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Table of Contents
4 Basic Kinematics of Constrained Rigid Bodies
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4.1 Degrees of Freedom of a Rigid Body
4.1.1 Degrees of Freedom of a Rigid Body in a Plane
The degrees of freedom (DOF) of a rigid body is definedas the number of independent movements it has. Figure 4-1shows a rigid body in a plane. To determine the DOF of this bodywe must consider how many distinct ways the bar can be moved. Ina two dimensional plane such as this computer screen, there are 3 DOF.The bar can be translated along the x axis, translatedalong the y axis, and rotated about its centroid.
Figure 4-1 Degrees of freedom of a rigid body in a plane
4.1.2 Degrees of Freedom of a Rigid Body in Space
An unrestrained rigid body in space has six degrees of freedom:three translating motions along the x, y and zaxes and three rotary motions around the x, y andz axes respectively.
Figure 4-2 Degrees of freedom of a rigid body in space
Two or more rigid bodies in space are collectively called a rigidbody system. We can hinder the motion of these independent rigidbodies with kinematic constraints. Kinematicconstraints are constraints between rigid bodies that result inthe decrease of the degrees of freedom of rigid body system.
The term kinematic pairs actually refers tokinematic constraints between rigid bodies. The kinematic pairsare divided into lower pairs and higher pairs, depending on how the twobodies are in contact.
4.2.1 Lower Pairs in Planar Mechanisms
There are two kinds of lower pairs in planar mechanisms: revolute pairs and prismatic pairs.
A rigid body in a plane has only three independent motions -- twotranslational and one rotary -- so introducing either a revolute pairor a prismatic pair between two rigid bodies removes two degrees offreedom.
Figure 4-3 A planar revolute pair (R-pair)
Figure 4-4 A planar prismatic pair (P-pair)
4.2.2 Lower Pairs in Spatial Mechanisms
There are six kinds of lower pairs under the category of spatial mechanisms. The types are: spherical pair, plane pair,cylindrical pair, revolutepair, prismatic pair, and screw pair.
Figure 4-5 A spherical pair (S-pair)
A spherical pair keeps two spherical centers together. Tworigid bodies connected by this constraint will be able torotate relatively around x, y and z axes,but there will be no relative translation along any of theseaxes. Therefore, a spherical pair removes three degrees of freedom inspatial mechanism. DOF = 3.
Figure 4-6 A planar pair (E-pair)
A plane pair keeps the surfaces of two rigid bodies together.To visualize this, imagine a book lying on a table where is can movein any direction except off the table. Two rigid bodies connected bythis kind of pair will have two independent translational motions inthe plane, and a rotary motion around the axis that is perpendicularto the plane. Therefore, a plane pair removes three degrees offreedom in spatial mechanism. In our example, the book would not beable to raise off the table or to rotate into the table. DOF =3.
Figure 4-7 A cylindrical pair (C-pair)
A cylindrical pair keeps two axes of two rigid bodiesaligned. Two rigid bodies that are part of this kind of system willhave an independent translational motion along the axis and a relativerotary motion around the axis. Therefore, a cylindrical pair removesfour degrees of freedom from spatial mechanism. DOF = 2.
Figure 4-8 A revolute pair (R-pair)
A revolute pair keeps the axes of two rigid bodiestogether. Two rigid bodies constrained by a revolute pair have anindependent rotary motion around their common axis. Therefore, arevolute pair removes five degrees of freedom in spatialmechanism. DOF = 1.
Figure 4-9 A prismatic pair (P-pair)
A prismatic pair keeps two axes of two rigid bodies align andallow no relative rotation. Two rigid bodies constrained by this kindof constraint will be able to have an independent translational motionalong the axis. Therefore, a prismatic pair removes five degrees offreedom in spatial mechanism. DOF = 1.
Figure 4-10 A screw pair (H-pair)
The screw pair keeps two axes of two rigid bodies aligned andallows a relative screw motion. Two rigid bodies constrained by ascrew pair a motion which is a composition of a translational motionalong the axis and a corresponding rotary motion around the axis.Therefore, a screw pair removes five degrees of freedom in spatialmechanism.
4.3 Constrained Rigid Bodies
Diary app windows. Rigid bodies and kinematic constraints are the basic components ofmechanisms. A constrained rigid body system can be a kinematic chain, a mechanism, a structure, or none of these.The influence of kinematic constraints in the motion of rigid bodieshas two intrinsic aspects, which are the geometrical and physicalaspects. In other words, we can analyze the motion of the constrainedrigid bodies from their geometrical relationships or using Newton's Second Law.
A mechanism is a constrained rigid body system in which one of thebodies is the frame. The degrees offreedom are important when considering a constrained rigid body systemthat is a mechanism. It is less crucial when the system is astructure or when it does not have definite motion.
New file menu 1 4 2018. Calculating the degrees of freedom of a rigid body system is straightforward. Any unconstrained rigid body has six degrees of freedom inspace and three degrees of freedom in a plane. Adding kinematicconstraints between rigid bodies will correspondingly decrease thedegrees of freedom of the rigid body system. We will discuss more onthis topic for planar mechanisms in the next section.
4.4 Degrees of Freedom of Planar Mechanisms
4.4.1 Gruebler's Equation
The definition of the degrees of freedom of a mechanismis the number of independent relative motions among the rigid bodies.For example, Figure 4-11 shows several cases of arigid body constrained by different kinds of pairs.
Figure 4-11 Rigid bodies constrained by different kinds of planar pairs
In Figure 4-11a, a rigid body is constrained by a revolute pair which allows only rotationalmovement around an axis. It has one degree of freedom, turning aroundpoint A. The two lost degrees of freedom are translational movementsalong the x and y axes. The only way the rigid body canmove is to rotate about the fixed point A.
In Figure 4-11b, a rigid body is constrained by a prismatic pair which allows onlytranslational motion. In two dimensions, it has one degree offreedom, translating along the x axis. In this example, thebody has lost the ability to rotate about any axis, and it cannot movealong the y axis.
In Figure 4-11c, a rigid body is constrained by a higher pair. It has two degrees offreedom: translating along the curved surface and turning about theinstantaneous contact point.
In general, a rigid body in a plane has three degrees of freedom.Kinematic pairs are constraints on rigid bodies that reduce thedegrees of freedom of a mechanism. Figure 4-11 shows the three kindsof pairs in planar mechanisms. Thesepairs reduce the number of the degreesof freedom. If we create a lower pair(Figure 4-11a,b), the degrees of freedom are reduced to 2. Similarly,if we create a higher pair (Figure4-11c), the degrees of freedom are reduced to 1.
Figure 4-12 Kinematic Pairs in Planar Mechanisms
Therefore, we can write the following equation:
(4-1)Where
- F = total degrees of freedom in the mechanism
- n = number of links (includingthe frame)
- l = number of lower pairs(one degree of freedom)
- h = number of higher pairs(two degrees of freedom)
This equation is also known as Gruebler's equation.
Example 1
Look at the transom above the door in Figure 4-13a. The opening andclosing mechanism is shown in Figure 4-13b. Let's calculate itsdegree of freedom.
Figure 4-13 Transom mechanism
n = 4 (link 1,3,3 and frame 4), l = 4 (at A, B, C, D), h = 0
(4-2)Note: D and E function as a same prismatic pair, so they only count as one lower pair.
Example 2
Calculate the degrees of freedom of the mechanisms shown in Figure 4-14b.Figure 4-14a is an application of the mechanism.
Figure 4-14 Dump truck
(4-3)Example 3
Calculate the degrees of freedom of the mechanisms shown in Figure 4-15.
Figure 4-15 Degrees of freedom calculation
For the mechanism in Figure 4-15a
n = 6, l = 7, h = 0
(4-4)For the mechanism in Figure 4-15b
n = 4, l = 3, h = 2
(4-5)Note: The rotation of the roller does not influence therelationship of the input and output motion of the mechanism. Hence,the freedom of the roller will not be considered; It is called apassive or redundant degree of freedom.Imagine that the roller is welded to link 2 when counting the degreesof freedom for the mechanism.
4.4.2 Kutzbach Criterion
The number of degrees of freedom of a mechanismis also called the mobility of the device. Themobility is the number of input parameters (usually pairvariables) that must be independently controlled to bring the deviceinto a particular position. The Kutzbach criterion,which is similar to Gruebler's equation,calculates the mobility.
In order to control a mechanism, the number of independent inputmotions must equal the number of degrees of freedom of the mechanism.For example, the transom in Figure 4-13ahas a single degree of freedom, so it needs one independent inputmotion to open or close the window. That is, you just push or pull rod 3to operate the window.
To see another example, the mechanism in Figure4-15a also has 1 degree of freedom. If an independent input isapplied to link 1 (e.g., a motor is mounted on joint A to drivelink 1), the mechanism will have the a prescribed motion.
4.5 Finite Transformation
Finite transformation is used to describe the motion of a point onrigid body and the motion of the rigid body itself.
4.5.1 Finite Planar Rotational Transformation
Figure 4-16 Point on a planar rigid body rotated through an angle
Suppose that a point P on a rigid body goes through a rotationdescribing a circular path from P1 toP2 around the origin of a coordinate system. We candescribe this motion with a rotation operatorR12:
(4-6)where
(4-7)4.5.2 Finite Planar TranslationalTransformation
Figure 4-17 Point on a planar rigid body translated through a distance
(4-8)where
(4-9)4.5.3 Concatenation of Finite Planar Displacements
Figure 4-18 Concatenation of finite planar displacements in space
(4-10)and
(4-11)We can concatenate these motions to get
(4-12)where D12 is the planar general displacement operator:
(4-13)4.5.4 Planar Rigid-Body Transformation
We have discussed various transformations to describe thedisplacements of a point on rigid body. Can these operators beapplied to the displacements of a system of points such as a rigidbody?
We used a 3 x 1 homogeneous column matrix to describe a vectorrepresenting a single point. A beneficial feature of the planar 3 x 3translational, rotational, and general displacement matrix operatorsis that they can easily be programmed on a computer to manipulate a 3x n matrix of n column vectors representing n points of a rigid body.Since the distance of each particle of a rigid body from every otherpoint of the rigid body is constant, the vectors locating each pointof a rigid body must undergo the same transformation when the rigidbody moves and the proper axis, angle, and/or translation is specifiedto represent its motion. (Sandor& Erdman 84). For example, the general planar transformationfor the three points A, B, C on a rigid body can be representedby
(4-14)4.5.5 Spatial Rotational Transformation
We can describe a spatial rotation operator for the rotationaltransformation of a point about an unit axis u passing through theorigin of the coordinate system. Suppose the rotational angle of the pointabout u is ,the rotation operator will be expressed by
(4-15)where
- ux, uy, uz are the othographicalprojection of the unit axis u on x, y, and z axes, respectively.
- s =sin
- c =cos
- v = 1 -cos
4.5.6 Spatial Translational Transformation
Suppose that a point P on a rigid body goes through atranslation describing a straight path from P1 toP2 with a change of coordinates of (x, y, z), we can describe thismotion with a translation operator T:
(4-16)4.5.7 Spatial Translation and Rotation Matrix for AxisThrough the Origin
Suppose a point P on a rigid body rotates with an angulardisplacement about an unit axis u passing through the origin ofthe coordinate system at first, and then followed by a translationDu along u. This composition of this rotationaltransformation and this translational transformation is a screwmotion. Its corresponding matrix operator, the screwoperator, is a concatenation of the translation operator in Equation 4-7 and the rotation operator in Equation 4-9.
4.6 Transformation Matrix Between Rigid Bodies
4.6.1 Transformation Matrix Between two ArbitrayRigid Bodies
For a system of rigid bodies, we can establish a local Cartesiancoordinate system for each rigid body. Transformation matrices areused to describe the relative motion between rigid bodies.
For example, two rigid bodies in a space each have local coordinatesystems x1y1z1 andx2y2z2. Let point P beattached to body 2 at location (x2, y2,z2) in body 2's local coordinate system. To find thelocation of P with respect to body 1's local coordinate system,we know that that the point x2y2z2can be obtained from x1y1z1 bycombining translation Lx1 along the x axis androtation z about zaxis. We can derive the transformation matrix as follows:
(4-18)If rigid body 1 is fixed as a frame, aglobal coordinate system can be created on this body. Therefore, theabove transformation can be used to map the local coordinates of apoint into the global coordinates.
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4.6.2 Kinematic Constraints Between Two RigidBodies
The transformation matrix above is a specific example for twounconstrained rigid bodies. The transformation matrix depends on therelative position of the two rigid bodies. If we connect two rigidbodies with a kinematic constraint, theirdegrees of freedom will be decreased. In other words, their relativemotion will be specified in some extent.
Suppose we constrain the two rigid bodies above with a revolute pair as shown in Figure 4-19. We canstill write the transformation matrix in the same form as Equation 4-18.
Figure 4-19 Relative position of points on constrained bodies
The difference is that the Lx1 is a constantnow, because the revolute pair fixes the origin of coordinate systemx2y2z2 with respect to coordinate systemx1y1z1. However, the rotationzis still a variable. Therefore, kinematic constraints specify thetransformation matrix to some extent.
4.6.3 Denavit-Hartenberg Notation
Denavit-Hartenberg notation (Denavit & Hartenberg 55) iswidely used in the transformation of coordinate systems of linkages and robot mechanisms. It can beused to represent the transformation matrix between links as shown inthe Figure 4-20.
Figure 4-20 Denavit-Hartenberg Notation
(4-19)The above transformation matrix can be denoted as T(ai,i, i, di)for convenience.
A linkage is composed of several constrained rigid bodies. Like amechanism, a linkage should have a frame. The matrix method can beused to derive the kinematic equations of the linkage. If all thelinks form a closed loop, the concatenation of all of thetransformation matrices will be an identity matrix. If the mechanismhas n links, we will have:
T12T23..T(n-1)n = I (4-20)
Table of Contents
Complete Table of Contents- 1 Introduction to Mechanisms
- 2 Mechanisms and Simple Machines
- 3 More on Machines and Mechanisms
- 4 Basic Kinematics of Constrained Rigid Bodies
- 4.1 Degrees of Freedom of a Rigid Body
- 4.1.1Degrees of Freedom of a Rigid Body in a Plane
- 4.1.2 Degrees of Freedom of a Rigid Body in Space
- 4.2 Kinematic Constraints
- 4.2.1 Lower Pairs in Planar Mechanisms
- 4.2.2 Lower Pairs in Spatial Mechanisms
- 4.3 Constrained Rigid Bodies
- 4.4 Degrees of Freedom of Planar Mechanisms
- 4.4.1 Gruebler's Equation
- 4.2.2 4.4.2 Kutzbach Criterion
- 4.5 4.5 Finite Transformation
- 4.5.1 Finite Planar Rotational Transformation
- 4.5.2 Finite Planar Translational Transformation
- 4.5.3 Concatenation of Finite Planar Displacements
- 4.5.4 Planar Rigid-Body Transformation
- 4.5.5 Spatial Rotational Transformation
- 4.5.6 Spatial Translational Transformation
- 4.5.7 Spatial Translation and Rotation Matrix for Axis Through the Origin
- 4.6 Transformation Matrix Between Rigid Bodies
- 4.6.1 Transformation Matrix Between two Arbitray Rigid Bodies
- 4.6.2 Kinematic Constraints Between Two Rigid Bodies
- 4.6.3 Denavit-Hartenberg Notation
- 4.6.4 Application of Transformation Matrices to Linkages
- 4.1 Degrees of Freedom of a Rigid Body
- 5 Planar Linkages
- 6 Cams
- 7 Gears
- 8 Other Mechanisms
- Index
- References
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