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Modeling piston/crank with position loop equation, excel plot

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Jul21-10, 12:55 PM
P: 1
1. The problem statement, all variables and given/known data

2. Relevant equations
I would like to know if my work is correct. I thought that the graph would be a sinusoidal, but there is a small irregularity in the middle.

Am I correct in using θAC = 90 deg and θAB = 36.8699 deg in solving for the constant angular velocity?

I attached the scratch paper portion because that contains my geometry work for determining the θAB value beyond the symbolic part, and I'm unsure if that's correct.

I've set up the position, velocity, and acceleration loop equations symbolically as shown below.

What concerned me about using θAC = 90 deg is that it causes θ'AB = 0 due to the cos term. This ends up making the max angular velocity -60 rad/s.....

3. The attempt at a solution

Scratch paper

Thanks in advance for helping out a clueless EE student!


Resized images......can resize further if the res is too big for your monitor. Also fixed broken link. Still missing the scratch paper image because I'm away from home. I'll upload when I get home.....but hopefully this should be enough for someone to check my basic equations and graph.
Attached Thumbnails
Problem.jpg   piston1.jpg   piston2.jpg  
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Jul23-10, 01:13 AM
P: 799
It's pretty difficult to follow your steps, so I cannot comment in detail on your work. I think we need not use so many variables (please excuse me if I misunderstand something). Two coordinates are enough: [tex]\phi[/tex] (angle between AC and AB) and x.

The only geometric condition relating the coordinates is: [tex]b^2=a^2+x^2-2axcos\phi[/tex]

Plug a=3 and b=5 in, then solve for x (x>0 as the origin is at A): [tex]x=\sqrt{9cos^2\phi +16}+3cos\phi[/tex]

Thus: [tex]\dot{x}=-(\frac{9sin\phi cos\phi}{\sqrt{9cos^2\phi +16}}+3sin\phi)\dot{\phi}[/tex]

From here, we can find the constant angular velocity of AC in the range [tex]0<\phi <\pi[/tex] and [tex]\dot{x}<180 in/s[/tex]. Then the rest is simple. The graph I obtained looks akin to yours.

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