Writing the equations in matrix form

Figure 2.68. Multidegree-of-freedom system: (a) multistory analytical model with lumped masses.


[M ] = the mass or inertia matrix {x} = the column vector of accelerations [K] = the structure stiffness matrix

{x} = the column vector of displacements of the structure If the effect of damping is included, the equations of motion would be of the form


[C] = the damping matrix {x} = the column vector of velocity {P} = the column vector of external forces

General methods of solutions of these equations are available, but tend to be cumbersome. Therefore, in solving seismic problems, simplified methods are used; the problem is first solved by neglecting damping. Its effects are later included by modifying the design spectrum to account for damping. The absence of precise data on damping does not usually justify a more rigorous treatment. Neglecting damping results in dropping the second term, and limiting the problem to free-vibrations results in dropping the right-hand side of Eq. (2.67). The resulting equations of motion will become identical to Eq. (2.66).

During free vibrations, the motions of the system are simple harmonic, which means that the system oscillates about the stationary position in a sinusoidal manner; all masses follow the same harmonic function, having similar angular frequency, w. Thus.

Xn = an sin Wnt or in matrix notation

{x}= [an] sin wnt where {an} represents the column vector of modal amplitudes for the nth mode, and wn the corresponding frequency. Substituting for {x} and its second derivative {x} in Eq. (2.66) results in a set of algebraic expressions.

Using a procedure known as Cramer's rule, the preceding expressions can be solved for determining the frequencies of vibrations and relative values of amplitudes of motion an, au, ♦ ♦ ♦, an. The rule states that nontrivial values of amplitudes exist only if the determinant of the coefficients of a is equal to zero because the equations are homogeneous, meaning that the right-hand side of Eq. (2.68) is zero. Setting the determinant of Eq. (2.68) equal to zero, we get


— w 2m11


- w2m12


— w2m13

■ k1n



- W2m21

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