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

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.

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

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

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.

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

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.

Hidden Attractors in a Model of a Bubble Contrast Agent Oscillating Near an Elastic Wall

A model describing the dynamics of a spherical gas bubble in a compressible viscous liquid is studied. The bubble is oscillating close to an elastic wall of finite thickness under the influence of an external pressure field which simulates a contrast agent oscillating close to a blood vessel wall. Here we investigate numerically the coexistence of chaotic and periodic attractors in this model. One of the tools applied for seeking coexisting attractors is the perpetual points method. This method can be helpful for localizing coexisting attractors, occurring in various physically realistic ranges of variation of the control parameters. We provide some examples of coexisting attractors to demonstrate the importance of the multistability problem for the applications.

Hyperchaos in a Conservative System with Nonhyperbolic Fixed Points

Chaotic dynamics exists in many natural systems, such as weather and climate, and there are many applications in different disciplines. However, there are few research results about chaotic conservative systems especially the smooth hyperchaotic conservative system in both theory and application. This paper proposes a five-dimensional (5D) smooth autonomous hyperchaotic system with nonhyperbolic fixed points. Although the proposed system includes four linear terms and four quadratic terms, the new system shows complicated dynamics which has been proven by the theoretical analysis. Several notable properties related to conservative systems and the existence of perpetual points are investigated for the proposed system. Moreover, its conservative hyperchaotic behavior is illustrated by numerical techniques including phase portraits and Lyapunov exponents.