- #31
whitejac
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I'm running out of variables here. The only other one not given is αm, which is something I currently do not know how to find specifically.haruspex said:
Kinematics Belt and Pulley Problem
- Thread starter whitejac
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SUMMARY
The forum discussion centers on solving a kinematics problem involving two pulleys connected by a belt, focusing on the relationship between torque, angular acceleration, and the time required to rotate through a specified angle. The key equations discussed include T = Jα and T = F*R, with specific values calculated for torque and angular acceleration. The participants emphasize the importance of understanding the relationship between the angular velocities and accelerations of the pulleys, particularly under the condition of no slip, leading to the conclusion that the acceleration must vary to minimize time.
PREREQUISITES- Understanding of rotational dynamics, specifically torque and angular acceleration.
- Familiarity with the equations of motion for rotating bodies, such as Θ = Θi + 0.5α*t².
- Knowledge of belt and pulley systems and their kinematic relationships.
- Basic algebra and calculus for differentiating and integrating motion equations.
- Study the dynamics of belt-driven systems, focusing on torque and angular acceleration relationships.
- Learn about piecewise functions in kinematics to model varying acceleration scenarios.
- Explore optimization techniques in physics to minimize time in rotational motion problems.
- Investigate the effects of moment of inertia on angular motion in complex pulley systems.
Mechanical engineers, physics students, and anyone involved in designing or analyzing belt and pulley systems in mechanical applications.
- #32
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Concentrate on how the time depends on R. Your equation in post #23 does not include time. You need an equation that relates R to time.whitejac said:
- #33
whitejac
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I know it doesn't, that's why taking derivatives isn't something I can currently do. I don't know where to put the t because it seems like I'd just be placing it arbitrarily. R isn't related to time as the pulley isn't growing or shrinking with relation to time. The Radius reduces the acceleration linearly, as shown in the relationship but I have not seen from the beginning how I can conjure a t and justify it as a logical relationship.haruspex said:
- #34
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The time taken would be different with a different R, so the two are related.whitejac said:
If the load accelerates from rest at αL, how long will it take to turn through angle θ/2?
- #35
whitejac
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I simply don't know.haruspex said:
- #36
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It's completely analogous to linear motion. If an object accelerates at constant rate a, how far will it go from rest in time t? Use SUVAT or just integrate.whitejac said:
- #37
whitejac
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haruspex said:
I've never heard of SUVAT, but this is what you mean? Integrating theta twice would yield the second equation regardless.
haruspex said:
Okay, a load accelerating from rest at αL would have the position equation:
θL = θ0L+ω0Lt+1/2αLt2
when θL =θL/2 and I.C.'s are applied:
θL = αLt2
And t = SQRT[θL/αL]
- #38
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Right. So minimising time means maximising αL, which is not surprising but worth checking.whitejac said:
In your post #23, you eliminated αL and kept αM. But αL is what we need to maximise. So you need to revisit that and obtain the equation that relates αL to τM, r, R, JL and JM.
- #39
whitejac
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haruspex said:
Okay, so I have done most of this but am out of town and away from the computer unail Sunday night.
I have rearranged Tm into an expression that has αL. Considering what you said about Radius R being the thing that changes the angular acceleration here, I took the derivative with respect to R and got an expression. To maximise alpha, I set the derivative equal to 0 and then solve for R. This became a heft fraction I cannot easily type on my phone.
Regardless, once I solve this, I will have the maximum apha and thus the Radius that minimizes the time required to go about a theta?
- #40
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Yes, that should work.whitejac said:
- #41
whitejac
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I've finally figured it out! I have to say, I am embarrassed how much I forgot of my multivariable calculator class. I thought I understood it better than this, but I suppose application is the real challenge.haruspex said:
Thank you so much for your patience. I will post back Sunday night with my attempt at part 2 and 3 but I believe it will not be too much of a challenge. It ought be similar in concept to this. The SUVAT has an expression with t, so I would relate it and the new found expression? Forgive me if this doesn't seem coherent. I'm not near paper to validate myself, but wanted to get the main idea straightened so I could mull it over during my trip.