压电复合材料结构的复杂非线性动力学与控制的研究
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摘要
压电材料是航空航天工程中的一种新型功能材料,具有比强度高、比刚度大和抗疲劳性能好等优点,并且易于加工,适合作为作动器和传感器嵌入到工程结构中组成压电复合材料结构提高结构的可控性。压电复合材料结构近年来被广泛应用于航天器、大型空间站等工程领域。最新研究成果表明,压电复合材料同样可用于可变体结构,为结构的变形产生驱动力,例如可变体机翼等变形结构。在这种可变体机翼中,压电复合材料板结构有可能产生大振幅的振动,导致压电复合材料板结构的非线性振动,从而降低结构的稳定性和可控性能。因此,研究压电复合材料结构的非线性振动、分叉和混沌动力学行为具有极为重要的理论和工程应用价值。本论文主要采用理论推导与数值计算相结合的方法对在参数激励和横向激励联合作用下压电复合材料层合梁和压电复合材料层合板的非线性振动、分叉和混沌动力学特性进行研究。论文的研究内容主要有以下几个方面。
(1)研究了简支压电复合材料层合梁在轴向、横向载荷共同作用下的非线性动力学、分叉和混沌动力学。基于von Karman理论和Reddy高阶剪切变形理论,推导出了压电复合材料层合梁的非线性动力学方程。利用Galerkin法离散偏微分方程,得到二个自由度非线性控制方程,并且利用多尺度法得到了得到了具有1:9内共振关系的平均方程。基于平均方程,研究了压电层合梁系统的动态分叉,分析了系统各种参数对倍周期分叉的影响及变化规律。结果表明,压电复合材料层合梁周期运动的稳定性和混沌运动对外激励的变化非常敏感,通过控制压电激励,可以控制压电复合材料层合梁的振动,保持系统的稳定性,即控制系统产生倍周期分叉解,从而阻止系统通过倍周期分叉进入混沌运动,并给出了控制分叉图。
(2)首次分析了简支压电复合材料层合板的非线性动力学。基于von Karman理论和Reddy高阶剪切变形理论,考虑压电载荷的作用,利用Hamilton原理推导了压电复合材料层合板的非线性动力学方程。采用Galerkin法对偏微分方程进行离散,得到包含外激励和参数激励的二自由度控制方程。考虑1:1,1:2,1:3内共振和主参数共振-1/2亚谐共振的情况,利用多尺度法得到压电复合材料层合板的四维平均方程。基于平均方程,采用数值方法研究了系统动态响应随面内激励和横向激励变化时的动态分叉图,分析了面内激励、横向激励与压电激励对系统分叉和混沌动力学的影响。结果表明压电复合材料层合板是存在周期和混沌运动。
(3)基于四边简支压电复合材料层合板的非线性动力学方程,利用Galerkin法对系统偏微分方程进行三阶模态离散,得到带有外激励和参数激励的三自由度控制方程。考虑1:2:3, 1:2:4内共振和主参数共振-1/2亚谐共振情况,利用多尺度法进行摄动分析得到六维平均方程。采用数值仿真方法分析横向激励、面内激励与压电激励对压电复合材料层合板的非线性振动、分叉与混沌动力学行为的影响。
(4)基于建立的压电复合材料层合板的非线性动力学方程,选取四阶模态,用Galerkin法对其进行离散,考虑1:2:9:9内共振和主参数共振-1/2亚谐共振情况,用多尺度法进行摄动分析得到八维的压电复合材料层合板的平均方程。基于平均方程研究了压电复合层合板的分叉与混沌动力学,采用数值仿真方法分析了横向激励、面内激励与压电激励对系统的分叉与混沌动力学的影响。结果表明压电复合材料层合板的低阶模态响应幅值大于高阶模态响应幅值。通过分析系统分叉图可知当系统的响应随着激励参数的变化而变化时,系统发生分叉并导致混沌,结果表明压电激励幅值是影响压电复合材料层合板结构运动形式的重要控制参数。
(5)首次利用全局摄动方法研究了压电复合材料层合板的全局分叉和Shilnikov型混沌动力学。在压电复合材料层合板四维平均方程的基础上,利用规范形理论对平均方程进行简化,得到了在一对双零特征值和一对纯虚特征值情况下压电复合材料层合板平均方程的规范形。在此基础上,利用Kovacic和Wiggins提出的全局摄动方法研究了压电复合材料层合板的Shilnikov型单脉冲同宿轨道和混沌动力学,理论分析表明在压电复合材料层合板中存在着Pitchfork分叉和Shilnikov型单脉冲同宿轨线,从而证明了在压电复合材料层合板系统中存在着由Shilnikov型单脉冲同宿轨线导致的Smale马蹄意义下的混沌。数值模拟进一步验证了理论分析的结果。
Piezoelectric materials, which include piezoelectric lead-zirconate-titanate (PZT) and piezoelectric polyvinylidene fluoride (PVDF), are new functional materials in engineering applications. Because of its properties of high strength, stiffness and durability, such materials can be used as the actuators and sensors in various engineering structures. For instance, composite laminated piezoelectric plates have been widely adopted in aerospace engineering over the last two decades, including the structural elements of the aircraft, large space station and shuttle. Besides, the morphing structures or morphing wings may be composed of laminated piezoelectric materials, which can undergo large oscillations of motion under the external excitations. These oscillating systems are thus nonlinear in nature. Research on the nonlinear dynamics of composite laminated piezoelectric plates plays a vital role in engineering applications. Heretofore, a few studies on the bifurcation and chaotic motions of composite laminated piezoelectric plates have been conducted. To the author’s best knowledge, it is the first-known solutions to reveal the bifurcation and chaotic motions of a simply supported symmetric cross-ply composite laminated piezoelectric beam and rectangular plate having the transverse and in-plane excitations. The global perturbation method is used to investigate the periodic and chaotic motions of composite laminated piezoelectric plates.
The major research scope and the innovative outcome of this dissertation are briefly summarized as follows:
(1) The nonlinear oscillation, bifurcation and chaotic dynamics of a simply supported laminated composite piezoelectric beam are analyzed. The beam with piezoelectric materials is forced by the axial and transverse loads. In accordance with the von Karman-type equations and Reddy’s third-order shear deformation plate theory, the nonlinear equations of motion for the laminated composite piezoelectric beam are derived. The Galerkin approach is employed to transform the governing partial differential equations to two-degree-of-freedom ordinary differential equations. Consider the resonant cases of 1:9 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the method of multiple scales is applied to yield the four-dimensional averaged equations. Making use of the averaged equations herein, the bifurcation and chaotic motions of the laminated composite piezoelectric beam are studied. The periodic and chaotic motions of beams are found by using the numerical simulation. It is concluded that the chaotic responses are sensitive to the piezoelectric excitations. By changing the piezoelectric excitations, we can control the nonlinear oscillation of laminated composite piezoelectric beams.
(2) The nonlinear dynamics of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate having the transverse and in-plane excitations are studied. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate are derived by Hamilton’s principle. The excitation loaded by piezoelectric layers is considered. The Galerkin approach is employed to deduce a two-degree-of-freedom nonlinear system under the combination of the parametric and external excitations from the governing partial differential equations. Consider the resonant cases of 1:1, 1:2 and 1:3 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the method of multiple scales is utilized to form the four-dimensional averaged equation of composite laminated piezoelectric rectangular plates. Numerical method is implemented to study the bifurcation and chaotic dynamics of composite laminated piezoelectric rectangular plates. The numerical results show the existence of the periodic and chaotic motions in the averaged equation. It is observed that the chaotic responses are especially sensitive to the forcing and parametric excitations. The influence of the transverse, in-plane and piezoelectric excitations on the bifurcation and chaotic behavior of composite laminated piezoelectric rectangular plates is investigated numerically.
(3) Based on Hamilton’s principle, the governing nonlinear equations of motion for composite laminated piezoelectric rectangular plates are derived, Selecting the appropriate mode functions satisfies the boundary conditions of composite laminated piezoelectric rectangular plates, the Galerkin approach is employed to reduce the partial differential governing equations to a three-degree-of-freedom nonlinear system under the combination of the parametric and external excitations. Consider the resonant cases of 1:2:3 and 1:2:4 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the method of multiple scales is utilized to obtain the six-dimensional averaged equation of composite laminated piezoelectric rectangular plates. Numerical method is implemented to study the bifurcation and chaotic dynamics of composite laminated piezoelectric rectangular plates. The influence of the transverse, in-plane and piezoelectric excitations on the bifurcation and chaotic behavior of composite laminated piezoelectric rectangular plates is investigated numerically.
(4) By virtue of Hamilton’s principle, the governing nonlinear equations of motion for the composite laminated piezoelectric rectangular plate are derived, Selecting the appropriate mode functions satisfies the boundary conditions of composite laminated piezoelectric rectangular plates, the Galerkin approach is employed to turn the partial differential governing equations into a four-degree-of-freedom nonlinear system under the combination of the parametric and external excitations. Consider the resonant cases of 1:2:9:9 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the method of multiple scales is utilized to achieve the eight-dimensional averaged equation of composite laminated piezoelectric rectangular plates. Numerical method is implemented to study the bifurcation and chaotic dynamics of composite laminated piezoelectric rectangular plates. The influence of the transverse, in-plane and piezoelectric excitations on the bifurcation and chaotic behavior of composite laminated piezoelectric rectangular plates is investigated numerically.
(5) Global bifurcation and Shilnikov type chaotic dynamics of composite laminated piezoelectric rectangular plates are primitively probed by means of the global perturbation method. According to the four dimensional averaged equations of composite laminated piezoelectric rectangular plates, the theory of normal form is applied to further reduce the explicit formulas to the simple one. The global perturbation method proposed by Kovacic and Wiggins are generalized herein to study the Shilnikov type single-pulse homoclinic orbit and chaotic dynamics of composite laminated piezoelectric rectangular plates. Theoretical analysis is not only to demonstrate the existence of Pitchfork bifurcation and Shilnikov type single-pulse homoclinic orbit, but also reveals the chaotic motion of the Smale horseshoe in the system. The numerical simulation of the multi-pulse orbits is presented to verify the analytical solutions.
(1)研究了简支压电复合材料层合梁在轴向、横向载荷共同作用下的非线性动力学、分叉和混沌动力学。基于von Karman理论和Reddy高阶剪切变形理论,推导出了压电复合材料层合梁的非线性动力学方程。利用Galerkin法离散偏微分方程,得到二个自由度非线性控制方程,并且利用多尺度法得到了得到了具有1:9内共振关系的平均方程。基于平均方程,研究了压电层合梁系统的动态分叉,分析了系统各种参数对倍周期分叉的影响及变化规律。结果表明,压电复合材料层合梁周期运动的稳定性和混沌运动对外激励的变化非常敏感,通过控制压电激励,可以控制压电复合材料层合梁的振动,保持系统的稳定性,即控制系统产生倍周期分叉解,从而阻止系统通过倍周期分叉进入混沌运动,并给出了控制分叉图。
(2)首次分析了简支压电复合材料层合板的非线性动力学。基于von Karman理论和Reddy高阶剪切变形理论,考虑压电载荷的作用,利用Hamilton原理推导了压电复合材料层合板的非线性动力学方程。采用Galerkin法对偏微分方程进行离散,得到包含外激励和参数激励的二自由度控制方程。考虑1:1,1:2,1:3内共振和主参数共振-1/2亚谐共振的情况,利用多尺度法得到压电复合材料层合板的四维平均方程。基于平均方程,采用数值方法研究了系统动态响应随面内激励和横向激励变化时的动态分叉图,分析了面内激励、横向激励与压电激励对系统分叉和混沌动力学的影响。结果表明压电复合材料层合板是存在周期和混沌运动。
(3)基于四边简支压电复合材料层合板的非线性动力学方程,利用Galerkin法对系统偏微分方程进行三阶模态离散,得到带有外激励和参数激励的三自由度控制方程。考虑1:2:3, 1:2:4内共振和主参数共振-1/2亚谐共振情况,利用多尺度法进行摄动分析得到六维平均方程。采用数值仿真方法分析横向激励、面内激励与压电激励对压电复合材料层合板的非线性振动、分叉与混沌动力学行为的影响。
(4)基于建立的压电复合材料层合板的非线性动力学方程,选取四阶模态,用Galerkin法对其进行离散,考虑1:2:9:9内共振和主参数共振-1/2亚谐共振情况,用多尺度法进行摄动分析得到八维的压电复合材料层合板的平均方程。基于平均方程研究了压电复合层合板的分叉与混沌动力学,采用数值仿真方法分析了横向激励、面内激励与压电激励对系统的分叉与混沌动力学的影响。结果表明压电复合材料层合板的低阶模态响应幅值大于高阶模态响应幅值。通过分析系统分叉图可知当系统的响应随着激励参数的变化而变化时,系统发生分叉并导致混沌,结果表明压电激励幅值是影响压电复合材料层合板结构运动形式的重要控制参数。
(5)首次利用全局摄动方法研究了压电复合材料层合板的全局分叉和Shilnikov型混沌动力学。在压电复合材料层合板四维平均方程的基础上,利用规范形理论对平均方程进行简化,得到了在一对双零特征值和一对纯虚特征值情况下压电复合材料层合板平均方程的规范形。在此基础上,利用Kovacic和Wiggins提出的全局摄动方法研究了压电复合材料层合板的Shilnikov型单脉冲同宿轨道和混沌动力学,理论分析表明在压电复合材料层合板中存在着Pitchfork分叉和Shilnikov型单脉冲同宿轨线,从而证明了在压电复合材料层合板系统中存在着由Shilnikov型单脉冲同宿轨线导致的Smale马蹄意义下的混沌。数值模拟进一步验证了理论分析的结果。
Piezoelectric materials, which include piezoelectric lead-zirconate-titanate (PZT) and piezoelectric polyvinylidene fluoride (PVDF), are new functional materials in engineering applications. Because of its properties of high strength, stiffness and durability, such materials can be used as the actuators and sensors in various engineering structures. For instance, composite laminated piezoelectric plates have been widely adopted in aerospace engineering over the last two decades, including the structural elements of the aircraft, large space station and shuttle. Besides, the morphing structures or morphing wings may be composed of laminated piezoelectric materials, which can undergo large oscillations of motion under the external excitations. These oscillating systems are thus nonlinear in nature. Research on the nonlinear dynamics of composite laminated piezoelectric plates plays a vital role in engineering applications. Heretofore, a few studies on the bifurcation and chaotic motions of composite laminated piezoelectric plates have been conducted. To the author’s best knowledge, it is the first-known solutions to reveal the bifurcation and chaotic motions of a simply supported symmetric cross-ply composite laminated piezoelectric beam and rectangular plate having the transverse and in-plane excitations. The global perturbation method is used to investigate the periodic and chaotic motions of composite laminated piezoelectric plates.
The major research scope and the innovative outcome of this dissertation are briefly summarized as follows:
(1) The nonlinear oscillation, bifurcation and chaotic dynamics of a simply supported laminated composite piezoelectric beam are analyzed. The beam with piezoelectric materials is forced by the axial and transverse loads. In accordance with the von Karman-type equations and Reddy’s third-order shear deformation plate theory, the nonlinear equations of motion for the laminated composite piezoelectric beam are derived. The Galerkin approach is employed to transform the governing partial differential equations to two-degree-of-freedom ordinary differential equations. Consider the resonant cases of 1:9 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the method of multiple scales is applied to yield the four-dimensional averaged equations. Making use of the averaged equations herein, the bifurcation and chaotic motions of the laminated composite piezoelectric beam are studied. The periodic and chaotic motions of beams are found by using the numerical simulation. It is concluded that the chaotic responses are sensitive to the piezoelectric excitations. By changing the piezoelectric excitations, we can control the nonlinear oscillation of laminated composite piezoelectric beams.
(2) The nonlinear dynamics of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate having the transverse and in-plane excitations are studied. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate are derived by Hamilton’s principle. The excitation loaded by piezoelectric layers is considered. The Galerkin approach is employed to deduce a two-degree-of-freedom nonlinear system under the combination of the parametric and external excitations from the governing partial differential equations. Consider the resonant cases of 1:1, 1:2 and 1:3 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the method of multiple scales is utilized to form the four-dimensional averaged equation of composite laminated piezoelectric rectangular plates. Numerical method is implemented to study the bifurcation and chaotic dynamics of composite laminated piezoelectric rectangular plates. The numerical results show the existence of the periodic and chaotic motions in the averaged equation. It is observed that the chaotic responses are especially sensitive to the forcing and parametric excitations. The influence of the transverse, in-plane and piezoelectric excitations on the bifurcation and chaotic behavior of composite laminated piezoelectric rectangular plates is investigated numerically.
(3) Based on Hamilton’s principle, the governing nonlinear equations of motion for composite laminated piezoelectric rectangular plates are derived, Selecting the appropriate mode functions satisfies the boundary conditions of composite laminated piezoelectric rectangular plates, the Galerkin approach is employed to reduce the partial differential governing equations to a three-degree-of-freedom nonlinear system under the combination of the parametric and external excitations. Consider the resonant cases of 1:2:3 and 1:2:4 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the method of multiple scales is utilized to obtain the six-dimensional averaged equation of composite laminated piezoelectric rectangular plates. Numerical method is implemented to study the bifurcation and chaotic dynamics of composite laminated piezoelectric rectangular plates. The influence of the transverse, in-plane and piezoelectric excitations on the bifurcation and chaotic behavior of composite laminated piezoelectric rectangular plates is investigated numerically.
(4) By virtue of Hamilton’s principle, the governing nonlinear equations of motion for the composite laminated piezoelectric rectangular plate are derived, Selecting the appropriate mode functions satisfies the boundary conditions of composite laminated piezoelectric rectangular plates, the Galerkin approach is employed to turn the partial differential governing equations into a four-degree-of-freedom nonlinear system under the combination of the parametric and external excitations. Consider the resonant cases of 1:2:9:9 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the method of multiple scales is utilized to achieve the eight-dimensional averaged equation of composite laminated piezoelectric rectangular plates. Numerical method is implemented to study the bifurcation and chaotic dynamics of composite laminated piezoelectric rectangular plates. The influence of the transverse, in-plane and piezoelectric excitations on the bifurcation and chaotic behavior of composite laminated piezoelectric rectangular plates is investigated numerically.
(5) Global bifurcation and Shilnikov type chaotic dynamics of composite laminated piezoelectric rectangular plates are primitively probed by means of the global perturbation method. According to the four dimensional averaged equations of composite laminated piezoelectric rectangular plates, the theory of normal form is applied to further reduce the explicit formulas to the simple one. The global perturbation method proposed by Kovacic and Wiggins are generalized herein to study the Shilnikov type single-pulse homoclinic orbit and chaotic dynamics of composite laminated piezoelectric rectangular plates. Theoretical analysis is not only to demonstrate the existence of Pitchfork bifurcation and Shilnikov type single-pulse homoclinic orbit, but also reveals the chaotic motion of the Smale horseshoe in the system. The numerical simulation of the multi-pulse orbits is presented to verify the analytical solutions.
引文
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