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Multiple Degree of Freedom Systems: Chapter 8 Examples

Example 8-1

The double pulley shown below (which is supported by a spring with stiffness k_2) has a moment of inertia J_O about its center O. The pulley supports a mass M by a light cable which has an effective stiffness k_1. The coordinates x and \theta as shown are used to describe the system.

  1. Determine the equations of motion for this system using Newton’s Law’s.
  2. If

        \begin{equation*} k_1 = k_2 = k, \qquad r_1 = r,\quad r_2 = 2r, \quad \text{and}\quad J_0 = \frac{1}{2} (5M) r_2^2, \end{equation*}

    the equations of motion become

        \begin{equation*} \begin{bmatrix} M & 0 \\ 0 & 10 M r^2 \\ \end{bmatrix} \biggl\{ \begin{array}{c} \ddot{x} \\ \ddot{\theta} \\ \end{array} \biggr\} + \begin{bmatrix} k & -k r \\ -k r & 5 k r^2 \\ \end{bmatrix} \biggl\{ \begin{array}{c} x \\ \theta \\ \end{array} \biggr\} = \biggl\{ \begin{array}{c} 0 \\ 0 \\ \end{array} \biggr\} . \end{equation*}

    Determine the resulting natural frequencies for this system and sketch the associated mode shapes.

Example 8-2

Determine expressions for the natural frequencies and mode shapes for the system shown. Use the coordinates x_1 and x_2 as indicated.

Example 8-3

Two simple pendulums of the same length are connected by a light spring with stiffness k as shown.

For this system, determine

  1. the equations of motion,
  2. the natural frequencies,
  3. the mode shapes,
  4. the total response of the system (for t>0) if the initial conditions at time t=0 are

        \begin{align*}{2}     \theta_1       =&\ \Phi_0, & \qquad \theta_2 =&\ 0,  \\     \dot{\theta}_1 =&\ 0,      & \qquad \dot{\theta}_2 =&\ 0. \end{align*}

For parts (b)–(d) assume that m_1 = m_2 = m.

Example 8-4

A rigid beam (mass m, length l, centroidal moment of inertia J_G) is supported by springs at its ends as shown. Determine the equations of motion, natural frequencies and mode shapes for this system.

Solve this problem using two different sets of coordinates:

  1. x_1 and x_2,
  2. x_G and \theta,

as illustrated below.


Be sure to identify any {nodes} in each mode shape.

Example 8-5

An asymmetric machine mounting is supported by both vertical and horizontal springs as shown below.

  1. Using the coordinates x, y, and \theta as illustrated, determine the equations of motion for this system.
  2. Explain how the equations of motion and mode shapes change in each of the following special cases:
    1. k_2 \, l_2 = k_1 \, l_1,
    2. h=0,
    3. k_2 \, l_2 = k_1 \, l_1 and h=0.

Example 8-6

The two degree of freedom system shown consists of a wheel (with mass m_1 and centroidal moment of inertia J_O = m_1 r^2) which rolls without slipping and a bar (with mass m_2 and centroidal moment of inertia J_G = \frac{1}{3} m_2 L^2) connected by three springs each of stiffness k.

  1. Using the coordinates x_1 (the horizontal position of the center of the wheel) and x_2 (the horizontal position of the lower end of the bar) show that if m_1 = m and m_2 = 3m, the equations of motion for this system become

        \begin{equation*}  \begin{bmatrix}    2 m & 0 \\     0  & m \\  \end{bmatrix}  \biggl\{      \begin{array}{c}        \ddot{x}_1 \\ \ddot{x}_2 \\     \end{array}  \biggr\}  +  \begin{bmatrix}   2 k & k \\     k & 2k \\  \end{bmatrix}  \biggl\{      \begin{array}{c}        x_1 \\ x_2 \\     \end{array}  \biggr\}  =  \biggl\{      \begin{array}{c}        0 \\ 0 \\     \end{array}  \biggr\}. \end{equation*}

  2. Determine the natural frequencies and mode shapes for this system.

Example 8-7

Determine the steady state response for the system shown.

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