Perpetual points in natural mechanical systems with viscous damping: A theorem and a remark

2020 ◽  
pp. 095440622093483
Author(s):  
Fotios Georgiades
Keyword(s):  
Rigid Body ◽  
Mechanical Systems ◽  
Dissipative Systems ◽  
Viscous Damping ◽  
Ongoing Research ◽  
Rigid Body Motions ◽  
Gyroscopic Effects ◽  
Nonlinear Mechanical Systems ◽  
Stiffness And Damping ◽  
Perpetual Points

Perpetual points have been defined recently and their role in the dynamics of mechanical systems is ongoing research. In this article, the nature of perpetual points in natural dissipative mechanical systems with viscous damping, but excepting any externally applied load, is examined. In linear dissipative systems, a theorem and its inverse are proven stating that the perpetual points exist if the stiffness and damping matrices are positive semi-definite and they coincide with the rigid body motions. In nonlinear dissipative natural mechanical systems with viscous damping excepting any external load, the existence of perpetual points that are associated with rigid body motions is shown. Also, an additional type of perpetual points due to the added dissipation is shown that exists, and this type of perpetual points, at least in principle can be used for identification of dissipation in nonlinear mechanical systems. Further work is needed to understand the nature of this additional type of perpetual points. In all the examined examples the perpetual points when they exist, they are not just a few points, but they are forming manifolds in state space, the Perpetual Manifolds, and their geometric characteristics worth further investigation. The findings of this article are applied in all mechanical systems with no gyroscopic effects on their motion, e.g. cars, airplanes, trucks, rockets, robots, etc. and can be used as part of the elementary studies for basic design of all mechanical systems. This work paves the way for new design processes targeting stable rigid body motions eliminating any vibrations in mechanical systems.

scholarly journals Exact Augmented Perpetual Manifolds: Corollary about Different Mechanical Systems with Exactly the Same Motions

2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
Fotios Georgiades
Keyword(s):  
Differential Equation ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Mechanical Engineering ◽  
Initial Conditions ◽  
Mechanical Systems ◽  
Perfect Agreement ◽  
Ongoing Research ◽  
Rigid Body Motions ◽  
Perpetual Points

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.

scholarly journals Augmented Perpetual Manifolds and Perpetual Mechanical Systems-Part II: Theorem and a Corollary for Dissipative Mechanical Systems Behaving as Perpetual Machines

2021 ◽  
Author(s):  
FOTIOS GEORGIADES
Keyword(s):  
Rigid Body ◽  
Mechanical System ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Internal Stresses ◽  
State Variables ◽  
Internal Damage ◽  
Ongoing Research ◽  
External Forces ◽  
Perpetual Points

Abstract Perpetual points in mathematics defined recently, and their significance in nonlinear dynamics and their application in mechanical systems is currently ongoing research. The perpetual points significance relevant to mechanics so far is that they form the perpetual manifolds of rigid body motions of mechanical systems. The concept of perpetual manifolds extended to the definition of augmented perpetual manifolds that an externally excited multi-degree of freedom mechanical system is moving as a rigid body. As a continuation of this work, herein the internal force’s and their associated energies, for a motion of multi-degree of freedom dissipative flexible mechanical system with solutions in the exact augmented perpetual manifolds, leads to the proof of a theorem that based on a specific decomposition with respect to their state variables dependence, all the internal forcing vectors are equal to zero. Therefore there is no energy storage as potential energy, and the process is internally isentropic. This theorem provides the conditions that a mechanical system behaves as a perpetual machine of a 2nd and a 3rd kind. Then in a corollary, the behavior of a mechanical system as a perpetual machine of third kind further on is examined. The developed theory leads to a discussion for the conditions of the violation of the 2nd law of thermodynamics for mechanical systems that their motion is described in the exact augmented perpetual manifolds. Moreover, the necessity of a reversible process to violate the 2nd thermodynamics law, which is not valid, is shown. The findings of the theorem analytically and numerically are verified. The energies of a perpetual mechanical system in the exact augmented perpetual manifolds for two types of external forces have been determined. Then, in two examples of mechanical systems, all the analytical findings with numerical simulations, certified with excellent agreement. This work is essential in physics since the 2nd law of thermodynamics is not admitting internally isentropic processes in the dynamics of dissipative mechanical systems. In mechanical engineering, the mechanical systems operating in exact augmented perpetual manifolds, with zero internal forces, there is no degradation of any internal part of the machine due to zero internal stresses. Also, the operation of a machine in the exact augmented perpetual manifolds is of extremely high significance to avoid internal damage, and there is no energy loss.

Augmented Perpetual Manifolds and Perpetual Mechanical Systems-Part I: Definitions, Theorem and Corollary for Triggering Perpetual Manifolds, Application in Reduced Order Modeling and Particle-Wave Motion of Flexible Mechanical Systems

2021 ◽  
Author(s):  
Fotios Georgiades
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Mechanical Engineering ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Reduced Order ◽  
Rigid Body Motions ◽  
Definition Of ◽  
External Forces ◽  
Perpetual Points

Abstract Perpetual points in mechanical systems defined recently. Herein are used to seek specific types of solutions of N-degrees of freedom systems, and their significance in mechanics is discussed. In discrete linear mechanical systems, is proven, that the perpetual points are forming the perpetual manifolds and they are associated with rigid body motions, and these systems are called perpetual. The definition of perpetual manifolds herein is extended to the augmented perpetual manifolds. A theorem, defining the conditions of the external forces applied in an N-degrees of freedom system lead to a solution in the exact augmented perpetual manifold of rigid body motions, is proven. In this case, the motion by only one differential equation is described, therefore forms reduced-order modelling of the original equations of motion. Further on, a corollary is proven, that in the augmented perpetual manifolds for external harmonic force the system moves in dual mode as wave-particle. The developed theory is certified in three examples and the analytical solutions are in excellent agreement with the numerical simulations. The outcome of this research is significant in several sciences, in mathematics, in physics and in mechanical engineering. In mathematics, this theory is significant for deriving particular solutions of nonlinear systems of differential equations. In physics/mechanics, the existence of wave-particle motion of flexible mechanical systems is of substantial value. Finally in mechanical engineering, the theory in all mechanical structures can be applied, e.g. cars, aeroplanes, spaceships, boats etc. targeting only the rigid body motions.

Dunkerley-Mikhlin Estimates of Gravest Frequency of a Vibrating System

1976 ◽  
Vol 43 (2) ◽  
pp. 345-348 ◽  
Author(s):  
J. E. Brock
Keyword(s):  
Rigid Body ◽  
Mechanical Systems ◽  
Vibrating System ◽  
Rigid Body Motions

Estimates are made of the smallest nonzero frequency of vibration of undamped linear mechanical systems having lumped and/or distributed mass and permitting rigid body motions. The approximations are smaller than the correct values but remarkable accuracy may be achieved. The procedures are based upon methods of S. Dunkerley and S. G. Mikhlin.

Dynamics of Nonlinear Mechanical Systems and Virtual Reality: An Educational Tool for Engineering

1997 ◽  
Author(s):  
Ilmar F. Santos ◽  
Alexandre P. Ferretti ◽  
Eduardo Schmidek
Keyword(s):  
Nonlinear Systems ◽  
Rigid Body ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Double Pendulum ◽  
Pendulum System ◽  
Practical Applications ◽  
Nonlinear Mechanical ◽  
System Of Particles ◽  
Nonlinear Mechanical Systems

Abstract This work has significant practical applications in the area of Dynamics of nonlinear systems. Here, one of these applications is presented: a methodology based on Dynamics principles and axioms and animation techniques, which yield the visualization of the dynamic behavior of nonlinear systems. Three theoretical and experimental examples are shown, illustrating mechanical systems modeled as systems of particles, as a rigid body and as multibody systems. Three laboratory prototypes were created: a double pendulum (system of particles), a top (rigid body) and a satellite with internal rotors (multibody system). The purpose of comparing the motions of these prototypes with the ones obtained by numerical simulations is to make learning easier and to show the physical meaning of equations of motion obtained through Dynamics principles and axioms formulated by Newton, Euler, D’Alembert, Jourdain and Lagrange.

scholarly journals Application of physical function model to state estimations of nonlinear mechanical systems

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Heisei Yonezawa ◽  
Itsuro Kajiwara ◽  
Shota Sato ◽  
Chiaki Nishidome ◽  
Takashi Hatano ◽  
...  
Keyword(s):  
Physical Function ◽  
Mechanical Systems ◽  
Function Model ◽  
Nonlinear Mechanical ◽  
State Estimations ◽  
Nonlinear Mechanical Systems

Design and structural performance measurements of a novel multi-cantilever foil bearing

2014 ◽  
Vol 229 (10) ◽  
pp. 1830-1838 ◽  
Author(s):  
Kai Feng ◽  
Xueyuan Zhao ◽  
Zhiyang Guo
Keyword(s):  
Dynamic Characteristics ◽  
Dynamic Load ◽  
Dynamic Stiffness ◽  
Excitation Frequency ◽  
Viscous Damping ◽  
Equivalent Viscous Damping ◽  
Foil Bearing ◽  
Load Tests ◽  
Stiffness And Damping ◽  
Motion Amplitude

With increasing need for high-speed, high-temperature, and oil-free turbomachinery, gas foil bearings (GFBs) have been considered to be the best substitutes for traditional oil-lubricated bearings. A multi-cantilever foil bearing (MCFB), a novel GFB with multi-cantilever foil strips serving as the compliant underlying structure, was designed, fabricated, and tested. A series of static and dynamic load tests were conducted to measure the structural stiffness and equivalent viscous damping of the prototype MCFB. Experiments of static load versus deflection showed that the proposed bearing has a large mechanical energy dissipation capability and a pronounced nonlinear static stiffness that can prevents overly large motion amplitude of journal. Dynamic load tests evaluated the influence of motion amplitude, loading orientation and misalignment on the dynamic stiffness and equivalent viscous damping with respect to excitation frequency. The test results demonstrated that the dynamic stiffness and damping are strongly dependent on the excitation frequency. Three motion amplitudes were applied to the bearing housing to investigate the effects of motion amplitude on the dynamic characteristics. It is noted that the bearing dynamic stiffness and damping decreases with incrementally increasing motion amplitudes. A high level of misalignment can lead to larger static and dynamic bearing stiffness as well as to larger equivalent viscous damping. With dynamic loads applied to two orientations in the bearing midplane separately, the dynamic stiffness increases rapidly and the equivalent viscous damping declines slightly. These results indicate that the loading orientation is a non-negligible factor on the dynamic characteristics of MCFBs.

Analytical models for the rigid body motions of skew bridges

1987 ◽  
Vol 15 (8) ◽  
pp. 923-944 ◽  
Author(s):  
Emmanuel A. Maragakis ◽  
Paul C. Jennings
Keyword(s):  
Rigid Body ◽  
Analytical Models ◽  
Skew Bridges ◽  
Rigid Body Motions
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