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Math Help - Normal Modes - One Dimensional Oscillating Systems

  1. #1
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    Normal Modes - One Dimensional Oscillating Systems

    Hi

    Can someone help me with the following questions please (I now understand why mathmaticians and physicists dont get along)?

    I really need some help on the following:

    i). Drawing a force diagram for each particle (I really hate drawing these).

    As a guess for m1, am I right in thinking that H and N point up and W points down? But do I also need to add m3 to W (or would it be half m3?)

    And for m3, would it be H1 and H2 up together with N, and W pointing downwards? Or would I also need to include m1 and m2?

    ii). Can someone help me indicate the changes in forces exerted by the springs when the particles are displaced from their equilibrium positions? (the forces need to be expressed as vectors in terms of stiffness and displacements.)

    iii) How do I derive the equation of motion of each particle?
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  2. #2
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    Quote Originally Posted by moolimanj View Post
    Hi

    Can someone help me with the following questions please (I now understand why mathmaticians and physicists dont get along)?

    I really need some help on the following:

    i). Drawing a force diagram for each particle (I really hate drawing these).

    As a guess for m1, am I right in thinking that H and N point up and W points down? But do I also need to add m3 to W (or would it be half m3?)

    And for m3, would it be H1 and H2 up together with N, and W pointing downwards? Or would I also need to include m1 and m2?

    ii). Can someone help me indicate the changes in forces exerted by the springs when the particles are displaced from their equilibrium positions? (the forces need to be expressed as vectors in terms of stiffness and displacements.)

    iii) How do I derive the equation of motion of each particle?
    m3 is going to oscillate without slanting huh? I guess we have to assume that one. If not you need to know how long m3 is, where it's center of mass is, etc.

    The FBD's are easy. I am assuming you need to draw the FBD's based on the initial displacements, otherwise you have a total of 8 possible sets of diagrams to deal with. (Two for each spring.)

    So for m1 we have a force from spring 1 downward, a force from spring 3 upward, and a weight downward, etc.

    Hooke's law states
    \vec{F} = -k \Delta \vec{x}
    and positive is always taken in the direction of stretching the spring. So if we displace m1 upward a distance d_1, then the change in force on m1 from equilibrium will be
    -k_1 \vec{d_1} + k_3 \vec{d_1}

    As far as solving this system is concerned, the simplest method is probably using the Lagrangian method. In the event you aren't up to using that, we need to use Newton's 2nd Law, as usual. Write Newton's 2nd for each mass. So
    \sum F_1 = -k_1d_1 + k_3d_1 + w_1 = ma_1
    etc. You will wind up with a system of three linear differential equations in three unknowns (d1, d2, and d3.) This will be "nontrivial" to solve.

    -Dan
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    So topsquark

    Am I right in thinking that the following is the force diagram for m1:
    Attached Thumbnails Attached Thumbnails Normal Modes - One Dimensional Oscillating Systems-m1.bmp  
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by moolimanj View Post
    So topsquark

    Am I right in thinking that the following is the force diagram for m1:
    I'm not certain of your notation but there are only three forces on m1: a force from spring 1, a force from spring 3, and a weight. You have what looks like four forces? (If that N is supposed to represent a normal force note that m1 is not resting on or contacting a surface so there will be no normal force. This is a common mistake to make.)

    -Dan
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  5. #5
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    So I have

    For mass m1:

    H1=-k1x1i

    H3=k3(x3-x1)i

    Hence, m1r1=-k1x1i+k3(x3-x1)

    For mass m2:

    H2= -k2x2i

    H4= k4(x3-x2)i

    Hence, m2r2=-k2x2i+k4(x3-x2)i
    For mass m3:

    Not sure: But does it look something like: m3r3=k3(x1-x3)i+k4(x2-x3)i. Should m3 be H1 + H2+H3+H4?

    Not sure about that - can you help?

    Then I can carry on from there as all i need to do is replace m1r1 by m1x1, collect rems and matrixise it.
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  6. #6
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    if you look at the matrix in next part of the question you will see the required finishing part of these equations. I agree with both you m1 and m2 equations but I get a different one for m3 which rearranges correctly for the matrix in the next part. Not sure if method is correct but I used the following:

    (delta)H5
    (delta)H6
    [(delta)h5 + (delta)h6]

    the h5 and h6 is affect of the springs separately on m3 and the last part is the combination of the both effects of the spring

    so i ended up with an equation of motion which i think in matrix form ends up at the bottom part of the matrix in the question.

    not sure if this correct like i said but seems to workout

    now if you could help me with that magnification question from question 1........
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    the way i did the magnification factor question was to subxtitute values for alpha, beta and w into the previous answer (A/a) and simplify. You will probably have to rearrange the expression for omega and r and m and then substitute. Not sure if its right though - scrathched my brain on it for ages. The other way is to work backwards. i.e. substitute beta, alpha and w into the equation given and see if you come up with the previous answer. Again, not sure if this is correct as i hate applied questions.
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