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Multiple Degree-of-Freedom Example

Consider the 3 degree-of-freedom system, There are 3 degrees of freedom in this problem since to fully characterize the system we must know the positions of the three masses (x1, x2, and x3). Three free body diagrams are needed to form the equations of motion. However, it is also possible to form the coefficient matrices directly, since each parameter in a mass-dashpot-spring system has a very distinguishable role.

Equations of Motion from Free Body Diagrams

The equations of motion can be obtained from free body diagrams, based on the Newton's second law of motion, F = m*a. The equations of motion can therefore be expressed as, In matrix form the equations become,

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Equations of Motion from Direct Matrix Formation

Observing the above coefficient matrices, we found that all diagonal terms are positive and contain terms that are directly attached to the corresponding elements. Furthermore, all non-diagonal terms are negative and symmetric. They are symmetric since they are attached to two elements and the effects are the same in these two elements (a condition known as Maxwell's Reciprocity Theorem). They are negative due to the relative displacements/velocities of the two attached elements. In summary,

1. Determine the number of degrees of freedom for the problem; this determines the size of the mass, damping, and stiffness matrices. Typically, one degree of freedom can be associated with each mass. 2. Enter the mass values (if associated with a degree of freedom) into the diagonals of the mass matrix; the exact ordering does not matter. All other values in the mass matrix are zero. 3. For each mass (associated with a degree of freedom), sum the damping from all dashpots attached to that mass; enter this value into the damping matrix at the diagonal location corresponding to that mass in the mass matrix. 4. Identify dashpots that are attached to two masses; label the masses as m and n. Write down the negative dashpot damping at the (m, n) and (n, m) locations in the damping matrix. Repeat for all dashpots. Any remaining terms in the damping matrix are zero. 5. For each mass (associated with a degree of freedom), sum the stiffness from all springs attached to that mass; enter this value into the stiffness matrix at the diagonal location corresponding to that mass in the mass matrix. 6. Identify springs that are attached to two masses; label the masses as m and n. Write down the negative spring stiffness at the (m, n) and (n, m) locations in the stiffness matrix. Repeat for all springs. Any remaining terms in the stiffness matrix are zero. 7. Sum the external forces applied on each mass (associated with a degree of freedom); enter this value into the force vector at the row location corresponding to the row location for that mass (in the mass matrix). 8. The resulting matrix equation of motion is,

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There are 3 degrees of freedom in this problem since to fully characterize the system we must know the positions of the three masses (x₁, x₂, and x₃).

Three free body diagrams are needed to form the equations of motion. However, it is also possible to form the coefficient matrices directly, since each parameter in a mass-dashpot-spring system has a very distinguishable role.