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

Authors: Fotios Georgiades
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

Perpetual points (PPs) have been defined in mathematics recently [1]. The PPs of a dynamical system are obtained by setting accelerations and jerks of the equations of motion of a mechanical system equal to zero for nonzero velocities. Currently, there are four research directions relevant to the perpetual points. The first research direction, including experiments, is strictly relevant to developing the perpetual point theory [1–4]. The second research direction is to identify hidden and chaotic attractors [5–13]. The third research direction is to identify dissipative systems [14–18], and the fourth direction is about their use in mechanics [19–22]. In [19, 20] with proved theorems, the PPs in linear unforced natural mechanical systems are associated with rigid body modes/motions, and they form the perpetual manifolds.Moreover, the perpetual manifolds of some nonlinear unforced systems which are associated with rigid body motions in [19, 20] are shown. Based on the correlation of the PPs with the rigid body modes/motions, some new definitions of mechanical systems in [22] are derived. The mechanical systems that admit rigid body modes as solutions are called perpetual mechanical systems [22]. The perpetual manifold’s definition extended in [22] to the augmented perpetual manifolds defined by the state space of the solutions of external forced mechanical systems motions that all the accelerations are equal but not necessarily zero. The state spaces, with the solutions that all generalized coordinates are equal and the velocities are equal, are called exact augmented perpetual manifolds. Based on these new definitions, a theorem written in [21] and proved in [22] defines the conditions for exact augmented perpetual manifolds solutions of mechanical systems. A corollary proved that, in exact augmented perpetual manifolds of harmonic excitation, the N-DOF mechanical system’s motion is particle wave. The theorem is applied in three examples, where, in one of them, explicit solution of two types of external forces in the exact augmented perpetual manifolds is defined.There are significant research efforts in examining rigid body equations’ validity in modeling elastic structures, for example, [23, 24]. The examination of a viscoelastic bar with end loads targeting application in rails, including more significant terms in modeling and further in the nonlinear dynamic analysis, is done in [23]. Also, in [24], the plane and antiplane dynamic problems of an elastic rectangle, with asymptotic and exact dynamic analysis, were examined to identify the limitations of the rigid body formalism in elastic structures.This article continues the work done in [22] by proving a corollary arising from the theorem in [21, 22]. The explicit analytical form of the exact augmented perpetual manifolds for several external forces is then given.In the numerical section, to validate the analytical results, three examples are examined.The first example is a two-DOF mechanical system in translational motion. A shaft in torsional motion with a nonlinear energy sink attachment, examined through finite element model, is the second example. The third example is a five-degrees-of-freedom mechanical system.


## 2. Theory

In Section2.1, the theory developed in [22], relevant to the corollary, is presented. In Section 2.2, there are the statement and the proof of corollary. The last section, Section 2.3, is about the augmented perpetual submanifold’s explicit analytical forms for specific external forces.


### 2.1. Preliminary

In this section, preliminary definitions and theory, developed in other articles, are presented. The perpetual points have been defined recently as the sets of points that arise when the accelerations and jerks of the equations describing the motion of a mechanical system, for nonzero velocities, are set equal to zero [1].In cases where the perpetual points are not just a few points but infinite, they form the perpetual manifolds [20]. A mechanical system that admits rigid body motions as a solution is called a perpetual mechanical system [22], and the rigid body motions with perpetual manifolds [19, 20] are associated. The concept of perpetual manifolds in [22] extended to augmented perpetual manifolds, which are defined when all mechanical systems’ accelerations are equal to each other but not necessarily equal to zero [22]. The augmented perpetual manifolds of a mechanical system, where all the generalized velocities are equal and the generalized displacements are equal, form the exact augmented perpetual manifolds [22]. Based on the exact augmented perpetual manifolds definition, the following theorem stated in [21] and proved in [22] arose:AnyN≥2-degrees-of-freedom discrete mechanical system with generalized coordinates qit can be written as a perpetual mechanical system with external forcing that is described by the following system of differential equations:(1)Mi,jt,qlt,q˙mt×q¨it+Ci,j×q˙it+Ki,j×qit+FiNLqnt,q˙ot=Fit,qpt,q˙qtfori=1,…,N,j=1,…,N,l,m,n,o,p,q∈1,2,…,Nqit,q˙it,q¨it∈ℝ3,and it admits unique solutions for the following matrices: Mi,j is a real N×N inertia matrix with elements that can be nonsmooth, nonlinear, and time- and state-dependent functions but having at least one nonzero sum of k-row for all time instants, Ki,j and Ci,j, are real N×N constant, stiffness and proportional to velocity vector, matrices, FiNL is an N×1 vector of nonlinear internal forces with elements that can be state-dependent nonlinear functions which can be nonsmooth but single-valued for rigid body motions, and FiNLqs,0=0 for qs∈ℝ, Fi is a real N×1 vector of external forces with elements that can be time- and state-dependent and may be nonlinear and nonsmooth functions, if the external forces Fi with the reference k-inertia external force Fk are related as follows:(2)Fit,qat,q˙at=∑j=1NMi,jt,qat,q˙at⋅Fkt,qat,q˙at∑j=1NMk,jt,qat,q˙at,fori,k∈1,2,…,N,andqat=qit,q˙at=q˙it,and then the solution of any of the following differential equations,(3)q¨at=Fkt,qat,q˙at∑j=1NMk,jt,qat,q˙at=Gt,qat,q˙at,with vector field G, for the following set of initial conditions at the time instant t0,(4a)qit0=qat0,fori=1,…,N,andqat0∈ℝ,(4b)q˙it0=q˙at0,fori=1,…,Nandqat0∈ℝ,is defining the generalized coordinates qi and their velocities in the exact augmented perpetual manifold, Xa=t,qat,…,qat,q˙at,…,q˙at,t,qat,q˙at∈ℝ3 (see (5)) [21, 22].The basis of the exact augmented perpetual manifold is the augmented perpetual submanifoldA, and the following triple defines it [22]:(5)A=t,qat,q˙at∈ℝ3.In the next section, based on the theorem above, there is the corollary.


### 2.2. Corollary

Corollary 1: .
Two externally forced different discrete perpetual mechanical systems, with an exact augmented perpetual manifold which arises by the same vector field, have the same general solution in the exact augmented perpetual manifold. If they also have the same initial conditions, then the functions defining their motions are coinciding.Proof: .
The exact augmented perpetual manifold solution by equation (3) arises with a vector field Gt,qat,q˙at. Therefore, the two systems with the same vector field in equation (3) have the same general solution, and if they start their motion from the same initial conditions, they have the same functions defining their motion.
Therefore, equation (3) defines the exact augmented perpetual manifold as the characteristic differential equation of the exact augmented perpetual manifold and applies to any perpetual mechanical system. Unless otherwise stated, all perpetual mechanical systems, following the theorem requirements ending up to a solvable characteristic differential equation (3), have the same general solution.
In this corollary, the equations might be the same for angular and translational generalized coordinates, meaning that the two systems’ inertia elements do not have the same motion, but the same functions define their motion.


### 2.3. Several Types of Exact Augmented Perpetual Manifolds

In this section, several types of external forces that fulfil the theorem’s requirements are considered. Each system’s motion is in an exact augmented perpetual manifold, described by equation (3) solutions.In the exact augmented perpetual manifolds, the responses/velocities of all masses are the same, which means that theN-DOF system is moving like a particle with absolute synchronization of all displacements.A time-dependent vector field in the exact augmented perpetual manifolds is given by(6)q¨at=Gt,qat,q˙at=Fkt,qat,q˙at∑j=1NMk,jt,qat,q˙at=fktm,m∈ℝ≠0.In case of constant inertia matrix, equation (6) takes the following form:(7)q¨at=Fkt,qat,q˙at∑j=1NMk,j=fkt,qat,q˙atm=Gt,qat,q˙at,and the denominator in the right-hand side is nonzero and positive for natural mechanical systems.In case of a state-dependent inertia matrix, equation (6) for specific types of external forces takes the following form:(8)q¨at=Fkt,qat,q˙at∑j=1NMk,jt,qat,q˙at=σ⋅∑j=1NMk,jt,qat,q˙at⋅fkt,qat,q˙at∑j=1NMk,jt,qat,q˙at==σ⋅fkt,qat,q˙at=fkt,qat,q˙atm=Gt,qat,q˙at.and they are applied in kth−mass. The theorem’s validity requires that the rest of applied forces in all the other masses should be correlated through equation (2). In the case of a vector field given by the right-hand side of equation (6), equation (2) that defines the rest of forces takes the following form:(9)Fit,qat,q˙at=∑j=1NMi,jt,qat,q˙at∑j=1NMk,jt,qat,q˙at⋅Fkt,qat,q˙at==∑j=1NMi,jt,qat,q˙at⋅σ⋅fkt,qat,q˙at,for=i,k∈1,2,…,N.Six types of external forces are considered, and, in Table1, the explicit forms of the displacements and velocities, for each considered type, in the exact augmented perpetual manifolds are provided. Irrespective of their original system equations of motion, these functions are the same for all mechanical systems with the same external forces in the exact augmented perpetual manifolds.Table 1
The external forces, the analytical displacements, the velocities, and the wave velocities in selected exact augmented perpetual manifolds.
rVector field,Grt,qat,q˙atAnalytical velocity,q˙a,rtAnalytical displacement,qa,rtMotion directed wave velocitywvrt1η⋅t+c/m, (10)η/2⋅m⋅t2−t02+c/m⋅t−t0+q˙a,1t0 , [22],(17a)η/6⋅m⋅t3−t03+c/2⋅m⋅t2−t02−η⋅t02/2⋅m+c⋅t0/m−q˙a,1t0⋅t−t0+qa,1t0 , [22],(18a)−2Aex⋅sinωex⋅t+θex/m,(11)−Aex/m⋅ωex⋅cosωex⋅t+θex+Aex⋅cosωex⋅t0+θex/m⋅ωex+q˙a,2t0 [22],(17b)−Aex/m⋅ωex2⋅sinωex⋅t+θex+Aex⋅cosωex⋅t0+θex/m⋅ωex+q˙a,2t0⋅t−t0+Aex⋅sinωex⋅t0+θex/m⋅ωex2+qa,2t0, [22],(18b)Aex⋅cosωex⋅t0+θex/m⋅ωex+q˙a,2t0, [22],(19a)3∑i=1NfAex,i⋅sinωex,i⋅t+θex,i/m, (12a)−∑i=1NfAex,i/m⋅ωex,i⋅cosωex,i⋅t+θex,i+∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i+q˙a,3t0,(17c)−∑i=1NfAex,i/m⋅ωex,i2⋅sinωex,i⋅t+θex,i+∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i+q˙a,3t0⋅t−t0++∑i=1NfAex,i⋅sinωex,i⋅t0+θex,i/m⋅ωex,i2+qa,3t0,(18c)∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i+q˙a,3t0,(19b)4∑i=1NfAex,i⋅sinωex,i⋅t+θex,i/m++V¨t/m, (13a)−∑i=1NfAex,i/m⋅ωex,i⋅cosωex,i⋅t+θex,i+V˙t/m−V˙t0/m+∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i+q˙a,4t0,(17d)−∑i=1NfAex,i/m⋅ωex,i2⋅sinωex,i⋅t+θex,i+Vt/m+∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i−V˙t0/m+q˙a,4t0⋅t−t0−Vt0/m+∑i=1NfAex,i⋅sinωex,i⋅t0+θex,i/m⋅ωex,i2+qa,4t0,(18d)V˙t/m−V˙t0/m+∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i+q˙a,4t0,(19c)5cd⋅q˙at+FT/m, (14a)q˙a,5t0+FT/cd⋅ecd/m⋅t−t0−FT/cd,(17e)m⋅q˙a,5t0/cd+FT/cd2⋅ecd/m⋅t−t0−FT/cd⋅t−t0+qa,5t0−m⋅q˙a,5t0/cd−m⋅FT/cd2,(18e)−6−μkdq˙at⋅FkN⋅signq˙at/m, (15a)1/δ⋅C1−C2⋅t−t0−1>0,(16h)−2⋅C1−C2⋅t−t03/3⋅δ⋅C2−1/δ⋅t−t0+2⋅C13/3⋅δ⋅C2+qa,6,+t0, (16k)−−1/δ⋅C3−C2⋅t−t0−1<0, (16p)2⋅C3−C2⋅t−t03/3⋅δ⋅C2+1/δ⋅t−t0−2⋅C33/3⋅δ⋅C2+qa,6,−t0, (16r)−More precisely, each type of forcefkr leads to a solution qa,rt through equation (11) or (24) and the triple t,qa,r,q˙a,r forms the augmented submanifold Ar. Depending on the number of degrees of freedom, using the augmented submanifolds Ar as a basis, the exact augmented manifold X is formed.The considered six external forces of Table1 lead easily to an explicit solution of equation (6) as follows:(1)
Linear time varying forcesfk1t:(10)q¨a,1t=fk1tm=η⋅t+cm=G1t,withη,c∈ℝ2.The solution of equation (10) is defined in [22] and is shown in Table 1. This solution leads to the augmented perpetual submanifold A1 and is given by equations (a)–(f) in Table 1.(2)
Single frequency harmonic forcesfk2:(11)q¨a,2t=fk2tm=Aex⋅sinωex⋅t+θexm=G2t,withAex,θex∈ℝ2andωex∈ℝ≥0,whereAex is the amplitude, ωex and θex are the external frequency and phase, respectively, and the other forces should be in the form defined by equation (9). The solution in [22] is given and in Table 1 is presented. The augmented perpetual submanifold A2 is defined through the general solution given by equations (17b)–(18b) in Table 1.(3)
Harmonic forces withNf− frequencies Nf∈ℕ∗, fk3t:(12a)G3t=fk3tm=∑i=1NfAex,i⋅sinωex,i⋅t+θex,im,withAex,i,θex,i∈ℝ2⋅Nfand,ωex,i∈ℝ≥0Nf,considering that the other forces are following equation (6), in the exact augmented perpetual manifolds, the motion is described by(12b)q¨a,3t=∑i=1NfAex,i⋅sinωex,i⋅t+θex,im=G3t,andq˙a,3t and qa,3t are given similarly with single and double integrations of equation (12b), in time, respectively. The general solution of equation (12b) is shown in Table 1. The equations ((c)-(h) presented in Table 1) can be used to define the augmented perpetual submanifold ANf3.(4)
Harmonic forces withNf− frequencies combined with a function V¨t: for example, V˙t is monotonic for the considered time interval of motion, and the form of the force fk4t is given by(13a)G4t=fk4tm=∑i=1NfAex,i⋅sinωex,i⋅t+θex,i+V¨tm,in case that the other forces follow equation (9), the motion is described by(13b)q¨a,4t=∑i=1NfAex,i,sinωex,i⋅t+θex,i+V¨tm=G4t,and, similarly,q˙a,4t and qa,4t, with direct integration in time of equation (13b), are given. The augmented perpetual submanifold ANf,V¨3 is defined through equations (d)-(i) in Table 1.(5)
Aerodynamic and time-dependent forcefk5t,q˙k,5t:(14a)fk5t=cd⋅q˙k,5t+FTt,withcd,FT∈ℝ2.For negativecdcd∈ℝ<0, the force is aerodynamic drag, and, in case of positive cdcd∈ℝ>0, it is a flutter. For positive q˙k,5t, positive FT is a throttle or negative FT is a break and vice versa for negative q˙k,5t. The other forces for the validity of theorem in the exact augmented perpetual manifold must follow equation (9) and must be given by(14b)fi5t,q˙a,5t=∑j=1NMi,jt,qat,q˙atm⋅cd⋅q˙a,5t+FTt.Then, in the exact augmented perpetual manifolds, the solution is provided through equation (6) which takes the following form:(14c)q¨a,5t=cd⋅q˙a,5t+FTtm=G5t,q˙at.ForFT=ctFT∈ℝ≠0, the following change of variables,(14d)zt=q˙a,5t+FTcd⇒q¨a,5=z˙t=cdm⋅zt,leads to the general solution of equations (14c) and (14d):(14e)zt=zt0⋅ecd/m⋅t−t0.Once the velocitiesq˙a,5t, using equation (14d), are defined, the displacements qa,5t, with direct integration of equation (14e) in time, can be obtained, and their forms are in Table 1. Equations (e) and (j) in Table 1 are used to define the augmented perpetual submanifold A5.(6)
A special type of considered external forcefk6 is dry friction nonsmooth force in the following form [25]:(15a)fk6t,q˙k,6t=−μkdq˙k,6t⋅FkN⋅signq˙k,6t,where FkN∈ℝ>0, is the normal force to the motion and μkd is the dynamic friction coefficient which is dependent on velocity and is defined as in [25]:(15b)μkdq˙k,6t=μks1+δ⋅q˙k,6t,withμks,δ∈ℝ>02.In the exact augmented perpetual manifold, equation (15a), considering equation (15b), takes the following form:(15c)fk6t,q˙a,6t=−μks1+δ⋅q˙a,6t⋅FkN⋅signq˙a,6t.The other forcesfi6 in the exact augmented perpetual manifolds must satisfy equation (6) and must be in the following form:(15d)fi6t,q˙a,6t=∑j=1NMi,jt,qat,q˙atm⋅−μks1+δ⋅q˙a,6t⋅FkN⋅signq˙a,6t.The existence of the system’s solutions is ensured when convexification of the external forces (equations (15c) and (15d)) takes place in all the equations, and then the system will be a Fillipov system that has solutions [25, 26].In this case, the dynamics are described by equation (6), which takes the following explicit form:(15e)q¨a,6t=1m⋅−μks1+δ⋅q˙a,6t⋅FkN⋅signq˙a,6t=G6t,q˙at.A change of variables,(16a)wt=q˙a,6t,leads to(16b)w˙t=−1m⋅μks1+δ⋅wt⋅FkN⋅signwt,and, after convexification of the vector field, equation (16b) takes the following form:(16c)w˙t∈−1m⋅μks1+δ⋅wt⋅FkN,wt>0,1m⋅μks⋅FkN⋅−1,1,wt=0,1m⋅μks1−δ⋅wt⋅FkN,wt<0.Equation (16c) with differential inclusions forms a Fillipov system, and the solution arises by considering initial conditions in the three individual vector fields as follows.In case of positive initial velocities in the mechanical systemwt0∈ℝ>0, as long as the velocity is positive, the motion is described by the first vector field of equation (16c) and in explicit form is given by(16d)w˙+t=−1m⋅μks1+δ⋅w+t⋅FkN.Consider a change of variables,(16e)vt=1+δ⋅w+t,where vt≠0, and then equation (16d) takes the following form:(16f)v˙t=−δ⋅μks⋅FkNm⋅1vt,which can be solved with integration and leads to(16g)vt=v2t0−2⋅δ⋅μks⋅FkNm⋅t−t0,or taking into account equations (16a) and (16e) in equation (16g) leads to(16h)q˙a,6,+t=1δ⋅C1−C2⋅t−t0−1>0,forq˙a,6,+t∈ℝ>0,where(16i)C1=1+δ⋅q˙a,6t02,thereforeC1∈ℝ>0,(16j)C2=2⋅δ⋅μks⋅FkNm,thereforeC2∈ℝ>0,since all the involved parameters in equation (16j) are positive.The displacements can be obtained with direct integration of equation (16h), and they are given by(16k)qa,6,+t=−2⋅C1−C2⋅t−t033⋅δ⋅C2−1δ⋅t−t0+2⋅C133⋅δ⋅C2+qa,6,+t0.In case that the initial velocities of the mechanical system are negativewt0∈ℝ<0, as long as the velocity is negative, the motion is described by the third vector field of equation (16c) and in explicit form is given by(16l)w˙−t=1m⋅μks1−δ⋅w−t⋅FkN,and then consider the following change of variables:(16m)ut=1−δ⋅w−t,where ut≠0, and then equation (16l) takes the following form:(16n)u˙t=−δ⋅FkN⋅μksm⋅1ut,which after integration leads to(16o)ut=u2t0−2⋅δ⋅FkN⋅μksm⋅t−t0,or taking into account equations (16a) and (16m) in equation (16o) leads to(16p)q˙a,6,−t=−1δ⋅C3−C2⋅t−t0−1,forq˙a,6,−t∈ℝ<0,where(16q)C3=1−δ⋅q˙a,6,−t02.The displacements can be obtained with direct integration of equation (16p) and they are given by(16r)qa,6,−t=2⋅C3−C2⋅t−t033⋅δ⋅C2+1δ⋅t−t0−2⋅C333⋅δ⋅C2+qa,6,−t0.In any of the two described ways, the velocities tend to zero. Therefore, the three ways where the velocities are becoming zero are as follows:(i)
Through equation (16h)(ii)
Through equation (16p),(iii)
Through initial zero velocities, and then the system is remaining with zero velocity, and all the masses are in stick condition.The exact augmented perpetual submanifoldA6 is defined by the general solution of equations (16d) and (16l) which is given by equations (16h), (16k), (16p), and (16r).The selected types of forces are some representative types of external forces in the perpetual mechanical system, which, of course, by no means are the types of the external forcing limited.The external forces can be any combination of them, or any other type and nature, that fulfil equations (2) and (9). They can also be thermomechanical, electromechanical, or aerodynamic forces, and so forth. In such cases, the perpetual mechanical system’s state space forms an exact augmented perpetual manifold, and the motion in the case of constant inertia matrices is described by the solution of the differential equations (3) and (6).


## 2.1. Preliminary

In this section, preliminary definitions and theory, developed in other articles, are presented. The perpetual points have been defined recently as the sets of points that arise when the accelerations and jerks of the equations describing the motion of a mechanical system, for nonzero velocities, are set equal to zero [1].In cases where the perpetual points are not just a few points but infinite, they form the perpetual manifolds [20]. A mechanical system that admits rigid body motions as a solution is called a perpetual mechanical system [22], and the rigid body motions with perpetual manifolds [19, 20] are associated. The concept of perpetual manifolds in [22] extended to augmented perpetual manifolds, which are defined when all mechanical systems’ accelerations are equal to each other but not necessarily equal to zero [22]. The augmented perpetual manifolds of a mechanical system, where all the generalized velocities are equal and the generalized displacements are equal, form the exact augmented perpetual manifolds [22]. Based on the exact augmented perpetual manifolds definition, the following theorem stated in [21] and proved in [22] arose:AnyN≥2-degrees-of-freedom discrete mechanical system with generalized coordinates qit can be written as a perpetual mechanical system with external forcing that is described by the following system of differential equations:(1)Mi,jt,qlt,q˙mt×q¨it+Ci,j×q˙it+Ki,j×qit+FiNLqnt,q˙ot=Fit,qpt,q˙qtfori=1,…,N,j=1,…,N,l,m,n,o,p,q∈1,2,…,Nqit,q˙it,q¨it∈ℝ3,and it admits unique solutions for the following matrices: Mi,j is a real N×N inertia matrix with elements that can be nonsmooth, nonlinear, and time- and state-dependent functions but having at least one nonzero sum of k-row for all time instants, Ki,j and Ci,j, are real N×N constant, stiffness and proportional to velocity vector, matrices, FiNL is an N×1 vector of nonlinear internal forces with elements that can be state-dependent nonlinear functions which can be nonsmooth but single-valued for rigid body motions, and FiNLqs,0=0 for qs∈ℝ, Fi is a real N×1 vector of external forces with elements that can be time- and state-dependent and may be nonlinear and nonsmooth functions, if the external forces Fi with the reference k-inertia external force Fk are related as follows:(2)Fit,qat,q˙at=∑j=1NMi,jt,qat,q˙at⋅Fkt,qat,q˙at∑j=1NMk,jt,qat,q˙at,fori,k∈1,2,…,N,andqat=qit,q˙at=q˙it,and then the solution of any of the following differential equations,(3)q¨at=Fkt,qat,q˙at∑j=1NMk,jt,qat,q˙at=Gt,qat,q˙at,with vector field G, for the following set of initial conditions at the time instant t0,(4a)qit0=qat0,fori=1,…,N,andqat0∈ℝ,(4b)q˙it0=q˙at0,fori=1,…,Nandqat0∈ℝ,is defining the generalized coordinates qi and their velocities in the exact augmented perpetual manifold, Xa=t,qat,…,qat,q˙at,…,q˙at,t,qat,q˙at∈ℝ3 (see (5)) [21, 22].The basis of the exact augmented perpetual manifold is the augmented perpetual submanifoldA, and the following triple defines it [22]:(5)A=t,qat,q˙at∈ℝ3.In the next section, based on the theorem above, there is the corollary.


## 2.2. Corollary

Corollary 1: .
Two externally forced different discrete perpetual mechanical systems, with an exact augmented perpetual manifold which arises by the same vector field, have the same general solution in the exact augmented perpetual manifold. If they also have the same initial conditions, then the functions defining their motions are coinciding.Proof: .
The exact augmented perpetual manifold solution by equation (3) arises with a vector field Gt,qat,q˙at. Therefore, the two systems with the same vector field in equation (3) have the same general solution, and if they start their motion from the same initial conditions, they have the same functions defining their motion.
Therefore, equation (3) defines the exact augmented perpetual manifold as the characteristic differential equation of the exact augmented perpetual manifold and applies to any perpetual mechanical system. Unless otherwise stated, all perpetual mechanical systems, following the theorem requirements ending up to a solvable characteristic differential equation (3), have the same general solution.
In this corollary, the equations might be the same for angular and translational generalized coordinates, meaning that the two systems’ inertia elements do not have the same motion, but the same functions define their motion.


## 2.3. Several Types of Exact Augmented Perpetual Manifolds

In this section, several types of external forces that fulfil the theorem’s requirements are considered. Each system’s motion is in an exact augmented perpetual manifold, described by equation (3) solutions.In the exact augmented perpetual manifolds, the responses/velocities of all masses are the same, which means that theN-DOF system is moving like a particle with absolute synchronization of all displacements.A time-dependent vector field in the exact augmented perpetual manifolds is given by(6)q¨at=Gt,qat,q˙at=Fkt,qat,q˙at∑j=1NMk,jt,qat,q˙at=fktm,m∈ℝ≠0.In case of constant inertia matrix, equation (6) takes the following form:(7)q¨at=Fkt,qat,q˙at∑j=1NMk,j=fkt,qat,q˙atm=Gt,qat,q˙at,and the denominator in the right-hand side is nonzero and positive for natural mechanical systems.In case of a state-dependent inertia matrix, equation (6) for specific types of external forces takes the following form:(8)q¨at=Fkt,qat,q˙at∑j=1NMk,jt,qat,q˙at=σ⋅∑j=1NMk,jt,qat,q˙at⋅fkt,qat,q˙at∑j=1NMk,jt,qat,q˙at==σ⋅fkt,qat,q˙at=fkt,qat,q˙atm=Gt,qat,q˙at.and they are applied in kth−mass. The theorem’s validity requires that the rest of applied forces in all the other masses should be correlated through equation (2). In the case of a vector field given by the right-hand side of equation (6), equation (2) that defines the rest of forces takes the following form:(9)Fit,qat,q˙at=∑j=1NMi,jt,qat,q˙at∑j=1NMk,jt,qat,q˙at⋅Fkt,qat,q˙at==∑j=1NMi,jt,qat,q˙at⋅σ⋅fkt,qat,q˙at,for=i,k∈1,2,…,N.Six types of external forces are considered, and, in Table1, the explicit forms of the displacements and velocities, for each considered type, in the exact augmented perpetual manifolds are provided. Irrespective of their original system equations of motion, these functions are the same for all mechanical systems with the same external forces in the exact augmented perpetual manifolds.Table 1
The external forces, the analytical displacements, the velocities, and the wave velocities in selected exact augmented perpetual manifolds.
rVector field,Grt,qat,q˙atAnalytical velocity,q˙a,rtAnalytical displacement,qa,rtMotion directed wave velocitywvrt1η⋅t+c/m, (10)η/2⋅m⋅t2−t02+c/m⋅t−t0+q˙a,1t0 , [22],(17a)η/6⋅m⋅t3−t03+c/2⋅m⋅t2−t02−η⋅t02/2⋅m+c⋅t0/m−q˙a,1t0⋅t−t0+qa,1t0 , [22],(18a)−2Aex⋅sinωex⋅t+θex/m,(11)−Aex/m⋅ωex⋅cosωex⋅t+θex+Aex⋅cosωex⋅t0+θex/m⋅ωex+q˙a,2t0 [22],(17b)−Aex/m⋅ωex2⋅sinωex⋅t+θex+Aex⋅cosωex⋅t0+θex/m⋅ωex+q˙a,2t0⋅t−t0+Aex⋅sinωex⋅t0+θex/m⋅ωex2+qa,2t0, [22],(18b)Aex⋅cosωex⋅t0+θex/m⋅ωex+q˙a,2t0, [22],(19a)3∑i=1NfAex,i⋅sinωex,i⋅t+θex,i/m, (12a)−∑i=1NfAex,i/m⋅ωex,i⋅cosωex,i⋅t+θex,i+∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i+q˙a,3t0,(17c)−∑i=1NfAex,i/m⋅ωex,i2⋅sinωex,i⋅t+θex,i+∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i+q˙a,3t0⋅t−t0++∑i=1NfAex,i⋅sinωex,i⋅t0+θex,i/m⋅ωex,i2+qa,3t0,(18c)∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i+q˙a,3t0,(19b)4∑i=1NfAex,i⋅sinωex,i⋅t+θex,i/m++V¨t/m, (13a)−∑i=1NfAex,i/m⋅ωex,i⋅cosωex,i⋅t+θex,i+V˙t/m−V˙t0/m+∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i+q˙a,4t0,(17d)−∑i=1NfAex,i/m⋅ωex,i2⋅sinωex,i⋅t+θex,i+Vt/m+∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i−V˙t0/m+q˙a,4t0⋅t−t0−Vt0/m+∑i=1NfAex,i⋅sinωex,i⋅t0+θex,i/m⋅ωex,i2+qa,4t0,(18d)V˙t/m−V˙t0/m+∑i=1NfAex,i⋅cosωex,i⋅t0+θex,i/m⋅ωex,i+q˙a,4t0,(19c)5cd⋅q˙at+FT/m, (14a)q˙a,5t0+FT/cd⋅ecd/m⋅t−t0−FT/cd,(17e)m⋅q˙a,5t0/cd+FT/cd2⋅ecd/m⋅t−t0−FT/cd⋅t−t0+qa,5t0−m⋅q˙a,5t0/cd−m⋅FT/cd2,(18e)−6−μkdq˙at⋅FkN⋅signq˙at/m, (15a)1/δ⋅C1−C2⋅t−t0−1>0,(16h)−2⋅C1−C2⋅t−t03/3⋅δ⋅C2−1/δ⋅t−t0+2⋅C13/3⋅δ⋅C2+qa,6,+t0, (16k)−−1/δ⋅C3−C2⋅t−t0−1<0, (16p)2⋅C3−C2⋅t−t03/3⋅δ⋅C2+1/δ⋅t−t0−2⋅C33/3⋅δ⋅C2+qa,6,−t0, (16r)−More precisely, each type of forcefkr leads to a solution qa,rt through equation (11) or (24) and the triple t,qa,r,q˙a,r forms the augmented submanifold Ar. Depending on the number of degrees of freedom, using the augmented submanifolds Ar as a basis, the exact augmented manifold X is formed.The considered six external forces of Table1 lead easily to an explicit solution of equation (6) as follows:(1)
Linear time varying forcesfk1t:(10)q¨a,1t=fk1tm=η⋅t+cm=G1t,withη,c∈ℝ2.The solution of equation (10) is defined in [22] and is shown in Table 1. This solution leads to the augmented perpetual submanifold A1 and is given by equations (a)–(f) in Table 1.(2)
Single frequency harmonic forcesfk2:(11)q¨a,2t=fk2tm=Aex⋅sinωex⋅t+θexm=G2t,withAex,θex∈ℝ2andωex∈ℝ≥0,whereAex is the amplitude, ωex and θex are the external frequency and phase, respectively, and the other forces should be in the form defined by equation (9). The solution in [22] is given and in Table 1 is presented. The augmented perpetual submanifold A2 is defined through the general solution given by equations (17b)–(18b) in Table 1.(3)
Harmonic forces withNf− frequencies Nf∈ℕ∗, fk3t:(12a)G3t=fk3tm=∑i=1NfAex,i⋅sinωex,i⋅t+θex,im,withAex,i,θex,i∈ℝ2⋅Nfand,ωex,i∈ℝ≥0Nf,considering that the other forces are following equation (6), in the exact augmented perpetual manifolds, the motion is described by(12b)q¨a,3t=∑i=1NfAex,i⋅sinωex,i⋅t+θex,im=G3t,andq˙a,3t and qa,3t are given similarly with single and double integrations of equation (12b), in time, respectively. The general solution of equation (12b) is shown in Table 1. The equations ((c)-(h) presented in Table 1) can be used to define the augmented perpetual submanifold ANf3.(4)
Harmonic forces withNf− frequencies combined with a function V¨t: for example, V˙t is monotonic for the considered time interval of motion, and the form of the force fk4t is given by(13a)G4t=fk4tm=∑i=1NfAex,i⋅sinωex,i⋅t+θex,i+V¨tm,in case that the other forces follow equation (9), the motion is described by(13b)q¨a,4t=∑i=1NfAex,i,sinωex,i⋅t+θex,i+V¨tm=G4t,and, similarly,q˙a,4t and qa,4t, with direct integration in time of equation (13b), are given. The augmented perpetual submanifold ANf,V¨3 is defined through equations (d)-(i) in Table 1.(5)
Aerodynamic and time-dependent forcefk5t,q˙k,5t:(14a)fk5t=cd⋅q˙k,5t+FTt,withcd,FT∈ℝ2.For negativecdcd∈ℝ<0, the force is aerodynamic drag, and, in case of positive cdcd∈ℝ>0, it is a flutter. For positive q˙k,5t, positive FT is a throttle or negative FT is a break and vice versa for negative q˙k,5t. The other forces for the validity of theorem in the exact augmented perpetual manifold must follow equation (9) and must be given by(14b)fi5t,q˙a,5t=∑j=1NMi,jt,qat,q˙atm⋅cd⋅q˙a,5t+FTt.Then, in the exact augmented perpetual manifolds, the solution is provided through equation (6) which takes the following form:(14c)q¨a,5t=cd⋅q˙a,5t+FTtm=G5t,q˙at.ForFT=ctFT∈ℝ≠0, the following change of variables,(14d)zt=q˙a,5t+FTcd⇒q¨a,5=z˙t=cdm⋅zt,leads to the general solution of equations (14c) and (14d):(14e)zt=zt0⋅ecd/m⋅t−t0.Once the velocitiesq˙a,5t, using equation (14d), are defined, the displacements qa,5t, with direct integration of equation (14e) in time, can be obtained, and their forms are in Table 1. Equations (e) and (j) in Table 1 are used to define the augmented perpetual submanifold A5.(6)
A special type of considered external forcefk6 is dry friction nonsmooth force in the following form [25]:(15a)fk6t,q˙k,6t=−μkdq˙k,6t⋅FkN⋅signq˙k,6t,where FkN∈ℝ>0, is the normal force to the motion and μkd is the dynamic friction coefficient which is dependent on velocity and is defined as in [25]:(15b)μkdq˙k,6t=μks1+δ⋅q˙k,6t,withμks,δ∈ℝ>02.In the exact augmented perpetual manifold, equation (15a), considering equation (15b), takes the following form:(15c)fk6t,q˙a,6t=−μks1+δ⋅q˙a,6t⋅FkN⋅signq˙a,6t.The other forcesfi6 in the exact augmented perpetual manifolds must satisfy equation (6) and must be in the following form:(15d)fi6t,q˙a,6t=∑j=1NMi,jt,qat,q˙atm⋅−μks1+δ⋅q˙a,6t⋅FkN⋅signq˙a,6t.The existence of the system’s solutions is ensured when convexification of the external forces (equations (15c) and (15d)) takes place in all the equations, and then the system will be a Fillipov system that has solutions [25, 26].In this case, the dynamics are described by equation (6), which takes the following explicit form:(15e)q¨a,6t=1m⋅−μks1+δ⋅q˙a,6t⋅FkN⋅signq˙a,6t=G6t,q˙at.A change of variables,(16a)wt=q˙a,6t,leads to(16b)w˙t=−1m⋅μks1+δ⋅wt⋅FkN⋅signwt,and, after convexification of the vector field, equation (16b) takes the following form:(16c)w˙t∈−1m⋅μks1+δ⋅wt⋅FkN,wt>0,1m⋅μks⋅FkN⋅−1,1,wt=0,1m⋅μks1−δ⋅wt⋅FkN,wt<0.Equation (16c) with differential inclusions forms a Fillipov system, and the solution arises by considering initial conditions in the three individual vector fields as follows.In case of positive initial velocities in the mechanical systemwt0∈ℝ>0, as long as the velocity is positive, the motion is described by the first vector field of equation (16c) and in explicit form is given by(16d)w˙+t=−1m⋅μks1+δ⋅w+t⋅FkN.Consider a change of variables,(16e)vt=1+δ⋅w+t,where vt≠0, and then equation (16d) takes the following form:(16f)v˙t=−δ⋅μks⋅FkNm⋅1vt,which can be solved with integration and leads to(16g)vt=v2t0−2⋅δ⋅μks⋅FkNm⋅t−t0,or taking into account equations (16a) and (16e) in equation (16g) leads to(16h)q˙a,6,+t=1δ⋅C1−C2⋅t−t0−1>0,forq˙a,6,+t∈ℝ>0,where(16i)C1=1+δ⋅q˙a,6t02,thereforeC1∈ℝ>0,(16j)C2=2⋅δ⋅μks⋅FkNm,thereforeC2∈ℝ>0,since all the involved parameters in equation (16j) are positive.The displacements can be obtained with direct integration of equation (16h), and they are given by(16k)qa,6,+t=−2⋅C1−C2⋅t−t033⋅δ⋅C2−1δ⋅t−t0+2⋅C133⋅δ⋅C2+qa,6,+t0.In case that the initial velocities of the mechanical system are negativewt0∈ℝ<0, as long as the velocity is negative, the motion is described by the third vector field of equation (16c) and in explicit form is given by(16l)w˙−t=1m⋅μks1−δ⋅w−t⋅FkN,and then consider the following change of variables:(16m)ut=1−δ⋅w−t,where ut≠0, and then equation (16l) takes the following form:(16n)u˙t=−δ⋅FkN⋅μksm⋅1ut,which after integration leads to(16o)ut=u2t0−2⋅δ⋅FkN⋅μksm⋅t−t0,or taking into account equations (16a) and (16m) in equation (16o) leads to(16p)q˙a,6,−t=−1δ⋅C3−C2⋅t−t0−1,forq˙a,6,−t∈ℝ<0,where(16q)C3=1−δ⋅q˙a,6,−t02.The displacements can be obtained with direct integration of equation (16p) and they are given by(16r)qa,6,−t=2⋅C3−C2⋅t−t033⋅δ⋅C2+1δ⋅t−t0−2⋅C333⋅δ⋅C2+qa,6,−t0.In any of the two described ways, the velocities tend to zero. Therefore, the three ways where the velocities are becoming zero are as follows:(i)
Through equation (16h)(ii)
Through equation (16p),(iii)
Through initial zero velocities, and then the system is remaining with zero velocity, and all the masses are in stick condition.The exact augmented perpetual submanifoldA6 is defined by the general solution of equations (16d) and (16l) which is given by equations (16h), (16k), (16p), and (16r).The selected types of forces are some representative types of external forces in the perpetual mechanical system, which, of course, by no means are the types of the external forcing limited.The external forces can be any combination of them, or any other type and nature, that fulfil equations (2) and (9). They can also be thermomechanical, electromechanical, or aerodynamic forces, and so forth. In such cases, the perpetual mechanical system’s state space forms an exact augmented perpetual manifold, and the motion in the case of constant inertia matrices is described by the solution of the differential equations (3) and (6).


## 3. Numerical Results

The theory developed in the previous section in three examples of mechanical systems is applied. In the first example of a 2-degrees-of-freedom (DOF) system, the four external forces (3–6 of Table1), with solutions determined in this article, are applied. In the second example of a shaft in torsional vibration with torsional nonlinear energy sink, modeled with finite element structural matrices, the first two types of forces with the analytical solution of [22] are applied, for the theory validation in many-degrees-of-freedom system. Finally, in the third example, a 5-DOF mechanical system, the first two types of external forces of the 2-DOF mechanical system are applied for comparison with the previous section’s results.


### 3.1. A Two-Degrees-of-Freedom System

In this section, an example of a 2-DOF mechanical system is examined. Figure1 shows the 2-DOF mechanical system in translational motion, and the equations of motion have the following form:(20)mx,0+x˙200my,0+y˙2×x¨y¨+k1−k1−k1k1×xy+c1−c1−c1c1×x˙y˙+k2⋅x−y3+c2⋅tanhb⋅x˙−y˙k2⋅y−x3+c2⋅tanhb⋅y˙−x˙=Fxt,x,y,x˙,y˙Fyt,x,y,x˙,y˙.Figure 1
Configuration of the 2-DOF mechanical system.The system is comprised of two masses (mx and my) that are velocity-dependent, coupled through linear (k1) and nonlinear springs (k2) and also through linear dashpot (c1) and nonlinear damping force (c2). In each, mx and my mass friction forces and external forces fx and fy are applied.The associated perpetual mechanical system without external forces is a smooth nonlinear system with unique solutions, but the existence of the solution of equations (20) depends on the external force’s form on the equation’s right-hand side. The theorem’s first condition is to examine if this mechanical system is a perpetual mechanical system. Initially, the unforced mechanical system has to be considered and eliminating the external forces (including the friction forces) in equations (20) leads to(21)mx,0+x˙200my,0+y˙2×x¨y¨+k1−k1−k1k1×xy+c1−c1−c1c1×x˙y˙+k2⋅x−y3+c2⋅tanhb⋅x˙−y˙k2⋅y−x3+c2⋅tanhb⋅y˙−x˙=00.In case of exact rigid body motions,(22a)xa=ya,(22b)x˙a=y˙a,and replacing equations (22a) and (22b) in the system of equations (21) leads to(23a)mx,0+x˙a2⋅x¨a=0⟺x¨a=0,(23b)my,0+y˙a2⋅y¨a=0⟺y¨a=0.The equations of jerks arise with direct differentiation of equations (21) in time, and they are given by(24)2⋅x˙⋅x¨002⋅y˙⋅y¨×x¨y¨+mx,0+x˙200my,0+y˙2×x⃛y⃛+k1−k1−k1k1×x˙y˙+c1−c1−c1c1×x¨y¨+3⋅k2⋅x−y2⋅x˙−y˙+b⋅c2⋅sech2b⋅x˙−y˙⋅x¨−y¨3⋅k2⋅y−x2⋅y˙−x˙+b⋅c2⋅sech2b⋅y˙−x˙⋅y¨−x¨=00.Replacing equations (22a), (22b), (23a), and (23b) in equation (24) leads to(25a)mx,0+x˙2⋅x⃛a=0⇔x⃛a=0,(25b)my,0+y˙2⋅y⃛a=0⇔y⃛a=0.The accelerations and jerks with exact rigid body motions are equal to zero, and, therefore, the system is a perpetual mechanical system.The second condition of the theorem is that the external forcing should comply with equation (10), where, in the exact augmented perpetual manifolds (EAPMs) of this system, it is given by(26a)∑j=12M1,jx˙a=mx,0+x˙a2,(26b)∑j=12M2,jx˙a=my,0+x˙a2,and considering equations (26a) and (26b) with equation (9) leads to(27)Fxt,xa,xa,x˙a,x˙amxx˙a=Fyt,xa,xa,x˙a,x˙amyx˙a⟺Fyt,xa,xa,x˙a,x˙a=my,0+x˙a2mx,0+x˙a2⋅Fxt,xa,xa,x˙a,x˙a,with reference mass mx.The analytical solution of the external forces applied to this system is derived in Section2.3, and it is given in Table 1. The type of vector field arising from the characteristic differential equation and the form of the external forces and the type of motion for each time interval are given in Table 2. Moreover, the equations associated with the analytical solutions, the wave velocity equations, and the associated type of motion are indicated in Table 2.Table 2
The external forces, the solutions, and the motions in EAPMs of the 2-DOF system.
iTime interval (s)GiFxi=fx+fd,xAnalytical velocityx˙a,iAnalytical displacementxa,iLongitudinal wave velocitywviType of motion1t∈0,1G23fx=σ1⋅mx,0+x˙a2⋅fk,23Equation (17c)Equation (18c)Equation (19b)Particle linear-two frequencies standing wave (using specific amplitudes)fd,x=02t∈1,2G1,fk14fx=σ2⋅mx,0+x˙a2⋅fk3+fk1Equation (17d)Equation (18d)Equation (19c)Particle-longitudinal wavefd,x=03t∈2,3G6fy=0Equation (16h)Equation (16k)—∗Particle decelerating (dry friction)fd,x=σ3⋅mx,0+x˙a2⋅fk64t∈3,4G5fx=σ4⋅mx,0+x˙a2⋅fk5Equation (17e)Equation (18e)—∗Particle exponential accelerating (throttle with flutter)fd,x=05t∈4,5G5fx=σ5⋅mx,0+x˙a2⋅fk5Equation (17e)Equation (18e)—∗Particle exponential decelerating (throttle with drag)fd,x=06t∈5,6G6fy=0Equation (16p)Equation (16r)—∗Particle decelerating (dry friction)fd,x=σ3⋅mx,0+x˙a2⋅fk6∗Not expected for this type of motion.The existence of nonsmooth forces (in the third- and sixth-time intervals) requires preliminary work in developing the numerical scheme based on the switch model algorithm of [26], and herein the algorithm developed in [22] is used.The chosen parameters for this example correspond to a car withmx,0=2000kg dragging a van with mass my,0=1000kg. The coupler has linear stiffness k1=106N/m and nonlinear stiffness k2=5⋅105N/m3. In the two-degrees-of-freedom-associated linear system, the linear natural frequencies ωi and mode shapes Vi areω1=0rad/s with normalized mode shape V1=0.8160.577,ω2=38.73rad/s with normalized mode shape V2=−0.5770.816.Therefore, when the second mode is excited, the two masses are vibrating in out-of-phase fashion, and although the main frequency of harmonic external forces would be this one of the second mode, this mode will not be excited due to the same direction of external forces (same sign as in equation (2)).Considering 1% damping ratio for the second mode, the damping coefficient isc1=516.398N⋅s/m and the nonlinear damping coefficient has same value c2=516.398N with b=106s/m (significantly high to approximate dry friction). In the developed theory, damping does not play any role, but the considered values are sensibly high enough, mainly for illustration purposes.The external forcing parameters are shown in Table3, with the associated initial conditions and the wave velocity for each type of external forcing.Table 3
The initial conditions, the external forcing parameters, and the wave velocities for each time interval of the 2-DOF system.
iTime interval (s)ICsExternal forcing parameterswvi (m/s)x˙a,it0 (m/s)xa,it0 (m)1t∈0,10.0000000.000000σ1=0.10Nf=2Aex,1=Aex=−5⋅105Nωex,1=ω2=38.73rad/sθex,1=0radAex,2=1⋅106Nωex,2=2⋅ω2=77.46rad/sθex,2=0rad2t∈1,21272.33096213.893890σ2=0.2500⋅t2−1.962⋅t−556.915η=5000N/sc=9.81NNf=1Aex,1ωex,1θex,13t∈2,3229.631038628.195850σ3=0.5−=μxs⋅FxNμxs⋅mx,0⋅g=1962N∗δ=3 s/mmy,0/mx,0⋅μxs⋅FxN=μxs⋅my,0⋅g=μxs⋅FyN∗4t∈3,4228.204655857.114437σ4=0.3−cd=3N⋅s/m,>0F1T=1000N5t∈4,51047.8272511434.472876σ5=0.4−cd=−5N⋅s/m<0F2T=−1000N6t∈5,6−31.1249461773.948974σ3=0.5−μxs⋅FxN=μxs⋅mx,0⋅g=1962N∗δ=3 s/mmy,0/mx,0⋅μxs⋅FxN=μxs⋅my,0⋅g=μxs⋅FyN∗∗In this example, without losing the generality and simplifying the frictional terms, the state-dependent inertia elements are considered which are not correlated with the weight of the vehicles.All the numerical simulations in this section, with 64-bit Scilab 6.1.0 [27] using “Adams” solver with time step dt=5⋅10−4 s and relative and absolute tolerance 6⋅10−14, have been performed. The parameter for the convexification of the vector field in the application of the switch algorithm of [22] is τ=dt⋅10−3=5⋅10−7 s.In Figures2(a)–2(f), the displacements xa obtained from theoretical solution in EAPMs (equations in Table 1) and those obtained from numerical simulations of the system with equations (20) are depicted, and it seems that they are in good agreement. This can be certified further by examining the maxima’s maximum of the absolute difference between the analytical and numerical displacements, which has the minimal value of 1.875 10−8 m.Figure 2
Numerically determinedx,y and analytically determined displacements xa of the 2-DOF mechanical system: (a) for the first time interval 0-1 s, with external forces fk,23; (b) for the 2nd time interval 1-2 s, with external forces fk3 and fk1; (c) for the 3rd time interval 2-3 s, with external forces fk6; (d) for the 4th time interval 3-4 s, with external forces fk5; (e) for the 5th time interval 4-5 s, with external forces fk5; and (f) for the 6th time interval with external forces fk6.
(a)(b)(c)(d)(e)(f)In Figure2(a), the displacements on the first time interval are depicted. The motion of the mechanical system is a particle standing wave with two frequencies (Nf=2), defined by equation (h) in Table 1, and it is associated with the vector field G23. In Figure 2(b), the displacements on the second time interval are depicted; the motion is a particle-longitudinal wave and is defned by equation (i) in Table 1. It is associated with the vector field G1,fk14, which indicates harmonic forcing with a single frequency combined with the associated vector field G1. The displacements, on the third time interval in Figure 2(c), are depicted, associated with the vector field G6 for positive velocities, and they are analytically determined by equation (16k). The motion is particle decelerating due to dry friction components. In Figures 2(d)–2(e), the displacements on the fourth and fifth time intervals are depicted, and both are associated with the vector field G5. They are analytically determined by equation (j) in Table 1. In the fourth time interval (Figure 2(d)), the positive value of cd is associated with flutter, which indicates particle-accelerating motion, and in the fifth time interval (Figure 2(e)), the negative value of cd is associated with drag, which indicates particle-decelerating motion. Finally, in Figure 3(f), the displacements on the sixth time interval are depicted, and the motion is decelerating, and it is associated with the vector field G6 for negative velocities, with displacements analytically determined by equation (16r).Figure 3
Numerically determinedx˙,y˙ and analytically determined velocities x˙a of the 2-DOF mechanical system: (a) for the first time interval 0-1 s, with external forces fk,23; (b) for the 2nd time interval 1-2 s, with external forces fk3 and fk1; (c) for the 3rd time interval 2-3 s, with external forces fk6; (d) for the 4th time interval 3-4 s, with external forces fk5; (e) for the 5th time interval 4-5 s, with external forces fk5; and (f) for the 6th time interval, with external forces fk6.
(a)(b)(c)(d)(e)(f)In Figures3(a)–3(f), the velocities obtained from the numerical simulations of equations (20) incorporating the analytical solutions x˙a are depicted, and it seems that they are in very good agreement. Moreover, this is certified by examining the maximum of the maxima of the absolute difference between the analytical and numerically determined velocities, which has the minimal value of 1.213 10−8 m/s.In Figure3(a), the velocities on the first time interval are depicted, with the analytical ones determined by equation (a) in Table 1, and they are associated with the vector field G23. Figure 3(b) depicts the velocities on the second time interval, and their increasing values indicate the nonzero wave velocity. The analytical velocity by equation (d) in Table 1 is given, and it is associated with the vector field G1,fk14.The decreasing velocity on the third time interval in Figure3(c) is depicted, and the analytical solution by equation (16h) is determined, which is associated with the vector field G6. The increasing velocities for the fourth time interval depicted in Figure 3(d) are associated with flutter. Equation (e) in Table 1 defines the analytical solution, and it is associated with the vector field G5. The decreasing velocities in the fifth time interval depicted in Figure 3(e) are associated with the vector field G5, for aerodynamic drag, and the analytical solution by equation (e) in Table 1 is obtained. Finally, in Figure 3(f), the velocities of the sixth time interval are depicted. They are decreasing due to dry friction, and they are associated with the vector field G6. The analytical solution of the velocity on the sixth time interval by equation (16p) is determined.The basis of each exact augmented perpetual manifold is the augmented perpetual submanifolds defined by equation (5), and it is associated with each vector field Gi of the characteristic differential equation (6). In Figures 4(a)–4(f), the augmented perpetual submanifolds, defined by the analytical solutions for each time interval, are depicted. In Figure 4(a), the shape of the augmented perpetual submanifold−A2,03 of the vector field−G24, defined by the analytical solution on the first time interval, is depicted with the first subscript indicating the number of frequencies and the second subscript indicating the wave velocity.Figure 4
Exact augmented perpetual submanifoldAr: (a) A2,03 that is associated with vector field G23, (b) A1,fk14 that is associated with the vector field G1,fk14, (c) A+6 that is associated with the vector field G6 for positive velocities, (d) A+5 that is associated with the vector field G5 for flutter, (e) A−5 that is associated with the vector field G5 for aerodynamic drag, and (f) A−6 that is associated with the vector field G6 for negative velocities.
(a)(b)(c)(d)(e)(f)In Figure4(b), using the analytical solution on the second time interval, the exact augmented perpetual submanifold−A1,fk14 that is associated with the vector field G1,fk14 is depicted, with the first subscript indicating the number of frequencies and the second subscript indicating the type of the nonharmonic function. In Figure 4(c), the exact augmented perpetual submanifold−A+6 that is associated with the vector field G6, for positive velocities indicated by the plus sign in the subscript, is depicted. The exact augmented perpetual submanifold−A+5 that is associated with the vector field G5 for flutter is depicted in Figure 4(d). In the case of aerodynamic drag, the exact augmented perpetual submanifold−A−5 associated with the vector field G5 is depicted in Figure 4(e). Finally, the exact augmented perpetual submanifold−A−6 that is associated with the vector field G6 for negative velocities is depicted in Figure 4(f).


### 3.2. A Shaft in Torsional Vibration with Torsional Nonlinear Energy Sink

In this example, a discretized, through finite element formulation, shaft in torsional vibration with free boundary condition in one end and dynamic boundary condition in the other end coupled with a nonlinear energy sink (NES) is considered, as indicated in Figure5. The shaft is of length Ls, with internal and external diameters of Di and of Do, respectively. The density is denoted as ρ and the shear modulus is denoted as G. The shaft with inertia I is coupled with an NES mass, Ines, through a linear torsional spring with stiffness kl,nes, a nonlinear torsional spring with stiffness knes, and a torsional dashpot (cnes) as indicated in Figure 5. The length of each element in finite element formulation is le=Ls/N for all divisions of the shaft.Figure 5
A discretized shaft in torsional vibration with NES attachment.The discrete model of the shaft can be easily obtained. The equation of motion is described by the same partial differential equations describing the motion of rod in axial vibration [28, 29], and they are given by(28)Is×q¨it+Cs×q˙it+Ks×qit+Js,NL,i=Js,ex,it,with structural matrices defined explicitly in Appendix A. In case of N elements discretization of the shaft, there are N+1 nodes in the shaft plus the NES DOF and the resulting mass, stiffness, and damping matrices dimensions being N+2×N+2 [28, 29].The nonlinear torque vector is given by(29)Js,NL,iT=knes⋅qN+1t−qNESt3⋅0…1−1.The external torque vector is obtained by considering only point loads at each node of the finite element formulation as follows:(30)Js,ex,itT=Js,ex,1tJs,ex,2t…Js,ex,NtJs,ex,N+1tJs,ex,N+2t.Initially, the mechanical system given by equation (28), by neglecting the vector forces on the right-hand side of the equations fulfilling the perpetual mechanical system’s requirements, is examined. In case of rigid body motion, the following is true:(31)Ks×1⋅qat=∑j=1N+2Ks,i,j⋅qat=0,which is easily obtained by summing up all the elements of the stiffness matrix, given by equation (A.3), which belongs to each row.Also, using equation (31) with equation (A.4) leads to(32)Cs×1⋅q˙at=β3⋅Ks×1⋅q˙at=0,and(33)Js,NL,aT=knes⋅qat−qat3⋅0…11=0.Therefore, taking into account equations (31)–(33) in case of unforced system, since the inertia matrix is positive-definite, for rigid body motions, all the accelerations are equal to zero in the system of equations (28).The equations of jerks for the unforced system are given by(34a)Is×q⃛it+Cs×q¨it+Ks×q˙it+J˙s,NL,i=0,with(34b)J˙s,NL,iT=3⋅knes⋅qN+1t−qNESt2⋅q˙N+1t−q˙NESt⋅0…1−1,where, for rigid body motion, they lead to(34c)J˙s,NL,aT=3⋅knes⋅qat−qat2⋅q˙at−q˙at⋅0…1−1=0T,and, also for rigid body motions, the following is true:(34d)Ks×q˙at=Ks×1⋅q˙at=0.Considering equations (34c) and (34d) with the fact that the accelerations are zero, in the equation that defines jerks (34a), one certifies that the jerks are equal to zero in the case of rigid body motions. Therefore, rigid body motions are acceptable perpetual manifolds for the shaft with a free end and NES attachment boundary conditions.Summing up the rows of the inertia matrix defined by equation (A.1) leads to(35a)∑j=1N+2Is,1,j=∑j=1N+2Is,N+1,j=I⋅le2,(35b)∑j=1N+2Is,k,j=I⋅le,fork∈2,3,…,N,(35c)∑j=1N+2Is,N+2,j=INs.The required condition for the external torques to lead to EAPMs solution is the following:(36)Js,ex,it,qa,q˙a∑j=1N+2Is,i,j=2⋅Js,ex,1t,qa,q˙aI⋅le=Js,exlt,qa,q˙aI⋅le=2⋅Js,ex,N+1t,qa,q˙aI⋅le=Js,ex,N+2t,qa,q˙aInes,fori∈1,2,…,N+2,andl∈2,3,…,N,which leads to several arrangements of the external torques and herein the following is considered:(37a)Js,ex,1t,qa,q˙a=Js,ex,N+1t,qa,q˙a,(37b)Js,ex,l=2⋅Js,ex,1t,qa,q˙a=2⋅Js,ex,1t,qa,q˙a,forl∈2,3,…,N,(37c)Js,ex,N+2=2⋅InesI⋅le⋅Js,ex,1t,qa,q˙a=2⋅InesI⋅le⋅Js,ex,1t,qa,q˙a.Considering the system of equations (28) and the external torques defined by equations (37a)–(37c), the solution is given by solving the following characteristic differential equation:(38)q¨at=Js,ex,it,qa,q˙a∑j=1NIs,i,j=Js,ex,1I⋅le.Four different types of excitation forces are applied in the shaft, as is shown in Table4. Also, in Table 4, the equations that determine the shaft’s motion for each time interval, the wave velocity where it is applicable, and the type of motion are given. Of the first, second, and fourth time intervals, the external forces are the same as those applied in the train as 5-DOF mechanical system model in [22].Table 4
Equations that describe the solutions in augmented perpetual manifolds with the expected motions at different time intervalsTch=1.52159874⋅10−2s.
iTime interval (s)Gr=Js,ex,1r/I⋅leAnalytical velocityq˙a,iAnalytical displacementqa,iLongitudinal wave velocitywviType of motion1t∈0,TchG2=fk2/I⋅le, Equation (11)Equation (17b)Equation (18b)Equation (19a)Particle-standing wave2t∈Tch,2⋅TchG1=fk1/I⋅le, Equation (10)Equation (17a)Equation (18a)—∗Particle curvilinear3t∈2⋅Tch,3⋅TchG23=fk,23/I⋅le, Equation (12a)Equation (17c)Equation (18c)Equation (19b)Particle-motion directed wave4t∈3⋅Tch,4⋅TchG2=fk2/I⋅le, Equation (11)Equation (17b)Equation (18b)Equation (19a)Particle-motion directed wave∗Not applicable on this type of motion.A stainless-steel solid shaft with externalro=0.03m, length of L=1m, density of ρ0=8000kg/m3, and shear modulus of G=76.9GPa, is considered. The geometrical and material properties of the shaft lead to an inertia coefficient I=5.0894⋅10−3kg/m3. The NES inertia coefficient is INES=5.0894⋅10−4kg/m2, linear torsional stiffness is kl,nes=2000N⋅m/rad, and nonlinear torsional stiffness is knes=2000N⋅m/rad3. The number of elements for the discretization of the shaft is N = 100. The first three natural frequencies of the underlying linear system, which are obtained through the eigenvalue solution, are ωs,1=0rad/sec, ωs,2=2064.67rad/sec, and ωs,3=9782.50rad/sec. The Rayleigh damping coefficients are α=0s−1 and β1=9.686798⋅10−6s which correspond to 1% damping ratio of the second natural frequency of the underlying linear system.The values of the external forcing parameters are given in Table5. Also, in this table, the initial conditions resulting from previous time interval motion associated with each time interval as well as the wave velocities are shown. After the first time interval, the initial conditions from the previous interval motion are given.Table 5
External forcing parameters of the shaft with the associated initial conditions for each time interval and the theoretical wave velocity.
iTime interval (s)ICsExternal forcing parameterswvi (t) (rad/s)q˙a,it0 (rad/s)qa,it0 (rad)1t∈0,0.0152−190.3335620.000000Aex,1=10N⋅m0.00ωex,1=ωs,2=2064.67rad/sθex=0rad2t∈0.01520,0.0304−190.3335620.000000η=1.5N⋅m/s−c=1N/m3t∈0.0304,0.0456428.0883761.791532Aex,1713.59ωex,1θexAex,1ωex,2=2⋅ωs,2=4129.33rad/sθex4t∈0.0456,0.0609428.08837612.649489Aex,1618.42ωex,1θexAll the numerical simulations in this section have been performed with 64-bit Scilab 6.6.1 [27] using “Adams” solver with time step 7.6080⋅10−6 s, as well as relative and absolute tolerance 5⋅10−15.In Figures6(a)–6(d), selected numerically determined angular displacements of the mechanical system, of the left edge of the shaft (q1), of the NES (q102), and of the middle q50 of the shaft are depicted, incorporating the analytical solution qa. They seem to be in good agreement and this can be certified further by considering the maximum of the maxima of the absolute differences between the analytical solution and each angular displacement of the mechanical system, which has the minimal value of 1.999 10−10 rad. In Figure 6(a), the angular displacements for the first time interval are depicted, which is a particle standing wave solution, and they are associated with the vector field G2 with analytical solution given by equation (g) in Table 1 and it has the same solution of the train example on the second time interval in [22]. The angular displacements for the second time interval in Figure 6(b) are depicted; they have particle curvilinear function motion, and they are associated with the vector field G1. The analytical solution is given by equation (f) in Table 1, and it has the same solution as the train example of [22] for the first time interval. In Figure 6(c), the angular displacements for the third time interval are depicted, which correspond to particle-motion directed two-frequency wave, and they are associated with the vector field G23. The analytical solution is given by equation (h) in Table 1 considering two frequencies of the external forcing. In Figure 6(d) the angular displacements for the fourth time interval are depicted, corresponding to particle-motion directed single-frequency wave and associated with the vector field G2. The analytical solution of displacements by equation (g) in Table 1 is given, and it has the same solution as the train example for the third time interval in [22].Figure 6
Selected numerically determined angular displacements of the FE model, of the left edge of the shaft (q1), of the NES (q102), and of the middle q50 of the shaft, incorporating the analytical solution (qa): (a) for the first time interval 0-1 s, with external forces fk2; (b) for the 2nd time interval 1-2 s, with external forces fk1; (c) for the 3rd time interval 2-3 s, with external forces fk3; and (d) for the 4th time interval 3-4 s, with external forces fk2.
(a)(b)(c)(d)In Figures7(a)–7(d), selected numerically determined angular velocities (the same ones depicted in the previous Figure 6) are depicted, incorporating the analytically determined angular velocities, and they seem to be in very good agreement. This is certified by examining the maximum of the maxima of the absolute differences of the analytical and each of the numerically determined velocities, which has the minimal value of 1.479 10−8 rad/s.Figure 7
Selected numerically determined angular velocities of the FE model, of the left edge of the shaft (q1), of the NES (q102), and of the middle q50 of the shaft, incorporating the analytical solution (qa): (a) for the first time interval 0-1 s, with external forces fk2; (b) for the 2nd time interval 1-2 s, with external forces fk1; (c) for the 3rd time interval 2-3 s, with external forces fk3; and (d) for the 4th time interval 3-4 s, with external forces fk2.
(a)(b)(c)(d)In Figure7(a), the angular velocities in the first time interval are depicted, and they are analytically determined by equation (b) in Table 1. The angular velocities for the second time interval in Figure 7(b) are depicted, and they are analytically determined by equation (a) in Table 1. In Figure 7(c), the angular velocities for the third time interval are depicted, and they are analytically determined by equation (c) in Table 1. The angular velocities for the last time interval in Figure 7(d) are depicted by equation (b) in Table 1.In Figures8(a)–8(d), the rest of exact augmented perpetual submanifolds defined by the vector fields of Table 1, which are not determined in the 2-DOF example of Section 3.1, are depicted. Figure 8(a) depicts the exact augmented perpetual submanifold A2,02 that is associated with the vector field G2 for particle-standing wave motion, and it has the same form as the train example for the second time interval of [22]. The exact augmented perpetual submanifold A1 in Figure 8(b), which is associated with the vector field G1, is depicted, and it has the same form as the train example for the first time interval of [22]. In Figure 8(c), the exact augmented perpetual submanifold A23 that is associated with the vector field G23 is depicted. The exact augmented perpetual submanifold A22 that is associated with vector field G2 is depicted in Figure 8(d), and it has the same form as the train example of [22] for the third time interval.Figure 8
Exact augmented perpetual submanifoldsAr: (a)A2,02 that is associated with the vector field G2, (b)A1 that is associated with the vector field G1, (c)A23 that is associated with the vector field G23, and (d)A22 that is associated with the vector field G2.
(a)(b)(c)(d)


### 3.3. A Five-Degrees-of-Freedom System

In this section, the train modeled as a five-degrees-of-freedom system in [22] is considered, and the motion is described by the following equation of motion:(39)Mi,j×x¨i+Ci,j×x˙i+Ki,j×xi+FiNLxn,x˙o=Fext,it,forn,o∈1,2,…,5,i=1,…,5,with mass matrix defined by(40a)Mi,j=m100000m200000m300000m400000m5,and mii=1,…,5 being positive constants; therefore all sums of rows are nonzero.The stiffness matrix is defined by(40b)Ki,j=k1−k1000−k1k1+k2−k2000−k2k2+k3−k3000−k3k3+k4−k4000−k4k4.The damping matrix is defined with Rayleigh damping and given by(40c)Ci,j=c1−c1000−c1c1+c2−c2000−c2c2+c3−c3000−c3c3+c4−c4000−c4c4=β⋅Ki,j.The nonlinear forces vector is given by(40d)FiNLxn,x˙o=knl,1⋅sinx1−x2+cnl,1⋅x˙1−x˙23−knl,1⋅sinx1−x2−cnl,1⋅x˙1−x˙23+knl,2⋅x2−x35+cnl,2⋅x˙2−x˙35−knl,2⋅x2−x35−cnl,2⋅x˙2−x˙35+knl,3⋅sinx3−x4+cnl,3⋅x˙3−x˙47−knl,3⋅sinx3−x4−cnl,3⋅x˙3−x˙47+knl,4⋅x4−x57+cnl,4⋅tanhb⋅x˙4−x˙5−knl,4⋅x4−x57−cnl,4⋅tanhb⋅x˙4−x˙5,forn,o∈1,2,…,5,and the theorem’s condition for the external forcing vector (equation (9)), considering the first mass as reference k-mass, leads to(40e)Fext,itT=1m2m1m3m1m4m1m5m1⋅Fext,1t.The considered external forces are the first two types of forces given by Table2 of the example of Section 3.1.The mechanical system in [22] is shown to form a perpetual mechanical system, and the solution in the exact augmented perpetual manifold is given by equation (6) with m=m1/σi.The nonlinear system of equations (38) has been solved numerically. All numerical simulations in this section have been performed, with 64-bit Scilab 6.6.1 [27] using “Adams” solver with time step 5⋅10−4 s, as well as relative and absolute tolerance 6⋅10−14.The system parameters, which define the matrices in equation (35), are shown in Table 6.Table 6
The parameters of the matrices defining the equations of motion of the train.
Mi,jKi,jCi,jFiNLm1=2000kgk1=1⋅105N/mc1=319.07N⋅s/mknl,1=1⋅104Ncnl,1=319.07N⋅s3/m3m2=1000kgk2=1.4⋅105N/mc2=446.70N⋅s/mknl,2=−1.5⋅104N/m5cnl,2=446.70N⋅s5/m5m3=1500kgk3=1.3⋅105N/mc3=414.80N⋅s/mknl,3=1.3⋅104Ncnl,3=414.80N⋅s7/m7m4=1200kgk4=1.2⋅105N/mc4=382.88N⋅s/mknl,4=1.2⋅104N/m7cnl,4=382.88Nm5=500kg————The external forces parameters are the same as those given in Table3 for the first two time intervals of the 2-DOF mechanical system in the previous example. These forces lead to the same characteristic differential equation, the first external force is multiplied with a constant σ1=200, and the second external force with a constant σ2=400, which leads to the same vector fields describing the motion for the first two time intervals of the 2-DOF mechanical system example of Section 3.1.In Figure9(a), selected numerically determined displacements of the 5-DOF system, of the two edges (x1,x5) and of the middle x3, are depicted incorporating the analytical solution xa and the numerical solution x−2dof of the 2-DOF system of the example of Section 3.1, and it seems that they are in good agreement. This is certified by examining (a) the maximum of the maxima of the absolute differences between the analytical solution and each of the displacements of the 5-DOF system, which has the minimal value of 4.669 10−8 m, and (b) the maximum of the maxima of the absolute differences between the numerically determined 2-DOF displacement and each of the displacements of the 5-DOF system, which has the minimal value of 4.204 10−8 m.Figure 9
(a) Selected numerically determined displacements of the 5-DOF mechanical system, of the two edges (x1,x3) and of the middle x5, incorporating the analytical solution xa and the 2-DOF mechanical system numerical solution x−2dof. (b) Selected numerically determined displacements of the 5-DOF mechanical system, of the two edges (x˙1,x˙3) and of the middle x˙5, incorporating the analytical solution x˙a and the 2-DOF mechanical system’s numerical solution x˙−2dof.
(a)(b)Some selected numerically determined velocities of the 5-DOF system of the two edges (x˙1,x˙5) and of the middle x˙3 are depicted in Figure 9(b), incorporating the analytical solution x˙a and the numerical solution x˙−2dof of the 2-DOF system of the example of Section 3.1, and it seems that they are in good agreement. This is certified by examining firstly the maximum of the maxima of the absolute differences between the analytical velocity and each of the velocities of the 5-DOF system, which has the minimal value of 1.84 10−7 m/s. Secondly, the maximum of the maxima of the absolute differences between the numerically determined 2-DOF velocity and each of the velocities of the 5-DOF system, which has the minimal value of 1.95 10−7 m/s.Therefore, the corollary is certified.


## 3.1. A Two-Degrees-of-Freedom System

In this section, an example of a 2-DOF mechanical system is examined. Figure1 shows the 2-DOF mechanical system in translational motion, and the equations of motion have the following form:(20)mx,0+x˙200my,0+y˙2×x¨y¨+k1−k1−k1k1×xy+c1−c1−c1c1×x˙y˙+k2⋅x−y3+c2⋅tanhb⋅x˙−y˙k2⋅y−x3+c2⋅tanhb⋅y˙−x˙=Fxt,x,y,x˙,y˙Fyt,x,y,x˙,y˙.Figure 1
Configuration of the 2-DOF mechanical system.The system is comprised of two masses (mx and my) that are velocity-dependent, coupled through linear (k1) and nonlinear springs (k2) and also through linear dashpot (c1) and nonlinear damping force (c2). In each, mx and my mass friction forces and external forces fx and fy are applied.The associated perpetual mechanical system without external forces is a smooth nonlinear system with unique solutions, but the existence of the solution of equations (20) depends on the external force’s form on the equation’s right-hand side. The theorem’s first condition is to examine if this mechanical system is a perpetual mechanical system. Initially, the unforced mechanical system has to be considered and eliminating the external forces (including the friction forces) in equations (20) leads to(21)mx,0+x˙200my,0+y˙2×x¨y¨+k1−k1−k1k1×xy+c1−c1−c1c1×x˙y˙+k2⋅x−y3+c2⋅tanhb⋅x˙−y˙k2⋅y−x3+c2⋅tanhb⋅y˙−x˙=00.In case of exact rigid body motions,(22a)xa=ya,(22b)x˙a=y˙a,and replacing equations (22a) and (22b) in the system of equations (21) leads to(23a)mx,0+x˙a2⋅x¨a=0⟺x¨a=0,(23b)my,0+y˙a2⋅y¨a=0⟺y¨a=0.The equations of jerks arise with direct differentiation of equations (21) in time, and they are given by(24)2⋅x˙⋅x¨002⋅y˙⋅y¨×x¨y¨+mx,0+x˙200my,0+y˙2×x⃛y⃛+k1−k1−k1k1×x˙y˙+c1−c1−c1c1×x¨y¨+3⋅k2⋅x−y2⋅x˙−y˙+b⋅c2⋅sech2b⋅x˙−y˙⋅x¨−y¨3⋅k2⋅y−x2⋅y˙−x˙+b⋅c2⋅sech2b⋅y˙−x˙⋅y¨−x¨=00.Replacing equations (22a), (22b), (23a), and (23b) in equation (24) leads to(25a)mx,0+x˙2⋅x⃛a=0⇔x⃛a=0,(25b)my,0+y˙2⋅y⃛a=0⇔y⃛a=0.The accelerations and jerks with exact rigid body motions are equal to zero, and, therefore, the system is a perpetual mechanical system.The second condition of the theorem is that the external forcing should comply with equation (10), where, in the exact augmented perpetual manifolds (EAPMs) of this system, it is given by(26a)∑j=12M1,jx˙a=mx,0+x˙a2,(26b)∑j=12M2,jx˙a=my,0+x˙a2,and considering equations (26a) and (26b) with equation (9) leads to(27)Fxt,xa,xa,x˙a,x˙amxx˙a=Fyt,xa,xa,x˙a,x˙amyx˙a⟺Fyt,xa,xa,x˙a,x˙a=my,0+x˙a2mx,0+x˙a2⋅Fxt,xa,xa,x˙a,x˙a,with reference mass mx.The analytical solution of the external forces applied to this system is derived in Section2.3, and it is given in Table 1. The type of vector field arising from the characteristic differential equation and the form of the external forces and the type of motion for each time interval are given in Table 2. Moreover, the equations associated with the analytical solutions, the wave velocity equations, and the associated type of motion are indicated in Table 2.Table 2
The external forces, the solutions, and the motions in EAPMs of the 2-DOF system.
iTime interval (s)GiFxi=fx+fd,xAnalytical velocityx˙a,iAnalytical displacementxa,iLongitudinal wave velocitywviType of motion1t∈0,1G23fx=σ1⋅mx,0+x˙a2⋅fk,23Equation (17c)Equation (18c)Equation (19b)Particle linear-two frequencies standing wave (using specific amplitudes)fd,x=02t∈1,2G1,fk14fx=σ2⋅mx,0+x˙a2⋅fk3+fk1Equation (17d)Equation (18d)Equation (19c)Particle-longitudinal wavefd,x=03t∈2,3G6fy=0Equation (16h)Equation (16k)—∗Particle decelerating (dry friction)fd,x=σ3⋅mx,0+x˙a2⋅fk64t∈3,4G5fx=σ4⋅mx,0+x˙a2⋅fk5Equation (17e)Equation (18e)—∗Particle exponential accelerating (throttle with flutter)fd,x=05t∈4,5G5fx=σ5⋅mx,0+x˙a2⋅fk5Equation (17e)Equation (18e)—∗Particle exponential decelerating (throttle with drag)fd,x=06t∈5,6G6fy=0Equation (16p)Equation (16r)—∗Particle decelerating (dry friction)fd,x=σ3⋅mx,0+x˙a2⋅fk6∗Not expected for this type of motion.The existence of nonsmooth forces (in the third- and sixth-time intervals) requires preliminary work in developing the numerical scheme based on the switch model algorithm of [26], and herein the algorithm developed in [22] is used.The chosen parameters for this example correspond to a car withmx,0=2000kg dragging a van with mass my,0=1000kg. The coupler has linear stiffness k1=106N/m and nonlinear stiffness k2=5⋅105N/m3. In the two-degrees-of-freedom-associated linear system, the linear natural frequencies ωi and mode shapes Vi areω1=0rad/s with normalized mode shape V1=0.8160.577,ω2=38.73rad/s with normalized mode shape V2=−0.5770.816.Therefore, when the second mode is excited, the two masses are vibrating in out-of-phase fashion, and although the main frequency of harmonic external forces would be this one of the second mode, this mode will not be excited due to the same direction of external forces (same sign as in equation (2)).Considering 1% damping ratio for the second mode, the damping coefficient isc1=516.398N⋅s/m and the nonlinear damping coefficient has same value c2=516.398N with b=106s/m (significantly high to approximate dry friction). In the developed theory, damping does not play any role, but the considered values are sensibly high enough, mainly for illustration purposes.The external forcing parameters are shown in Table3, with the associated initial conditions and the wave velocity for each type of external forcing.Table 3
The initial conditions, the external forcing parameters, and the wave velocities for each time interval of the 2-DOF system.
iTime interval (s)ICsExternal forcing parameterswvi (m/s)x˙a,it0 (m/s)xa,it0 (m)1t∈0,10.0000000.000000σ1=0.10Nf=2Aex,1=Aex=−5⋅105Nωex,1=ω2=38.73rad/sθex,1=0radAex,2=1⋅106Nωex,2=2⋅ω2=77.46rad/sθex,2=0rad2t∈1,21272.33096213.893890σ2=0.2500⋅t2−1.962⋅t−556.915η=5000N/sc=9.81NNf=1Aex,1ωex,1θex,13t∈2,3229.631038628.195850σ3=0.5−=μxs⋅FxNμxs⋅mx,0⋅g=1962N∗δ=3 s/mmy,0/mx,0⋅μxs⋅FxN=μxs⋅my,0⋅g=μxs⋅FyN∗4t∈3,4228.204655857.114437σ4=0.3−cd=3N⋅s/m,>0F1T=1000N5t∈4,51047.8272511434.472876σ5=0.4−cd=−5N⋅s/m<0F2T=−1000N6t∈5,6−31.1249461773.948974σ3=0.5−μxs⋅FxN=μxs⋅mx,0⋅g=1962N∗δ=3 s/mmy,0/mx,0⋅μxs⋅FxN=μxs⋅my,0⋅g=μxs⋅FyN∗∗In this example, without losing the generality and simplifying the frictional terms, the state-dependent inertia elements are considered which are not correlated with the weight of the vehicles.All the numerical simulations in this section, with 64-bit Scilab 6.1.0 [27] using “Adams” solver with time step dt=5⋅10−4 s and relative and absolute tolerance 6⋅10−14, have been performed. The parameter for the convexification of the vector field in the application of the switch algorithm of [22] is τ=dt⋅10−3=5⋅10−7 s.In Figures2(a)–2(f), the displacements xa obtained from theoretical solution in EAPMs (equations in Table 1) and those obtained from numerical simulations of the system with equations (20) are depicted, and it seems that they are in good agreement. This can be certified further by examining the maxima’s maximum of the absolute difference between the analytical and numerical displacements, which has the minimal value of 1.875 10−8 m.Figure 2
Numerically determinedx,y and analytically determined displacements xa of the 2-DOF mechanical system: (a) for the first time interval 0-1 s, with external forces fk,23; (b) for the 2nd time interval 1-2 s, with external forces fk3 and fk1; (c) for the 3rd time interval 2-3 s, with external forces fk6; (d) for the 4th time interval 3-4 s, with external forces fk5; (e) for the 5th time interval 4-5 s, with external forces fk5; and (f) for the 6th time interval with external forces fk6.
(a)(b)(c)(d)(e)(f)In Figure2(a), the displacements on the first time interval are depicted. The motion of the mechanical system is a particle standing wave with two frequencies (Nf=2), defined by equation (h) in Table 1, and it is associated with the vector field G23. In Figure 2(b), the displacements on the second time interval are depicted; the motion is a particle-longitudinal wave and is defned by equation (i) in Table 1. It is associated with the vector field G1,fk14, which indicates harmonic forcing with a single frequency combined with the associated vector field G1. The displacements, on the third time interval in Figure 2(c), are depicted, associated with the vector field G6 for positive velocities, and they are analytically determined by equation (16k). The motion is particle decelerating due to dry friction components. In Figures 2(d)–2(e), the displacements on the fourth and fifth time intervals are depicted, and both are associated with the vector field G5. They are analytically determined by equation (j) in Table 1. In the fourth time interval (Figure 2(d)), the positive value of cd is associated with flutter, which indicates particle-accelerating motion, and in the fifth time interval (Figure 2(e)), the negative value of cd is associated with drag, which indicates particle-decelerating motion. Finally, in Figure 3(f), the displacements on the sixth time interval are depicted, and the motion is decelerating, and it is associated with the vector field G6 for negative velocities, with displacements analytically determined by equation (16r).Figure 3
Numerically determinedx˙,y˙ and analytically determined velocities x˙a of the 2-DOF mechanical system: (a) for the first time interval 0-1 s, with external forces fk,23; (b) for the 2nd time interval 1-2 s, with external forces fk3 and fk1; (c) for the 3rd time interval 2-3 s, with external forces fk6; (d) for the 4th time interval 3-4 s, with external forces fk5; (e) for the 5th time interval 4-5 s, with external forces fk5; and (f) for the 6th time interval, with external forces fk6.
(a)(b)(c)(d)(e)(f)In Figures3(a)–3(f), the velocities obtained from the numerical simulations of equations (20) incorporating the analytical solutions x˙a are depicted, and it seems that they are in very good agreement. Moreover, this is certified by examining the maximum of the maxima of the absolute difference between the analytical and numerically determined velocities, which has the minimal value of 1.213 10−8 m/s.In Figure3(a), the velocities on the first time interval are depicted, with the analytical ones determined by equation (a) in Table 1, and they are associated with the vector field G23. Figure 3(b) depicts the velocities on the second time interval, and their increasing values indicate the nonzero wave velocity. The analytical velocity by equation (d) in Table 1 is given, and it is associated with the vector field G1,fk14.The decreasing velocity on the third time interval in Figure3(c) is depicted, and the analytical solution by equation (16h) is determined, which is associated with the vector field G6. The increasing velocities for the fourth time interval depicted in Figure 3(d) are associated with flutter. Equation (e) in Table 1 defines the analytical solution, and it is associated with the vector field G5. The decreasing velocities in the fifth time interval depicted in Figure 3(e) are associated with the vector field G5, for aerodynamic drag, and the analytical solution by equation (e) in Table 1 is obtained. Finally, in Figure 3(f), the velocities of the sixth time interval are depicted. They are decreasing due to dry friction, and they are associated with the vector field G6. The analytical solution of the velocity on the sixth time interval by equation (16p) is determined.The basis of each exact augmented perpetual manifold is the augmented perpetual submanifolds defined by equation (5), and it is associated with each vector field Gi of the characteristic differential equation (6). In Figures 4(a)–4(f), the augmented perpetual submanifolds, defined by the analytical solutions for each time interval, are depicted. In Figure 4(a), the shape of the augmented perpetual submanifold−A2,03 of the vector field−G24, defined by the analytical solution on the first time interval, is depicted with the first subscript indicating the number of frequencies and the second subscript indicating the wave velocity.Figure 4
Exact augmented perpetual submanifoldAr: (a) A2,03 that is associated with vector field G23, (b) A1,fk14 that is associated with the vector field G1,fk14, (c) A+6 that is associated with the vector field G6 for positive velocities, (d) A+5 that is associated with the vector field G5 for flutter, (e) A−5 that is associated with the vector field G5 for aerodynamic drag, and (f) A−6 that is associated with the vector field G6 for negative velocities.
(a)(b)(c)(d)(e)(f)In Figure4(b), using the analytical solution on the second time interval, the exact augmented perpetual submanifold−A1,fk14 that is associated with the vector field G1,fk14 is depicted, with the first subscript indicating the number of frequencies and the second subscript indicating the type of the nonharmonic function. In Figure 4(c), the exact augmented perpetual submanifold−A+6 that is associated with the vector field G6, for positive velocities indicated by the plus sign in the subscript, is depicted. The exact augmented perpetual submanifold−A+5 that is associated with the vector field G5 for flutter is depicted in Figure 4(d). In the case of aerodynamic drag, the exact augmented perpetual submanifold−A−5 associated with the vector field G5 is depicted in Figure 4(e). Finally, the exact augmented perpetual submanifold−A−6 that is associated with the vector field G6 for negative velocities is depicted in Figure 4(f).


## 3.2. A Shaft in Torsional Vibration with Torsional Nonlinear Energy Sink

In this example, a discretized, through finite element formulation, shaft in torsional vibration with free boundary condition in one end and dynamic boundary condition in the other end coupled with a nonlinear energy sink (NES) is considered, as indicated in Figure5. The shaft is of length Ls, with internal and external diameters of Di and of Do, respectively. The density is denoted as ρ and the shear modulus is denoted as G. The shaft with inertia I is coupled with an NES mass, Ines, through a linear torsional spring with stiffness kl,nes, a nonlinear torsional spring with stiffness knes, and a torsional dashpot (cnes) as indicated in Figure 5. The length of each element in finite element formulation is le=Ls/N for all divisions of the shaft.Figure 5
A discretized shaft in torsional vibration with NES attachment.The discrete model of the shaft can be easily obtained. The equation of motion is described by the same partial differential equations describing the motion of rod in axial vibration [28, 29], and they are given by(28)Is×q¨it+Cs×q˙it+Ks×qit+Js,NL,i=Js,ex,it,with structural matrices defined explicitly in Appendix A. In case of N elements discretization of the shaft, there are N+1 nodes in the shaft plus the NES DOF and the resulting mass, stiffness, and damping matrices dimensions being N+2×N+2 [28, 29].The nonlinear torque vector is given by(29)Js,NL,iT=knes⋅qN+1t−qNESt3⋅0…1−1.The external torque vector is obtained by considering only point loads at each node of the finite element formulation as follows:(30)Js,ex,itT=Js,ex,1tJs,ex,2t…Js,ex,NtJs,ex,N+1tJs,ex,N+2t.Initially, the mechanical system given by equation (28), by neglecting the vector forces on the right-hand side of the equations fulfilling the perpetual mechanical system’s requirements, is examined. In case of rigid body motion, the following is true:(31)Ks×1⋅qat=∑j=1N+2Ks,i,j⋅qat=0,which is easily obtained by summing up all the elements of the stiffness matrix, given by equation (A.3), which belongs to each row.Also, using equation (31) with equation (A.4) leads to(32)Cs×1⋅q˙at=β3⋅Ks×1⋅q˙at=0,and(33)Js,NL,aT=knes⋅qat−qat3⋅0…11=0.Therefore, taking into account equations (31)–(33) in case of unforced system, since the inertia matrix is positive-definite, for rigid body motions, all the accelerations are equal to zero in the system of equations (28).The equations of jerks for the unforced system are given by(34a)Is×q⃛it+Cs×q¨it+Ks×q˙it+J˙s,NL,i=0,with(34b)J˙s,NL,iT=3⋅knes⋅qN+1t−qNESt2⋅q˙N+1t−q˙NESt⋅0…1−1,where, for rigid body motion, they lead to(34c)J˙s,NL,aT=3⋅knes⋅qat−qat2⋅q˙at−q˙at⋅0…1−1=0T,and, also for rigid body motions, the following is true:(34d)Ks×q˙at=Ks×1⋅q˙at=0.Considering equations (34c) and (34d) with the fact that the accelerations are zero, in the equation that defines jerks (34a), one certifies that the jerks are equal to zero in the case of rigid body motions. Therefore, rigid body motions are acceptable perpetual manifolds for the shaft with a free end and NES attachment boundary conditions.Summing up the rows of the inertia matrix defined by equation (A.1) leads to(35a)∑j=1N+2Is,1,j=∑j=1N+2Is,N+1,j=I⋅le2,(35b)∑j=1N+2Is,k,j=I⋅le,fork∈2,3,…,N,(35c)∑j=1N+2Is,N+2,j=INs.The required condition for the external torques to lead to EAPMs solution is the following:(36)Js,ex,it,qa,q˙a∑j=1N+2Is,i,j=2⋅Js,ex,1t,qa,q˙aI⋅le=Js,exlt,qa,q˙aI⋅le=2⋅Js,ex,N+1t,qa,q˙aI⋅le=Js,ex,N+2t,qa,q˙aInes,fori∈1,2,…,N+2,andl∈2,3,…,N,which leads to several arrangements of the external torques and herein the following is considered:(37a)Js,ex,1t,qa,q˙a=Js,ex,N+1t,qa,q˙a,(37b)Js,ex,l=2⋅Js,ex,1t,qa,q˙a=2⋅Js,ex,1t,qa,q˙a,forl∈2,3,…,N,(37c)Js,ex,N+2=2⋅InesI⋅le⋅Js,ex,1t,qa,q˙a=2⋅InesI⋅le⋅Js,ex,1t,qa,q˙a.Considering the system of equations (28) and the external torques defined by equations (37a)–(37c), the solution is given by solving the following characteristic differential equation:(38)q¨at=Js,ex,it,qa,q˙a∑j=1NIs,i,j=Js,ex,1I⋅le.Four different types of excitation forces are applied in the shaft, as is shown in Table4. Also, in Table 4, the equations that determine the shaft’s motion for each time interval, the wave velocity where it is applicable, and the type of motion are given. Of the first, second, and fourth time intervals, the external forces are the same as those applied in the train as 5-DOF mechanical system model in [22].Table 4
Equations that describe the solutions in augmented perpetual manifolds with the expected motions at different time intervalsTch=1.52159874⋅10−2s.
iTime interval (s)Gr=Js,ex,1r/I⋅leAnalytical velocityq˙a,iAnalytical displacementqa,iLongitudinal wave velocitywviType of motion1t∈0,TchG2=fk2/I⋅le, Equation (11)Equation (17b)Equation (18b)Equation (19a)Particle-standing wave2t∈Tch,2⋅TchG1=fk1/I⋅le, Equation (10)Equation (17a)Equation (18a)—∗Particle curvilinear3t∈2⋅Tch,3⋅TchG23=fk,23/I⋅le, Equation (12a)Equation (17c)Equation (18c)Equation (19b)Particle-motion directed wave4t∈3⋅Tch,4⋅TchG2=fk2/I⋅le, Equation (11)Equation (17b)Equation (18b)Equation (19a)Particle-motion directed wave∗Not applicable on this type of motion.A stainless-steel solid shaft with externalro=0.03m, length of L=1m, density of ρ0=8000kg/m3, and shear modulus of G=76.9GPa, is considered. The geometrical and material properties of the shaft lead to an inertia coefficient I=5.0894⋅10−3kg/m3. The NES inertia coefficient is INES=5.0894⋅10−4kg/m2, linear torsional stiffness is kl,nes=2000N⋅m/rad, and nonlinear torsional stiffness is knes=2000N⋅m/rad3. The number of elements for the discretization of the shaft is N = 100. The first three natural frequencies of the underlying linear system, which are obtained through the eigenvalue solution, are ωs,1=0rad/sec, ωs,2=2064.67rad/sec, and ωs,3=9782.50rad/sec. The Rayleigh damping coefficients are α=0s−1 and β1=9.686798⋅10−6s which correspond to 1% damping ratio of the second natural frequency of the underlying linear system.The values of the external forcing parameters are given in Table5. Also, in this table, the initial conditions resulting from previous time interval motion associated with each time interval as well as the wave velocities are shown. After the first time interval, the initial conditions from the previous interval motion are given.Table 5
External forcing parameters of the shaft with the associated initial conditions for each time interval and the theoretical wave velocity.
iTime interval (s)ICsExternal forcing parameterswvi (t) (rad/s)q˙a,it0 (rad/s)qa,it0 (rad)1t∈0,0.0152−190.3335620.000000Aex,1=10N⋅m0.00ωex,1=ωs,2=2064.67rad/sθex=0rad2t∈0.01520,0.0304−190.3335620.000000η=1.5N⋅m/s−c=1N/m3t∈0.0304,0.0456428.0883761.791532Aex,1713.59ωex,1θexAex,1ωex,2=2⋅ωs,2=4129.33rad/sθex4t∈0.0456,0.0609428.08837612.649489Aex,1618.42ωex,1θexAll the numerical simulations in this section have been performed with 64-bit Scilab 6.6.1 [27] using “Adams” solver with time step 7.6080⋅10−6 s, as well as relative and absolute tolerance 5⋅10−15.In Figures6(a)–6(d), selected numerically determined angular displacements of the mechanical system, of the left edge of the shaft (q1), of the NES (q102), and of the middle q50 of the shaft are depicted, incorporating the analytical solution qa. They seem to be in good agreement and this can be certified further by considering the maximum of the maxima of the absolute differences between the analytical solution and each angular displacement of the mechanical system, which has the minimal value of 1.999 10−10 rad. In Figure 6(a), the angular displacements for the first time interval are depicted, which is a particle standing wave solution, and they are associated with the vector field G2 with analytical solution given by equation (g) in Table 1 and it has the same solution of the train example on the second time interval in [22]. The angular displacements for the second time interval in Figure 6(b) are depicted; they have particle curvilinear function motion, and they are associated with the vector field G1. The analytical solution is given by equation (f) in Table 1, and it has the same solution as the train example of [22] for the first time interval. In Figure 6(c), the angular displacements for the third time interval are depicted, which correspond to particle-motion directed two-frequency wave, and they are associated with the vector field G23. The analytical solution is given by equation (h) in Table 1 considering two frequencies of the external forcing. In Figure 6(d) the angular displacements for the fourth time interval are depicted, corresponding to particle-motion directed single-frequency wave and associated with the vector field G2. The analytical solution of displacements by equation (g) in Table 1 is given, and it has the same solution as the train example for the third time interval in [22].Figure 6
Selected numerically determined angular displacements of the FE model, of the left edge of the shaft (q1), of the NES (q102), and of the middle q50 of the shaft, incorporating the analytical solution (qa): (a) for the first time interval 0-1 s, with external forces fk2; (b) for the 2nd time interval 1-2 s, with external forces fk1; (c) for the 3rd time interval 2-3 s, with external forces fk3; and (d) for the 4th time interval 3-4 s, with external forces fk2.
(a)(b)(c)(d)In Figures7(a)–7(d), selected numerically determined angular velocities (the same ones depicted in the previous Figure 6) are depicted, incorporating the analytically determined angular velocities, and they seem to be in very good agreement. This is certified by examining the maximum of the maxima of the absolute differences of the analytical and each of the numerically determined velocities, which has the minimal value of 1.479 10−8 rad/s.Figure 7
Selected numerically determined angular velocities of the FE model, of the left edge of the shaft (q1), of the NES (q102), and of the middle q50 of the shaft, incorporating the analytical solution (qa): (a) for the first time interval 0-1 s, with external forces fk2; (b) for the 2nd time interval 1-2 s, with external forces fk1; (c) for the 3rd time interval 2-3 s, with external forces fk3; and (d) for the 4th time interval 3-4 s, with external forces fk2.
(a)(b)(c)(d)In Figure7(a), the angular velocities in the first time interval are depicted, and they are analytically determined by equation (b) in Table 1. The angular velocities for the second time interval in Figure 7(b) are depicted, and they are analytically determined by equation (a) in Table 1. In Figure 7(c), the angular velocities for the third time interval are depicted, and they are analytically determined by equation (c) in Table 1. The angular velocities for the last time interval in Figure 7(d) are depicted by equation (b) in Table 1.In Figures8(a)–8(d), the rest of exact augmented perpetual submanifolds defined by the vector fields of Table 1, which are not determined in the 2-DOF example of Section 3.1, are depicted. Figure 8(a) depicts the exact augmented perpetual submanifold A2,02 that is associated with the vector field G2 for particle-standing wave motion, and it has the same form as the train example for the second time interval of [22]. The exact augmented perpetual submanifold A1 in Figure 8(b), which is associated with the vector field G1, is depicted, and it has the same form as the train example for the first time interval of [22]. In Figure 8(c), the exact augmented perpetual submanifold A23 that is associated with the vector field G23 is depicted. The exact augmented perpetual submanifold A22 that is associated with vector field G2 is depicted in Figure 8(d), and it has the same form as the train example of [22] for the third time interval.Figure 8
Exact augmented perpetual submanifoldsAr: (a)A2,02 that is associated with the vector field G2, (b)A1 that is associated with the vector field G1, (c)A23 that is associated with the vector field G23, and (d)A22 that is associated with the vector field G2.
(a)(b)(c)(d)


## 3.3. A Five-Degrees-of-Freedom System

In this section, the train modeled as a five-degrees-of-freedom system in [22] is considered, and the motion is described by the following equation of motion:(39)Mi,j×x¨i+Ci,j×x˙i+Ki,j×xi+FiNLxn,x˙o=Fext,it,forn,o∈1,2,…,5,i=1,…,5,with mass matrix defined by(40a)Mi,j=m100000m200000m300000m400000m5,and mii=1,…,5 being positive constants; therefore all sums of rows are nonzero.The stiffness matrix is defined by(40b)Ki,j=k1−k1000−k1k1+k2−k2000−k2k2+k3−k3000−k3k3+k4−k4000−k4k4.The damping matrix is defined with Rayleigh damping and given by(40c)Ci,j=c1−c1000−c1c1+c2−c2000−c2c2+c3−c3000−c3c3+c4−c4000−c4c4=β⋅Ki,j.The nonlinear forces vector is given by(40d)FiNLxn,x˙o=knl,1⋅sinx1−x2+cnl,1⋅x˙1−x˙23−knl,1⋅sinx1−x2−cnl,1⋅x˙1−x˙23+knl,2⋅x2−x35+cnl,2⋅x˙2−x˙35−knl,2⋅x2−x35−cnl,2⋅x˙2−x˙35+knl,3⋅sinx3−x4+cnl,3⋅x˙3−x˙47−knl,3⋅sinx3−x4−cnl,3⋅x˙3−x˙47+knl,4⋅x4−x57+cnl,4⋅tanhb⋅x˙4−x˙5−knl,4⋅x4−x57−cnl,4⋅tanhb⋅x˙4−x˙5,forn,o∈1,2,…,5,and the theorem’s condition for the external forcing vector (equation (9)), considering the first mass as reference k-mass, leads to(40e)Fext,itT=1m2m1m3m1m4m1m5m1⋅Fext,1t.The considered external forces are the first two types of forces given by Table2 of the example of Section 3.1.The mechanical system in [22] is shown to form a perpetual mechanical system, and the solution in the exact augmented perpetual manifold is given by equation (6) with m=m1/σi.The nonlinear system of equations (38) has been solved numerically. All numerical simulations in this section have been performed, with 64-bit Scilab 6.6.1 [27] using “Adams” solver with time step 5⋅10−4 s, as well as relative and absolute tolerance 6⋅10−14.The system parameters, which define the matrices in equation (35), are shown in Table 6.Table 6
The parameters of the matrices defining the equations of motion of the train.
Mi,jKi,jCi,jFiNLm1=2000kgk1=1⋅105N/mc1=319.07N⋅s/mknl,1=1⋅104Ncnl,1=319.07N⋅s3/m3m2=1000kgk2=1.4⋅105N/mc2=446.70N⋅s/mknl,2=−1.5⋅104N/m5cnl,2=446.70N⋅s5/m5m3=1500kgk3=1.3⋅105N/mc3=414.80N⋅s/mknl,3=1.3⋅104Ncnl,3=414.80N⋅s7/m7m4=1200kgk4=1.2⋅105N/mc4=382.88N⋅s/mknl,4=1.2⋅104N/m7cnl,4=382.88Nm5=500kg————The external forces parameters are the same as those given in Table3 for the first two time intervals of the 2-DOF mechanical system in the previous example. These forces lead to the same characteristic differential equation, the first external force is multiplied with a constant σ1=200, and the second external force with a constant σ2=400, which leads to the same vector fields describing the motion for the first two time intervals of the 2-DOF mechanical system example of Section 3.1.In Figure9(a), selected numerically determined displacements of the 5-DOF system, of the two edges (x1,x5) and of the middle x3, are depicted incorporating the analytical solution xa and the numerical solution x−2dof of the 2-DOF system of the example of Section 3.1, and it seems that they are in good agreement. This is certified by examining (a) the maximum of the maxima of the absolute differences between the analytical solution and each of the displacements of the 5-DOF system, which has the minimal value of 4.669 10−8 m, and (b) the maximum of the maxima of the absolute differences between the numerically determined 2-DOF displacement and each of the displacements of the 5-DOF system, which has the minimal value of 4.204 10−8 m.Figure 9
(a) Selected numerically determined displacements of the 5-DOF mechanical system, of the two edges (x1,x3) and of the middle x5, incorporating the analytical solution xa and the 2-DOF mechanical system numerical solution x−2dof. (b) Selected numerically determined displacements of the 5-DOF mechanical system, of the two edges (x˙1,x˙3) and of the middle x˙5, incorporating the analytical solution x˙a and the 2-DOF mechanical system’s numerical solution x˙−2dof.
(a)(b)Some selected numerically determined velocities of the 5-DOF system of the two edges (x˙1,x˙5) and of the middle x˙3 are depicted in Figure 9(b), incorporating the analytical solution x˙a and the numerical solution x˙−2dof of the 2-DOF system of the example of Section 3.1, and it seems that they are in good agreement. This is certified by examining firstly the maximum of the maxima of the absolute differences between the analytical velocity and each of the velocities of the 5-DOF system, which has the minimal value of 1.84 10−7 m/s. Secondly, the maximum of the maxima of the absolute differences between the numerically determined 2-DOF velocity and each of the velocities of the 5-DOF system, which has the minimal value of 1.95 10−7 m/s.Therefore, the corollary is certified.


## 4. Conclusions

As a continuation of the main theorem defining the conditions for a mechanical system’s motion to belong in the exact augmented perpetual manifolds, some theoretical developments with the definition of a corollary and its proof are presented. More precisely, all the perpetual mechanical systems with the same characteristic differential equation have the same general solution. Moreover, the analytical solutions in the exact augmented perpetual manifolds and their augmented perpetual submanifolds for six types of forces are determined. The analytical results are examined with numerical simulations in three examples, and they are in very good agreement.The significance of this outcome is that the many different mechanical systems, in the exact augmented perpetual manifolds with the same characteristic differential equation, have the same general solution, independent of the type of system, and upon the external forcing can lead to particle-single-frequency or multiple frequencies’ standing wave or motion directed single-frequency or multiple frequencies’ wave motions.This work is significant in mathematics for unifying specific solutions of nonautonomous systems of differential equations. Also, in physics/mechanics, the particle-wave motions are of high significance. In mechanical engineering, the rigid body motions without any vibrations are the ultimate motions for some flexible mechanical systems.Further work can be done incorporating the Part-II article of the augmented perpetual manifolds outcome, which is associated with the energies of the mechanical systems in the exact augmented perpetual manifolds.



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