基于振形正交性的弹性板动力分析法

【作者】 郭杰；

【导师】 许琪楼；

【作者基本信息】 郑州大学 ， 结构工程， 2004， 硕士

【摘要】 弹性薄板广泛应用于工程各领域，因此研究弹性薄板动力分析方法具有很高的理论意义及实用价值。在现有的各种解法中除差分法、能量法、有限元法、集中质量法等近似方法外，各种理论解法都存在一定的局限性。有的或者不满足振动微分方程。有的或者不全满足边界条件或者二者兼有之，即使上述条件均可满足的解法，其振形又不具有正交性的特征。 振形是正交的，这是振动理论中最重要、最基本的原理。基于振形的正交性。我们提出主振方向排序的动力分析方法。其基本思路是：在两个振动方向上任选一个振动主方向。为了保证振形的正交性，主振方向的主振波形数是唯一的，但为了保证振形曲线满足全部边界条件，在另一振动方向上振动波形数又不是唯一的。在两个振动方向上所采用的振动曲线形态要与相应的边界条件所限定的变形形态相吻合。由此建立的振形函数表达式可以满足振动微分方程，全部边界条件，满足边界条件的振形曲线又具有振形的正交性。 由板的边界条件可以建立相应的齐次线形方程组。为了保证主振波形具有非零解。相应的代定系数不能全为零，由此得振动频率方程。从而可以确定板的振动频率及相应的振动曲线。 一边简支一柱支角点支承的矩形板不仅具有简支边、固定边、自由角点等经典边界，还有柱支角点这一特殊边界条件。在板振动过程中，柱支角点要产生相应周期变化支反力。同时该角点处位移为零，由于边界条件的复杂性，使这类板的动力分析具有一定的难度。本文采用主振方向排序的动力分析法成功地解决了这一问题，并取得满意的结果。 从而说明主振方向排序的动力分析方法具有普遍的适用性。更多还原

【Abstract】 Elastic thin plate is widely used in all kinds of engineering fields. It is of high theoretical significance and high practical valve to researsh a dynamic analysis method for elastic thin plate. In different kinds of solution , besides approximate methods of calculus of differences , energy approach, finite-element method, lumped mass method, all sorts of theory solution have definite limitation. Some dissatisfy the vibration differential equation, some dissatisfy boundary conditions , or others have both shortcomings. Even if forementioned conditions are satisf iable , the mode of vibration don’t possess character of orthogonality . The mode of vibration is orthogonal .which is the most basic and important theory in vibration theory. Based on orthogonality of mode of vibration ,we put forward arrangement method on the main mode direction.Its basic way is to elect a main vibration direction at two vibration direction. In this text , we adopt this method to analyse the rectangular plate of one side freedom and one point.For guaranteeing orthogonality of mode , the waveform number is only. But the same time for guaranteeing mode curve satisfy the whole boundary condition , at other vibration direction the waveform number isn’ t only. The vibration curve shape we adopt should fit the homologous boundary condition shape. Based on this mode function expression could satisfy the whole boundary condition of vibration differential equation and possess orthogonality of mode. By plate’s boundary condition we can found the homologous homogeneous linear system of equations. Guaranteeing main mode to have nonzero solution, the homologous coefficient are not zero all. So we can get the vibration frequency equation, and ascertain plate’ s vibration frequency and homologous vibration .The plate of one side simply supported and one angular point supported not only possess simply supported edge , clamped edge, free elevator angle point , but also have angular point of post supported. During the plate vibrationcourse , angular point generate the homologize periodic change counterforce .At the same time the angular point displacement is zero , owing to boundary condition complexity, the plate ’ s dynamical analysis have some hardness. In this text we take arrangement method on the main mode direction’s dynamical analysis to successfully solve the question and acquire content result.The arrangement method on the main mode direction possess universal applicability. It not only work out classic boundary constituent rectangular plate’ s vibrate, but also dispose prsent plate of post supported angular point . In the text we take this method to have dynamical analysis for plate of one side simple supported and one post supported angular point , and procure the content outcome.更多还原