Lecture 04 [Rigid Mechanisms]-1

7/30/2019 Lecture 04 [Rigid Mechanisms]-1

1/23

2.000: Rigid mechanisms

Laptop checkup~ Internet capable?

~ Notes downloadable?

Mechanisms/linkages

~ Linkage fundamentals

~ Examples

~ Position synthesis and analysis

www.physikinstrumente.com

Mechanisms

Actuators

Electronics

Sensors

Machines


2/23

Announcements

Problem set 1

Make sure you get a camera


3/23

What is a linkage?

x

yInput or Output

Grounded(Held motionless)

Input or Output

Input or Output

What is a linkage?

A mechanism used to define motion (kinematics)&/or transfer energy using links & joints

Link

Joint


4/23

Ubiquitous mechanical modules

http://smithstuff.net/albums/

www.p

hotogarry.c

om

Generic 4 bar mechanism

Ubiquitous building block

Aim to combine nano 4 bars

to create complex nano devices

Complex motion example


5/23

Compliant mechanism: All/part of motion due tocompliance

Given that we can engineer

compliant 4 bars more

complex machines can

be built by combining 4 bars

Major types of mechanisms

www.p

hotogarry.com

Linear

Rotary

Conventional rigid mechanism Compliant mechanism


6/23

Linkage components

F1 =f(F2) what is it?

Input or Output Input or Output

Input or Output

Linkage components:~ Links = rigid components or groups of rigidly connected

components between joints

~ 2D joints can either be pinned, sliding or rolling contact

x

y

F1

F2Grounded(Held motionless)


7/23

Linkage classification

Input or Output

Grounded(Held motionless)

Input or Output

Input or Output

Linkages are classified by:~ The number of links (we will deal with 4-bar linkages)

~ Type of links (we will deal with bars and sliders)

~ Connection between links (we will deal with pinned, spherical and

sliding joints)

x

y


8/23

Example: 4 bar door damper linkage

1

1= Wall or Link 1 This is the grounded (held still)

2

2 = Bar 2 or Link 2

3

3 = Bar 3 or Link 3

4

4 = Door or Link 4

Pin or rotary joints


9/23

Mechanisms relationship to the 5 Fs

Actual machine design

Nominal machine design

Modeling

Dominant

Application

Fysics

Limiting

Momentum

Information

Mass

Flows

Energy

Who?

Where?

What?

Function

Why?

Motion

Constraints

Geometry

Form

Interfaces

Quality

Cost

Processes

Fabrication

Rate

A. What is

engineered?

B. What is

realized?

Modeling

Dominant

Application

Fysics

Limiting

Momentum

Information

Mass

Flows

Energy

Who?

Where?

What?

Function

Why?

Motion

Constraints

Geometry

Form

Interfaces


10/23

Example: Stewart platform

Grounded (Constrained in 6 DOF)

How to identify links and jointsStep 1: Find the link that is not moving (ground)

Step 2: Find a feature that experiences relative motion (joint)

Step 3: The next link lies between this joint and the next pt ofrelative motion

Step x: Continue through links/joints to shortest path to ground

This sounds easy, but


11/23

Example: Automotive suspension

Tie rod

Rotor

Spherical joint

Control arm

Frame

SteeringKnuckle

Rack Pinion

Spherical joint


12/23


x

y


13/23


x

y


14/23

Effective 4-bar linkage

x

y

Appears that there are only 3 linksLink 4 is an effective link

This linkage can be modeled as a 4 bar linkage

Length of link 1 varies with linkage position

An effective link is characterized by a rigid non-changing lengththat is needed to define the motion of the mechanism

1

23

4

1

23

4


15/23

Common 4-bar linkage types

1

2

4

3

1

2

4

3

4 BAR SLIDER-CRANK:

Consists of 3 links and slider:

- 1 = Ground- 2 = Crank- 3 = Coupler- 4 = Slider

Crank is link driven by Force/torque

Link # 4 can be:- A pin, block, or tube

Link # 4 slides in/on one of these:- Another part, rod or tube

Links can cross each other

4 BAR COUPLER:

Consists of 4 links:- 1 = Ground- 2 = Crank or follower- 3 = Coupler- 4 = Crank or follower

Crank is the link driven by force/torque

Links can cross each other

x

y


16/23

4-bar pin slider linkages

24

3

Pull

32

4

11

x

y


17/23

Simulation: Working model

Backhoe Piston assembly

Automotive suspension

Credit: http://workingmodel.com/simulations.htmlx

y

1

2 3

4


18/23

Graphical synthesis of 4-bar, 2 positions

Situation: Want coupler to hit 2 specific positions~ Q: Location of ground link and length of crank/follower

9Know points A & A and B & B

~

A: Obtain these by following this procedure9 1. Draw lines (chords) between points A & A and B & B

9 2. Connect A & A and B & B ends with desired arc of motion

9 3. Construct perpendicular bisector to each chord

9 4. Joints to ground lie on bisectors

9 5. Ground joint positions defined by tangent to arcs

9Note: Your choice of arc radius affects position!!!

1

24

3

A

A B

B


19/23

Graphical synthesis of 4-bar, 2 positions

A

A B

B


20/23

Analytic method: Vector analysis

1Rv

2Rv

3Rv

4Rv

Analytic position/velocity method~ Represent links as vectors between joints

~ Works for 2D and 3D, we will use 2D to illustrate the point

~ Vectors form a closed loop; vector sum = 0

~

jiRRRR 0004321 +==+++vvvvv


21/23

Analytic method: Vector analysis (2D)

jiRRRR 0004321 +==+++ vvvvv

( ) ( ) jRiRR sincos 11111 +=vvv

( ) ( ) jRiRR sincos 22222 += vvv



( ) ( ) ( ) ( ) iRRRRi coscoscoscos0 44332211 +++=vvvv

( ) ( ) ( ) ( )( ) jRRRRj sinsinsinsin0 44332211 +++=vvvv

in x:

in y:

jiRRRR 0004321 +==+++ vvvvv

1Rv

2Rv

3Rv

4Rv


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Parallel link 4 bar mechanism

Parallel 4 bar linkage:~ Opposing links have equal lengths

~ 1 = 180o |R1| = |R3| |R2| = |R4|

~ Opposing links remain parallel

x

y


23/23

Linkage competition [ 10 pts ]

In groups of 4, use LeggosTM to experiment~ Design a linkage(s) which inverts a sketch (turns it upside down)

~ Hand in a sketch of your linkage geometry, the sketches it made

and an few sentence explanation before the bell sounds

~ Describe the logic (bulleted points and/or mathematics) that explain

why the device works

8

0

412

Minutes

2

6

14

10

Lecture 04 [Rigid Mechanisms]-1

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