Explore BrainMass

Explore BrainMass

    Stiffness matrices and displacements in spring systems

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    For the spring system shown in the attached diagram, find:
    a) The stiffness matrix for each element
    b) The unconstrained global stiffness matrix
    c) The constrained (reduced) stiffness matrix
    d) The displacements of node 2 and node 3
    e) The reaction forces at node 1 and node 4

    For the given spring system in the second attachment, find:
    a) The stiffness matrix for each element (symbolically)
    b) The unconstrained global stiffness matrix (symbolically)
    c) The constrained (reduced) stiffness matrix (symbolically)

    Take k1 = 100N/mm, k2 = 200N/mm, k3 = 300N/mm, k4 = 400 N/mm, P1 = 1000N, P3 = 3000N
    d) The displacements of nodes 1, 2, 3 (numerically)
    e) The reaction forces at node 4 and node 5 (numerically)

    © BrainMass Inc. brainmass.com December 24, 2021, 5:46 pm ad1c9bdddf
    https://brainmass.com/engineering/mechanical-engineering/stiffness-matrices-displacements-spring-systems-69435

    Attachments

    Solution Preview

    Problem 1

    a.) k1 -k1 100 -100
    = (stiffness matrix for element 1)
    - k1 k1 -100 100

    k2 -k2 200 -200
    = (stiffness matrix for element 2)
    - k2 k2 -200 200

    k3 -k3 100 -100
    = (stiffness matrix for element 3)
    - k3 k3 -100 100

    b) k1 -k1 u1 f1ele1
    X = (1)
    - k1 k1 u2 f2ele1

    k2 -k2 u2 f2ele2
    X = (2)
    - k2 k2 u3 f3ele2

    k3 -k3 u3 f3ele3
    X = (3)
    - k3 k3 u4 f4ele3

    P1= f1ele1 = k 1u 1-k1u 2 (4)
    P2= f2ele1 + f2ele2 = - k1u1 +u2(k1 +k2) -k2u3 (5)

    P3= f3ele2 + f3ele3 = - k2u2 +u3(k2 +k3) -k3u4 (6)

    P4 = f4ele3 = k3u4 -k3u3 (7)

    The above equations can be represented as

    k1 -k1 0 0 u1 P1

    - k1 k2 +k1 -k2 0 u2 P2
    x = (8)
    0 -k2 k2+k3 -k3 u3 P3

    0 0 -k3 ...

    Solution Summary

    The solution calculates stiffness matrices for different elements of springs systems as well as displacements and reaction forces.

    $2.49

    ADVERTISEMENT