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:
We can rewrite this in matrix format:
A more compact form of this matrix equation can be written as:
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:
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 20s 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)
“Complete courage and absolute cowardice are extremes that very few men fall into. The vast middle space contains all the intermediate kinds and degrees of courage; and these differ as much from one another as mens faces or their humors do.”
—François, Duc De La Rochefoucauld (16131680)
“The freedom to make a fortune on the Stock Exchange has been made to sound more alluring than freedom of speech.”
—John Mortimer (b. 1923)
“In all systems of theology the devil figures as a male person.... Yes, it is women who keep the church going.”
—Don Marquis (18781937)
“If Thought is capable of being classed with Electricity, or Will with chemical affinity, as a mode of motion, it seems necessary to fall at once under the second law of thermodynamics as one of the energies which most easily degrades itself, and, if not carefully guarded, returns bodily to the cheaper form called Heat. Of all possible theories, this is likely to prove the most fatal to Professors of History.”
—Henry Brooks Adams (18381918)
“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 assignd,
This beauteous form assures a piteous mind.”
—John Donne (15721631)