Lecture 27 Analytical Modal Analysis

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Rose-Hulman Institute of Technology Mechanical Engineering Vibrations Today’s Objectives : Students will be able to: a) Find the homogeneous or steady state solution using analytical modal analysis Analytical Modal Analysis Note: Bring your laptop to class tomorrow!

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modal analysis

Transcript of Lecture 27 Analytical Modal Analysis

  • Rose-Hulman Institute of TechnologyMechanical Engineering

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    Todays Objectives:

    Students will be able to:

    a) Find the homogeneous or steady state solution using analytical modal analysis

    Analytical Modal Analysis

    Note: Bring your laptop to class tomorrow!

  • Rose-Hulman Institute of TechnologyMechanical Engineering

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    Overview (from Pete Avitabile)

  • Rose-Hulman Institute of TechnologyMechanical Engineering

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    Analytical Modal Analysis(MDOF free and forced vibration)

    EOM (1)

    Solve the eigenvalue problem

    For eigenvalues:

    For eigenvectors:

    Mass normalize the eigenvectors:

    Form the modal matrix:

    [ ]{ } [ ]{ } [ ]{ } { }FxKxCxM =++ &&&

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    Modal analysis (cont.)

    Apply a coordinate transformation

    Into (1) gives:

    Premultiply by []T gives:

    Gives:

    Notes: we get n uncoupled differential equations! qi(t) is called a principle coordinate of, if [] is mass normalized, a

    normal coordinate

    [ ][ ]{ } [ ][ ]{ } { }FqKqM =+ &&[ ] [ ][ ]{ } [ ] [ ][ ]{ } [ ] { }FqKqM TTT =+ &&

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    Modal analysis (cont.)

    Then uncoupled equations are:

    We know how to solve this!!

    How do we fine Ai and Bi?

    iiii Qqq =+ 2&&

    ( ) ( ) ( )tqtqtqPH iii

    +=Particular solution(depends on RHS)

    Homogeneous solution=Aicosit + Bisinit

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    Initial conditions for normal coordinates

    We need

    Recall:

    But if we mass normalize the modes we have

    So

    So we get

    One we find {q(t)} we can find {x(t)} by transforming back

    ( ) ( )0 and 0 q q &

    ( ){ } = 0q( ){ } = 0q&

    ( ){ } [ ] ( ){ }tqtx =

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    Summary of Analytical Modal Analysis

    1. Find EOM2. Solve the eigenvalue problem to find frequencies and modes

    Mass normalize the modes

    3. Apply coordinate transformation (basically any motion can be considered as a superposition of the normal modes) Obtain the decoupled equations of motion:

    Find the generalized force: Qi(t) = []T{F(t)} Find the initial conditions for {q(t)}:

    4. Solve the decoupled equations and apply ICs to find qi(t)5. Transform back to find {x(t)}

    [ ]{ } [ ]{ } { }FxKxM =+&&[ ]{ } [ ]{ }XKXM =2 [ ] [ ] and

    [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] so == KIM TT( ){ } [ ] ( ){ }tqtx =

    ( ){ } ( ){ }0 and 0 qq &iiii Qqq =+ 2&&

    ( ){ } [ ] ( ){ }tqtx =

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    Example

    The 3-DOF system shown is found to have the natural frequencies

    0.4

    0.3

    0.1

    23

    22

    21

    ===

    and the mass normalized modal matrix:

    [ ]

    =

    4083.05.02886.04083.005774.0

    4083.05.02886.0 and mass matrix [ ]

    =

    200020002

    M

    a) Using modal analysis determine the time response of each mass if the system is given the initial conditions

    0)0()0()0()0( 3211 ==== xxxx &&& and 1)0()0( 32 == xx b) What is the stiffness matrix for this problem?

    x1 x2 x3