Gears and Cams. Optimal Problem Formulation Lecture 6 (3)

8/10/2019 Gears and Cams. Optimal Problem Formulation Lecture 6 (3)

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Due 03/18/14, Tuesday (submit in Titanium, submit a hard copy in class)

Late submissions will not be accepted

Presentations: 03/18/14 and 03/20/14, Tuesday and Thursday

Goal: Disassemble, analyze and evaluate an existing real-world mechanism. Formulatethe problem as optimization problem and identify future work. Each student is to

prepare 3-5 slides generated in the format provided.

The analysis should include one of the following mechanisms:Slider Crank, Inverted Slider Crank,

Four Bar Linkage, Planar Robots, Simple Gears and Gear Trains,Cam and Follower Systems

Individual Projects: Device Analysis. Optimal Design Problem Formulation


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Gears and Gear TrainsOptimization Problem Formulation


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Spur Gears: Parallel Axes


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Bevel Gears: Perpendicular Axes


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Worm and Worm Gears: Non-Parallel Non-Intersecting Axes


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Gear Ratio


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Example

Two parallel shafts, separated by a distance of 3.5 in are to be connected bya gear set so that the output shaft rotates at 40% of the speed of the input

shaft. Design a gear set to fit this situation, i.e. find the radii of the pitch circlesR2 and R3, as well as the number of teeth on the gears.

Which gear is the input and which is the output?


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Example: Solution(1) R2 + R3 = 3.5 in

(2) ! 2/ ! 3 = R3/R2 = 0.40 (output shaft rotates at 40% of the speed of the input shaft)

Solve (1) and (2) simultaneously to find R2 = 1 in, R 3 = 2.5 in

Now, chose the size of the teeth by tentatively picking the diametral pitch: P = 10 teeth/in

Use the equation for the diametral pitch: P = N/2R (R is the pitch circle radius in [in]) to find:

N2 = 2P*R 2 = 20 teeth and N 3 = 2P*R 3 = 50 teeth

Note: The choice of P for tooth size must later be checked for strength and wear.

! 2/ ! 3 = R3/R2, i.e. as gears go through their mesh the pitch point must remain stationary on the line of centers for the speed ratio to remain constant.


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Simple Gear Trains: Gears Mounted Individually to

Shafts and in Series


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Compound Gear Trains: Two or More

Gears Mounted on the Same Shaft


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Mechanical Advantage


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Design Synthesis of a Nine Speed Gear Drive:

Optimization Problem Formulation


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Design Synthesis of a Nine Speed Gear Drive:



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Cam and Follower Systems



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Cam and Follower Systems


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Cam and Follower Systems: From Internal Combustion Enginesto Prosthetic Assistive Devices


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The Cam Joint

A connection between two links that is formed by general surfaces in contact is called a cam-joint.The input link is called the cam and the output is called the follower .

Contact is defined as (i) a point A 1 in the cam and a point A 2 on the profiles of the cam and

follower positioned so they have the same coordinates A c in the world frame W , and such that (ii) the tangent vectors T 1 and T 2 of cam and follower profiles are the same T c at A c.

The cam joint has two degrees of freedom, because the relative configuration the cam B1 and follower B2 are defined by specifying contact of a point A 1 on the profile of B1 (one degree of freedom) with a point A 2 on the profile of B2 (the second degree of freedom).


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Mobility

Let c be the number of cam joints in a collection of n rigid bodies and j the number of hinges and sliders, then the mobility formula becomes

M=3(N -1) - 2j1 j2,where N is the number of links; N=n+1 includes the ground frame; j1 is the number of single dof joints;

j2 is the number of two dof pairs.

This allows us to determine the dof of the Cam system as: M= 3(3 1) 2(2) 1 = 6 4 1 = 1.

Let C denote the cam-joint, then we have the combinations RCR--Radial cam and oscillating follower, RCP--Radial cam and translating (reciprocating) follower, and PCP--Translating cam and translating follower.


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Followers

The shape of the cam and follower combine to define the displacement function. Followers are usually selected to have standard shapes so it is the cam that requires careful shaping operations.

The primary follower shapes are (i) the knife-edge, (ii) the flat-face, and (iii) the roller follower. The curved shoe is spherical solid that has the circular cross-section of a roller follower.

The different shapes of these followers require the cam profile to be different in order to define the same displacement function.


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Displacement Functions

The profiles of the cam and follower are shaped to provide a specific displacement function.

RCR(radial cam, oscillating follower): ! (out) = f( " (in)),

RCP(radial cam, translating follower): s(out) = f( " (in)),

PCP(translating cam, translating follower): t(out) = f(s(in)).

Displacement functions consists of three basic segments: the rise , dwell and the return .

The displacement function also defines the follower velocity and acceleration:

Assume the angular velocity of the cam is a constant ! , then velocity and acceleration of the follower are seen in the displacement function:


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Radial Cam and Translating Followers: DisplacementDiagrams

Displacement function: s = {dwell from 0 to 90deg,

3in simple harmonic rise from 90 to 180deg,3in simple harmonic return form 180 to 360deg.}


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Cam Nomenclature: Relation Between the Cam Profile andthe Displacement Diagram

The cam surface is developed by holding the cam stationary and rotating the follower from station 0 through stations 1, 2, 3, etc.

LEFT: ROLLER FOLLOWER RIGHT: FLAT FACE FOLLOWER

1. IDENTIFY STATION NUMBERS AROUND THE PRIME CIRCLE2. THE CENTER LINE OF THE FOLLOWER IS CONSTRUCTED FOR EACH STATION3. WE CAN IDENTIFY THE CAM PROFILE FROM A GIVEN DISPLACEMENT DIAGRAM AND VICE VERSA


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Parabolic Rise

Parabolic Rise/Return:b is the radius of the base circle,h0 is the radius at the start of the return,h is the amount of rise," 0 is the starting angle of the rise,# = " 1 - " 0 is the angular range of the rise.

Accelerating segment:

Decelerating segment:


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Simple Harmonic Rise and Return

Simple Harmonic Rise:b is the radius of the base circle,h is the amount of rise," 0 is the starting angle of the rise,#= " 1 - " 0 is the angular range of the rise..

Simple Harmonic Return:h0 is the radius at the start of the return,h is the amount of return," 1 is the starting angle of the return,#= " 2 - " 1 is the angular range of the return..


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Cycloidal Rise and Return

Cycloidal Rise:b is the radius of the base circle,h is the amount of rise," 0 is the starting angle of the rise,# = "

1 - "

0 is the angular range of the rise.

Cycloidal Return:h0 is the radius at the start of the return,h is the amount of return," 1 is the starting angle of the return,# = " 2 - " 1 is the angular range of the return.


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Cam Profile for a Knife Edge Follower

The Cam ProfileThe cam profile is a plane curve of the form A = R cos" i + R sin" j,where i and j are the unit vectors in the x and y directions of the frame B attached to the cam.

Let A i be the points of contact between the cam and follower at different angular positions " i of the follower relative to the

cam.

The coordinates of the points of contact define the cam:A i = R cos" i i + R sin" i j.

The functions R and " that define the cam profile are derived from the follower geometry and the displacement function s=f( " ).

For a knife-edge follower these functions are particularly simple.


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Summary

A cam-follower mechanism has a two degree of freedom cam joint that connects the input and output links. The relative shape of the cam and follower define the displacement function of the mechanism.

Displacement functions for radial cams are periodic functions consisting of sequences of dwell, rise and return segments.

The geometry of the follower is usually simplified to a point, line or circle, and combines with the displacement function to define the cam profile .


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Example

Problem : Cam Design

A knife-edge cam-follower is to:1. dwell for 30 ,2. rise 2 inches with cycloidal motion from 30 to 100 ,3. dwell from 100 to 180 ,4. return 1 inch with simple harmonic motion from 180 to 230 ,

5. dwell from 230

to 270

, and then6. return 1 inch with simple harmonic motion from 270 to 330 ,7. dwell for the remaining 30 of cam rotation.The cam's base radius is 2 inches and the angular velocity is constant at 1 rad/s.

1. Formulate and sketch the displacement function forthe follower as a function of the cam rotation angle " .2. Formulate and sketch the velocity of the followerversus " .3. Formulate and sketch the acceleration of the followerversus " .4. Sketch the cam profile.


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Example

Problem : Cam Design

A knife-edge cam-follower is to:1. dwell for 30 ,2. rise 2 inches with cycloidal motion from 30 to 100 ,3. dwell from 100 to 180 ,4. return 1 inch with simple harmonic motion from 180 to 230 ,

5. dwell from 230

to 270

, and then6. return 1 inch with simple harmonic motion from 270 to 330 ,7. dwell for the remaining 30 of cam rotation.The cam's base radius is 2 inches and the angular velocity is constant at 1 rad/s.

1. Formulate and sketch the displacement function forthe follower as a function of the cam rotation angle " .2. Formulate and sketch the velocity of the followerversus " .3. Formulate and sketch the acceleration of the followerversus " .4. Sketch the cam profile.


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Simple Harmonic

Cycloidalb=do=2 is the radius of the base circle,h=2 is the amount of rise," 0 = 30

is the starting angle of the rise,# = " 1 - " 0 = 100

-30 is the angular range of the rise

h0 =4 is the radius at the start of the return,h=1 is the amount of return," 1 =180

is the starting angle of the return,# = "

2 - "

1 = 230 -180

is the angular range of the return.

1. dwell for 30 ,2. rise 2 inches with cycloidal motion from 30 to 100 ,3. dwell from 100 to 180 ,4. return 1 inch with simple harmonic motion from 180 to 230 ,5. dwell from 230 to 270 , and then6. return 1 inch with simple harmonic motion from 270 to 330 ,7. dwell for the remaining 30 .The cam's base radius is 2 inches.


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1. dwell for 30 ,2. rise 2 inches with cycloidal motion from 30 to 100 ,3. dwell from 100 to 180 ,4. return 1 inch with simple harmonic motion from 180 to 230 ,5. dwell from 230 to 270 , and then6. return 1 inch with simple harmonic motion from 270 to 330 ,7. dwell for the remaining 30 .The cam's base radius is 2 inches.


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Cam Profile

1. dwell for 30 ,2. rise 2 inches with cycloidal motion from 30 to 100 ,3. dwell from 100 to 180 ,4. return 1 inch with simple harmonic motion from 180 to 230 ,5. dwell from 230 to 270 , and then6. return 1 inch with simple harmonic motion from 270 to 330 ,

7. dwell for the remaining 30

.The cam's base radius is 2 inches.


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Cam Profile Optimization for a New Cam Drive:



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Representation of the Cam Profiles for the Intake and Exhaust Cams by a Four Piece General Sixth Order

Polynomial


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Objective Function


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Equality Constraints


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Inequality Constraints


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Summary


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HW 5 Due 03/25

Documents

Gears and Cams. Optimal Problem Formulation Lecture 6 (3)