Learning with Purpose January 28, 2013Learning with Purpose January 28, 2013

22.322 Mechanical Design II

Spring 2013

Learning with Purpose January 28, 2013

Ultimately we would like to choose a mathematical expression that will allow the follower to exhibit a desired motion. Easiest approach is to “unwrap” the cam from its circular shape and consider it as a function plotted on Cartesian axes.The motion of the cam is analyzed using a S V A J diagram• S = displacement• V = velocity (first derivative)• A = acceleration (second derivative)• J = jerk (third derivative)

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S V A J Diagrams

Learning with Purpose January 28, 2013

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Consider the specifications for a four-dwell cam that has eight segments:

These function characteristics can be easily investigated with program DYNACAM• Generate data and plots

Learning with Purpose January 28, 2013

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4/5

y=4/5x

-2/5

+∞3/2

-∞

3parabolic

zero

Learning with Purpose January 28, 2013

Consider the following cam timing diagram. There are two dwells and we would like to design the cam such that there is good motion in the rise and the return.

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Learning with Purpose January 28, 2013

Because we’re new at designing cams, let’s just try a linear function between the low and high dwells:The cam designer should be more concerned with the higher derivatives!

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Infinite acceleration requires an infinite force (F=ma)

Infinite forces are not possible to achieve, but the dynamic forces will be very large at the boundaries and will cause high stresses and rapid wear. Separation between the cam and follower may occur.

Learning with Purpose January 28, 2013

Any cam designed for operation at other than very low speeds must be designed such that:• a) The cam function must be continuous through the first and

second derivatives of displacement across the entire 360o interval of motion

• b) The jerk must be finite across the entire 360o interval. The displacement, velocity, and acceleration function must have no discontinuities in them.

Large or infinite jerks cause noise and vibration. To obey the Fundamental Law of Cam Design, the displacement function needs to be at least a fifth order polynomial.• V=4th order, A=3rd order, J=2nd order and finite

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Fundamental Law of Cam Design

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Sinusoids are continuously differentiable:On repeated differentiation, sine becomes cosine, which becomes negative sine, which becomes negative cosine, etc.If we apply a simple harmonic motion rise and return to our cam timing diagram:

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Simple Harmonic Motion

continuous

Piecewise continuous

Discontinuous leads to infinite jerk (bad cam)

Learning with Purpose January 28, 2013

Better approach is to start with consideration of higher derivatives (acceleration)Cycloidal displacement = sinusoidal accelerationIf we apply the cycloidal displacement function to our cam timing diagram:Disadvantage: high level of peak acceleration

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Cycloidal Displacement

Cam motion is continuous through acceleration; jerks are finite = acceptable cam

Learning with Purpose January 28, 2013

Square wave (constant acceleration) function best minimizes peak magnitude of accelerationHowever, this function is not continuous unacceptableSquare wave’s discontinuities can be removed by simply “knocking the corners off” the square wave function trapezoidal accelerationDisadvantage: discontinuous jerk functionSolution: Modified Trapezoidal Acceleration

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Finite jerk

Learning with Purpose January 28, 2013

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Learning with Purpose January 28, 2013

Modified trapezoidal function is one of many combined functions created for cams by piecing together various functions, while being careful to match the values of the s, v, and a curves at all the interfaces between the joined functions.The modified sinusoidal acceleration function is a combination of cycloidal displacement (smoothness) and modified trapezoidal (minimize peak acceleration).The combination results in lower peak velocity.Made up of two sinusoids with two different frequencies.

Lecture 3

Modified Sinusoidal Acceleration