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:

    “There is a continual exchange of ideas between all minds of a generation. Journalists, popular novelists, illustrators, and cartoonists adapt the truths discovered by the powerful intellects for the multitude. It is like a spiritual flood, like a gush that pours into multiple cascades until it forms the great moving sheet of water that stands for the mentality of a period.”
    —Auguste Rodin (1849–1917)

    “When a thought of Plato becomes a thought to me,—when a truth that fired the soul of Pindar fires mine, time is no more. When I feel that we two meet in a perception, that our two souls are tinged with the same hue, and do as it were run into one, why should I measure degrees of latitude, why should I count Egyptian years?”
    —Ralph Waldo Emerson (1803–1882)

    “[T]here is a Wit for Discourse, and a Wit for Writing. The Easiness and Familiarity of the first, is not to savour in the least of Study; but the Exactness of the other, is to admit of something like the Freedom of Discourse, especially in Treatises of Humanity, and what regards the Belles Lettres.”
    —Richard Steele (1672–1729)

    “People stress the violence. That’s the smallest part of it. Football is brutal only from a distance. In the middle of it there’s a calm, a tranquility. The players accept pain. There’s a sense of order even at the end of a running play with bodies stewn everywhere. When the systems interlock, there’s a satisfaction to the game that can’t be duplicated. There’s a harmony.”
    —Don Delillo (b. 1926)

    “The body sins once, and has done with its sin, for action is a mode of purification. Nothing remains then but the recollection of a pleasure, or the luxury of a regret.”
    —Oscar Wilde (1854–1900)

    “No, no; but as in my idolatry
    I said to all my profane mistresses,
    Beauty, of pity, foulness only is
    A sign of rigour: so I say to thee,
    To wicked spirits are horrid shapes assign’d,
    This beauteous form assures a piteous mind.”
    —John Donne (1572–1631)