Fig. 1 Largest Lyapunov exponent and phase portraits of system (31) , for α = 0.98 ,   ρ = 1 2 ,   t ∈ [ 1 , 60 ] , p  =   10, q  =   123, r = 8 3 , ( x 1 ( 1 ) , x 2 ( 1 ) , x 3 ( 1 ) ) T = ( 5 , ... More

Fig. 2 Largest Lyapunov exponent and phase portraits of system (31) , for α = 0.98 , ρ  = 4, t ∈ [ 1 , 2 ] , p  =   10, q  =   123, r = 8 3 , ( x 1 ( 1 ) , x 2 ( 1 ) , x 3 ( 1 ) ) T = ( 5 , 3 , 9 ) T... More

Fig. 3 Largest Lyapunov exponent and phase portraits of system (33) , for α = 0.98 , ρ  = 2, t ∈ [ 1 , 3 ] , p  =   10, q  =   125, r = 8 3 , ( x 1 ( 1 ) , x 2 ( 1 ) , x 3 ( 1 ) ) T = ( 5 , 3 , 9 ) T... More

Fig. 4 Largest Lyapunov exponent and phase portraits of system (33) , for α = 0.98 , ρ  = 10, t ∈ [ 1 , 1.5 ] , p  =   10, q  =   125, r = 8 3 , ( x 1 ( 1 ) , x 2 ( 1 ) , x 3 ( 1 ) ) T = ( 5 , 3 , 9 ) ... More

Fig. 1 A cantilever beam More

Fig. 2 Zero-crossing of χ ˙ at any particular Gauss point within a time-step. For details on associated quantities shown, see the main text. More

Fig. 3 Tip displacement for γ h = 3000 and n h = 0.5 shows an approximate equivalent damping of about 1.6%, as per Eq. (19) . Here 10 beam elements were used, with three hysteresis Gauss points per element. Time integration was done using matlab 's ode15 s with error ... More

Fig. 4 The long term oscillation in the beam tip response shows very slow power law decay when n h = 1 2 . A small portion of the solution is shown zoomed within the left subplot. The aim of this simulation is to show that the model and numerical integration method together reta... More

Fig. 5 ( a ) Variation of T min with n e and ( b ) Frequency content of the transient tip displacement response when the first three modes are disturbed in a uniform FE model with 100 elements with n h = 0.5 More

Fig. 6 Tip displacement response calculated using different time steps (compared to ode15 s ) with n h = 0.5 More

Fig. 7 Time step versus RMS error (128 equispaced points in time) for n e = 10 and 30 with n h = 0.5 More

Fig. 8 Hysteresis loop for the z 1 driven by χ 1 (at Gauss point “1” near the fixed end) More

Fig. 9 Error at t  =   1, for FE models with 10 and 30 elements with n h = 0.5 More

Fig. 10 Tip displacement for γ h = 0.3 and n h = 1.5 shows an approximate equivalent damping of about 1.5%, as per Eq. (19) . Here 10 beam elements were used, with 3 hysteresis Gauss points per element. Time integration was done using Matlab's ode15 s with error tolera... More

Fig. 11 Time step versus RMS error (128 equispaced points in time) for n e = 10 and 30 with n h = 1.5 . The linear fit for 10 elements has slope 2.0. The linear fit for 30 elements has slope 1.9; another line with slope 2 is shown for comparison. More

Fig. 12 Comparison between ROM and full model, for two cases (see text for details). Tip displacements (left) and hysteresis curve (right). In subplots (b) and (d), the retained hysteresis Gauss point closest to the fixed end of the beam has been selected for display. More

Fig. 13 Error convergence of ROM with increasing number of Gauss points retained. For comparison, if y tip ( ROM ) ( m ) ( t ) is set identically to zero, the error measure obtained is 0.006 for both models. Thus, for reasonable accuracy, m ≥ 50 may be needed. More