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Math Help - Constant Torque Mechanism

  1. #1
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    Constant Torque Mechanism

    Hi.

    I have a problem designing a mechanism intended to deliver constant output torque. One end of a tension spring is located at any of seven locations around a circular arc. The other end of the spring is attached to the start of a curved cheek. As the spring is relocated to more distant points, it wraps around the cheek, thereby lengthening the spring. The spring tension thus increases, but the requirement is that the output torque about a defined point below the cheek should be a constant. I'm trying to determine the shape of the curved cheek to achieve this.

    The following link explains in detail and, most importantly, includes a drawing. Explaining (and understanding) the mechanism without the drawing is near impossible and I can't see how to post a drawing directly here.

    Here's the link: https://docs.google.com/viewer?a=v&p...OGM1ZWZlMzZlOQ

    Can anyone define the shape of the curved cheek, please? I'll eventually need to show how the calculations are done, if you could include those, please.

    Many thanks for any help.
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  2. #2
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    Re: Constant Torque Mechanism

    I choose to use polar coordiantes centred on P.

    Because the spring is tangent to the surface at the seperation point, for a small rotation dTheta the stretchy part of the spring increases in length by dL where:

    dl = r d \theta

    So the change in force is:

    \frac{dF}{d \theta} = rk

    Now:
     P = Fr

    And

     \frac{dP}{d \theta}= \frac{dr}{d \theta}F+r \frac{dF}{d \theta}=0

    Substututing for F and df/d theta

     \frac{dP}{d \theta}= \frac{dr}{d \theta}\frac Pr+r^2k=0

    So

     \frac{dr}{d \theta}=\frac {-r^3k}{P}

    Or

     \frac{dr}{r^3}=\frac {-kd \theta}{P}

    Integrating:

     \frac{-1}{2r^2}=\frac {-k \theta}{P}

    So

     r^2=\frac {P}{2k \theta}

    Finally

     r=\sqrt{\frac {P}{2k \theta}}
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  3. #3
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    Re: Constant Torque Mechanism

    This is exactly the sort of thing I was hoping for.

    Many thanks Kiwi_Dave - brilliant!
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  4. #4
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    Re: Constant Torque Mechanism

    Problem:

    Just been trying to apply this and realised that there's an assumption that the point of separation (and, therefore, tangency) of the spring from the cheek curve is the same as the point on the spring from which a perpendicular to the spring passes through P. Unfortunately, they aren't the same point.

    The diagram in the link given in the first post illustrates the difference. A perpendicular to the spring from the point of separation from the curve would miss point P (which is why it hasn't been drawn - it would have no immediate purpose and would merely clutter the diagram). The point on the spring from which a perpendicular passes through P is defined where the red torque arm intersects the cheek curve. The two perpendiculars are not the same line - they are two separate lines.

    Last edited by foglefogle; November 29th 2013 at 10:43 PM.
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  5. #5
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    Re: Constant Torque Mechanism

    Good spotting!

    I think that \frac{dF}{d \theta} = rk is still correct, but  P = Fr is not.

    I can't see an easy way to resolve this (but haven't given up yet). It looks like we need to say something like:

     P = F (r_0 + \int^\beta_0 dr) where \beta is the angle between the separation point and the perpendicular. But now we have a more complicated differential equation and a new variable beta the value of which we don't know. This could drive us to a numerical solution?

    Do you mind telling me where this problem has come from? The context sometimes tells us what techniques are likely to be needed. e.g. real world engineering problems often need numerical techniques. Also if this is a real world thing then the original solution might be accurate enough. If this comes from the real world then calculate the shape and check to see how close to constant the torque is. If this has come out of university study then it would be interesting to know what you are studying at the moment.

    Oh, and in my original "solution" I would replace  \theta with  \theta + \theta_0 because I forgot to add a constant of integration when I integrated.

    A really interesting problem!
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  6. #6
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    Re: Constant Torque Mechanism

    Thanks Kiwi_Dave. Very interesting proposals.

    This is both a purely theoretical and a real world engineering research problem, applicable to what is only allowed to be a purely clockwork device requiring constant torque. The requirements are to not only create a working, optimised mechanism, but also include evidence of how it was optimised and explain the theory behind the design of the cheek profile. No electricity can be supplied to the device. A small coiled spring (like an old-fashioned clock spring, which we'll call spring 'S2') feeding the device is rewound every few seconds by a much larger coiled spring (call that one 'S1'). Spring S1 delivers more torque to spring S2 when S1 is fully wound than when S1 is almost completely unwound, but that doesn't matter at all. However, spring S2 also delivers more torque to the device when S2 is fully wound than when it is almost unwound, and that definitely does matter - it needs to be an absolutely constant torque beyond that point. So the curved cheek device, which also rewinds every few seconds, smoothes out the varying torque from spring S2. The requirement is for as close to perfectly constant torque as possible, because the mechanism spring S2 supplies energy to is a mechanical oscillator, which is extremely sensitive to variations in torque.

    Regrettably, your original solution wouldn't be accurate enough, but the general approach (calculus; elegant; universal; very speedy to execute) would be exactly what I'm looking for. An accurate numerical technique would be fine, given the power of computers these days, which would render the process of calculation quick and easy. For this problem I've had little success in that direction myself, but my experince of such techniques is limited. Speedy calculation is as important as accuracy, because I need to design many of these cheek devices, for research purposes. Sadly, my very, very slow graphical technique I mentioned in an earlier post has also just revealed a technical flaw, which I became aware of after seeing the (remarkably similar) flaw in your first method (basically, I made the same mistake). So I'm right back to square one now. Your work so far helped me spot the flaw in my graphical method, so many thanks for that, which potentially avoided a lot of wasted manufacturing.

    As you so rightly say, a really interesting problem. I've been at it for weeks and have enjoyed the mental challenge, despite the lack of success. However, I've come to a point where I need to complete this task and I've had to admit that more skilled help is needed.
    Last edited by foglefogle; November 30th 2013 at 10:38 PM.
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  7. #7
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    Re: Constant Torque Mechanism

    Hi Foglefogle

    Are you still about? did you resolve this problem?

    This really caught my interest and exercised my brain for a couple of months! I went down many blind paths and ultimately was unable to come up with a decent solution.
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