45 On the Behaviour of Structures with Many Nonlinear Elements 511 Fig. 45.1 n-DoF linear chain system which forms the basis for the analysis Response Function (FRF) plots were generated by running 200 parallel simulations each for 100 s at 10 kHz with a different realisation of the random forcing. The FRF of each of the 200 time domain signals was computed after discarding the first 10 s of time domain data to remove transient behaviour. The FRF values were then averaged across the 200 realisations. As a basis for investigating the effect of nonlinearities on high DoF systems, first a linear chain system (Fig. 45.1) was generated with a unit mass matrix, tri-diagonal stiffness matrix and damping matrix proportional to the mass matrix. All springs were set to have stiffness coefficient ki Dk D10 4 Nm 1, damping coefficients were all set to ci Dc D5Nsm 1 andmasses mi DmD1kg. This prescription gives a damping ratio D0:025for an SDoF system which is 2:5% of critical damping. The system matrices are thus Œm D ŒI , Œc D 5 ŒI (where ŒI is the identity matrix) and Œk is a tridiagonal matrix with the diagonal values equal to 2k and the off diagonal non-zero elements equal to k. The linear system is well understood and its natural frequencies, mode shapes and frequency response can be computed straightforwardly. The integration method was checked, initially, by numerically generating FRFs for known linear systems and comparing them with analytical predictions. First, a low DoF system was considered. For a three DoF system, generated with the parameters described previously, the numerical and theoretical results can be seen to be in excellent agreement for both the magnitude and phase components of the FRF (Fig. 45.2). When the number of simulated DoFs was increased to 30, the same agreement could be seen (Fig. 45.3). The next benchmark assessed the ability of the modeller to handle nonlinear equations of motion. For this exercise, the chain system with three degrees of freedom was considered. A single cubic spring was added as discussed in [9]; for this system, the effect of the position of the cubic spring can be classified in three ways. System Type A is an asymmetric system, obtained when the nonlinearity is placed between mass one and ground, mass three and ground, mass one and two or mass two and three. In this case all modes will exhibit nonlinear behaviour. System Type B occurs when the nonlinear spring is between mass two and ground; in this case mode two is linear since mass two remains stationary in that mode. Type C is obtained if the nonlinear spring connects masses one and three; in this case modes one and three remain linear since masses one and three move in phase in these modes. The simulation results for these systems can be seen in Fig. 45.4. A further set of simulations considered the effects of harmonic forcing at 4 Hz on the systems with a single DoF or 30 DoF; the linear systems and those with a single cubic spring connecting mass one to ground were considered. When spectra were computed for the responses (Figs. 45.5–45.8), the expected variation in harmonic content with level of excitation was observed. This exercise concluded the benchmarking of the numerical solver. 45.3 Effect of Nonlinear Elements With the conviction that the numerical modeller accurately represents the motion of the systems of interest, a routine was developed to investigate the effect of cubic nonlinearities being introduced into the system in multiple random locations. This allowed the number of nonlinearities in the system to be controlled by automatically generating a matrix with a given number of nonlinear springs equal in size to the linear stiffness matrix. The first set of investigations considered the effect of inserting a single cubic spring into the linear chain system.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==