基于基础结构法的柔顺机构拓扑优化设计研究
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摘要
柔顺机构是通过其部分或全部具有柔性构件的弹性变形来传递力和运动的机构。它具有结构简单、容易制造、无摩擦磨损、容易装配、高精度、高可靠性、轻质量及实现微型化等优点。它已经在微机电系统器件设计、生物工程显微操作、光纤对接和航空航天等领域得到广泛应用。采用拓扑优化方法设计柔顺机构,只需给定设计域和指定输入输出位置,无需从一个已知的刚性机构出发,且所得机构具有优化的力-位移输入输出关系,因而引起了人们的重视。本文基于基础结构法,对柔顺机构动力学、可靠性及几何非线性拓扑优化设计进行了深入的研究,主要研究内容如下:
(1)基于基础结构法进行了多输入多输出柔顺机构多目标拓扑优化。建立了柔顺机构多目标拓扑优化的数学模型,目标函数以应变能最小和互应变能最大来满足机构的刚度和柔度需求。在此基础上,推导了描述多输入多输出柔顺机构柔性的互应变能公式和描述机构刚性的应变能公式,给出了抑制输出耦合效应的计算公式,建立了考虑抑制输出耦合效应时多输入多输出柔顺机构的多目标拓扑优化数学模型。通过数值算例验证该模型的有效性。
(2)基于基础结构法进行了柔顺机构动力学拓扑优化。以动力放大系数最大化与应变能最小化的加权函数为目标,采用标准化方法定义多目标拓扑优化的目标函数,满足在动态条件下柔顺机构具有足够柔度和刚度;并且对目标函数进行规一化避免不同性质目标函数在数量级上的差异。通过数值算例分析了不同外激励频率、不同结构阻尼和不同输出刚度对拓扑优化结果影响。
(3)基于基础结构法进行了柔顺机构可靠性拓扑优化设计。在确定性柔顺机构多目标拓扑优化数学模型基础上,将作用荷载及几何尺寸视为随机变量,机构的失效模式视为应变能和互应变能双模式的串联系统,采用一次可靠度方法计算串联系统的失效概率,建立了柔顺机构可靠性多目标拓扑优化数学模型。数值算例表明基于可靠性拓扑优化方法所得机构比确定性拓扑优化更合理可靠。并以确定性和可靠性拓扑优化机构为基础,加入最小尺寸约束,采用线切割加工工艺,研制确定性和可靠性设计反向器原型,对其进行了位移性能测试,结果表明反向器的实验结果与理论分析结果基本吻合,达到设计要求。
(4)基于基础结构法进行了柔顺机构几何非线性拓扑优化。建立增量形式的平衡方程,采用随转坐标的全拉格朗日法有限元描述方法和迭代增量混合法来求解获得平面框架单元结构的几何非线性响应。推导描述柔顺机构柔度的几何增益公式和描述机构刚度的应变能公式,建立了适合求解几何非线性的多目标拓扑优化数学模型,目标函数以几何增益最大化和应变能最小化来满足柔顺机构柔度和刚度。与线性结果比较,利用非线性理论获得的机构具有更好的性能,同时也说明了进行几何非线性拓扑优化的必要性。
Compliant mechanisms achieve force and motion transmission through elastic deformation of relative flexibility of its members. Compared with rigid-body mechanisms, compliant mechanisms have many advantages such as a simple structure, simplified manufacturing processes, reduced friction, reduced assembly time, reduced weight, increased precision, reduced weight and miniaturization. Thus, it has been applied widely in the micro-electro-mechanical systems device design, biological engineering micro-manipulation, fiber alignment and aerospace. Topology optimization of compliant mechanisms has drawn more and more attentions because it only needs to designate a design domain and the positions of the inputs and outputs. In this paper, dynamic topology optimization of compliant mechanisms, reliability-based topology optimization of compliant mechanisms and topology optimization of compliant mechanisms with geometrical nonlinearity using the structure approach are investigated deeply. The main contributions of this thesis are listed as follows:
A methodology for multi-objective topology optimization of multiple inputs and multiple outputs compliant mechanisms using the ground structure approach is presented. The multi-objective is developed by the minimum strain energy and the maximum mutual potential energy to design a mechanism, which meets both stiffness and flexibility requirements, respectively. Based on this, the multi-objective function of topology optimization of multiple inputs multiple outputs compliant mechanism is also developed by the strain energy and the mutual potential energy. The suppression strategy of output coupling terms is studied,and the expression of the output coupling terms is further developed. Numerical simulations are presented to show that the proposed optimization model is valid.
A methodology for dynamic topology optimization of compliant mechanisms using the ground structure approach is presented. The function is developed by the maximum dynamic magnification factor and the minimum strain energy to design a mechanism which meets both stiffness and flexibility requirements under harmonic excitation, respectively. The objective function is normalized to eliminate magnitude difference of the objectives. Some numerical examples with different driving frequencies, different structural damping factors and different output spring stiffness are presented to illustrate the effect of driving frequency, damping factor and output spring stiffness.
A methodology for reliability-based topology optimization of compliant mechanisms using the ground structure approach is presented. The applied load and the structural geometry size are considered as the uncertain variables. That the strain energy and mutual potential energy have upon system reliability are evaluated by regarding as a series systems. The first-order reliability method is adopted to solve the failure probability of the series system. The numerical example is simulated to show that the proposed method is correct and effective because it helps to obtain mechanisms with higher performance than those obtained by the deterministic topology optimization. Based on the topology mechanisms and the restriction of fabrication technology, the prototypes of the compliant inverters are determined. The prototypes are manufactured by means of wire cutting technology. The displacements of the compliant inverters are measured by using the measurement system. The experimental results approximately agree with the numerical results. It shows that properties of compliant inverters can meet the designing demands.
A topology optimization method of compliant mechanisms with geometrical nonlinearity under displacement loading is presented in this paper. Geometrically nonlinear plane frame structural response is solved using the co-rotational Total Lagrange finite element formulation and the equilibrium is solved using the incremental scheme combined with Newton-Raphson iteration. The multi-objective function is developed by the minimum strain energy and maximum geometric advantage to design a mechanism which meets both stiffness and flexibility requirements. Compared with the result using linear formulation, the benefits of the optimal mechanisms obtained by nonlinear formulation are illustrated by the numerical example.
(1)基于基础结构法进行了多输入多输出柔顺机构多目标拓扑优化。建立了柔顺机构多目标拓扑优化的数学模型,目标函数以应变能最小和互应变能最大来满足机构的刚度和柔度需求。在此基础上,推导了描述多输入多输出柔顺机构柔性的互应变能公式和描述机构刚性的应变能公式,给出了抑制输出耦合效应的计算公式,建立了考虑抑制输出耦合效应时多输入多输出柔顺机构的多目标拓扑优化数学模型。通过数值算例验证该模型的有效性。
(2)基于基础结构法进行了柔顺机构动力学拓扑优化。以动力放大系数最大化与应变能最小化的加权函数为目标,采用标准化方法定义多目标拓扑优化的目标函数,满足在动态条件下柔顺机构具有足够柔度和刚度;并且对目标函数进行规一化避免不同性质目标函数在数量级上的差异。通过数值算例分析了不同外激励频率、不同结构阻尼和不同输出刚度对拓扑优化结果影响。
(3)基于基础结构法进行了柔顺机构可靠性拓扑优化设计。在确定性柔顺机构多目标拓扑优化数学模型基础上,将作用荷载及几何尺寸视为随机变量,机构的失效模式视为应变能和互应变能双模式的串联系统,采用一次可靠度方法计算串联系统的失效概率,建立了柔顺机构可靠性多目标拓扑优化数学模型。数值算例表明基于可靠性拓扑优化方法所得机构比确定性拓扑优化更合理可靠。并以确定性和可靠性拓扑优化机构为基础,加入最小尺寸约束,采用线切割加工工艺,研制确定性和可靠性设计反向器原型,对其进行了位移性能测试,结果表明反向器的实验结果与理论分析结果基本吻合,达到设计要求。
(4)基于基础结构法进行了柔顺机构几何非线性拓扑优化。建立增量形式的平衡方程,采用随转坐标的全拉格朗日法有限元描述方法和迭代增量混合法来求解获得平面框架单元结构的几何非线性响应。推导描述柔顺机构柔度的几何增益公式和描述机构刚度的应变能公式,建立了适合求解几何非线性的多目标拓扑优化数学模型,目标函数以几何增益最大化和应变能最小化来满足柔顺机构柔度和刚度。与线性结果比较,利用非线性理论获得的机构具有更好的性能,同时也说明了进行几何非线性拓扑优化的必要性。
Compliant mechanisms achieve force and motion transmission through elastic deformation of relative flexibility of its members. Compared with rigid-body mechanisms, compliant mechanisms have many advantages such as a simple structure, simplified manufacturing processes, reduced friction, reduced assembly time, reduced weight, increased precision, reduced weight and miniaturization. Thus, it has been applied widely in the micro-electro-mechanical systems device design, biological engineering micro-manipulation, fiber alignment and aerospace. Topology optimization of compliant mechanisms has drawn more and more attentions because it only needs to designate a design domain and the positions of the inputs and outputs. In this paper, dynamic topology optimization of compliant mechanisms, reliability-based topology optimization of compliant mechanisms and topology optimization of compliant mechanisms with geometrical nonlinearity using the structure approach are investigated deeply. The main contributions of this thesis are listed as follows:
A methodology for multi-objective topology optimization of multiple inputs and multiple outputs compliant mechanisms using the ground structure approach is presented. The multi-objective is developed by the minimum strain energy and the maximum mutual potential energy to design a mechanism, which meets both stiffness and flexibility requirements, respectively. Based on this, the multi-objective function of topology optimization of multiple inputs multiple outputs compliant mechanism is also developed by the strain energy and the mutual potential energy. The suppression strategy of output coupling terms is studied,and the expression of the output coupling terms is further developed. Numerical simulations are presented to show that the proposed optimization model is valid.
A methodology for dynamic topology optimization of compliant mechanisms using the ground structure approach is presented. The function is developed by the maximum dynamic magnification factor and the minimum strain energy to design a mechanism which meets both stiffness and flexibility requirements under harmonic excitation, respectively. The objective function is normalized to eliminate magnitude difference of the objectives. Some numerical examples with different driving frequencies, different structural damping factors and different output spring stiffness are presented to illustrate the effect of driving frequency, damping factor and output spring stiffness.
A methodology for reliability-based topology optimization of compliant mechanisms using the ground structure approach is presented. The applied load and the structural geometry size are considered as the uncertain variables. That the strain energy and mutual potential energy have upon system reliability are evaluated by regarding as a series systems. The first-order reliability method is adopted to solve the failure probability of the series system. The numerical example is simulated to show that the proposed method is correct and effective because it helps to obtain mechanisms with higher performance than those obtained by the deterministic topology optimization. Based on the topology mechanisms and the restriction of fabrication technology, the prototypes of the compliant inverters are determined. The prototypes are manufactured by means of wire cutting technology. The displacements of the compliant inverters are measured by using the measurement system. The experimental results approximately agree with the numerical results. It shows that properties of compliant inverters can meet the designing demands.
A topology optimization method of compliant mechanisms with geometrical nonlinearity under displacement loading is presented in this paper. Geometrically nonlinear plane frame structural response is solved using the co-rotational Total Lagrange finite element formulation and the equilibrium is solved using the incremental scheme combined with Newton-Raphson iteration. The multi-objective function is developed by the minimum strain energy and maximum geometric advantage to design a mechanism which meets both stiffness and flexibility requirements. Compared with the result using linear formulation, the benefits of the optimal mechanisms obtained by nonlinear formulation are illustrated by the numerical example.
引文
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