Pages from Theory_of_machines_COMPLETE_2.pdf

Theory of MachinesCourse # 1

Ayman NadaAssistant Professor

Jazan University, [email protected]

March 29, 2010

1Sucess is not coming in a day

Chapter 1

INTRODUCTION

1.1 Introduction

Mechanisms may be dened as the division of machine design concerned with the kinematicdesign of linkages, cams, gears, and gear trains. Kinematic design is the design on the basis ofmotion requirements in contrast to the design on basis of strength requirements. The study ofmechanisms and machines is an applied science used to understand the relation between themotions of their elements and the forces producing these motions within some geometricalconstraints. With the continuous advances in designing instruments and automated systems,the study of mechanisms becomes of great importance. This chapter is concerned with thestudy of simple mechanisms topological structure, kinematic diagram, inversions, mobilityindex, degrees of freedom, geometric constraints, and geometry of motion. The functions ofmany important mechanisms are also included. These items are important for the study ofmechanism motion. The chapter is organized in ve main sections:

(a) Basic denitions and mechanism elements

(b) Kinematic chain, kinematic diagram and mechanism inversions

(c) Examples of important mechanisms

(d) Mobility index, degrees of freedom, geometric constraints, redundancy, and exibility

(e) Mechanism topology and geometry of motion

The chapter also includes 4 solved examples and ends by a set of problems.

1.2 Basic Denitions

To better understand the mechanics of mechanisms, it is necessary to keep in mind thefollowing denitions:

Mechanism: A combination of rigid and/or exible bodies connected in such awayto do work and there are denite constrained relative motions between them.

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4 CHAPTER 1. INTRODUCTION

Structure: The same denition of mechanism, but its purpose is not to do work andthere is no relative motion between its parts.

Machine: An arrangement of parts and/or mechanisms for doing work and there areconstrained relative motions between its parts.

Statics: The part of mechanics, which deals with the action of forces on bodies atrest.

Kinematics: Study of motion without reference to the forces causing the motion.

Kinetics: Relates the action of forces on bodies to their resulting motions.

Dynamics: The part of mechanics, which deals with the action of forces on bodies inmotion.

Mechanics: Deals with the action of forces on bodies at rest and in motion.

Figure 1.1: Mechanism

Figure 1.2: A 3D truss structure

1.3. MECHANISM ELEMENTS AND CLASSIFICATION 5

1.3 Mechanism Elements and Classication

A Mechanism is composed of three main elements: links, pairing elements, and a driveor drives. The links are connected together with kinematic pairs, called joints, to permittheir constraints relative motions. A mechanism is normally driven through a transmissionsystem, which may include belts, ropes, chains, and/or gears, by a motor. Mechanism linksmay be rigid, uidics, or exible. For the sake of simplicity, links are assumed rigid andjoints have perfect geometry with no clearance through out this text. In many mechanismssprings are used for restoring forces and do not aect their kinematics.Mechanism Links: Links through out the text are considered rigid and the number of

joints on each link gives its type. In other words, a binary link is that having 2 joints (Fig.1.3.a), a ternary link is that having 3 joints (Fig. 1.3.b) and a quaternary link is that having4 joints (Fig. 1.3.c). A well-known ternary link is the bell crank shown in Fig. 1.4. Othernames are given to mechanism links such as: input link, output link, driving link, drivenlink, initial link, frame, base, bar, rocker, coupler, sliding block, slider, guide, crosshead,ram, connecting rod, and many other names. This class of links makes the so-called linkagemechanisms such as the crank-slider mechanism (Fig. 1.5) and the 4-bar linkage (Fig. 1.6).

Figure 1.3: Types of links (a) binary, (b) ternary, (c) quaternary

Cams and followers are another class of mechanism links, which make the so-called cam-follower mechanisms as those shown in Fig. 1.7.Mechanism Joints: There are two types of connecting pairing elements: lower pairs

and higher pairs. Lower pairs have surface contact between mating elements and higher pairshave line or point contact. The contact surface of a shaft in a bearing and that of the wristpin joining the piston and connecting rod as well as the surface between the piston and thecylinder are some examples of lower pairs. Lower pairs include spherical (S) , revolute (R),cylindrical (C) and prismatic (P) joints which represented in Fig.1.8. The contact betweena cam and a follower or between two meshing gear teeth is examples of higher pairs.Table 1.2: Classication of linkage jointsMechanism Classications: There are three types of mechanisms: planar, spherical,

and spatial. In planar mechanisms, all particles describe plane parallel curves in space whilein spatial mechanisms there is no restrictions on the relative motions of particles. In sphericalmechanisms, each link has some point, which remains stationary as the linkage moves. Thestationary points of all links lie at the same location in space. Hooks or universal joint


Figure 1.4: Bell crank

Figure 1.5: Slider crank mechanism

Figure 1.6: Four Bar Mechanism

1.3. MECHANISM ELEMENTS AND CLASSIFICATION 7

Figure 1.7: Cam-follower mechanism

Figure 1.8: Mechanical Joints


Table 1.1: Generalized coordinates and position of an arbitrary point

used in automobile is an example of spherical mechanisms. If spherical mechanisms haveonly revolute joints they are called spherical linkages. Mechanisms may form closed loops,open loops or combination. Crank-slider mechanism and 4-bar linkage are of closed looptype while robot arms are of open loop. Another mechanism classication based on the typeof their links is linkage, cam-follower, and gearset or gear train mechanisms. In practice amechanism may be a combination of all these types such as the engine mechanism.

1.4 Examples of Important Mechanisms

1.4.1 Slider-Crank Mechanisms

Slider-Crank Mechanism: The sketch of linkage arranged as shown in Fig.1.9 is knownas the slider-crank mechanism. Link 1 is a stationary base or a frame, link 2 is the crank,link 3 is the connecting rod, and link 4 is the slider. The line of slider stroke passes throughthe center of crank rotation.Oset Slider-Crank Mechanism: The slider crank can be oset as shown in Fig.1.10.

This oset produces a quick return motion for the slider. However, the amount of quickreturn is very slight, the mechanism would be only used where space is limited.Scotch Yoke Mechanism: This mechanism is sketched in Fig.1.11. It consists of the

same elements as slider-crank mechanism and is early used in steam pumps and in computingmachines as a harmonic generator. Recently, it is used as a mechanism on a test machine toproduce vibrations.

1.4.2 Four-Bar Linkage

The 4-bar linkage consists of 4 pin-connected rigid links as shown in Fig.1.12. There aremany types and names of the 4-bar linkage depending on the mechanism dimensions. Theseinclude double crank, crank-rocker, drag link, double-rocker, and crossover-piston or change-point mechanisms. For crank-rocker type, link 1 is the frame, which is stationary, link 2 isthe crank, which makes complete revolutions, link 3 is the coupler, and link 4 is the rocker,

1.4. EXAMPLES OF IMPORTANT MECHANISMS 9

Figure 1.9: Slider-Crank Mechanisms

Figure 1.10: Slider-Crank Mechanisms with oset


Figure 1.11: Scotch Yoke Mechanism

which performs the desired task. This simple mechanism is important, as it is the base ofmany other mechanisms. For these reasons it will be studied in all details through out thetext.

Figure 1.12: Four Bar Mechanism

1.4.3 Quick-Return Mechanisms

Several types of quick-return mechanisms QRM are in use in machine tools. The QRM givequick return strokes and slow cutting strokes for constant angular velocities of the drivingcranks. These mechanisms are combinations of simple linkages such as the 4-bar linkageand the slider-crank mechanism. An inversion of the slider crank in combination with theconventional slider crank is also used. All known QRM are described after.

1.4. EXAMPLES OF IMPORTANT MECHANISMS 11

Figure 1.13: Crank-Shaper Mechanism


1.5 MOBILITY OF MECHANISMS

The mobility of a mechanism is its number of degrees of freedom. This translates into anumber of independent input motions leading to a single follower motion. A single uncon-strained link (Figure 1.14.a) has three DOF in planar motion: two translational and onerotational. Thus, two disconnected links (Figure 1.14.b) will have six DOF. If the two linksare welded together (Figure 1.14.c), they form a single link having three DOF. A revolutejoint in place of welding (Figure 1.14.d) allows a motion of one link relative to another, whichmeans that this joint introduces an additional (to the case of welded links) DOF. Thus, thetwo links connected by a revolute joint have four DOF. One can say that by connectingthe two previously disconnected links by a revolute joint, two DOF are eliminated. Similarconsiderations are valid for a prismatic joint.

Figure 1.14: Various congurations of links with two revolute joints

Since the revolute and prismatic joints make up all low-pair joints in planar mechanisms,the above results can be expressed as a rule: a low-pair joint reduces the mobility of amechanism by two DOF.These results are generalized in the following formula, which is called Kutzbachs criterion

of mobility

M = 3(n 1) 2j1 j2where n is the number of links, j1 is the number of low-pair joints, and j2 is the number

of high-pair joints. Note that 1 is subtracted from n in the above equation to take intoaccount that the mobility of the frame is zero.In Figure 1.15 the mobility of various congurations of connected links is calculated. All

joints are low-pair ones. Note that the mobility of the links in Fig.1.15.a is zero, whichmeans that this system of links is not a mechanism, but a structure. At the same time, thesystem of interconnected links in Fig.1.15.d has mobility 2, which means that any two linkscan be used as input links (drivers) in this mechanism. Look at the eect of an additionallink on the mobility. This is shown in Fig.1.16, where a four-bar mechanism (Figure 1.16.a)is transformed into a structure having zero mobility (Figure 1.16.b) by adding one link, and

1.6. GRASHOFS LAW FOR A FOUR-BAR MECHANISM 13

then into a structure having negative mobility (Figure 1.16.c) by adding one more link. Thelatter is called an overconstrained structure.

Figure 1.15: Mobility of various congurations of connected links:(a) n = 3; j1 = 3; j2 =0;m = 0; (b) n = 4; j1 = 4; j2 = 0;m = 1;(c) n = 4; j1 = 4; j2 = 0;m = 1;(d) n = 5; j1 =5; j2 = 0;m = 2:

In compound mechanisms, there are links with more than two joints. Kutzbachs criterionis applicable to such mechanisms provided that a proper account of links and joints is made.Consider a simple compound mechanism shown in Fig.1.17, which is a sequence of two four-bar mechanisms. In this mechanism, joint B represents two connections between three links.In other words, it should be taken into account that there are, in fact, two revolute jointsat B. The axes of these two joints may not necessarily coincide. According to Kutzbachsformula M = 3 5 2 7 = 1.

1.6 GRASHOFS law for a Four-Bar mechanism

The fourbar linkage has been shown above to be the simplest possible pin-jointed mechanismfor single degree of freedom controlled motion. It also appears in various disguises such asthe slider-crank and the cam-follower. It is in fact the most common and ubiquitous deviceused in machinery. It is also extremely versatile in terms of the types of motion which it cangenerateSimplicity is one mark of good design. The fewest parts that can do the job will usually

give the least expensive and most reliable solution. Thus the fourbar linkage should be amongthe rst solutions to motion control problems to be investigated. The Grashof condition isa very simple relationship which predicts the rotation behavior or rotatability of a fourbarlinkages inversions based only on the link lengths.LetS =length of shortest link


Figure 1.16: Eect of additional links on mobility: (a)M = 1; (b)M = 0; (c)M = 1:

Figure 1.17: An example of a compound mechanism with coaxial joints at B

1.6. GRASHOFS LAW FOR A FOUR-BAR MECHANISM 15

L = length of longest linkP = length of one remaining linkQ =length of other remaining linkThen if :

S + L P +Qthe linkage is Grashof and at least one link will be capable of making a full revolution withrespect to the ground plane. This is called a Class I kinematic chain.If the inequality is not true, then the linkage is non-Grashof and no link will be capable ofa complete revolution relative to any other link. This is a Class II kinematic chain.Note that the above statements apply regardless of the order of assembly of the links.

That is, the determination of the Grashof condition can be made on a set of unassembledlinks. Whether they are later assembled into a kinematic chain in S, L, P , Q, or S, P , L,Q or any other order, will not change the Grashof condition.The motions possible from a fourbar linkage will depend on both the Grashof condition

and the inversion chosen. The inversions will be dened with respect to the shortest link.The motions are:For the Class I case,

S + L < P +Q

Ground either link adjacent to the shortest and you get a crank-rocker, in which theshortest link will fully rotate and the other link pivoted to ground will oscillate.Ground the shortest link and you will get a double-crank, in which both links pivoted to

ground make complete revolutions as does the coupler.Ground the link opposite the shortest and you will get a Grashof double-rocker, in which

both links pivoted to ground oscillate and only the coupler makes a full revolution.Determine the mobility index and the degrees of freedom of each of the plane mechanisms

shown in Fig. 1.37. All joints are of R type.

Chapter 2

Kinematic Analysis of Mechanisms

There are various methods of performing kinematic analysis of mechanisms, including graph-ical, analytical, and numerical. The choice of a method depends on the problem at handand on available computational means.Kinematic analysis of a mechanical system means the computation, at any time instant,

of the mechanism congurations, positions, displacements, linear velocities and accelerationsof its interesting points as well as the angular velocities and accelerations of its links. Thischapter deals with the kinematics of planar linkage mechanisms using graphical and ana-lytical methods. More emphasis is given to analytical methods in order to simplify the useof computers for mechanism animation and simulation. The analysis of the 4-bar linkage,slider-crank mechanism, and the shaper quick return mechanism is used through out thechapter to illustrate the used methods for linkage kinematic analysis. The chapter ends witha set of interesting problems.

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2.1. COORDINATE SYSTEMS AND VECTOR REPRESENTATION 17

2.1 Coordinate Systems and Vector Representation

For planar mechanisms, two coordinate systems are used: rectangular (x; y) and polar (r; )as shown in Fig. ( ). The choice of the coordinate system is arbitrary and must be selectedto suit the situation. After dening the mechanism working space, reference frame, andtime instant, its kinematic analysis is possible using graphical or analytical methods. In thistext, only planar mechanisms are considered and vectors are represented either in Cartesiancoordinates as x and y components or in polar coordinates by its magnitude r and phaseangle or by complex numbers. Rotation in planar motion is always represented by a vectornormal to the plane of motion i.e. the z-axis.

Figure 2.1: Coordinate System

Position Vector: The vector ~r or ~rP dening the absolute position of point P; Fig. ( )is represented in polar coordinates by its magnitude and phase angle or by complex numbersas:

~r = ~r p = r\= r ej

= r (cos + j sin )

where j is the imaginary number.Velocity Vector: The rst time derivative of the position vector r denes the absolute

velocity of point P; ~vP as:

18 CHAPTER 2. KINEMATIC ANALYSIS OF MECHANISMS

Figure 2.2: Velocity vector

~vP =~rp

= _r ej + j r _ ej

= _r ej + r _ ej (+=2)

If r is constant in magnitude, the absolute velocity of point P is given by:

~vP = r _ ej (+=2)

= r !\ ( + =2)

where, _ is the angular velocity of the vector OP, _ = !:Acceleration Vector: The second time derivative of the position vector ~r denes the

~aP =~rp = ~a =r e

j (+=2) + r !ej (+)

= r ej (+=2) + r !2ej (+)

= r \ ( + =2) + r!2\ ( + )= ~aP

~a = ~at + ~ar


where

~at = r \ ( + =2)

~ar = r!2\ ( + )

Figure 2.3: Acceleration vectors

Where is the angular acceleration of the vector OP, and ~at and ~ar are respectively theacceleration components.


Example 1 Slider Crank Mechanism

The general linkage conguration and terminology for a slider-crank linkage with oestare shown in Figure (2.4). The link lengths and the values of 2 , !2 and 2 are dened inthe table. For the row(s) assigned, draw the linkage to scale and nd the velocities of thepin joints A and B and the velocity of slip at the sliding joint using a graphical method.

Figure 2.4: Conguration and terminology

row Link 2 Link 3 Oset 2 !2 2f. 3 13 0 100 -45 50e. 5 20 -5 225 -50 10g. 7 25 10 330 100 18


(f.)

Figure 2.5: Example : (f.) Position

Results:

row vB !3 aB 3f. 2:5 50 = 125 vB=A

r3= 0:650

13= 2:307 2:47 1000 = 2470 aB=A t

r3= 6:061000

13= 466:15


Figure 2.6: Example : (f.) Velocity & Acceleration


(e.)

Figure 2.7: Example : (e.) Position

Results:

row vB !3 aB 3e. 1:91 100 = 191 vB=A

r3= 1:77100

20= 8:85 3:31 2000 = 6620 aB=A t

r3= 4:492000

20= 449


Figure 2.8: Example : (e.) Velocity & Acceleration

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