轴向运动粘弹性梁的横向振动分析
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摘要
轴向运动梁是一种重要的工程元件,在动力传送带、磁带、纸带、纺织纤维、带锯、空中缆车索道、高楼升降机缆绳、单索架空索道等多种工程系统中都有着广泛的应用,因而轴向运动连续体横向振动及其控制的研究有着重要的实际应用价值。同时,轴向运动连续体作为典型陀螺连续系统,由于陀螺项的存在,对其振动的分析也有着重要的理论意义。
轴向运动梁控制方程中的非线性项是由梁的大变形引起,梁的弯曲变形引起轴向应力的变化,这种非线性项即所谓几何非线性。Wickert提出准静态假设,认为因梁弯曲变形而引起的应力变化,沿梁的轴向近似均匀分布,应力取梁应力的一个平均值,得到了轴向运动梁非线性振动的积分-偏微分方程。在本文中,我们分析梁上微单元的受力情况,利用牛顿第二定律得到梁非线性振动的偏微分方程,在这种非线性模型,梁的轴向应力在梁的整个轴是不再是一个静态值,而是与轴向坐标有关的一个变量。
在本文的轴向运动梁振动的分析中,我们还要考虑梁材料的粘弹性。这种粘弹性阻尼的存在对运动梁振动的幅频响应、受迫振动以及受激励运动梁的稳定性有非常明显的作用。
对于带有小扰动的轴向运动梁的非线性振动,摄动法是解决问题的有效途径。由于连续介质为无穷维的系统,对其离散必造成误差。传统的对离散化方程做摄动法有一定的局限性。本文中,将要利用直接多尺度法来分析轴向运动梁的振动问题,把多尺度法直接应用于梁的控制偏微分方程,然后根据可解性条件求解。
利用多尺度法,我们对带有参数激励或外激励的轴向运动梁非线性振动的幅频响应做出了详细的分析。用这种方法研究了次谐波共振及组合共振时的分岔行为和稳定性以及受迫共振的跳跃现象和稳定性问题对跳跃的影响。对于线性系统的小扰动情况,我们还用平均法分析了共振所引发的失稳现象,讨论了多种参数,比如轴向速度,刚度,粘弹性阻尼等对失稳区域的影响。
多尺度方法是解决微分方程有效的方法,人们往往用一阶近似来讨论问题。我们用二阶多尺度方法发现了梁粘弹性阻尼对梁自然频率的影响,而这是用一阶多尺度方法无法得到这种结果的。
以前文献所假设的边界条件,多认为运动两端为铰支或固支,而实际上,这种假设过于理想化。本文中,我们将研究一种新的边界条件,即两端带有扭转弹簧的铰支支承条件,这种边界条件更符合工程实际中的真实情况。可以证明,铰支边界假设低估了梁的自然频率,而固定支承计算所得固有频率结果值偏大。
Galerkin截断方法常常用于求解偏微分方程,它可以用来分析强非线性及高
The class of systems with axially moving materials involves power transmission chains, band saw blades, aerial cableways and paper sheets during processing. Transverse vibration of such systems is generally undesirable although characteristic of operation at high transport speeds. The study of the vibration response of the axially moving materials is of great significance. Through a convective acceleration component, the governing equations of motion for axially moving materials are skew-symmetric in the state space formulation. The research of the transverse vibration in that case may pay contribution to the context of continuous gyroscopic systems.The nonlinear effect cannot be neglected if the transverse displacement of the axially moving beam is rather large. When transverse motion is treated for axially moving beams, there are two types of nonlinear models, a partial-differential equation or an integro-partial-differential equation. The partial-differential equation is derived from considering the transverse displacement only, and the integro-partial-differential equation is traditionally derived from decoupling the governing equation of coupled longitudinal and transverse motion under the quasi-static stretch assumption that supposes the influence of longitudinal inertia can be neglected.The modeling of dissipative mechanisms is an important research topic of axially moving material vibrations. Viscoelasticity is an effective approach to model the damping mechanism. In present investigation, the Kelvin viscoelastic model will be adopted in the studying of the free vibration, parametric resonance, and the forced vibration of the axially moving beam.Vibrations of continuous systems are always modeled in the form of a partial differential equation with small nonlinear or perturbed terms. The perturbation methods may be applied directly to the partial differential equation system. This approach is called direct-perturbation method. The direct-perturbation method produces more accurate results than the discretization method because the eigenfunctions represent the real system better in the case of the direct-perturbation method.In fact, many real systems could be represented by the axially moving materials with pulsating speed. That is, the axial transport speed is a constant mean velocity with small periodic fluctuations. In some other case, if the foundations supporting the axially moving materials are not motionless, the forced transverse vibration must be considered. The method of multiple scales can be used in those governing equations. The amplitude
轴向运动梁控制方程中的非线性项是由梁的大变形引起,梁的弯曲变形引起轴向应力的变化,这种非线性项即所谓几何非线性。Wickert提出准静态假设,认为因梁弯曲变形而引起的应力变化,沿梁的轴向近似均匀分布,应力取梁应力的一个平均值,得到了轴向运动梁非线性振动的积分-偏微分方程。在本文中,我们分析梁上微单元的受力情况,利用牛顿第二定律得到梁非线性振动的偏微分方程,在这种非线性模型,梁的轴向应力在梁的整个轴是不再是一个静态值,而是与轴向坐标有关的一个变量。
在本文的轴向运动梁振动的分析中,我们还要考虑梁材料的粘弹性。这种粘弹性阻尼的存在对运动梁振动的幅频响应、受迫振动以及受激励运动梁的稳定性有非常明显的作用。
对于带有小扰动的轴向运动梁的非线性振动,摄动法是解决问题的有效途径。由于连续介质为无穷维的系统,对其离散必造成误差。传统的对离散化方程做摄动法有一定的局限性。本文中,将要利用直接多尺度法来分析轴向运动梁的振动问题,把多尺度法直接应用于梁的控制偏微分方程,然后根据可解性条件求解。
利用多尺度法,我们对带有参数激励或外激励的轴向运动梁非线性振动的幅频响应做出了详细的分析。用这种方法研究了次谐波共振及组合共振时的分岔行为和稳定性以及受迫共振的跳跃现象和稳定性问题对跳跃的影响。对于线性系统的小扰动情况,我们还用平均法分析了共振所引发的失稳现象,讨论了多种参数,比如轴向速度,刚度,粘弹性阻尼等对失稳区域的影响。
多尺度方法是解决微分方程有效的方法,人们往往用一阶近似来讨论问题。我们用二阶多尺度方法发现了梁粘弹性阻尼对梁自然频率的影响,而这是用一阶多尺度方法无法得到这种结果的。
以前文献所假设的边界条件,多认为运动两端为铰支或固支,而实际上,这种假设过于理想化。本文中,我们将研究一种新的边界条件,即两端带有扭转弹簧的铰支支承条件,这种边界条件更符合工程实际中的真实情况。可以证明,铰支边界假设低估了梁的自然频率,而固定支承计算所得固有频率结果值偏大。
Galerkin截断方法常常用于求解偏微分方程,它可以用来分析强非线性及高
The class of systems with axially moving materials involves power transmission chains, band saw blades, aerial cableways and paper sheets during processing. Transverse vibration of such systems is generally undesirable although characteristic of operation at high transport speeds. The study of the vibration response of the axially moving materials is of great significance. Through a convective acceleration component, the governing equations of motion for axially moving materials are skew-symmetric in the state space formulation. The research of the transverse vibration in that case may pay contribution to the context of continuous gyroscopic systems.The nonlinear effect cannot be neglected if the transverse displacement of the axially moving beam is rather large. When transverse motion is treated for axially moving beams, there are two types of nonlinear models, a partial-differential equation or an integro-partial-differential equation. The partial-differential equation is derived from considering the transverse displacement only, and the integro-partial-differential equation is traditionally derived from decoupling the governing equation of coupled longitudinal and transverse motion under the quasi-static stretch assumption that supposes the influence of longitudinal inertia can be neglected.The modeling of dissipative mechanisms is an important research topic of axially moving material vibrations. Viscoelasticity is an effective approach to model the damping mechanism. In present investigation, the Kelvin viscoelastic model will be adopted in the studying of the free vibration, parametric resonance, and the forced vibration of the axially moving beam.Vibrations of continuous systems are always modeled in the form of a partial differential equation with small nonlinear or perturbed terms. The perturbation methods may be applied directly to the partial differential equation system. This approach is called direct-perturbation method. The direct-perturbation method produces more accurate results than the discretization method because the eigenfunctions represent the real system better in the case of the direct-perturbation method.In fact, many real systems could be represented by the axially moving materials with pulsating speed. That is, the axial transport speed is a constant mean velocity with small periodic fluctuations. In some other case, if the foundations supporting the axially moving materials are not motionless, the forced transverse vibration must be considered. The method of multiple scales can be used in those governing equations. The amplitude
引文
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