5.1 A 501-DOF system with nonlinear springs A generic system is considered which consists of 501 masses connected by linear springs and dampers, see Figure 1. Figure 1: 501-DOF system The system is supported at the left hand side ( 1u 0= ), and is loaded by a step load 501 f on the right hand side. Thus, the set of boundary coordinates bu includes the coordinates 1u and 501 u . All other coordinates are interior coordinates and assembled in the set iu . Additionally to the linear springs, nonlinear springs are added to the system in the area between mass 400 up to mass 450, see Figure 2. The load deflection relation of the additional nonlinear springs has the form ( )3 nl nl,1 nl,2 f k u k u = Δ + Δ , (26) where uΔ is the relative displacement between two involved masses. However, the nonlinear springs are considered on the right hand side of the equation of motion by the state-dependent force vector kf . Figure 2: 501-DOF system Beside the two interface constraint modes, the first three fixed-interface normal modes are assembled to the Craig-Bampton transformation matrix cb Φ . From the 50 existing mass-orthogonal local Ritzvectors only the first 12 are assembled to the matrix j ˆΨ . Figure 3 shows the eigenfrequencies of both, the retained normal modes and the first three massorthogonal local Ritz-vectors (m.-o. JIM 1, m.-o. JIM 2, m.-o. JIM 3). Figure 3 indicates that the eigenfrequencies of the mass-orthogonal local Ritz-vectors are significantly higher as those of the retained fixed-interface normal modes. 43
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