非线性能量阱的力学特性与振动抑制效果研究
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摘要
航天器在实际任务中面临的振动环境十分复杂，振动对航天器的结构及某些敏感载荷都会造成威胁，因此振动抑制在航天领域一直是重点研究问题。本文研究的非线性能量阱（Nonlinear energy sink, NES）是一种可实现靶能量传递（Targeted energy transfer, TET）的非线性被动吸振器，在航天器局部减振方面具有良好的应用前景。NES的力学特性与振动抑制效果的研究是理论研究的重点，对NES在实际工程中的应用具有重要意义。
NES在抑制结构的自由振动时有两个特点:其一、振动抑制效率同输入能量的大小有关；其二、靶能量传递的实现需要NES与主结构的质量之比大于一定值。目前对以上两点只有定性的结论，本文对其作了进一步的定量分析。通过分析得到了保守系统中振动能量在待减振部件与NES间完全传递的解析条件，并得到了实现最优靶能量传递所需的NES与主结构之间质量比的最小值。而后将能量完全传递时立方刚度与初始能量的关系扩展至含有小阻尼的非保守系统中，进而提出了在小阻尼情况下，依据输入能量大小设计NES的立方刚度的方法，并验证了该方法的正确性。
NES能否在不同振动峰值处同时实现良好的振动抑制效果是衡量其性能优劣的一个重要方面，因此，本文用两种方法研究了NES进行两自由度线性系统的强迫振动抑制问题：在激励频率固定的条件下，使用复变量—平均法推导出了系统的慢变方程及解耦的平衡点计算公式，得到了系统的周期解，并用Lyapunov理论分析了周期解的稳定性；对激励频率可变的情况，使用增量谐波平衡法（Incremental Harmonic Balance Method, IHB）得到了系统的周期解，并用Floquet理论判断了周期解的稳定性。两种方法从不同角度揭示了系统的鞍结分岔和Hopf分岔现象,并揭示了NES同时抑制两个不同振动峰值的有效性。解析分析的结论均通过数值解法得到了验证。
针对单自由度NES在抑制结构的自由振动时，因振动能量大小的变化，振动抑制效率起伏较大的问题，本文提出了一种两自由度的NES，可以在不增加NES总质量的前提下缓解该问题，通过数值仿真说明了这种两自由度NES的优越性。并且，使用Hilbert-Huang变换研究了系统产生靶能量传递所需的内共振条件，进而用打靶法及复变量—平均法分析了系统在主共振附近的力学特性，支持了上述结论。
鉴于前述两自由度NES对自由振动抑制的优越性，本文进一步研究了用其抑制结构在简谐荷载下的强迫振动的问题。当激励频率等于结构的固有频率时，推导了系统的慢变方程，对平衡点的个数及位置进行了解析分析，并判断了平衡点的稳定性；当激励频率可变时，用IHB法计算出系统的周期响应，并判断了系统响应的稳定性。以上研究揭示了系统的局部分岔现象，并说明了系统在产生何种响应类型时对振动抑制更为有利。最后比较了两自由度NES同单自由度NES在简谐荷载下的振动抑制效果。
Since spacecraft faces complicated vibration environment which poses greatthreat to the structure and sensitive equipment on it during practical missions,vibration suppression is always a focus in the field of spacecraft design. Thisresearch is focus on a kind of nonlinear vibration absorber which terms as nonlinearenergy sink. It’s a method for vibration suppression using the principle of targetedenergy transfer, and it processes good application foreground for local vibrationreduction of spacecrafts. Research of the dynamics and efficiency of vibrationsuppression of NES is the key point of theoretical research, which is also importantfor application in actual projects.
NES has two features while used for free vibration suppression: First, NESbecomes most efficient when the vibration energy of the system reaches a certainvalue; Second, for realizing targeted energy tranfer, the mass ratio between the NESand the main structure must beyond a certain value. The above two features arequalitative conclusions, and will be analyzed quantitatively in this research. Theanalytic conditions for vibration energy completely transfers between the mainstrucure and the NES is derived for the conservative system, and the minimum massratio between the NES and the main structure for realizing optimal targeted energytransfer is also abtained. Then, the relation between cubic stiffness and intial energyis generalized to non-conservative systems which contain small damping. Based onthe above conclusions, the method for designing the cubic stiffness according to theinitial input energy to the main structure is developed on the cases of small damping,and the method is verified numerically.
The ability of suppressing two different vibration peaks is an important aspectto evaluate the efficiency of the NES. So, vibration mitigation of atwo-degree-of-freedom linear system under harmonic excitation by means of NES isstudied with two methods: the decoupled formula of equilibrium points of the slowdynamics are derived, and the periodic response of the system is obtained usingcomplex-averaging method when the excitation frequency is fixed. The stability ofthe equilibium points are studied based on Lyapunov’s theorem; The periodicresponse of the system is studied using incremental harmonic balance method whenthe excitation frequency is tuning, and the stability of the periodic response isstudied based on Floquet’s theorem. Sadle-node bifurcation and Hopf bifurcation arerevealed from different aspects with the two methods, and the effectiveness ofvibration suppression for the two different energy peaks is also revealed. Analyticalconclusions are verified using numerial simulations.
The effeciency of vibration suppression varies drastically with the changing ofinitial energy of the main structure when the NES is used for free vibration control.It is proposed that the problem could be reliefed with the application of a kind oftwo-degree-of-freedom NES. The advantage of the-two-degree-of-freedom NES isverified numerically. The phenomenon of resonance capture when targeted energytranfer occurs is studied using Hilbert-Huang transform. Then, dynamics of thesystem is analyzed using shooting method and complexification-averaging method,and the analytical results support the frontal conclusions.
A fruther study of the two-degree-of-freedom NES is carried out for forcedvibration suppression under harmonic excitation since the superiority of thetwo-degree-of-freedom NES is found in chapter4. The slow dynamics of the systemis derived and the equilibrium points of the slow dynamics is studied analyticallywhen the excitation frequency is equal to the natural frequency of the main structure.The stability of the equilibrium points is also studied. The periodic response of thesystem is studied using incremental harmonic balance method when the excitationfrequency is tuning, and the stablility of the periodic response is given. Localbifurcations of the system are revealed and verified using numerical method, and itis also revealed which response regime is more favorable for vibration suppression.Finally, the efficiency of vibration suppression under harmonic force of thetwo-degree-of-freedom NES is compared with the single-degree-of-freedom NES.
NES在抑制结构的自由振动时有两个特点:其一、振动抑制效率同输入能量的大小有关；其二、靶能量传递的实现需要NES与主结构的质量之比大于一定值。目前对以上两点只有定性的结论，本文对其作了进一步的定量分析。通过分析得到了保守系统中振动能量在待减振部件与NES间完全传递的解析条件，并得到了实现最优靶能量传递所需的NES与主结构之间质量比的最小值。而后将能量完全传递时立方刚度与初始能量的关系扩展至含有小阻尼的非保守系统中，进而提出了在小阻尼情况下，依据输入能量大小设计NES的立方刚度的方法，并验证了该方法的正确性。
NES能否在不同振动峰值处同时实现良好的振动抑制效果是衡量其性能优劣的一个重要方面，因此，本文用两种方法研究了NES进行两自由度线性系统的强迫振动抑制问题：在激励频率固定的条件下，使用复变量—平均法推导出了系统的慢变方程及解耦的平衡点计算公式，得到了系统的周期解，并用Lyapunov理论分析了周期解的稳定性；对激励频率可变的情况，使用增量谐波平衡法（Incremental Harmonic Balance Method, IHB）得到了系统的周期解，并用Floquet理论判断了周期解的稳定性。两种方法从不同角度揭示了系统的鞍结分岔和Hopf分岔现象,并揭示了NES同时抑制两个不同振动峰值的有效性。解析分析的结论均通过数值解法得到了验证。
针对单自由度NES在抑制结构的自由振动时，因振动能量大小的变化，振动抑制效率起伏较大的问题，本文提出了一种两自由度的NES，可以在不增加NES总质量的前提下缓解该问题，通过数值仿真说明了这种两自由度NES的优越性。并且，使用Hilbert-Huang变换研究了系统产生靶能量传递所需的内共振条件，进而用打靶法及复变量—平均法分析了系统在主共振附近的力学特性，支持了上述结论。
鉴于前述两自由度NES对自由振动抑制的优越性，本文进一步研究了用其抑制结构在简谐荷载下的强迫振动的问题。当激励频率等于结构的固有频率时，推导了系统的慢变方程，对平衡点的个数及位置进行了解析分析，并判断了平衡点的稳定性；当激励频率可变时，用IHB法计算出系统的周期响应，并判断了系统响应的稳定性。以上研究揭示了系统的局部分岔现象，并说明了系统在产生何种响应类型时对振动抑制更为有利。最后比较了两自由度NES同单自由度NES在简谐荷载下的振动抑制效果。
Since spacecraft faces complicated vibration environment which poses greatthreat to the structure and sensitive equipment on it during practical missions,vibration suppression is always a focus in the field of spacecraft design. Thisresearch is focus on a kind of nonlinear vibration absorber which terms as nonlinearenergy sink. It’s a method for vibration suppression using the principle of targetedenergy transfer, and it processes good application foreground for local vibrationreduction of spacecrafts. Research of the dynamics and efficiency of vibrationsuppression of NES is the key point of theoretical research, which is also importantfor application in actual projects.
NES has two features while used for free vibration suppression: First, NESbecomes most efficient when the vibration energy of the system reaches a certainvalue; Second, for realizing targeted energy tranfer, the mass ratio between the NESand the main structure must beyond a certain value. The above two features arequalitative conclusions, and will be analyzed quantitatively in this research. Theanalytic conditions for vibration energy completely transfers between the mainstrucure and the NES is derived for the conservative system, and the minimum massratio between the NES and the main structure for realizing optimal targeted energytransfer is also abtained. Then, the relation between cubic stiffness and intial energyis generalized to non-conservative systems which contain small damping. Based onthe above conclusions, the method for designing the cubic stiffness according to theinitial input energy to the main structure is developed on the cases of small damping,and the method is verified numerically.
The ability of suppressing two different vibration peaks is an important aspectto evaluate the efficiency of the NES. So, vibration mitigation of atwo-degree-of-freedom linear system under harmonic excitation by means of NES isstudied with two methods: the decoupled formula of equilibrium points of the slowdynamics are derived, and the periodic response of the system is obtained usingcomplex-averaging method when the excitation frequency is fixed. The stability ofthe equilibium points are studied based on Lyapunov’s theorem; The periodicresponse of the system is studied using incremental harmonic balance method whenthe excitation frequency is tuning, and the stability of the periodic response isstudied based on Floquet’s theorem. Sadle-node bifurcation and Hopf bifurcation arerevealed from different aspects with the two methods, and the effectiveness ofvibration suppression for the two different energy peaks is also revealed. Analyticalconclusions are verified using numerial simulations.
The effeciency of vibration suppression varies drastically with the changing ofinitial energy of the main structure when the NES is used for free vibration control.It is proposed that the problem could be reliefed with the application of a kind oftwo-degree-of-freedom NES. The advantage of the-two-degree-of-freedom NES isverified numerically. The phenomenon of resonance capture when targeted energytranfer occurs is studied using Hilbert-Huang transform. Then, dynamics of thesystem is analyzed using shooting method and complexification-averaging method,and the analytical results support the frontal conclusions.
A fruther study of the two-degree-of-freedom NES is carried out for forcedvibration suppression under harmonic excitation since the superiority of thetwo-degree-of-freedom NES is found in chapter4. The slow dynamics of the systemis derived and the equilibrium points of the slow dynamics is studied analyticallywhen the excitation frequency is equal to the natural frequency of the main structure.The stability of the equilibrium points is also studied. The periodic response of thesystem is studied using incremental harmonic balance method when the excitationfrequency is tuning, and the stablility of the periodic response is given. Localbifurcations of the system are revealed and verified using numerical method, and itis also revealed which response regime is more favorable for vibration suppression.Finally, the efficiency of vibration suppression under harmonic force of thetwo-degree-of-freedom NES is compared with the single-degree-of-freedom NES.
引文
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