4 Effect of Structural Parameters on the Nonlinear Vibration of L-Shaped Beams 29 (a) (b ) Fig . 4. 2 Comparison of the frequency response with ANSYS for Case 1 at (a) x1/L1 = 1, (b) x2/L2 = 1 Table 4.2 Controlled variables and their values Controlled variable Case 2: ρAL1 Case 3: ρAL2 Case 4: M1 Case 5: M2 Case 6: .LM1/L1 Case 7: .LM2/L2 Case 8: EI/L1 3 Value 0.19625 kg 0.19625 kg 0.5 kg 0.5 kg 1 0.5 666.67 N/m Table 4.3 Natural frequencies for each case ANSYS Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 ω1 (rad/s) 55.365 55.37 29.57 89.557 54.812 35.75 53.796 62.223 39.152 ω2 (rad/s) 150.36 150.44 118.25 295.91 143.77 106.95 145.64 170.97 106.38 4.3.2 Case Studies After verifying the linear model, several case studies are performed. Case 1 is used as the basis for the system parameters and for each study, one of the parameters is selected as the controlled variable, while others are kept constant in order to observe its effects on the nonlinear response of the system. The selected controlled variables and their values are given in Table 4.2. Natural frequencies of the linear system for each case are presented in Table 4.3. In the first case study, in order to observe the effect of forcing amplitude on the nonlinear response of the model, different excitation forcing amplitudes are considered and the normalized responses with respect to forcing amplitudes are compared with each other. It can be seen from Fig. 4.3 that when the forcing is very small, the response is very similar to the response of the linear system. When a forcing with an amplitude of 0.05 N is applied, cubic stiffness nonlinearity starts to take effect. As the forcing amplitude increases, the resonance frequency shifts to the right, which is the outcome of the cubic stiffness nonlinearity dominant in this system. Moreover, when the forcing amplitude is 2 N, the squared nonlinearity becomes effective as seen from the notch at the first resonance. For the second case study, the effect of the masses of the first and the second beams (ρAL1 and ρAL2) on the vibration characteristics are compared. These are labeled as Case 2 and Case 3, respectively. Since densities and areas of each beam are equal to each other, changing these parameters results in the change of the ratio of the beam lengths. The steady-state frequency responses at x1/L1 = 1 and x2/L2 =1 are given in Fig. 4.4. It can be seen from the figures that decreasing the mass of the first beam causes a decrease in the resonance frequencies, whereas decreasing the mass of the second beam causes an increase in the resonance frequencies in contrast to Case 2. Additionally, decrease in the L2/L1 ratio makes cubic stiffness more dominant at the first resonance frequency. Moreover, increasing the mass of the first beam amplifies the resonance response at the tip of the beam; similarly, increase in the mass of the second beam amplifies the response at the tip point of the second beam. The third case study investigates the effect of the concentrated masses connected to the first and second beams (M1 and M2) on the vibration characteristics. These are labeled as Case 4 and Case 5, respectively. The frequency responses at x1/L1 = 1 and x2/L2 = 1 are obtained and given in Fig. 4.5. It can be clearly seen that increase in the mass of the
RkJQdWJsaXNoZXIy MTMzNzEzMQ==