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Finding the force of constraint--compound pendulum on spring

  1. Oct 31, 2016 #1
    1. The problem statement, all variables and given/known data
    From Fetter and Walecka 5.1:

    Consider the compound pendulum in FIg 28.1 (mass M, moments of inertia Iij relative to the center of mass, which is a distance L from the point of support Q) but with Q attached to the bottom of a vertical spring (force constant k) and constrained to move vertically. The top of the spring is fixed to a rigid support.
    (a) Construct the Lagrangian for this system in terms of the generalized coordinates η (the downward displacement of Q) and φ. Derive Lagrange's equations.
    (b) Obtain the same equations from the general principles of rigid-body dynamics concerning the motion of the center of mass and rotations about it (don't forget the force of constraint)
    https://lh3.googleusercontent.com/8K4utS2xNx0v9MV9P7Nhg-wnYFxzLVknSHNo1OI7pbzBpL8B9LqbpT9ODbnKp2jlMtDT0WYQzJF5y3cdueMftuaZSBG3PoR0WbzK-fOI0deYmKe9JVqxgevpVgBKvf1hR5FCJsYnVJVKlWYggq3Pg3Juy0vTQD5MGmEZSssJOUq1uD7tjEELzDu24Obg0F2FOepDH3dBFUBxxx4ucVHUE7up5g16qkq1k8TtvmZZLUNv7d2qC9Tu3hvCHB5yH-s7AJTBXLTwNyzRdEVQWyev69i-H19gyhFElV-nEKtbkb_-zTmHWBd37ZSYCzN6Wi0hoJ2xTh5MxLnIgs2wdiAeIxmzN9lnsx4g7wbIsPgbQMcxm4v2y-mjoSCTOVmeRx-ABsuKK0cnnPS-N14xfpuoemzzdquUsRvLhcKCKcxTTov6g6b7qhb3JpJpuXZPPABni7Zia3QcF3ZZ5mkIrHYL1ilAb7W-4jots6SbihTd61WIAPavW7tZUCg2b2jGF0-D7rIDhA-ok53B-8l_0LWdhICHj8mHcSV2SWTvTdjIAvQnavDxO2-F6QAlHM8O7w1gJiVjD7O4Npz5GlocePLAUpLDxnWPQVsJQTpDx_LYFSVZFIlHQg=w345-h612-no
    2. Relevant equations
    Euler's equations:
    I11/dt =ω2ω3(I2-I3) +Γ1(e) and similar permutations of indexes
    where
    Γ1(e)= torque⋅e^1
    3. The attempt at a solution
    I've done part (a) getting η⋅⋅= g- k/m η and φ⋅⋅ = -mgl sin(φ)/(I33,cm+ml2)

    For part(b)
    I initially used the euler equations and the torque about the point Q ΓextQ=R×mg= - mlg sinφ, which results in the equation I had from part (a). However since the question specifically asks to use rotations about the center of mass I'm not sure if I would be allowed to do this.

    However bigger issue is in trying to find the torque about the center of mass I don't know how to go about finding the force of constraint from the suspension point that contributes. Nor am I sure of what the ωs which are needed to use the Euler equation would be in a reference frame centered on the center of mass, since the rotation is taking place around Q and not cm. I imagine I may have to use the relations of each reference frame in terms of the other? It seems as those the torque about the center of mass is needed in order to find the η EOM.

    Γcmspring = R×(-kη)= -lkη sin(φ) e^[\SUP]3
     
  2. jcsd
  3. Nov 5, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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