Learning with Purpose February 1, 2013Learning with Purpose February 1, 2013

22.322 Mechanical Design II

Spring 2013

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A cam converts one motion into another formA follower follows the cam profile

Summary

Lectures 1-4

Think of a cam-follower system as a linkage with variable-length “effective” linksBecome familiar with basic cam terminology:• Translating vs. rotating

followers• Follower types:

• Flat-faced• Roller• Knife-edge• Curved

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Follower displacement curves: • Rise, dwell, return/fall

Graphical Design• Based on a displacement curve, generate the cam contour (HW #1)• Understand what these are:

• Base Circle• Pressure Angle• Pitch Point• Throw/stroke/travel

• Start with your base circle,divide it into q intervalsaccording to displacementcurve, plot thecorresponding points, maketangent lines and drawcam contour tangent tothose lines

Summary

Lectures 1-4

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Summary

HW #1 Solution

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SVAJ Diagrams (HW #2)• When designing a cam-follower system, consider the higher derivatives of displacement• Satisfy the Fundamental Law of Cam Design

• s, v, a must be continuous • j must be finite

SCCA Family of Functions• Acceleration curves that will change

the peak magnitude of velocity, acceleration, jerk

• Curves are defined by the same set ofequations but by varying parametersand coefficients• Don’t worry about the derivation of those

equations/parameters programs likeDYNACAM do that for you!

• Different values lead to different curvesdepending on the design specifications

e.g. modified sine acceleration functionleads to a low peak velocityIf the follower mass is large, you want tominimize its velocity

Summary

Lectures 1-4

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Kloomok and Muffley method of combining displacement curves

Summary

Lectures 1-4

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Summary

C-1

H-2P-2

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Lecture 5

Polynomial Functions

The general form of a polynomial function is:

s = Co + C1x + C2x2 + C3x3 + C4x4 + … + Cnxn

where s is the follower displacement, x is the independent variable (q/b or time t)

C coefficients are unknown and depend on design specificationThe number of terms in the polynomial will be equal to the number of boundary conditions on the s v a j diagrams.

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Lecture 5

For the double-dwell problem, we can define six BCs:

Since we have 6 BCs, we need a polynomial with 6 coefficients:

Differentiate with respect to q to get V and a:

Now we can apply the BC’s to solve for C0, C1, C2, C3, C4, C5

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Lecture 5

We now have 3 equations and 3 unknowns and can solve for the unknown coefficients!

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Lecture 5

𝑠=h[10 (𝜃𝛽 )3

−15( 𝜃𝛽 )4

+6 (𝜃𝛽 )5]

This is called a 3-4-5 polynomial because of the exponents

Velocity and acceleration are continuous but jerk is not it was left unconstrained.

In order to constrain the jerk, we need to add two additional coefficients, C6 and C7, and use two more BCs:

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Lecture 5

If we repeat the same analysis but add the two extra BCs and coefficients, we will result in the following 4-5-6-7 polynomial for the displacement function:

This 4-5-6-7 polynomial will have a smoother jerk but a higher acceleration than the 3-4-5 polynomial.

Learning with Purpose February 1, 2013

Many applications require a single dwell cam profile (rise-fall-dwell)Example: the cam that opens valves in an engine The valve opens on the rise, closes on the return, and remains shut while combustion and compression take place.The cam profiles for the double dwell case may work but will not be optimal (e.g. cycloidal displacement)

Lecture 5

Single-Dwell Cams

Acceleration is negative in rise and fall regions…why go to 0?

Subsequently leads to discontinuous jerk

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Although the jerk is finite, it is discontinuous. A better approach is to use the double harmonic function.

Lecture 5

Single-Dwell Cams

Double harmonic should never be used for double-dwell case non-zero acceleration at one end of interval.

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If we used a polynomial function to design the single dwell cam, we could use only one function to describe the rise and the return. The BC’s are:

With 7 BC’s, we need 7 coefficients. If we solve as we have previously, the function turns out to be a 3-4-5-6 polynomial:

Lecture 5

Single-Dwell Cams

𝑠=h[64 ( 𝜃𝛽 )3

−192 (𝜃𝛽 )4

+192( 𝜃𝛽 )5

−64( 𝜃𝛽 )6]

Learning with Purpose February 1, 2013

Note that the acceleration is reduced compared to the double harmonic, but the jerk is discontinuous at the ends.We could have also set but that would increase the order of the polynomial by two.

Lecture 5

Single-Dwell Cams

A high-degree function may have undesirable oscillations between its BCs!