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Mechanical Design II Spring 2013

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Lecture 3 S V A J Diagrams 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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Lecture 3 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

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Lecture 3 y=4/5x parabolic 3 4/5 -2/5 +∞ 3/2 zero -∞

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Lecture 3 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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Lecture 3 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! 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.

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**Fundamental Law of Cam Design**

Lecture 3 Fundamental Law of Cam Design 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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**Discontinuous leads to infinite jerk (bad cam)**

Lecture 3 Simple Harmonic Motion 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: continuous Piecewise continuous Discontinuous leads to infinite jerk (bad cam)

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**Cycloidal Displacement**

Lecture 3 Cycloidal Displacement Better approach is to start with consideration of higher derivatives (acceleration) Cycloidal displacement = sinusoidal acceleration If we apply the cycloidal displacement function to our cam timing diagram: Disadvantage: high level of peak acceleration Cam motion is continuous through acceleration; jerks are finite = acceptable cam

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Lecture 3 Square wave (constant acceleration) function best minimizes peak magnitude of acceleration However, this function is not continuous unacceptable Square wave’s discontinuities can be removed by simply “knocking the corners off” the square wave function trapezoidal acceleration Disadvantage: discontinuous jerk function Solution: Modified Trapezoidal Acceleration Finite jerk

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Lecture 3

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**Modified Sinusoidal Acceleration**

Lecture 3 Modified Sinusoidal Acceleration 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.

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Mechanics of Machines Dr. Mohammad Kilani

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