Vibration - Multiple Degrees of Freedom Systems and Mode Shapes

Multiple Degrees of Freedom Systems and Mode Shapes

The simple mass–spring damper model is the foundation of vibration analysis, but what about more complex systems? The mass–spring–damper model described above is called a single degree of freedom (SDOF) model since we have assumed the mass only moves up and down. In the case of more complex systems we need to discretize the system into more masses and allow them to move in more than one direction – adding degrees of freedom. The major concepts of multiple degrees of freedom (MDOF) can be understood by looking at just a 2 degree of freedom model as shown in the figure.

The equations of motion of the 2DOF system are found to be:

m_1 \ddot{x_1} + { (c_1+c_2) } \dot{x_1} - { c_2 } \dot{x_2}+ { (k_1+k_2) } x_1 -{ k_2 } x_2= f_1,
m_2 \ddot{x_2} - { c_2 } \dot{x_1}+ { (c_2+c_3) } \dot{x_2} - { k_2 } x_1+ { (k_2+k_3) } x_2 = f_2. \!

We can rewrite this in matrix format:

\begin{bmatrix}m_1 & 0\\ 0 & m_2\end{bmatrix}\begin{Bmatrix}\ddot{x_1}\\ \ddot{x_2}\end{Bmatrix}+\begin{bmatrix}c_1+c_2 & -c_2\\ -c_2 & c_2+c_3\end{bmatrix}\begin{Bmatrix}\dot{x_1}\\ \dot{x_2}\end{Bmatrix}+\begin{bmatrix}k_1+k_2 & -k_2\\ -k_2 & k_2+k_3\end{bmatrix}\begin{Bmatrix} x_1\\ x_2\end{Bmatrix}=\begin{Bmatrix} f_1\\ f_2\end{Bmatrix}.

A more compact form of this matrix equation can be written as:

\begin{bmatrix}M\end{bmatrix}\begin{Bmatrix}\ddot{x}\end{Bmatrix}+\begin{bmatrix}C\end{bmatrix}\begin{Bmatrix}\dot{x}\end{Bmatrix}+\begin{bmatrix}K\end{bmatrix}\begin{Bmatrix} x\end{Bmatrix}=\begin{Bmatrix} f \end{Bmatrix}

where and are symmetric matrices referred respectively as the mass, damping, and stiffness matrices. The matrices are NxN square matrices where N is the number of degrees of freedom of the system.

In the following analysis we will consider the case where there is no damping and no applied forces (i.e. free vibration). The solution of a viscously damped system is somewhat more complicated.

This differential equation can be solved by assuming the following type of solution:

\begin{Bmatrix} x\end{Bmatrix}=\begin{Bmatrix} X\end{Bmatrix}e^{i\omega t}.

Note: Using the exponential solution of is a mathematical trick used to solve linear differential equations. If we use Euler's formula and take only the real part of the solution it is the same cosine solution for the 1 DOF system. The exponential solution is only used because it easier to manipulate mathematically.

The equation then becomes:

Since cannot equal zero the equation reduces to the following.

Read more about this topic:  Vibration

Famous quotes containing the words multiple, degrees, freedom, systems, mode and/or shapes:

    “... the generation of the 20’s was truly secular in that it still knew its theology and its varieties of religious experience. We are post-secular, inventing new faiths, without any sense of organizing truths. The truths we accept are so multiple that honesty becomes little more than a strategy by which you manage your tendencies toward duplicity.”
    —Ann Douglas (b. 1942)

    “So that the life of a writer, whatever he might fancy to the contrary, was not so much a state of composition, as a state of warfare; and his probation in it, precisely that of any other man militant upon earth,—both depending alike, not half so much upon the degrees of his WIT—as his RESISTANCE.”
    —Laurence Sterne (1713–1768)

    “That is coupled to foul thraldom.
    But if he had assayed it,
    Then all perquer he should it wit;
    And should think freedom more to prize
    Than all the gold in world that is.”
    —John Barbour (1316?–1395)

    “The only people who treasure systems are those whom the whole truth evades, who want to catch it by the tail. A system is just like truth’s tail, but the truth is like a lizard. It will leave the tail in your hand and escape; it knows that it will soon grow another tail.”
    —Ivan Sergeevich Turgenev (1818–1883)

    “Happiness is a matter of one’s most ordinary everyday mode of consciousness being busy and lively and unconcerned with self. To be damned is for one’s ordinary everyday mode of consciousness to be unremitting agonising preoccupation with self.”
    —Iris Murdoch (b. 1919)

    “All know that all the dead in the world about that place are stuck
    And that should mother seek her son she’d have but little luck
    Because the fires of Purgatory have ate their shapes away;
    I swear to God I questioned them and all they had to say
    Was fol de rol de rolly O.”
    —William Butler Yeats (1865–1939)