Research on the Dynamic Behaviors of the Vortex Induced Vibration Power Generation System Under Nonlinear Restoring Forces
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摘要: 利用线性弹簧斜向布置的几何非线性产生非线性恢复力，提出了引入非线性恢复力的水下涡激振动(VIV)发电系统. 该系统通过单向轴承、齿轮齿条机构、增速箱和转子发电机，将钝体横向往复运动转变为发电机的单向旋转运动. 建立了综合考虑流-固-电耦合的水下涡激振动发电系统动力学方程，利用非线性振动理论，获得了钝体非线性振动的静态平衡点分岔和不同稳态运动的区间，重点研究了PF-2SN和2PF-2SN两种静态分岔情况下钝体的非线性动力学行为，获得了不同流速下钝体振动的Poincaré映射、相图和幅频图，分析了钝体在单周期小幅运动、大幅混沌运动和准周期大幅运动等运动模式下的振动行为及运动规律，并计算了在钝体处于不同稳态运动时的发电机功率. 结果表明：在PF-2SN分岔方式中，系统处于二稳态运动时的振动和发电具有明显优势，平均振幅比为2.18、发电功率最大值为24.45 W. 而在2PF-2SN分岔方式中，系统处于三稳态运动时的振动和发电更具优势，平均振幅比为1.98、发电功率最大值为18.32 W.Abstract: An underwater vortex-induced vibration power generation system under nonlinear restoring forces was proposed. The nonlinear restoring force was generated by means of the geometrical nonlinearity of linear springs arranged obliquely. The lateral reciprocating motion of the oscillator was transformed into a unidirectional rotary motion of the generator by dint of unidirectional bearings and gear-rack mechanisms, a booster box and a rotor generator. The dynamic flow-structure-electricity coupling equations for the vortex-induced vibration power generation system were established. Then the static equilibrium point bifurcation of the nonlinear vibration of the oscillator and the ranges of different stable state motions were obtained under the nonlinear vibration theory. The nonlinear dynamic behaviors of the oscillator under the conditions of PF-2SN and 2PF-2SN bifurcations were studied mainly. The bifurcation graphs, phase graphs and Poincaré mappings of the system were achieved. The vibration behaviors and motion laws of the oscillator under the conditions of single-period small motion, large chaos motion and quasi-periodic large motion were analyzed. Then, the generation power values of the generator for different stable state motions of the oscillator were also calculated. The results show that, in the PF-2SN bifurcation mode, the system has obvious advantages in vibration and power generation in the bi-stable motion, with an average amplitude ratio of 2.18 and a maximum power of 24.45 W. While in the 2PF-2SN bifurcation mode, the vibration and power generation of the system are more superior in the tri-stable motion, with an average amplitude ratio of 1.98 and a maximum power generation of 18.32 W.
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图 1 引入非线性恢复力的涡激振动发电装置
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 1. The vortex-induced vibration power generator with nonlinear restoring forces
图 4 机械传动及发电装置力学模型
Figure 4. The mechanical model for the transmission and power generation system
图 5 不同安装长度a下的系统静态分岔图
Figure 5. Static bifurcation diagrams of the system under different installation lengths a
图 6 不同安装长度b下的系统静态分岔图
Figure 6. Static bifurcation diagrams of the system under different installation lengths b
图 7 不同弹簧安装长度下的钝体位移与洋流流速的分岔图
Figure 7. Bifurcation plots of blunt body displacements vs. flow speeds under different installation lengths of springs
图 8 不同流速下的钝体非线性振动响应(a=0.1 m，b=0.1 m)
Figure 8. Nonlinear vibration responses of the blunt body at different flow speeds(a=0.1 m, b=0.1 m)
图 9 不同流速下的钝体非线性振动响应(a=0.1 m，b=0.2 m)
Figure 9. Nonlinear vibration responses of the blunt body at different flow speeds(a=0.1 m, b=0.2 m)
图 10 不同流速下的钝体非线性振动响应(a=0.1 m，b=0.25 m)
Figure 10. Nonlinear vibration responses of the blunt body at different flow speeds(a=0.1 m, b=0.25 m)
图 11 不同稳态下的钝体振幅比和发电功率对比(PF-2SN分岔)
Figure 11. Vibration amplitude ratios of the oscillator and harvesting power values under different stable states (PF-2SN bifurcation)
图 12 不同弹簧安装长度下的钝体位移与洋流流速u的分岔图
Figure 12. Bifurcation plots of blunt body displacements vs. current flow velocity u for different spring installation lengths
图 13 不同流速下的钝体非线性振动响应(a=0.1 m，b=0.15 m)
Figure 13. Nonlinear vibration responses of the blunt body for different flow speeds(a=0.1 m, b=0.15 m)
图 14 不同流速下的钝体非线性振动响应(a=0.2 m，b=0.15 m)
Figure 14. 14 Nonlinear vibration responses of the blunt body at different flow speeds(a=0.2 m, b=0.15 m)
图 15 不同稳态下的钝体振幅比和发电功率对比(2PF-2SN分岔)
Figure 15. Vibration amplitude ratios of the oscillator and harvesting power values under different stable states (2PF-2SN bifurcation)
表 1 系统结构及物理参数表
Table 1. Structural and physical parameters of the system
parameter symbol value parameter symbol value oscillator mass ms/kg 50 spring length Ls/m 0.4 structural damping cs/(N·s·m－1) 20 seawater density ρ/(kg·m－3) 1 040 additional quality factor Cm[21] 1 oscillator length L/m 1.6 viscous force coefficient γ[21] 0.8 oscillator diameter D/m 0.3 fluid-structure coupling parameter A[20] 12 Strouhal number Sr[21] 0.2 (left/right) gear radius r/m 0.05 total moment of inertia J/(kg·m2) 0.625 booster ratio nb 3.3 Van der Pol parameter ε[21] 0.3 generator internal resistance Ra/Ω 0.25 flywheel moment of inertia If/(kg·m2) 0.04 generator external resistance RL/Ω 20 generator moment of inertia Ig/(kg·m2) 0.01 voltage constant Kg/(V·s·rad－1) 0.26 torque coefficient Kt/(N·m·A－1) 0.45 equivalent damping coefficient Cs[22] 0.057 2 static cylinder lift amplitude CL0[23] 0.3 
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表 2 钝体运动状态汇总(a=0.1 m，b=0.1 m)
Table 2. Summary of the motion of the oscillator(a=0.1 m, b=0.1 m)
flow speed motion of the oscillator cross potential wells (yes or no) u＜0.6 m/s small single-cycle motion in the potential well no 0.6 m/s≤u＜0.99 m/s large chaotic motion between potential wells yes 0.99 m/s≤u＜1.56 m/s large 3-cycle motion between potential wells yes 1.56 m/s≤u＜1.77 m/s large quasiperiodic and chaotic motion between potential wells yes 1.77 m/s≤u＜1.92 m/s small 2-cycle motion in the potential well no u≥1.92 m/s small single-cycle motion in the potential well no
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表 3 钝体运动模式汇总(a=0.1 m，b=0.2 m)
Table 3. Summary of motion modes of the oscillator(a=0.1 m, b=0.2 m)
flow speed motion of the oscillator cross potential wells (yes or no) u＜0.36 m/s small single-cycle motion in the potential well no 0.36≤u＜0.85 m/s large chaotic motion between potential wells yes 0.85≤u＜1.01 m/s 3-cycle large-scale periodic motion between potential wells yes 1.01≤u＜1.12 m/s large quasi-periodic motion between potential wells yes 1.12≤u＜1.34 m/s small 2-cycle motion in the potential well no u≥1.34 m/s small single-cycle motion in the potential well no
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表 4 钝体运动模式汇总(a=0.1 m，b=0.15 m)
Table 4. Summary of motion modes of the oscillator (a=0.1 m, b=0.15 m)
flow speed motion of the oscillator cross potential wells (yes or no) u＜0.48 m/s small single-cycle motion in the potential well no 0.48 m/s≤u＜0.93 m/s large chaotic motion between potential wells yes 0.93 m/s≤u＜1.13 m/s quasi 3-cycle large-scale motion between potential wells yes 1.13 m/s≤u＜1.15 m/s large 6-cycle motion between potential wells yes 1.15 m/s≤u＜1.26 m/s large 3-cycle motion between potential wells yes 1.26 m/s≤u＜1.46 m/s large quasiperiodic and chaotic motion between potential wells yes 1.46 m/s≤u＜1.61 m/s small 2-cycle motion in the potential well no u≥1.61 m/s small single-cycle motion in the potential well no
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表 5 钝体运动模式汇总(a=0.2 m，b=0.15 m)
Table 5. Summary of motion modes of the oscillator (a=0.2 m, b=0.15 m)
flow speed motion of the oscillator cross potential wells (yes or no) u＜0.28 m/s small single-cycle motion in the potential well no 0.28 m/s≤u＜0.72 m/s large single-cycle motion between potential wells yes 0.72 m/s≤u＜0.92 m/s large quasi-periodic motion between potential wells yes 0.92 m/s≤u＜0.96 m/s large 5-cycle motion between potential wells yes 0.96 m/s≤u＜1.01 m/s large quasi-periodic motion between potential wells yes 1.01 m/s≤u＜1.03 m/s small quasi-periodic motion in the potential well no 1.03 m/s≤u＜1.11 m/s small 2-cycle motion in the potential well no u≥1.11 m/s small single-cycle motion in the potential well no
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