Figure 3: Natural frequencies of both, the retained normal modes and the first three modified local Ritz-vectors The parameters used for this example are given by { 1 2 500 1 2 500 nl,1 nl,2 3 1 2 500 k j 501 N N N k k k 100 , m m m 1kg, k 1000 , k 1000 m m m kg 0N: t 0s c c c 1500 , t01000s, m 3, m 25, f 1N: t 0s s = =…= = = =…= = = = ≤ = =…= = = … = = = > (27) Due to fact that this system includes dampers, the damping has to be considered in the reduced model as well. The reduced damping matrix red C may be written as: T FE red cb j cb j ˆ ˆ C C ª º ª º = Φ Ψ Φ Ψ ¬ ¼ ¬ ¼ , (28) where FE C is the damping matrix of the non-reduced system. The reference solution (I) has been computed using the entire non-reduced system. This is a nonlinear system of differential equations with 501 DOF of the form FE FE FE ext k M u C u K u f f (u) + + = + . (29) An approximation of the solution (II) is achieved by solving Eq.(21). Thus, the transformation matrix comprises 17 modes and consequently a nonlinear system of differential equations with 17 DOF has to be solved. Finally, another approximation of the solution (III) is obtained by solving the nonlinear differential-algebraic system of Eq.(25). There, the system of differential equations has 5 DOF and the algebraic side-condition has 12 DOF. For all systems the time integration has been performed with the MATLAB solver ODE15s. It is worth noting that MATLAB ODE solver accept only first-order differential equations. Therefore, all three systems have been rewritten to equivalent systems of first-order nonlinear differential equations. Figure 4 shows the displacement-time graph of mass 425 (left figure of Figure 4) and the relative displacement uΔ between mass 424 and mass 425 with respect to time (right figure of Figure 4). The displacement field of all 501 masses at t = 1000 s can be seen in Figure 5. 44
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