Week 1 Lecture Slides.pdf

Week 1

Dr Oliver Kennedy - Office: 8.109E-mail: [email protected]

A/Prof Weihua Li – Office: 8.110 E-mail: [email protected]

1 Week 1

Objectives• Analyse rigid bodies in plane motion• Classify plane mechanisms, determine

mobility and carry out basic mechanism synthesis

• Understand the kinematics of plane mechanisms

• Analyse the forces acting on plane mechanisms

• Solve a kinematic or kinetic problem using analytical, graphical or software tools

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Week 1

Topics• Mechanism synthesis• Kinematics (motion) & Kinetics (force) of

simple mechanisms in Plane Motion• graphical and analytical velocity and

acceleration analysis• Superposition• Virtual work• Energy method• CAD mechanism design & analysis• Balancing rotating masses

3 Week 1

Subject “mind map”

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Week 1

Subject “mind map”

5 Week 1

Text/Reference Books• Text book

R L Norton - Design of Machinery, 5th

Edition

• References:Hibbler - Engineering Mechanics -Dynamics (or similar)Mabie & Reinholtz – Mechanisms and Dynamics of Machinery, 4th edition (call no 621.8/51)

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Week 1

MECH226 – reference mat’ls

• Books in the library with call numbers around 621.81 may be of interest…

• Kinematic design of machines and mechanisms – H D Eckhardt

• Dynamics of machinery - J Hirschorn• Ingenious mechanisms for designers and

inventors – F D Jones• Mechanisms and mechanical devices

sourcebook – ed. Chironis & Slater7 Week 1

Lecture/Tutorial Times• Lecture Time

Thursday 10.30 – 12.30 20.3

• Tutorial Times Group 1: Thursday 14.30 – 16.30 19.G024Group 2: Thursday 14.30 – 16.30 8.G10Group 3: Friday 08.30 – 10.30 1.G05Group 4: Friday 13.30 – 15.30 24.103

You can register and check your tutorial group via SOLs

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Week 1

Labs (details to be finalised)• Lab 1 (6-G09)

weeks 3 & 4 (groups of 4, 20 per session)

• Lab 2 (6-G09) week 7 (groups of 3, 30 per session)

• Lab 3 (6-G09) weeks 10 & 11 (groups of 3, 30 per session)

Labs 1 & 2 are 1 hour duration, Lab 3 runs for 2 hours.

9 Week 1

LabsFive Practical classes have been timetabled for this subjectPrac 1: Thur 13.30-14.30Prac 2: Thur 16.30-17.30Prac 3: Fri 10.30-11.30Prac 4: Fri 11.30-12.30Prac 5: Fri 12.30-13.30

Still sorting out arrangements for these – will set up self enrolling on Sols – email me if you have any specific needs / timetable clashes

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Week 1

LabsIf you are repeating the subject you may be permitted to carry over previous marks for Labs 1 and 2 and be exempt those labs this time.Please see and/or email me if you want to consider this option.For difficulties regarding enrolling in tutes or labs please see me.

11 Week 1

MECH226 - Assessment

• 1 mid session quiz (of tutorial type problems), held in the lecture timeslot (week 8) – 16%

• 6 tutorial handins – weeks 2, 4, 6, 9, 11 & 13 – 6%• 3 labs:

Inertia Determination week 3 & 4 5%

Cam Profile week 7 10%

Engine Dismantling weeks 10 & 11 18%

• Final Exam - 45%12

Week 1

MECH226 - general• NB PC’s no longer exist - may get WS (possible

PS after supp) if close to a pass• TF criteria not applied in this subject this time• Penalties for late submission (5% of max mark

per calendar day late)• Safety (in labs) – you will not be permitted to

work in labs with “open” footwear (thongs, sandals …)

• Students with disabilities• Plagiarism• Supplementary Examinations...

13 Week 1

MECH226 - generalEmail EtiquetteWhen sending formal email to the subject coordinator, lecturers or tutors, please ensure that you include; a useful descriptive heading, address the recipient in a professional manner and include a detailed succinct email outlining your question/issue/problem and finally sign off with your full name and student number. Emails from email accounts other than UoW will not be replied to for privacy/security reasons. Also, please allow adequate time for a reply to your email to be returned (generally within 2-3 days).

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Week 1

Weekly program

15 Week 1

Weekly program

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Tutorial Quizzes, Mid session and final exams are closed book

Week 1

Past Results - 2007

F PC P C D HD enrolled 7610.5% 3.9% 48.7% 26.3% 6.6% 2.6% ungraded 1

MECH226 2007: Average Mark: 58.74%

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10152025303540

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17 Week 1

Past Results - 2008

F PC P C D HD enrolled 8312.0% 9.6% 31.3% 20.5% 16.9% 7.2% ungraded 2

Machine Dynamics Average Mark: 58.22%

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F PC P C D HD

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Week 1

Past Results

0%

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HDDCPPCFTFOverall - 2731

20102009

19 Week 1

Week 1

Introduction, Kinematics Fundamentals

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Week 1

Overall PurposeTo develop our ability to design variable

mechanism solutions to real, unstructured engineering problems by using a design process

• Synthesis of mechanisms to accomplish desired motions or tasks

• Analysis of mechanisms to determine their rigid-body dynamic behavior

• The above relates to kinematics and kinetics

21 Week 1

Kinematics and Kinetics• Kinematics: the study of motion

without regard to forcesdisplacement, velocity, and acceleration

• Kinetics: the study of forces in mechanisms in motion

Newton’s second law: F = maplus the transmitted forces

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Week 1

Mechanisms and machines• Mechanism: A means of transmitting,

controlling, or constraining relative movement (cams, gears, belts, chains, etc.)

23 Week 1

Mechanisms and machines• Machine: A machine contains mechanisms

that are designed to provide significant forces and transmit significant power (car, bike, tank, etc.)

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Week 1

Mechanism types• There is an infinite variety of mechanisms,

but only a limited number of mechanism types we need to be concerned with

• “3 bar” mechanisms (such as cam & follower systems and gear pairs)

• “4 bar” mechanisms of several varieties• Additional driving linkages are often

required – resulting in 6 bar and higher orders – however these can usually be understood and analysed as assemblies of 3 and 4 bar mechanisms

25 Week 1

3 bar mechanisms

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Week 1

4 bar mechanisms

27 Week 1

Kinematics Fundamentals• Degree of freedom (DOF)• Types of motions: rotational and

translation. planar (2-D) kinematic systems

• Links, joints, kinematic chains• Mobility (degree of freedom) in

planar mechanisms

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Week 1

Degree of freedom (DOF)• Degree of freedom: the number of independent

parameters needed to uniquely define system’s position in space at any instant of time. A rigid body in space has 6 DOF. A rigid body in a plane has 3 DOF

29 Week 1

Degree of freedom (DOF)

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Week 1

Links• A link is a rigid body that possesses at least two

nodes that are points for attachment to other linksBinary link: one with two nodesTernary link: one with three nodesQuaternary link: one with four nodes

31 Week 1

Types of Links• Crank: a link that makes a complete

revolution and is pivoted to the ground• Rocker: a link that has oscillatory (back and

forth) rotation and is pivoted to ground• Coupler (connecting rod): a link that has

complex motion and is not pivoted to ground• Ground: any link or links that are fixed

(nonmoving) with respect to the reference frame

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Week 1

Joints• A joint is a connection between two or more

links (at their nodes). Joints also called Kinematic pairs

• Joint types:– Lower Pair (surface contact in joint)– Higher Pair (rolling, point or line contact in

the joint)– Joints may be form closed (closed by its

geometry)– Or force closed (required external force

to keep it together or closed)

33 Week 1

Joints

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Week 1

Joints

35 Week 1

Kinematic Chain

• Is an assembly of links connected by means of pairs (joints)

– Locked chain (no motion is possible –structure)

– A constrained chain (relative motion between the links is the same at the same phase of any cycle)

– An unconstrained chain (no repeated motion cycle is guaranteed)

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Week 1

Mobility (or Degrees of Freedom)

• The mobility of a mechanism is the number of degrees of freedom it possesses – or the minimum number of independent parameters (or inputs/drives) required to completely specify the location of every link within the mechanism.

• Gruebler’s Equation (2.1c in text, p.40) may be used to determine the mobility

37 Week 1

Mobility (Gruebler’s Eqn)

Where:M = mobilityL = number of links (incl. the ground)J1 = number of 1 degree of freedom jointsJ2 = number of 2 degrees of freedom joints (or half joints)

NB: where “k” links connect at a single joint, it must be counted as “k-1” joints

21213 JJLM

38

Week 1

Mobility Examples

L=4, J1=4

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

L=3, J1=3

M=3*(3-1)-2*3=0

L=2, J1=2

M=3*(2-1)-2*2=-1

39 Week 1

Mobility Examples

L=3, J1=2, J2 =1

M=3*(3-1)-2*2-1=1

L=4, J1=4

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

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Week 1

Mobility Examples

L=8, J1=10

M=3*(8-1)-2*10=1

L=6, J1=7, J2=1

M=3*(6-1)-2*7-1=041 Week 1

Summary: Mobility (Gruebler’s Eqn)M > 1 may be an unconstrained mechanism

(has M deg of freedom, needs M inputs)M = 1 a constrained mechanismM = 0 a statically determinate structureM < 0 a statically indeterminate (or preloaded)

structure

{Caution needs to be exercised, as there are numerous exceptions (paradoxes – section 2.8 of text) to Gruebler’s Eqn resulting from special geometry (e.g. parallel links ..), need to use your engineering intuition}

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Week 1

Paradoxes

43 Week 1

Synthesis of mechanisms

• Most of the theoretical work in this subject (mech226) focuses on the analysis of a given mechanism.

• Important in determining whether the motion is satisfactory, magnitude of forces, design of components etc.

• However, SYNTHESIS is the important step that comes before this.

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Week 1


• Can be a mix of inspiration (aided by referring to some source materials .. ) and theory.

• Some mechanisms may be designed to carry out computational tasks (Function Generation, e.g. the integrator in section 2.16 of M&R) and rely heavily on theory to develop the dimensions of the links.

• Other approaches use prescribed locations of the “output link” to develop the rest of the mechanism (Path and Motion Generation)

45 Week 1


• We will only consider some simple cases of graphical (and analytical) design of 4 bar mechanisms for the 2 and 3-precision-position cases (path & motion generation ..)

• Dealt with in considerably more detail in Chapters 3 & 5 of the text (and sections 11.5 to 11.7 incl. of M & R)

Closing door problem …

46

Week 1

Synthesis – 2 Position

One simple case concerns the control of the range of angular motion experienced by the output link.Examples: ………………………………………

47 Week 1


It is often desirable that this range of motion be driven by a completely rotating crank (say on an electric motor)

required range of motionfor output link

Link 4

48

Week 1


This can be arranged by placing the crank pivot somewhere along the line passing through the extreme positions of the output link. The crank size is determined from the radius of the circle passing through the same extreme positions.


Link 4

Link 3Link 2

49 Week 1


NB: links 2 and 3, together, are sometimes called a “dyad”, and can be set up in a similar fashion to this in other mechanisms to cause a mechanism to oscillate between required limits.The user is free to locate the pivot point of link 2 anywhere along the dotted line, the choice dictates the length of link 3.Link 3 may also be called the “coupler”.


Link 4

Link 3Link 2

50

Week 1


IF it is necessary to control the position of the pivot of the driving dyad, a similar but slightly more complex method can be used to find the required lengths of links 2 & 3. It is again based on the min & max distances to the extreme locations of the output link.Link 2 = (R2 – R1)/2 ….. Link 3 = (R1 + R2)/2


Link 4

R1 65.03

101.32R2

For a pre-determinedfixed pivot position

51 Week 1


Sample solution for the previous problem … NB this arrangement means different times for the forward and reverse motions of the output link (for a constant speed of the crank … link 2) – may be desirable in some cases.

required range of motion for output link

Link 4


83.2 83.2

R18.15

52

Length of link 3

Length of link 2

Week 1

Synthesis – Limiting Conditions (section 3.3 of text)

In designing a mechanism there are a couple of simple limiting conditions that should be considered:Toggle positions (limit, or stationary point) – where 2 connected links become collinear. The preceding mechanism has 2, however in that case it doesn’t prevent link 2 from completing its rotation. In many cases a toggle position locks the mechanism (and can be used to advantage).Transmission angle – the smaller angle between the coupler and the driven link (usually links 3 and 4 respectively). To avoid excessive force and friction being generated at the pin, this angle should be kept between 45 and 135 degrees.

53 Week 1

Toggle Points & Transmission Angles

required range of motion for output link

Link 4


83.2 83.2

R18.15

54

Toggle Points

TransmissionAngles

Week 1

Synthesis – 2 Position – coupler link

The motion of the coupler is usually the most complex and “interesting” of the 3 moving links, (not being confined to pure circular arcs) and hence maybe what the designer wants to control (& exploit).

Required 2 positionsof link 3, or coupler

55 Week 1

Synthesis – 2 Position – coupler linkThis motion can be achieved by bisecting the intended pivot points (A & B) on this link, and pivoting the connecting links somewhere along the respective centrelines..


A1

A2

B1 B2

In this case one simple solution could be to place a single link pivoting at the intersection of both centrelines

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Week 1


One possible solution (not really a “coupler link”)


A1

A2

B1 B2

57 Week 1


An alternate solution, NB links 3 & 4 are in a “toggle position” at state 1. The mechanism is locked, and can’t move further to the left.


A1

A2

B1 B2Link 2

Link 3

Link 4

Hence Link 2 can’t be used to drive this mechanism.58

Week 1


A “dyad” can be used to drive link 3 between the 2 limits, as shown above. NB many other dyads could be used, incl. fixed pivot…

A1

A2

B1 B2Link 2

Link 3

Link 4

59 Week 1


The next level of complexity is introduced when the position of the coupler link is specified in 3 locations.The same geometrical procedure is used (i.e. bisecting the lines connecting pivot point locations) – however in this case 2 lines are generated for each point.The intersection of these 2 lines gives a uniquesolution in this case.

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Week 1


One possible situation where 3 points are defined - where it is desired develop a mechanism to close a hatch with a final motion close to vertical.

A desired sequence ofmoves for a door (withconnection pointsnominated)

61 Week 1


The red and blue “dashed” construction lines indicate how the length and location of the pivot point points for the hinge links can be determined.


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Week 1


This analysis guarantees that mechanism can be assembled in the 3 nominated positions.However this does not mean it will be capable of moving smoothly between the 3 positions.


A 4 bar mechanism can be assembled in one of 2 different branches. It is quite possible that one of the three chosen positions is actually on a different branch.

Something to be aware of and check for (build a model, use a computer simulation)

63 Week 1

Synthesis of mechanismsProvided the coordinates of the chosen pivot points on the coupler link having the defined motion are known,

the equations presented in section 11.7 of M & R (Eqs 11.23 and 11.24) may be set up in a spreadsheet and solved simultaneously to determine the required geometry of the mechanism.

There are some similar equations provided in the text (ch. 5) based on complex number representation of the linkages.

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Week 1

Synthesis – 3 Position – coupler linkControlling the location of the Fixed Pivots

The previous analysis starts with a selection of pivot points on the moving “coupler” link, and results in finding the location of the fixed pivots and the lengths of the connecting links.Sometimes it is necessary to choose the fixed pivot locations first, then find out where the connecting points on the coupler link should be located.An “inversion” of the previous analysis can be used to give this result.

65 Week 1


The red & blue pivots indicate desired fixed pivot locations in this case.The analysis is based on finding the apparent motion of these points, as seen by someone moving with the “hatch”


Chosen positionsof Fixed Pivots

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Week 1


These figures show the apparent motion of the fixed pivot points with respect to the hatch.The 3rd figure shows the “ground” as a brown bar, moving with respect to a fixed hatch (or door).

A1

A2

A3

B1

B2

B3

A1

A2

A3

B1

B2

B3

67 Week 1


The blue and red construction lines show the new pivot locations required “on” the door.Some extensions to the door would be required to make this possible (as well as branch compatibility being checked).

A1

A2

A3

B1

B2

B3

68

Week 1


A possible outcomeNB it looks like this mechanism will lock …ie: continuous motion between these positions will not be possible –different fixed positions need to be selected


Chosen positionsof Fixed Pivots

69 Week 1

Example 3-7: p. 113-116

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Week 1

Example 3-7: p. 113-116

71 Week 1

Transmission of Motion• A fundamental characteristic of

mechanisms is that they transmit motion from one link (input link) to another (output link).

• This may occur via direct contact between the input and output links (pair of gears, or cam and follower) or via a third intermediate link (connecting rod, belt or chain)

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Week 1

Transmission of Motion• May be via direct contact (gears, cam &

follower)– The transmission line is common normal

where the components contact• Via an intermediate link (link 3 in a 4-bar

linkage)– The transmission line between links 2 &

4 is defined by the line of link 3• Via a flexible link (belt or chain)

– The transmission line is the line of the belt or chain

73 Week 1

Line of action• There is a very simple, but important,

principle related to the line of action along which this motion transfer takes place

• Basically, for the parts to stay in contact (cam & follower)

• or the mechanism to not break any links (four bar mechanism),

• the velocity component of the input & output links along the line of action MUST BE IDENTICAL

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Week 1

Cam & follower

• The line of action for a cam and follower is along the common normal.

• The velocity component of both these parts at the point of contact along this line (the common normal) must be the same if contact is to be maintained.

75 Week 1

Cam & follower

76

Week 1

Cam & follower

• Where the cam and follower are in contact (point “P”):

• the velocity of the point on the cam (link 2) is perpendicular to O2P

• the velocity of the point on the follower (link 3) is perpendicular to O3P

• the component of both these velocities in the direction of the common normal (P N) must be the same for the surfaces to remain in contact

77 Week 1

4 Bar Mechanism

• In this case, the line of action is defined by the straight line between the points of connection of the intermediate link (normally designated as link 3)

• The velocity component of both these points along this line must be the same, otherwise link 3 would be either compressing or stretching

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Week 1

4 Bar Mechanism

79 Week 1

4 Bar Mechanism• The actual velocities of points A & B are

perpendicular to the lines O2A and O4B respectively

• The components of these 2 velocities (VA and VB) that lie along link 3 (shown in red as VLOA) must be identical if link 3 is not to be deformed in the process of the mechanism moving

• Note the point “K”80

Week 1

Some calculations

Section 1.8 of Mabie & Reinholtzdiscusses these concepts, and also presents some equations for determining the ratio between the angular velocity of the input and output links based on this approach (given in following slides)

81 Week 1

Some calculationsThe procedure is based on:a “similar triangles” analysis

the line of action for motion transmissionthe line defined by the pivot points for the input and output links (O2, O3, O4)and the intersection between these 2 lines (point K)the length from this intersection point (K) and the pivot points for the input and output links

82

Week 1

Some calculationsCam & Follower (3 bar mechanisms)

KOKO

3

2

2

3

KOKO

42

2

4

(Eq. 1.1 M & R)

(Eq. 1.2 M & R)

Four Bar Mechanism

83 Week 1

Transmission of Motion• Velocity along the common normal (line of

action) must be equal• Velocity of sliding is the relative velocity along

the common tangent• For pure rolling the point of contact must lie

on the line of centres• For a constant angular velocity ratio to be

maintained between 2 links, the common normal must intersect the line between the centres at a fixed point (for all phases of the cycle)

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Week 1

Next Week

Browse through sections 8.0 to 8.3 and familiarise yourself with concepts relating to cams…NB: Tutorial Handin Problem in the tute next week.

85

Documents

Week 1 Lecture Slides.pdf