theory of machinessite.iugaza.edu.ps/mawad/wp-content/uploads/iugaza...the theory of machines by...

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 1

Theory of Machines

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012

Syllabus and Course Outline

2

SAT 09:30 – 11:00 Q412

MON 09:30 – 11:00 Q412

Faculty of EngineeringDepartment of Mechanical Engineering

EMEC 3302, Theory of Machines

Instructor: Dr. Anwar Abu-ZarifaOffice: IT Building, Room: I413 Tel: 2821eMail: [email protected]: http://site.iugaza.edu.ps/abuzarifaOffice Hrs: see my website


Text Book: R. L. Norton, Design of Machinery “An Introduction to the Synthesis and Analysis of Mechanisms and Machines”, McGraw Hill Higher Education; 3rd edition

Reference Books:

John J. Uicker, Gordon R. Pennock, Joseph E. Shigley, Theory of Machines and Mechanisms

R.S. Khurmi, J.K. Gupta,Theory of Machines Thomas Bevan, The Theory of Machines The Theory of Machines by Robert Ferrier McKay Engineering Drawing And Design, Jensen ect., McGraw-Hill Science, 7th

Edition, 2007 Mechanical Design of Machine Elements and Machines, Collins ect., Wiley,

2 Edition, 2009


Grading:Attendance 5%Design Project 25%Midterm 30%Final exam 40%

Course Description:

The course provides students with instruction in the fundamentals of theory ofmachines. The Theory of Machines and Mechanisms provides the foundationfor the study of displacements, velocities, accelerations, and static anddynamic forces required for the proper design of mechanical linkages, cams,and geared systems.


Course Objectives:

Students combine theory, graphical and analytical skills to understand the Engineering Design. Upon successful completion of the course, the student will be able:

To develop the ability to analyze and understand the dynamic(position, velocity, acceleration, force and torque) characteristics ofmechanisms such as linkages and cams.

To develop the ability to systematically design and optimizemechanisms to perform a specified task.

To increase the ability of students to effectively present written,oral, and graphical solutions to design problems.

To increase the ability of students to work cooperatively on teamsin the development of mechanism designs.


Chapter 1Introduction


Definitions

7

The subject Theory of Machines may be defined as that branch ofEngineering-science, which deals with the study of relative motionbetween the various parts of a machine, and forces which act onthem. The knowledge of this subject is very essential for anengineer in designing the various parts of a machine.

Kinematics: The study of motion without regard to forces

More particularly, kinematics is the study of position, displacement, rotation, speed, velocity, and acceleration.


Kinetics: The study of forces on systems in motion

A mechanism: is a device that transforms motion to some desirable patternand typically develops very low forces and transmits little power.

A machine: typically contains mechanisms that are designed to providesignificant forces and transmit significant power.


Application of Kinematics

9

Any machine or device that moves contains one or more kinematic elements suchAs linkages, … gears…. belts and chains.

Bicycle is a simple example of a kinematic system that contains a chain drive to provide Torque.


An Automobile contains many more examples of kin-systems…

the transmission is full of gears….


Chapter 2DEGREES OF FREEDOM (MOBILITY)


Degrees of Freedom (DOF) or Mobility

• DOF: Number of independent parameters (measurements) needed to uniquely define position of a system in space at any instant of time.

• A mechanical system’s mobility (M) can be classified according to the number of degrees of freedom (DOF).

• DOF is defined with respect to a selected frame of reference (ground).


Rigid body in a plane has 3 DOF: x,y,z Rigid body in 3D-space has 6 DOF, 3 translations & 3

rotations three lengths (x, y, z), plus three angles (θ, φ, ρ).

The pencil in these examples represents a rigid body, or link.


Types of Motion

• Pure rotation: the body possesses one point (center of rotation) that has no motion with respect to the “stationary” frame of reference. All other points move in circular arcs.

• Pure translation: all points on the body describe parallel (curvilinear or rectilinear) paths.

• Complex motion: a simultaneous combination of rotation and translation.


Excavator


Slider-Crank Mechanism


Links, joints, and kinematic chains

Linkage design: Linkages are the basic building blocks of all mechanisms All common forms of mechanisms (cams, gears, belts, chains)

are in fact variations on a common theme of linkages.• Linkages are made up of links and joints.

• Links: rigid member having nodes• Node: attachment points


1. Binary link: 2 nodes2. Ternary link: 3 nodes3. Quaternary link: 4 nodes

Joint: connection between two or more links (at theirnodes) which allows motion;

(Joints also called kinematic pairs)


Joint Classification

Joints can be classified in several ways:1.By the type of contact between the elements, line, point, or surface.2.By the number of degrees of freedom allowed at the joint.3.By the type of physical closure of the joint: either force or form closed.4.By the number of links joined (order of the joint).

A more useful means to classify joints (pairs) is by the number of degrees of freedom that they allow between the two elements joined.


A joint with more than one freedom may also be a higher pair

• Type of contact: line, point, surface• Number of DOF: full joint=1DOF, half joint=2DOF• Form closed (closed by geometry) or Force closed

(needs an external force to keep it closed)• Joint order

Joint order = number of links-1


lower pair to describe joints with surface contact

The six lower pairs


The half joint is also called a roll-slide jointbecause it allows both rolling and sliding

Form closed (closed by geometry) or Force closed


A joint (also called kinematic pair) is a connection between two ormore links at their nodes, which may allow motion between the links.

A lower pair is a joint with surface contact; a higher pair is a joint withpoint or line contact.

A full joint has one degree of freedom; a half joint has two degreesof freedom. Full joints are lower pairs; half-joints are higher pairs andallow both rotation and translation (roll-slide).

A form-closed joint is one in which the links are kept together form byits geometry; a force-closed joint requires some external force tokeep the links together.

Joint order is the number of links joined minus one (e.g. 1st ordermeans two links).

Terminology of Joints


Kinematic chains, mechanisms,machines, link classification

• Kinematic chain: links joined together for motion• Mechanism: grounded kinematic chain• Machine: mechanism designed to do work• Link classification:

Ground: any link or links that are fixed, nonmoving withrespect to the reference frame

Crank: pivoted to ground, makes complete revolutions Rocker: pivoted to ground, has oscillatory motion Coupler: link has complex motion, not attached to ground


Elements:0: Ground (Casing, Frame)1: Rocker2: Coupler3: Crank

crank mechanism


When studying mechanisms it is very helpful to establish a fixed reference frame by assigning one of the links as “ground”.

The motion of all other links are described with respect to the ground link.

For example, a fourbar mechanism often looks like a 3-bar mechanism, where the first “bar” is simply the ground link.

The “Ground” Link


Drawing kinematic Diagrams


Determining Degrees of Freedom


Determining Degrees of Freedom

Two unconnected links: 6 DOF(each link has 3 DOF)

When connected by a full joint: 4 DOF(each full joint eliminates 2 DOF)

Gruebler’s equation for planar mechanisms: DOF = 3L-2J-3GWhere:L: number of linksJ: number of full jointsG: number of grounded links


Determining DOF’s

• Gruebler’s equation for planar mechanisms

• WhereM = degree of freedom or mobilityL = number of linksJ = number of full joints (half joints count as 0.5)G = number of grounded links =1

3 1 2M L J Kutzbach’s modification of Gruebler’s equation

M= 3L-2J-3G


The Cylindrical (cylindric) joint - two degrees of freedomIt permits both angular rotation and an independent sliding motion (C joint)


The Spherical (spheric) - Three degree of freedomIt permits rotational motion about all three axes, a ball-and-socket joint (S joint)


Example


Gruebler’s Equation

Gruebler’s Equation

DOF = mobilityL = number of linksJ = number of revolute joints or

prismatic jointsG = number of grounded links

DOF (M) = 3*L – 2* J – 3 *G= 3 (L-1) – 2 * J

L = 2J = 1G = 1

DOF = 1

Gruebler’s equation can be used to determine the mobility of planar mechanisms.

Link 13 DOF

Link 23 DOF

1 DOF


Mobility of Vise Grip Pliers

L = 5J = 4 (revolute)J = 1 (screw)G = 1 (your hand)

DOF = 3*5 - 2*5 - 1*3 = 2

1

23

4

1

2

3

4

This example applies Gruebler’s equation to the determine the mobility of a vise grip plier.

5

Each revolute joint removes two DOF.The screw joint removes two DOF.

The mobility of the plier is two. Link 3 can be moved relative link1 when you squeeze your hand and the jaw opening is controlled by rotating link 5.


Punch Press

Slider-Crank Mechanism

As designated in the figure, there are four links link 1, link 2, link 3 and link 4. Link 1 acts as a crank. Link 2 acts as connecting link, link 3 is the slider and link 4 is ground.

Joint Number Formed between links Joint type

1 Link 4 and Link 1 Revolute (or Pin)



4 Link 3 and Link 4Translatio

nal or (Slider)


Mechanisms and Structures

If DOF > 0, the assembly of links is a mechanism and will exhibit relative motion

If DOF = 0, the assembly of links is a structure and no motion is possible.

If DOF < 0,then the assembly is a preloaded structure, no motion is possible, and in general stresses are present.


Paradoxes

• Greubler criterion does not include geometry, so it can give wrong prediction

• We must use inspection

E-quintetL=5J=6G=1M=3*5-2*6-3*1=0


Rolling cylinders even without slip (The joint between the two wheels can bepostulated to allow no slip, provided that sufficient friction is available) is anexample in which the ground link is exactly the same length as the sum of twoother links.If no slip occurs, then this is a one-freedom, or full, joint that allows only relative angular motion (Δθ) between the wheels.With that assumption, there are 3 links and 3 full joints,The equation predicts DOF = 0 (L=3,J1=3), but the mechanism has DOF = 1.

Others paradoxes exist, so the designermust not apply the equation blindly.


Chapter 3Linkage


History

• Leonardo da Vinci (1452, 1519), Codex Madrid I. • Industrial Revolution was the boom age of linkages: cloth

making, power conversion, speed regulation, mechanical computation, typewriting and machining


Linkages Today

In many applications linkages have been replaced by electronics.

Still linkages can have a cost advantage over electronic solutions: Couple different outputs by a mechanism rather than using one motor per output and electronics to achieve the coupling.

Current applications: Sports Equipment, Automotive (HVAC modules), Precision Machinery (Compliant Mechanisms), Medical Devices

44


Mechanical linkages are usually designed to transform a given input force and movement into a desired output force and movement.

Transmission System

Gear Linkage

consistent translationlinear transfer function

Inconsistent translationnon-linear transfer function


transfer function

consistent translationlinear transfer function

Gearbox


crank drive = Linkage Inconsistent translationnon-linear transfer function


The pushing movement of the piston (crank mechanism) is transferred into a swinging movement of the shovel.

Bagger


Fourbar MechanismTwobar has -1 degrees of freedom

(preloads structure)Threebar has 0 degrees of freedom

(structure)Fourbar has 1 degree of freedom The fourbar linkage is the simplest

possible pin-jointed mechanism for single degree of freedom controlled motion

One link is grounded in each case


The fourbar linkage is the simplest possible pin-jointed mechanism for controlled motion with one degree of freedom.

Changing the relative lengths of the links can create a wide variety of motions.


4-Bar Nomenclature

• Ground Link• Links pivoted to ground:

– Crank– Rocker

• Coupler

Ground Link

Coupler

Link 1, length d

Pivot 02 Pivot 04

A

B

CrankRocker


Linkages of more than 4 bars

• Provide more complex motion• See Watt’s sixbar and Stephenson’s sixbar mechanisms in the textbook


The Grashof Condition Grashof condition predicts behavior of linkage based

only on length of links S=length of shortest link L=length of longest link P,Q=length of two remaining links

If S+L ≤ P+Q the linkage is Grashof :at least one link is capable of making a complete revolution

Otherwise the linkage is non-Grashof : no link is capable of making a complete revolution


I. If S + L < P + Q (Class I), the linkage is Grashof and at least one link will be capable of making a full revolution with respect to ground.

II. If S + L > P + Q (Class II), the linkage is non-Grashof and no link will be capable of making a full revolution with respect to any other link.

III. If S + L = P + Q (Class III), the linkage is special-case Grashof and although at least one link will be capable of making a full revolution.

Grashof-Type Rotatability Criteria for Higher-Order Linkages

Rotatability is defined as the ability of at least one link in a kinematic chain to make a full revolution with respect to the other links and defines the chain as Class I, II or III.

Revolvability refers to a specific link in a chain and indicates that it is one of the links that can rotate.


Crank-Slider

The crank-slider (right) is a transformation of the fourbar crankrocker, by replacing the revolute joint at the rocker pivot by ajoint, maintaining the same one degree of freedom.


Cam Follower

A cam follower is a mechanism that appears to have only two moving links (apart from ground), but it has 1 DOF.

It has a fourbar equivalent if the coupler (Link 3) is viewed as a link of variable length.


Practical Considerations

Pin Joints versus Sliders and Half Joints

A. Pin Joint Easy to lubricate ( with hydrodynamic lubrication) Can use relatively inexpensive bearings

B. Slider Requires carefully machined straight slot or rod Custom made bearings Lubrication is difficult to maintain

There are many factors that need to be considered to create good-quality designs.

The choice of joint type can have a significant effect on the ability to provide good, clean lubrication over the lifetime of the machine.

pin joint is the clear winner


Sleeve or journal bearing, the geometry of pin-in-hole traps a lubricant film within its annular interface by capillary action and promotes a condition called hydrodynamic lubrication in which the parts are separated by a thin film of lubricant .

Seals can easily be provided at the ends of the hole, wrapped around the pin, to prevent loss of the lubricant.


Relatively inexpensive ball and roller bearings are commercially available in a large variety of sizes for revolute joints.

Their rolling elements provide low-friction operation and good dimensional control.


For revolute joints pivoted to ground, several commercially available bearing types, Pillow blocks and flange-mount bearings.


MOTORS AND DRIVERS Unless manually operated, a mechanism will require some type of

driver device to provide the input motion and energy.

A motor is the logical choice to create the input.

Motors come in a wide variety of types. The most common energy source for a motor is electricity, but compressed air and pressurized hydraulic fluid are also used to power air and hydraulic motors.

Electrical Motors AC DC Servo Stepping


Chapter 4Design of Linkage Systems


Engineering Design involves

1. Synthesis

2. Analysis

Design a mechanism to obtain a specified motion or force.


Mechanism Synthesis

•Type Synthesis given the required performance, what type of mechanism is suitable? Linkages, gears, cam and follower, belt and pulley and chain and sprocket.

•Number Synthesis How many links should the mechanism have? How many degrees of freedom are desired?

deals with determining the length of all links, gear diameter, cam profile.

•Dimensional Synthesis


QUALITATIVE SYNTHESIS

• The creation of potential solutions in the absence of awell-defined algorithm which configures or predicts thesolution and also judge its quality.

• Several tools and techniques exist to assist you in thisprocess. The traditional tool is the drafting board, onwhich you layout, to scale, multiple orthographic viewsof the design, and investigate its motions by drawingarcs, showing multiple positions, and usingtransparent, movable overlays.

• Commercially available programs such as SolidWorkand Working Model allow rapid analysis of a proposedmechanical design. The process then becomes one ofqualitative design by successive analysis which isreally an iteration between synthesis and analysis.


TYPE SYNTHESIS

• The definition of the proper type of mechanism bestsuited to the problem and is a form of qualitativesynthesis.

• This is perhaps the most difficult task for the student asit requires some experience and knowledge of thevarious types of mechanisms which exist and whichalso may be feasible from a performance andmanufacturing standpoint.

• An engineer can do, with one dollar, what any fool cando for ten dollars. Cost is always an importantconstraint in engineering design.


DIMENSIONAL SYNTHESIS

• The determination of the proportions (lengths) of thelinks necessary to accomplish the desired motions andcan be a form of quantitative synthesis if an algorithmis defined for the particular problem, but can also be aform of qualitative synthesis if there are more variablesthan equations.


MECHANISM SYNTHESIS: TWO APPROACHES

CAD program SolidWorks


LIMITING CONDITIONS

• Once a potential solution is found, it must beevaluated for its quality. There are many criteria whichmay be applied. However, one does not want toexpend a great deal of time analyzing, in great detail,a design which can be shown to be inadequate bysome simple and quick evaluations.


TOGGLE: One important test consist in to check that the linkage can infact reach all of the specified design positions without encountering alimit or toggle position, also called a stationary configuration.

The toggle positions are determined by the colinearity of two of the moving links.


TRANSMISSION ANGLE: The transmission angle μ isdefined as the angle between the output link and thecoupler.

It is usually taken as the absolute value of the acute angle ofthe pair of angles at the intersection of the two links.


It is a measure of the quality of force transmission atthe joint.

Radial component only increases friction at pivot O4.

Tangential (normal to Link 4) produces torque.– μ = 90o is optimal.– In design, keep μ > 40o

To promote smooth running and good force transmission.

Ideally, as close to 90° as possible


Position analysis for Crank-Rocker mechanism

• The calculation of out-put angle


4.5 ALGEBRAIC POSITION ANALYSIS OF LINKAGES -Additional


ALGEBRAIC POSITION ANALYSIS OF LINKAGES



Excel or other program


O2

O44. Select two fixed pivot points, O2and O4, anywhere on the two midnormals.

Graphical Synthesis –Motion Generation Mechanism

Two positions, coupler as the output

A1 A2

B1

B2

1. Draw the link AB in its two desired positions, A1B1 and A2B2

5. Measure the length of all links,

O2A = link 2, AB = link 3,

O4B = link 4 and O2 O4 = link 1

2. Connect A1 to A2 and B1 to B2.

3. Draw two lines perpendicular to A1 A2 and B1B2 at the midpoint (midnormals).


O4O2

Graphical Synthesis – Motion Generation MechanismThree positions, coupler as the output

A1

A2

A3

B1

B2

B3

Same procedure as for two positions.

1. Draw the link AB in three desired positions.

2. Draw the midnormals to A1A2 and A2A3, the intersection locates the fixed pivot point O2. Same for point B to obtain second pivot point O4.

3. Check the accuracy of the mechanism, Grashof condition and the transmission angle.

4. Change the second position of link AB to vary the locations of the fixed points


Graphical Synthesis –Motion Generation Mechanism

Two positions Grashof 4-Bar mechanism with rocker as the output

D1

C1 C2

D2O2

5. Connect B1 to B2 and extend. Select any location on this line for fixed pivot point O2.

O2A = B1B2 / 2

7. Measure the length of all links, O2A = link 2, AB = link 3, O4CD = link 4 and O2 O4 = link 1

1. Draw the link CD in its two desired positions, C1D1 and C2D2

2. Connect C1 to C2 and D1 to D2 and draw two midnormals to C1C2 and D1D2

O4

3. The intersection of the two midnormals is the fixed pivot point O4.

B1 B2

4. Select point B1 anywhere on link O4C1 and locate B2 so O4B1= O4B2

A2

6. Draw a circle with radius B1 B2 / 2, point A is the intersection of the circle with the B1 B2 extension.


DIMENSIONAL SYNTHESIS - Solution


Coupler Curves

85

A coupler in a linkage in general has complex motion and provides the greatest variety of paths that can be traced.

The Hrones and Nelson Atlas of Fourbar Coupler Curves is a good reference


Chapter 5Velocity Analysis


Velocity

90

Definitions


iPA peR

Velocity of a point

Link in pure rotation

RPA as a complex number in polar formP is the scalar lengthJ is the complex operator (constant)

Position of Point P

Velocity of Point P

jjPAPPA

jepdtdpje

RVV


cos sinire r i

sin cosiire r i

cosr sinr

r

Real

Imaginary

cosr

sinr

Vector r can be written as:

Multiplying by i gives:

Multiplying by i rotates a vector 90°

Euler's formula


If point A is moving (Relative Velocity)


Velocity Analysis of a 4-Bar Linkage

94

Given 2. Find 3 and 4


Analytical Velocity Analysis of Fourbar Linkage

Numerical Example


• Numerical Example

Position Analysis


• Plot of Output Velocity versus Input Variable


Chapter 6Acceleration Analysis


Definition of Acceleration

• Acceleration is the rate of change of velocity with respect to time.

99

Linear acceleration

Angular acceleration

VRA

dtd


2

2

( )

iP

i iPA

i iPA PA

i i

R pe

V pe i pe i

A V pe i pe i

pe i pe

Acceleration of a point

Acceleration has 2 components: normal & tangential

PA

tAPA

nA

A link PA in pure rotation, pivoted at point A in the xy plane


Acceleration Difference / Relative Acceleration

If point A is moving

2

P A PA

i iA

A A A

A pe i pe


Analytical Acceleration Analysis (4bar)


Fourbar Pin-Jointed Linkage

103

Given 2. Find 3 and 4


Numerical Example


Human Tolerance for Acceleration

108

Humans are limited in the level of acceleration they cantolerate.

Machines are limited by the stressesin the parts, e.g. automobile piston40g’s at idle,700g’s at highway 2000g’s peak.

g defined as the acceleration due to gravityg= 9.8 m/sec2


Jerk

109

The time derivative of acceleration is called jerk, pulse or shock.

linear jerk:

angular jerk:

dtdAVRJ

dtd

High jerkHigh acceleration

Controlling and minimizing jerk in machine design is often of interest, especially if low vibration is desired.

roller coaster !!!


Chapter 7Gears


Mechanical Transmissions

111

Chains Belts Gears


Rolling Cylinders

112

Gear analysis is based on rolling cylinders External gears rotate in opposite directions Internal gears rotate in same direction


External Set: Opposite Movement

Internal Set: Movement in the same

direction

Internal and external gears:Two gears together are called a gearset


Gear Types

Spur Gears Bevel Gears Helical Gears Worm Gears Rack and Pinion

Rack and Pinion


Spur Gears

115

Straight teeth Noisy since all of the tooth

contacts at one time Low Cost High efficiency (98-99%)


Helical Gears

• Slanted teeth to smooth contact• Axis can be parallel or crossed• Has a thrust force• Efficiency of 96-98% for parallel and

50-90% for crossed


Bevel Gears

117

Based on rolling cones Bevel gears are most often

mounted on shafts that are 90 degrees apart, but can be designed to work at other angles as well.


Worm Gears

118

High gear ratio Impossible to back drive 40-85% efficient


Rack and Pinion

119

Generates linear motion Teeth are straight


Stage automatic transmission

120

Source: ZF Friedrichshafen AG


Fundamental Law of Gearing

121

The angular velocity ratio between 2 meshing gears remains constant throughout the mesh

Angular velocity ratio (mV) Torque ratio (mT) is mechanical advantage (mA)

v ωrin in out outω r ω r

Output

Input

in

out

in

out

out

inT

out

in

out

in

in

outV

dd

rr

ωωm

dd

rr

ωωm

Pinion

GearThe positive or negative sign accounts for internal or external cylinder sets


v ωrin in out outω r ω r

Output

Input Pinion

Gear

out

in

out

in

in

outV N

Nrr

m

out = angular velocity of output gearin = angular velocity of input gearrin = pitch radius of input gear

rout= pitch radius of output gearNin = number of teeth on input gearNout = number of teeth on output gear

This means that torque is exchanged for velocity in

out

in

out

out

in

VA N

Nrr

mm

1

Gear Ratio, mG, is what is commonly referred to when specifying gear trains

It is the magnitude of either the velocity ratio or torque ratio, whichever is > 1. VG mm


Meshing Action

123


The common normal of the tooth profiles, at all contact points within themesh, must always pass through a fixed point on the line of centers, calledthe pitch point.


Circular Pitch: pc=d/N Diametral Pitch (in 1/inch): pd=N/d=/pc Module (in mm): m=d/N


Compound Train Design

inω

outω

2

3 4

5

2 4

3 5in outN Nω ωN N

If N2=N4 and N3=N5

2

2

3in outNω ωN

2

3

2

in

out

ω Nω N

Reduction ratio

2 stages

Will be used to determine the no. of stages given a reduction ratio


Reverted Compound Train

• Input and output shafts are aligned

• For reverted gear trains:R2+R3=R4+R5D2+D3=D4+D5N2+N3=N4+N5

• Gear ratio is

Commercial three stage reverted compound train

5

4

3

2

NN

NN

ωω

in

out


Planetary or Epicyclic Gears

• Conventional gearset has one DOF• If you remove the ground at gear 3, it has two DOF• It is difficult to access 3


Planetary Gearset with Ring Gear Output

• Two inputs (sun and arm) and one output (ring) all on concentric shafts

theory of machinessite.iugaza.edu.ps/mawad/wp-content/uploads/iugaza...the theory of machines by...

Documents