basic of kinematics

8/7/2019 Basic of Kinematics

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Fundamentals of Kinematics of Machines

Mechanics: It is that branch of scientific analysis which deals with motion, time and force.

Kinematics is the study of motion, without considering the forces which produce that motion.

Kinematics of machines deals with the study of the relative motion of machine parts. It involves

the study of position, displacement, velocity and acceleration of machine parts.

Dynamics of machines involves the study of forces acting on the machine parts and the motionsresulting from these forces.

Plane motion: A body has plane motion, if all its points move in planes which are parallel to

some reference plane. A body with plane motion will have only three degrees of freedom. I.e.,

linear along two axes parallel to the reference plane and rotational/angular about the axisperpendicular to the reference plane. (eg. linear along X and Z and rotational about Y.)The

reference plane is called plane of motion. Plane motion can be of three types. 1) Translation

2) rotation and 3) combination of translation and rotation.

Translation: A body has translation if it moves so that all straight lines in the body move toparallel positions. Rectilinear translation is a motion wherein all points of the body move in

straight lie paths. Eg. The slider in slider crank mechanism has rectilinear translation. (link 4 infigure1)

Figure -1

Translation, in which points in a body move along curved paths, is called curvilinear translation.The tie rod connecting the wheels of a steam locomotive has curvilinear translation. (link 3 infigure.2)

Figure - 2

Rotation: In rotation, all points in a body remain at fixed distances from a line which isperpendicular to the plane of rotation. This line is the axis of rotation and points in the body

describe circular paths about it. (Eg. link 2 in Figure -1 and links 2 & 4 in Figure - 2)

Translation and rotation: It is the combination of both translation and rotation which is

exhibited by many machine parts. (Eg. link 3 in Figure -1)

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Link or element: It is the name given to any body which has motion relative to another. All

materials have some elasticity. A rigid link is one, whose deformations are so small that they can

be neglected in determining the motion parameters of the link.

Figure - 3

Binary link: Link which is connected to other links at two points. (Figure - 3 a)

Ternary link: Link which is connected to other links at three points. (Figure - 3 b)

Quaternary link: Link which is connected to other links at four points. (Figure - 3 c)

Pairing elements: the geometrical forms by which two members of a mechanism are joined

together, so that the relative motion between these two is consistent are known as pairing

elements and the pair so formed is called kinematic pair. Each individual link of a mechanismforms a pairing element.

Figure - 4 Kinematic pair Figure - 5

Degrees of freedom (DOF): It is the number of independent coordinates required to describe the

position of a body in space. A free body in space (figure - 5) can have six degrees of freedom.i.e., linear positions along x, y and z axes and rotational/angular positions with respect to x, y and

z axes.

In a kinematic pair, depending on the constraints imposed on the motion, the links may loose

some of the six degrees of freedom.

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Basic Kinematics of Constrained Rigid Bodies

Degrees of Freedom of a Rigid Body in a Plane

The degree of freedom (DOF) of a rigid body is defined as the

number of independent movements it has. Figure 6 shows a rigid

body in a plane. To determine the DOF of this body we must

consider how many distinct ways the bar can be moved. In a twodimensional plane such as shown in figure 6, there are 3 DOF. The

bar can be translated along the x axis, translated along the y axis, and

rotated about its centroid.Figure 6 Degrees of freedom of a

rigid body in a plane

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 zaxesand three rotary motions around the x, y and z axes

respectively as shown in figure 7

Figure 7 Degrees of freedom of a

rigid body in space

Kinematic Constraints

Two or more rigid bodies in space are collectively called a rigid body system. One can prevent

the motion of these independent rigid bodies with kinematic constraints. Kinematic constraints

are constraints between rigid bodies that result in the decrease of the degrees of freedom of rigidbody system.

The term kinematic pair actually refers to kinematic constraints between rigid bodies. The

kinematic pairs are divided into lower pairs and higher pairs, depending on how the two bodiesare in contact.

Lower Pairs in Planar Mechanisms

There are two kinds of lower pairs in planar mechanisms: revolute pairs figure 8 and prismaticpairs figure 9

A rigid body in a plane has only three independent motions two translational and one rotary

so introducing either a revolute pair or a prismatic pair between two rigid bodies removes twodegrees of freedom.

Figure -8 A planar revolute pair (R-pair) Figure - 9 A planar prismatic pair (P-pair)

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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, revolute pair, prismatic pair, and screw pair.

A spherical pair (S-pair) Figure 10A spherical pair keeps two spherical centers together. Two rigid

bodies connected by this constraint will be able to rotate relatively

around x, y and zaxes, but there will be no relative translation alongany of these axes. Therefore, a spherical pair removes three degrees

of freedom in spatial mechanism. DOF = 3.Figure 10

A planar pair (E-pair) Figure 11

A plane pair keeps the surfaces of two rigid bodies together. Tovisualize this, imagine a book lying on a table where is can move in

any direction except off the table. Two rigid bodies connected by this

kind of pair will have two independent translational motions in the

plane, and a rotary motion around the axis that is perpendicular to theplane. Therefore, a plane pair removes three degrees of freedom in

spatial mechanism. In our example, the book would not be able to raiseoff the table or to rotate into the table. DOF = 3. Figure 11

A cylindrical pair (C-pair) Figure 12A cylindrical pair keeps two axes of two rigid bodies aligned. Two

rigid bodies that are part of this kind of system will have an

independent translational motion along the axis and a relative rotary

motion around the axis. Therefore, a cylindrical pair removes fourdegrees of freedom from spatial mechanism. DOF = 2.

Figure 12

A revolute pair (R-pair) Figure 13A revolute pair keeps the axes of two rigid bodies together. Two

rigid bodies constrained by a revolute pair have an independent

rotary motion around their common axis. Therefore, a revolute pairremoves five degrees of freedom in spatial mechanism. DOF = 1.

Figure 13

A prismatic pair (P-pair) Figure 14A prismatic pairkeeps two axes of two rigid bodies align and allow no

relative rotation. Two rigid bodies constrained by this kind of

constraint will be able to have an independent translational motion

along the axis. Therefore, a prismatic pair removes five degrees of

freedom in spatial mechanism. DOF = 1.Figure 14

A screw pair (H-pair) Figure 15The screw pair keeps two axes of two rigid bodies aligned and

allows a relative screw motion. Two rigid bodies constrained by a

screw 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 spatial mechanism. Figure 15

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Constrained Rigid Bodies

Rigid bodies and kinematic constraints are the basic components of mechanisms. A constrained

rigid body system can be a kinematic chain, a mechanism, a structure, or none of these. Theinfluence of kinematic constraints in the motion of rigid bodies has two intrinsic aspects, which

are the geometrical and physical aspects. In other words, we can analyze the motion of the

constrained rigid bodies from their geometrical relationships or using Newton's Second Law.

A mechanism is a constrained rigid body system in which one of the bodies is the frame. Thedegrees of freedom are important when considering a constrained rigid body system that is a

mechanism. It is less crucial when the system is a structure or when it does not have definite

motion.

Calculating the degrees of freedom of a rigid body system is straight forward. Any unconstrainedrigid body has six degrees of freedom in space and three degrees of freedom in a plane. Adding

kinematic constraints between rigid bodies will correspondingly decrease the degrees of freedom

of the rigid body system.

Degrees of Freedom of Planar Mechanisms

Gruebler's Equation

The definition of the degrees of freedom of a mechanism is the number of independent relative

motions among the rigid bodies. For example, Figure 16 shows several cases of a rigid body

constrained by different kinds of pairs.

Figure 16 Rigid bodies constrained by different kinds of planar pairs

In Figure 16a, a rigid body is constrained by a revolute pair which allows only rotational

movement around an axis. It has one degree of freedom, turning around point A. The two lost

degrees of freedom are translational movements along the x and y axes. The only way the rigidbody can move is to rotate about the fixed point A.

In Figure 16b, a rigid body is constrained by a prismatic pairwhich allows only translational

motion. In two dimensions, it has one degree of freedom, translating along the x axis. In this

example, the body has lost the ability to rotate about any axis, and it cannot move along the y

axis.

In Figure 16c, a rigid body is constrained by a higher pair. It has two degrees of freedom:

translating along the curved surface and turning about the instantaneous contact point.

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In general, a rigid body in a plane has three degrees of freedom. Kinematic pairs are constraints

on rigid bodies that reduce the degrees of freedom of a mechanism. Figure 17 shows the three

kinds of pairs in planar mechanisms. These pairs reduce the number of the degrees of freedom. If

we create a lower pair(Figure 17a, b), the degrees of freedom are reduced by 2. Similarly, if wecreate a higher pair(Figure 17c), the degrees of freedom are reduced by 1.

Figure 17 Kinematic Pairs in Planar Mechanisms

Therefore, we can write the following equation: M = 3(n-1) 2J1 J2 ----------------- (1)

Wherem = total degrees of freedom in the mechanism

n = number oflinks (including the frame)J1 = number oflower pairs or Joints (one degree of freedom)

J2 = number oflower pairs orhigher pairs or Joints (two degrees of freedom)

For a planar mechanism with only lower pair which are one degree of freedom only ie. M=1 and

J2 = 0 the above equation (1) becomes

1 = 3(n-1) 2J1 =

3n 2J1 - 4 = 0 ------------------ (2) This equation is known as Gruebler's equation.

3-DIMENSIONAL DEGREES OF FREEDOM.

Kutzbach Criterion

The number ofdegrees of freedom of a mechanism is also called the mobility of the device. The

mobility is the number of input parameters (usually pair variables) that must be independently

controlled to bring the device into a particular position. The Kutzbach criterion, which is similarto Gruebler's equation, calculates the mobility.

In order to control a mechanism, the number of independent input motions must equal the

number of degrees of freedom of the mechanism. For example, the transom in Figure10a has asingle degree of freedom, so it needs one independent input motion to open or close the window.That is, you just push or pull rod 3 to operate the window.

To see another example, the mechanism in Figure 12a also has 1 degree of freedom. If an

independent input is applied to link 1 (e.g., a motor is mounted on joint A to drive link 1), the

mechanism will have then a prescribed motion.

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Kutz-Bach Criterion is a 3D form of Grublers equation. Grublers equation as we said earlier

deals only with mobility of planar systems. In a 3D system each link now has a possible 6

degrees of freedom 2 in each plane. Again one link must be fixed giving a total 6(n-1) DOF.

Each joint type removes a different number of degrees of freedom from the system giving theequation below:

M =6(n-1) 5J1 4J2 3J3 2J4 J5 ---------------------- (3)

Where:

J1 = Number of 1 DOF jointsJ2 = Number of 2 DOF joints

J3 = Number of 3 DOF jointsJ4 = Number of 4 DOF joints

J5 = Number of 5 DOF joints

MOBILITY:

Mobility is the minimum number of independent parameters required to specify the location of

every link within a mechanism ie. mobility of a mechanism is the number of input parametersusually pair variable) which must be controlled independently in order to bring the mechanism

into a particular positionMobility of a mechanism can be determined from the count of the number of links, number andtype of joints which it includes

Figure 18

In the Figure 18 of a planar system there are initially two links with 3 degrees of freedom each

Link 1 has 3DOF x1, y1, 1,

Link 2 has 3DOF x2, y2, 2Thus two non-connected links in a two dimensional system have 6DOF in total

If 1, 2 joined using a 1 DOF joint (rotational motion only allowed) then x1, y1, 1, 2, are thevariables describing the system giving DOF = 4

For this system then two links with 3DOF each have 4DOF whenlinked.

A 1 DOF joint removes 2 DOF from a planar system and a 2DOF joint removes 1 DOF from the same system See a 2 DOF

joint in Figure 19

Figure 19

A planar system contains n links in total, each of which has 3 DOF. For it to be a mechanismone link must be fixed or connected to the ground otherwise motion cannot be described

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M = Mobility = Number DOF = 3n 3 = 3(n-1) (As one link is the ground it has no DOF)

The following equation is used to describe mobility in 2D or planar systems:

M = 3(n-1) 2 J1 J2Where: n = total number of links

M = DOF or MobilityJ1 = number of 1DOF joints

J 2 = number of 2 DOF joints

This is also known as GRUBLERS EQUATION and is for mobility of planar systems.

M = 0 Motion impossible - statically determinate

M = 1 Single input /monitoring necessary

M = 2 Double input/output necessary

M = -1 Statically indeterminate structure

STATICALLY INDETERMINATE: This is a body, or rigid body of elements that has more

external supports, ties or constraints than are needed to maintain equilibrium it cannot moveand more than one element is responsible for locking up the mechanism in this way. It is

therefore not possible to say which element is doing what proportion of the locking up. The

body of elements will behave as a truss and not as a mechanism it cannot move.

STATICALLY DETERMINATE BODIES: Are ones that are supported by the minimum

number of constraints necessary to ensure an equilibrium configuration, i.e., if you remove one

support the truss will become a mechanism, capable of movement.

REDUNDANT SUPPORTS: These are supports that can be removed from a statically

indeterminate structure without destroying the equilibrium condition of the body.

A joint connecting more than two links is special and is counted as X 1 joint where X is the

number of links.

MINIMUM NUMBER OF BINARY LINKS IN A CONSTRAINED MECHANISM WITH

LOWER PAIRS

In a mechanism there cannot be any singular link. Thus let

n2 = number of binary linksn3 = number of ternary links

n4 = number of quaternary links and so on

Then the total number of links in a mechanism will be equal to n = n1 + n2 + n3 + ------ ni -- (1)

Since each simple hinge or joint consists of two elements. Thus the total number of elements in a

mechanism will be equal to e = 2j ------ (2)

A binary link has two joints or elements and ternary link has three joints or elements Thus

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e = 2n2 + 3n3 + 4n4 ---------- +ini ---- (3)

Thus from equation (2) we have e = = 2j = 2n2 + 3n3 + 4n4 ---------- +ini ---- (4)

To satisfy Grublers criteria given by 3n 2j 4 = 0

We have ( 2n2 + 3n3 + 4n4 ---------- +ini ) 3(n1 + n2 + n3 + ------ ni ) - 4 = 0 ---- (5)

We get after simplifying n2 = 4 + n4 + 2n5 + -------+ (i-3)nii

i.e. n2 = 4 + (i-3)ni ------------ (6)

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Thus, the minimum number of binary links is four hence the four bar linkage is the simplestmechanism. The maximum number of hinges or joints on one link in a constrained mechanism

with n link will be

imax = n/2 --------- (7)

CONSTRAINED MOTION OF A KINEMATIC CHAIN OF N LINKS WITH SIMPLE

HINGES OR JOINTS

By Grublers criterion 3n-2j -4 = 0 or n = 2/3(j+2) thus the number of links must be even.

Simplest mechanism is a four bar linkage mechanism the next higher order starts with n = 6

then we have imax = n/2 = 6/2 = 3 i.e the highest order of the links will be ternary link hencefrom equation (1) we have

n = n2 + n3 i.e n2 + n3 = 6 ------ (8)

From Grublers Criterion 3n -2j 4 = 0 i.e 3x6 -2j 4 = 0 or j = 7

But from equation (4) 2j = 2n2 + 3n3 we have 2n2 + 3n3 = 14 ----- (9)

Thus solving equation (8) and (9) we get n2 = 4 and n3 = 2

Thus in a six link mechanism with constrained motion there will be four binary links and two

ternary links.

Example 1 Figure 20

M = 3(N-1) 2 J1 J2N = 3

J1 = 3

J2 = 0

M = 3(N-1) 2 J1 J2

M = 3(3-1) 2(3) = 0 Statically determinate Figure 20

Example 2 Figure 21

M = 3(N-1) 2 J1 J2N = 4J1 = 5

J2 = 0

M = 3(N-1) 2 J1 J2

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M = 3(4-1) 2(5) = 1

Statically indeterminate Figure 21

Example 3 Figure 22

M = 3(N-1) 2 J1 J2N = 4J1 = 4

J2 = 0M = 3(N-1) 2 J1 J2M = 3(4-1) 2(4) = 1

Mechanism Figure 22

Example 4 Figure 23

M = 3(N-1) 2 J1 J2N = 5

J1 = 5J2 = 0M = 3(N-1) 2 J1 J2M = 3(5-1) 2(5) = 2

Mechanism but requires a double input or output Figure 23

Example 5 Figure 24

M = 3(N-1) 2 J1 J2N = 3

J1 = 2J2 = 1

M = 3(N1) 2 J1 J2M = 3(3 1) 2(2) 1 = 1

Mechanism Figure 24

Example 6 Figure 25

M = 3(N-1) 2 J1 J2N = 4

J1 = 3J2 = 1

M = 3(N1) 2 J1 J2M = 3(4 1) 2(3) 1 = 2

Figure 25

Mechanism with 2DOF assuming cam both slips androlls. If the cam rolled only with no slippage then the

mechanism would have one degree of freedom. Alternatively if there was no rotation and

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slippage only, as in a slider-crank mechanism, then it would also be a one degree of freedom

system

Example 7 Figure 26

M = 3(N-1) 2 J1 J2N = 7

J1 = 8

J2 = 0

M = 3(7 1) 2(8) 0 = 2

Mechanism with 2DOF.Centre joint A is counted twice. Figure 26

Example 8

Look at the transom above the door in Figure 27a. The opening and closing mechanism is shownin Figure 27b. Let's calculate its degree of freedom.

Figure 27 Transom Mechanism

n = 4 (link 1,2,3 and frame 4), l = 4 (at A, B, C, D), h = 0

Note: D and E function as a same prismatic pair, so they only count as one lower pair.

Example 9

Calculate the degrees of freedom of the mechanisms shown in Figure 28b. Figure 28a is an

application of the mechanism.

n = 4, l = 4 (at A, B, C, D), h = 0

Figure 28 Dump truck

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Example 10

Calculate the degrees of freedom of the mechanisms shown in Figure 29.For the mechanism in Figure 24a

n = 6, l = 7, h = 0

For the mechanism in Figure 29b

n = 4, l = 3, h = 2

Figure 29

Degrees of freedom calculation

Note: The rotation of the roller does not influence the relationship of the input and output motionof 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 degrees of freedom for the mechanism.

MOBILITY EQUATION INCONSISTENCY

No general rule can be given as to when the mobility equation is incorrect but problems arise

when several links in a mechanism are parallel.

Figure 30M = 3(N-1) 2 J1 J2

N = 6, J1 = 7, J2 = 0 Figure 30M = 3(6 1) 2(7) 0 = 1

Mechanism with 1 DOF as centre joint counted

twice is correct

Figure 316

M = 3(N-1) 2 J1 J2N = 5, J1 = 6, J2 = 0 Figure 31

M = 3(5 1) 2(6) 0 = 0 Mechanism with no DOF which is INCORRECT as it

has 1 DOF

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basic of kinematics

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