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)

    “No sooner met but they looked; no sooner looked but they loved; no sooner loved but they sighed; no sooner sighed but they asked one another the reason; no sooner knew the reason but they sought the remedy; and in these degrees have they made a pair of stairs to marriage, which they will climb incontinent, or else be incontinent before marriage.”
    —William Shakespeare (1564–1616)

    “Before anything else, we need a new age of Enlightenment. Our present political systems must relinquish their claims on truth, justice and freedom and have to replace them with the search for truth, justice, freedom and reason.”
    —Friedrich Dürrenmatt (1921–1990)

    “Not out of those, on whom systems of education have exhausted their culture, comes the helpful giant to destroy the old or to build the new, but out of unhandselled savage nature, out of terrible Druids and Berserkirs, come at last Alfred and Shakespeare.”
    —Ralph Waldo Emerson (1803–1882)

    “The method of scientific investigation is nothing but the expression of the necessary mode of working of the human mind. It is simply the mode in which all phenomena are reasoned about, rendered precise and exact.”
    —Thomas Henry Huxley (1825–95)

    “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)