Open Educational Resources

Multiple Degree of Freedom Systems: Forced Vibrations of Undamped Two Degree of Freedom Systems

The general form of the equations of motion for an undamped two degree of freedom system
subjected to external loading are

    \[ \begin{bmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \\ \end{bmatrix}\!\! \biggl\{\!\!\! \begin{array}{c} \ddot{x}_1 \\ \ddot{x}_2 \\ \end{array} \!\!\!\biggr\} + \begin{bmatrix} k_{11} & k_{12} \\ k_{21} & k_{22} \\ \end{bmatrix}\!\! \biggl\{\!\!\! \begin{array}{c} x_1 \\ x_2 \\ \end{array} \!\!\!\biggr\} = \biggl\{\!\!\! \begin{array}{c} F_1(t) \\ F_2(t)\\ \end{array} \!\!\!\biggr\} . \]

The solution to this general form of the equations of motion is beyond the scope of this course (but only slightly). However, we will consider a special case in which the forcing functions F_1(t) and F_2(t) are harmonic, at the same frequency and in-phase with each other. In such a case the equations of motion become

(8.34)   \[ \begin{bmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \\ \end{bmatrix}\!\! \biggl\{\!\!\! \begin{array}{c} \ddot{x}_1 \\ \ddot{x}_2 \\ \end{array} \!\!\!\biggr\} + \begin{bmatrix} k_{11} & k_{12} \\ k_{21} & k_{22} \\ \end{bmatrix}\!\! \biggl\{\!\!\! \begin{array}{c} x_1 \\ x_2 \\ \end{array} \!\!\!\biggr\} = \biggl\{\!\!\! \begin{array}{c} F_1 \\ F_2 \\ \end{array} \!\!\!\biggr\} \ensuremath{\sin\left(\omega t + \phi\right)}, \]

or

    \[ \ensuremath{\bigl[m\bigr]}\!\ensuremath{\bigl\{\ddot{x}\bigr\}} + \ensuremath{\bigl[k\bigr]}\!\ensuremath{\bigl\{x\bigr\}} = \ensuremath{\bigl\{F\bigr\}} \ensuremath{\sin\left(\omega t + \phi\right)}. \]

To find the steady–state response of the system, we again look for solutions in which the masses are undergoing simple simultaneous harmonic motion at the forcing frequency

    \[ \biggl\{\!\!\! \begin{array}{c} x_1 \\ x_2 \\ \end{array} \!\!\!\biggr\} = \biggl\{\!\!\! \begin{array}{c} \mathbb{A}_1 \\ \mathbb{A}_2 \\ \end{array} \!\!\!\biggr\} \sin{(\omega t + \phi)} \text{ or } {x} = {\mathbb{A}}\sin{(\omega t + \phi)} \]

The equations of motion become

    \[ -\omega^2[m]{\mathbb{A}}\sin{(\omega t + \phi)} + [k]{A}\sin{(\omega t + \phi)} =  {F}\sin{(\omega t + \phi)} \]

which to be valid for all times t requires

(8.35)   \[ \Bigl[ \ensuremath{\bigl[k\bigr]} -\omega^2 \ensuremath{\bigl[m\bigr]} \Bigr] \ensuremath{\bigl\{\!\ensuremath{\mathbb{A}}\!\bigr\}} = \ensuremath{\bigl\{F\bigr\}}, \]

or

    \[ \ensuremath{\bigl[\mathcal{Z}\bigr]}\!\!\ensuremath{\bigl\{\!\ensuremath{\mathbb{A}}\!\bigr\}} = \ensuremath{\bigl\{F\bigr\}}, \nonumber \]

where \ensuremath{\bigl[\mathcal{Z}\bigr]} is defined as

    \[ \ensuremath{\bigl[\mathcal{Z}\bigr]} = \ensuremath{\bigl[k\bigr]} - \omega^2 \ensuremath{\bigl[m\bigr]} = \begin{bmatrix} \mathcal{Z}_{11} & \mathcal{Z}_{12} \\ \mathcal{Z}_{21} & \mathcal{Z}_{22} \\ \end{bmatrix} = \begin{bmatrix} k_{11} - \omega^2 m_{11} & k_{12} - \omega^2 m_{12} \\ k_{21} - \omega^2 m_{21} & k_{22} - \omega^2 m_{22} \\ \end{bmatrix}. \]

Unlike the free vibration situation, the RHS of equation 8.35 is not zero, so that a unique solution for the amplitudes of each of the masses can be found as

(8.36)   \[ \ensuremath{\bigl\{\!\ensuremath{\mathbb{A}}\!\bigr\}} = \ensuremath{\bigl[\mathcal{Z}\bigr]\!\!\rule[5mm]{0pt}{0pt}^{-1}}\!\!\ensuremath{\bigl\{F\bigr\}} \]

which is valid provided \det \ensuremath{\bigl[\mathcal{Z}\bigr]} \neq 0. Since \ensuremath{\bigl[\mathcal{Z}\bigr]} is a 2\times2 matrix, the inverse is

    \[ \ensuremath{\bigl[\mathcal{Z}\bigr]\!\!\rule[5mm]{0pt}{0pt}^{-1}} = \frac{1}{\mathcal{Z}_{11}\mathcal{Z}_{22}-\mathcal{Z}_{12}\mathcal{Z}_{21}} \biggl[\!\! \begin{array}{rr} \mathcal{Z}_{22} & -\mathcal{Z}_{12} \\ -\mathcal{Z}_{21} & \mathcal{Z}_{11} \\ \end{array} \!\!\biggr] \]

so we see from equation (8.36) that the steady–state amplitudes of the response are

(8.37)   \[ \biggl\{\!\!\! \begin{array}{c} \mathbb{A}_1 \\ \mathbb{A}_2 \\ \end{array} \!\!\!\biggr\} = \frac{1}{\mathcal{Z}_{11}\mathcal{Z}_{22}-\mathcal{Z}_{12}\mathcal{Z}_{21}} \biggl\{\!\!\! \begin{array}{c} \mathcal{Z}_{22}F_1 - \mathcal{Z}_{12}F_2 \\  -\mathcal{Z}_{21}F_1 + \mathcal{Z}_{11}F_2\\ \end{array} \!\!\!\biggr\} \]

with

     \begin{align*} \mathcal{Z}_{11} &= k_{11} - m_{11} \omega^2, \\ \mathcal{Z}_{12} &= k_{12} - m_{12} \omega^2, \\ \mathcal{Z}_{21} &= k_{21} - m_{21} \omega^2, \\ \mathcal{Z}_{22} &= k_{22} - m_{22} \omega^2. \\ \end{align*}


Determine the steady state response for the system shown.


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