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Exact Augmented Perpetual Manifolds: Corollary about Different Mechanical Systems with Exactly the Same Motions
Mathematical Problems in Engineering ( IF 1.430 ) Pub Date : 2021-09-10 , DOI: 10.1155/2021/6031142
Fotios Georgiades 1
Affiliation  

Perpetual points have been defined in mathematics recently, and they arise by setting accelerations and jerks equal to zero for nonzero velocities. The significance of perpetual points for the dynamics of mechanical systems is ongoing research. In the linear natural, unforced mechanical systems, the perpetual points form the perpetual manifolds and are associated with rigid body motions. Extending the definition of perpetual manifolds, by considering equal accelerations, in a forced mechanical system, but not necessarily zero, the solutions define the augmented perpetual manifolds. If the displacements are equal and the velocities are equal, the state space defines the exact augmented perpetual manifolds obtained under the conditions of a theorem, and a characteristic differential equation defines the solution. As a continuation of the theorem herein, a corollary proved that different mechanical systems, in the exact augmented perpetual manifolds, have the same general solution, and, in case of the same initial conditions, they have the same motion. The characteristic differential equation leads to a solution defining the augmented perpetual submanifolds and the solution of several types of characteristic differential equations derived. The theory in a few mechanical systems with numerical simulations is verified, and they are in perfect agreement. The theory developed herein is supplementing the already-developed theory of augmented perpetual manifolds, which is of high significance in mathematics, mechanics, and mechanical engineering. In mathematics, the framework for specific solutions of many degrees of freedom nonautonomous systems is defined. In mechanics/physics, the wave-particle motions are of significance. In mechanical engineering, some mechanical system’s rigid body motions without any oscillations are the ultimate ones.

中文翻译:

精确增强的永久流形:具有完全相同运动的不同机械系统的推论

最近在数学中定义了永点,它们是通过将非零速度的加速度和急动设置为零而产生的。永久点对机械系统动力学的重要性正在持续研究中。在线性自然、非受力机械系统中,永恒点形成永恒流形并与刚体运动相关联。扩展永久流形的定义,通过考虑相同加速度,在强制机械系统中,但不一定为零,解决方案定义了增广永久流形。如果位移相等且速度相等,则状态空间定义在定理条件下获得的精确增广永流形,特征微分方程定义解。作为这里定理的延续,推论证明了不同的机械系统,在精确的增广恒流形中,具有相同的通解,并且在相同的初始条件的情况下,它们具有相同的运动。特征微分方程导致定义增广永续子流形的解和导出的几种特征微分方程的解。对少数机械系统的理论进行了数值模拟验证,两者完全吻合。这里发展的理论是对已经发展的增广恒流形理论的补充,这在数学、力学和机械工程中具有重要意义。在数学中,定义了许多自由度非自治系统的特定解的框架。在力学/物理学中,波粒运动很重要。在机械工程中,一些机械系统没有任何振荡的刚体运动是极限运动。
更新日期:2021-09-10
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