angle or angular velocity of a link about one joint, the configuration and motion of the entire linkage will be determined. This predictability makes the four-bar link very suitable for many engineering and industrial applications. It is unarguably one of the most widely used basic linkages used in the industry. 2-10

11 Example 7. A crank-slider mechanism as shown infigure There are four links including the ground link, 3 pin joints and one full sliding joint. The total DOF is Y B A α β C Figure 2-15: A crank slider has 1DOF. Like the four bar links, cranker-slider is widely used in the industry thanks to its 1DOF or one-to-one correspondence between input and output. For example, in an internal combustion engine, this mechanism is used to convert the linear motion of the piston to the rotation of the crankshaft. Meanwhile, in an air compressor, the motion is traveled the opposite way from the rotational motion of a link connected to an electric motor output shaft to the linear motion of the piston in the compressing chamber. Example 8. A V-2 engine. There are 6 links, 5 pin joints, and 2 full sliding joints. The total DOF is (Add a V2 engine figure here. ) Add a V6 engine simulation here. From this example, we can see that it is achievable to obtain a coordinated motion from a slightly complicated linkage design. If we stack 3 or 4 such V2 engine along a common shaft, and adjust each pair so they will reach the limit positions sequentially with predetermined interval, then we have our schematic design of a V6 or V8 engine used in many performance motor vehicles. 2.3 Grashof Mechanism 2-11

12 2.3.1 Grashof condition Let us compare the following four mechanisms: Insert a Crank link to simulation video Rocker figure with a Rocker crank Insert a double Crank link to simulation video figure with a couple Rocker We notice that some links will rock back and forth, while some will just rotate continuously. However, how do we know that which one will rock and which one will rotate? Someone may just cut the length by chance and let the luck fly, but as a trained engineer, we cannot live by chance. Fortunately, a German engineer named Franz Grashof helped us to solve the problem. It is called the Grashof s criterion for four bar links. Given a four bar linkage, we can name each of them after their lengths: s: length of the shortest link l: length of the longest link p & q: lengths of the other two links. Theorem. Grashof s criterion: Given a four bar linkage, it will have at least one revolving or cranking link (usually the shortest link), if. The link is called Grashof. have no revolving or cranking link, if. The link is called non-grashof Grashof Mechanism Some definitions: -Crank: one pin is grounded, can make full revolution about grounded pin Rocker: one pin is grounded, does not make full revolution about grounded pin Coupler: neither pin is grounded. Experiences complex motion There are several common situations that we encounter with the Grashof condition. 2-12

2.3.2.1 Crank-Rocker. The shortest link s is the crank. The shortest link s is the crank. The longest link l is grounded.

The shortest link s is the ground link, or the frame. A B Crank Y Crank O C Franz Grashof (7.11.1826 ~ 10.26.1893), was a professor of Applied Mechanics at the Technische Hochschule Karlsruhe.

13 Crank-Rocker. The shortest link s is the crank. The shortest link s is the crank. The longest link l is grounded. The intermediate link opposite to the shortest link is the rocker that bounces back and forth. B Y Coupler A p Rocker q Crank s O l Figure 2-16: A crank-rocker Double crank. The shortest link s is the ground link, or the frame. A B Crank Y Crank O C Franz Grashof ( ~ ), was a professor of Applied Mechanics at the Technische Hochschule Karlsruhe. His name also appeared in the field of fluid dynamics and heat transfer with the dimensionless Grashof number Gr. Figure 2-17: A double crank Double rocker. The shortest link s is opposite to the frame. It is both a coupler and a crank. Unlike the crank in the previous examples, where the cranks rotates about a fixed joint, the crank in a double rocker will not rotate about a fixed center. On the contrary, it will follow a combination of rotation about its center and a complicate movement of the center. 2-13

A Crank B Y Rocker Rocker O C Figure 2-18: A double rocker. 2.3.2.4 Special Case: S + L = P + Q Coupler Cranks Crank Rocker Cranks Parallelogram Form Antiparallelogram Form Figure 2-19: Three possible cases.

14 A Crank B Y Rocker Rocker O C Figure 2-18: A double rocker Special Case: S + L = P + Q Coupler Cranks Crank Rocker Cranks Parallelogram Form Antiparallelogram Form Figure 2-19: Three possible cases. In the special case where S + L = P + Q there are three possible configurations, as shown above. The parallelogram linkage may switch (uncontrollably!) to the anti-parallelogram linkage if care is not taken to prevent this. Figure 2-20: A parallelogram steering linkage. 2-14

15 A parallelogram steering linkage can be used in a steering system. The pitman arm, track rod, idler arm, and the car frame (as ground) form a parallelogram. When the steering wheel turns the pitman arm, it moves the two tie rods simultaneously, which in turn, push one end of the steering arm. The steering arm will then turn the wheels The Non-Grashof Linkage If S + L > P + Q then the linkage is non-grashof, and all permutations are double-rockers Practical Considerations Driver dyad Motor Motor Figure 2-21: Input for a Grashof and a non-grashof mechanism can be different. In many practical situations we will use a motor (AC or DC) to drive the linkage. This is simple in the case of a Grashof linkage we attach the motor to the crank, and the linkage can spin forever. In the non-grashof case, we must either use a servomotor or stepper motor, or, where we desire to use a simple AC or DC motor (if we need continuous motion), we can attach a driver dyad as shown in the figure above. Driver dyad Figure 2-22: Windshield Wiper Linkage for 1998 Ford Escort. In the linkage shown in Figure 2-22, the wipers have rocker motion, but are driven by 12DC motor, which runs continuously. The linkage is approximately a special-case (parallelogram) Grashof linkage, and both blades are driven by one motor. 2-15

16 Design Projects Project 1: Folding chair. Locate a folding chair, be it a lawn chair or a beach chair. 1. Draw a schematic diagram of the folding chair using the visualization method. 2. Identify the links and joints used in the folding chair. 3. Find the #DOF of the chair. 4. Think about 3 ways to achieve the same effect using different methods. 5. Measure the dimension of each link of the chair. 6. Draw a 3D computer model of the folding chair using a CAD software such as SolidWorks, AutoCAD, or ProE. 7. Vary the length of some links and examine the results. Project 2: Automobile windshield wiper. Locate an automobile windshield wiper. A movie about the intermittent windshield wiper was released in The title of the movie is Flash of Genius. Think about the social, technical and economical benefit of this seemly simple mechanism. 1. Draw a schematic diagram of the windshield wiper using the visualization method. 2. Identify the links and joints used in the windshield wiper. 3. Find the #DOF of the linkage. 4. Research patents on various types of windshield wipers. 5. Measure the dimension of each link of the targeted windshield wiper. 6. Draw a 3D computer model of the windshield wiper using a CAD software such as SolidWorks, AutoCAD, or ProE. 7. Generate a simulation of the windshield wiper by connecting it to an electric motor. 8. Vary the length of some links and examine the results. Other possible projects: Casement window opening mechanism, umbrella, etc. 2-16

Chapter 4. Mechanism Design and Analysis

Chapter 4. Mechanism Design and Analysis All mechanical devices containing moving parts are composed of some type of mechanism. A mechanism is a group of links interacting with each other through joints