Impact modes in discrete vibrating systems with rigid barriers

A class of strongly non-linear vibrating systems composed of linear elastic structures under absolutely rigid constraints condition is considered. Impact mode exact solutions are expressed through a saw-tooth time argument and, as a result, represented in a closed unit form. Based on this special representation, su $ cient conditions of existence and also non-existence for the impact modes are formulated and interpreted on the spectral axes. In particular, it was shown that impact modes exist when their basic frequencies are shifted into the right small neighborhood of any natural frequency of the linearized (no barriers) system. The frequencies of the localized impact mode solutions are located at the right of the linear spectrum and have no upper b oundary.


Introduction
In many practically important cases, the displacements of a vibrating mechanical system can be spatially bounded by rigid barriers. This can be caused by the system design and/or some gained clearances due to deterioration of joints. Both situations were considered by many investigators. The related engineering applications were summarized in research monographs quite completely, see for example [1}3]. If the total energy of the system is su$cient to reach the barriers, the system immediately becomes strongly non-linear even when the di!erential equations of motion between the barriers are linear. As a result, even structurally simple models can perform a very complex dynamical behavior [4,5]. This is why general analytical solutions for many degrees of freedom vibroimpact systems are rather impossible. However, an analytical insight into the multidimensional impacting systems can be reached by imposing certain restrictions on the class of motions. In this paper, we consider special vibrating regimes when all the system's particles synchronously vibrate as a single oscillator. In linear cases, such assumption leads to the normal mode solutions, and hence, gives a complete information about vibrating systems due to the linear superposition principle. In non-linear cases, this principle does not work, however, special non-linear normal mode solutions appear to be very important. An overview of the related analytical methods as well as some original material can be found in [6]. An important physical feature is that non-linear modes frequently perform a`trenda to spatial localization [7]. From the point of view of applied mechanics, a literature survey and an overview of di!erent analytical methods can be found in [6,8] Spatial symmetries, stability, and localization of non-linear modes in the two-and three degrees of freedom systems with bilateral perfectly rigid barriers were considered in [9] combining the special transformation of coordinates [10] and the idea of averaging. Recently, the non-linear normal modes concept was also adapted for two degrees of freedom non-linear systems with nonlinear regular (polynomial) and singular (potential barriers) components in [11].
If clearances between the barriers are su$ciently small, the di!erential equations of motion between the barriers can be assumed linear. Although the system still remains strongly non-linear due to the constraints condition, one can employ the temporal symmetry of the periodic motion in order to match di!erent pieces of the linear solution between the barriers, and thus, obtain some kind of exact normal mode solutions [12]. Exact unit-form periodic solutions were obtained in recent work [13] for a two degrees of freedom`vibroimpacta system. In principal, the methodology of work [13] is based on the saw-tooth transformation of time adapted for a class of impulsively excited systems [14] (see Conclusion for related remarks).
In certain sense, the present work combines the ideas of works [12,13]. As a result, exact unit-form analytical solutions for a multidegrees of freedom system with rigid barriers are obtained. In particular, conditions of existence and localization of impact modes, and also their physical meaning are considered. The unit form solutions could be suitable for analytical manipulations with them in different perturbations schemes, including averaging techniques.
In this work, the impact modes of a discrete vibrating system with bilateral rigid barriers are periodic motions when some or may be all particles of the system interact with the barriers twice per one period.
It will be shown that the case of unilateral constraints can be considered as well.

An introductory example
Let us consider a one degree of freedom free harmonic oscillator between two absolutely rigid barriers. A mechanical model of the oscillator can be represented as a massive particle attached to the free end of a massless cantilever beam as it is shown in Fig. 1a.
The interaction with the barriers at x"$ is assumed to be perfectly elastic, and the system is represented in the form Since the normal mode regimes are periodic by their dexnition, the reaction of constraints can be treated as a periodic series of external impulses acting on the masses of the system.
Applying this remark to the one degree of freedom system as it is shown in Fig. 1, the related A periodic series of points, + : ( )"$1,, at which "rst derivative ( ) has a step wise discontinuity, may give an exemption. However, it is not important because the di!erential equations of motion must be treated in terms of distributions whenever the Dirac -function is involved in the equations (see [20]). di!erential equation of motion is written in the linear form where ( ) is Dirac function; 2p, , and will be kept as arbitrary constants; ( ) is a standard sawtooth sine This piece-wise linear periodic function can be also represented in the unit form ( )" (2/ )arcsin sin( /2). The related amplitude and the period are normalized in such a manner that the expression [ ( )]"1 holds at least for almost all 3(!R, R). For further convenience, the right-hand side of Eq. (2) is expressed through second-order generalized derivative of the saw-tooth sine, where the derivative is taken with respect to the whole argument, t# . The parameter will be called a frequency parameter; though the standard circular frequency is expressed as "( /2) . In this paper, both parameters, and , will be used. In contrast to system (1), the auxiliary system (2) is linear. Representing unknown steady-state periodic solution in the form, x"X( ), " ( t# ) (4) one obtains the boundary value problem with no singular terms, and the related solution is represented in the sawtooth time form [14] x" p This solution can be veri"ed by direct substitution of expression (6) into the equation of motion (2).
A connection between solution (6) and vibration of the original system with the rigid constraints is established by imposing the conditions (Fig 1): E The impulses at the right-hand side of Eq. (2) must act at those time instances at which the mass is in contact with the constraints, i.e., x"$ when "$1 O0 (7) E The system cannot penetrate through the barriers, i.e., Substituting solution (6) into condition (7), one determines the impulses parameter p, p" cot( / ) (9) for which solution (6) takes the "nal form Solution (10) satis"es condition (7) automatically. The related parameter p (9) will further be treated as an`eigen-valuea of the non-linear (impact) problem.
The parameters, and , can be expressed through the initial conditions. Let us assume that x(0)"0, i.e., "0, and hence, the total energy of the oscillator per unit mass is expressed through the initial velocity as E"[x (0)]/2. Substitution (10) into the last equality gives expression for the frequencies ratio, The right-hand side gives a sequence of real numbers when the total energy, E, is su$ciently large, E*E H " /2, so that the oscillator can reach the constraints.
Generally speaking, not all the magnitudes of the parameter given by sequence (11) lead to real motions of the original system with rigid barriers. Indeed, since the auxiliary system of type (2) does not have any constraints for the coordinate, x(t), hence condition (7) does not guarantee that this coordinate will remain inside the admissable region, "x") , during the whole period of vibration. That is why another condition (8) must be veri"ed as well. Such a veri"cation implemented for solution (10) shows that condition (8) is satis"ed only for the smallest root in set (11). Fig. 2 illustrates the temporal mode shapes corresponding to the "rst two roots / . It is seen that the second solution, which is shown in fragment (b), violates condition (8), although it satis"es condition (7). Such`phantoma solutions, however, can be provided with a certain physical meaning if we imagine removable barriers appearing periodically twice per one period and re#ecting the particle into the exterior of the region "x") .
In this paper, the case of unilateral barriers will not be considered specially. The related investigation could be implemented, however, in the same way. Let us remove, for example, the left barrier and consider the oscillator (1) under the unilateral constraint condition x) . In this case, the boundary conditions in (5) should be modi"ed as X( )" O! "$p \. Such periodic change of sign e!ectively switches the directions of positivepulses to opposite at the right-hand side of Eq. (2).
As a result, the solution takes the form, where the period of the solution, x(t), and the related basic frequency will be ¹"2/ and " , respectively, due to evenness of cosine. Analysis of the temporal mode shapes for di!erent frequency ratio / is not a di$cult task.

Impact modes
Let us consider N-degrees of freedom conservative system described by the coordinates vector We suppose that one of the system coordinates, say the ath one, is restricted as "x ? ") ? . In the vector notation, this can be rewritten as "I2 ?
where I ? is the ath vector of the basis of the system coordinates, I ?
Inside the domain (13), the di!erential equations of motion are assumed to be linear, where M and K are respectively constant mass and sti!ness N;N-matrices.
A mass-spring model of the system (13) and (14) is shown in Fig. 3.
Note that the form of matrix equation (14) does not suppose the system to be necessarily a massspring chain, see Example 1 later.
To obtain an explicit analytical solution for the impact modes, the essentially non-linear system (13) and (14) is replaced by an impulsively forced linear system under no constraints condition where p is an unknown`eigenvaluea, the constants and are going to be kept as arbitrary parameters. Thus one seeks a two parameter family of periodic solutions.
A family of periodic solutions of the period 4/ can be found as a linear superposition of solutions (6) for each of the N modes of system (14) with a relevant replacement of the parameters [14], where " ( t# ) is the sawtooth temporal argument, e H and H are the jth normal mode and the natural frequency of linear system (3.2). The linear normal modes are normalized as e2 is the Kronecker symbol. We suppose that G ( H when i(j. The impulses must act at those time instances when the ath mass interacts with the constraints. Such spatially-temporal`location of the impactsa can be assigned by condition
Substituting (16) into (17), one obtains the related eigenvalue,a p" ? , where e2 H I ? is simply the ath component of the jth linear mode vector. Substituting (18) into (16) gives a two parameter family of the periodic solutions for the impact modes. The parameter is an arbitrary phase shift, whereas the frequency parameter implies some restrictions due to condition (13).
E For all G such that iOj the linear frequencies Under these conditions, solution (16), (18), satis-"es inequality (13), and hence, describes a two parameter family of the real impact mode regimes, x"x(t; , ).
An idea of the proof is to "nd out such cases when x is a monotonic function of on the interval !1) )1, and hence condition (17) at the boundaries guarantees that inequality (13) holds inside the whole interval as well, see Appendix for further details of the proof. The above conditions are suzcient but not necessary. Namely, if the conditions are true then the impact mode exists. Existence of an impact mode does not guarantee, however, that the conditions hold. For example, some impact modes can exist in the resonance case G / H "2k!1, when the parameter, is su$ciently small. This parameter shows a relative weight of the non-monotonic term which appears in the expression due to the resonance (see appendix). Example 2 below shows that the impact modes may possess a quite complicated spectral structure. Thus, the global investigation is needed to formulate necessary and suzcient conditions. On the other hand, a suzcient condition of non-existence can be formulated by making use the physical meaning of the impact interaction parameter p. Assuming that another angle, , is su$ciently small as well, one can take the di!erential equations of motion between the barriers in the linear form where "1#m /m , overdot denotes di!erentiation with respect to a new time parameter, tM "(g/l)t, and l is the length of the rods.
In this case, the linear modes and natural frequencies are, respectively, Taking into account that I ? "I , ? " , and N"2, the formal solution (16) and (18)  In this case, the parameter given by expression (20) is evaluated exactly as "1. This value is relatively small (see appendix) so that the existence of impact modes is not a!ected, if condition (19) holds.
It is important to note that the "rst impact mode basic frequency must be su$ciently close to the "rst linear frequency , so that it is not approaching the left neighborhood of the next frequency, . Since there are no more frequencies after , hence its right`neighborhooda has no upper boundary Note that this is just a suitable notation since the (N#1)th frequency does not physically exist. and can be extended to in"nity, ( (R. As a result, the highest impact mode becomes spatially localized when the parameter is growing. The localization can be estimated explicitly by asymptotic ( \P0) expansion It is seen that ( / )" O P!1 as PR, i.e., the horizontal displacement of the bottom mass becomes negligibly small whereas the impacting mass oscillates between the barriers. Fig. 5 represents the related trend of the mode shape.

Example 2. Let us consider a mass-spring chain of
n"1, 2 , N; x "x ,> "0 under the bilateral constraints condition (see Fig. 3), where k and m are sti!ness of the springs and mass of the particles, respectively. A set of no impact modes and frequencies can be found explicitly, since the system can be viewed as a "nite element model of a continuous string. The jth mode vector and related frequency of the linear system are given by the expressions where notation ,> "2(k/m has been introduced, the basis vectors are normalized such that e2 H e G " HG . The ath component of the jth normal mode vector is Let us show that the impact mode periodic solution (16) and (18)  In this particular example, the sums can easily be evaluated. Taking into account expression (24) and the standard trigonometric sums [15], one obtains Later on, we will need also the sum Substituting (26) into the asymptotic expression (25), one obtains This expression shows that the ath particle of the chain vibrates according to the saw-tooth temporal mode shape with an in"nitely large frequency, whereas all other particles remains at rest. Thus, the impact mode becomes spatially localized as PR. The localization tendency of the impact modes is illustrated by Fig. 5a.  Fig. 5. Localized impact modes of the mass}spring chain (n"25) for (a) single constrained particle (b) two constrained particle and the same frequency " #0.01. The same case of two constrained particles when increasing the frequency is illustrated by (b) and (c) " #0.2.   6 provides Propositions 1 and 2 with some additional illustration on the spectral axes of the chain consisting of N"5 particles. The "gure represents diagrams of the impulsive force parameter, p, as a function of the basic circular frequency of vibration, "( /2) , for di!erent masses under the constraints, a"1, 2, and 3. It is seen that each of the frequencies of the linear spectrum possesses its right neighborhood where the parameter p is positive, and hence the impact modes can exist (see fragments (a)}(c)). At the same time, the parameter p is not necessarily negative on the left of the frequencies , 2 , as it is seen from the fragments (b) and (c) (see, respectively, the frequency and the frequencies and ). The diagrams structure depends on the particle under the constraint condition. An interesting feature of the diagrams is that they are more complicated around the lowest frequency. The parameter p can change its sign many times in the domain between the "rst two frequencies, ( ( .

Systems with multiple impacting particles
Let us consider the case of two constrained particles. It will be seen that a generalization on multiple constrained particles is a quite trivial task. For example, let us put the constraints condition on the ath and bth coordinates (aOb) of the general linear system (14). These conditions are "x ? ") ?
and "x @ ") @ or in the vector notations In this case, the impulsive excitation on the right-hand side of auxiliary equation must act on the both ath and bth particles, so that the equation takes the form MxK #Kx"(p ?
where p ? and p @ are parameters to be determined. The related solution includes some items related to p ? and p @ , and can be represented in the form Following the normal modes ideology, we assume that the impact mode periodic regime is accompanied by synchronous impact interactions of the both impacting particles with the constraints according to conditions
(32) Substitution (31) into (32) gives linear algebraic equations with respect to p ? and p @ in the form where Expressions (31) and (33) give a formal impact mode solution. The impact mode (31) exists for those frequencies at which the determinant of system (33) is non-zero, and also condition (29) holds. Solution (31) can be viewed as a strongly non-linear superposition of the two simplest impact modes with a single impact pair. The result of the superposition is shown in Figs. 5b and c for two di!erent magnitudes of the frequency, .
Let us consider the higher frequency domain for the above example of mass}spring system, when < , . In this case, one has tan( H / )'0 for all j"1, 2 , N, and hence, the coe$cients k ?@ (34) create the Gram matrix with non-zero determinant [16]. The asymptotic estimation below con"rms this conclusion.
Indeed, for < , all ratios H / are small and thus an asymptotic expansion holds as Substituting this expression into solution (31) and coe$cients (34), one obtains the asymptotic solution for arbitrary cth particle in the form and " (2 t/ # ). The impact interaction parameters, P ? and P @ , are given by the linear algebraic system with the coe$cients I.e., it gives zero contribution into the related integrals of the theory of distributions.
The parameters and the coe$cients have been rescaled as follows: It is seen that K ?@ P ?@ as PR, and Eqs. (37) give the solution P ?
" ? and P @ " @ . In this limit, according to (36), the vibration becomes localized on the two particles which vibrate between the barriers performing the saw-tooth temporal shape, This is the leading-order asymptotic approach. All other particles, cOa and cOb, are at rest. However, they oscillate with small amplitudes of di!erent orders of \ when the parameter is bounded. Asymptotic expansion (36) includes the terms of order \. It is seen that the particles numbered c"a$1 and c"b$1 have the amplitudes of the lowest order, \. These particles are the nearest to the impacting ones. Some analysis of expansion (36) shows that the temporal mode shapes of the impacting particles, c"a and c"b, are non-smooth and close to the saw-tooth sine wave as it is expected to be due to the interactions with the constraints. On the other hand, the temporal mode shapes of the nearest particles, c"a$1 and c"b$1, are estimated by the combination of the saw-tooth sine, ! /3, with the "rst two continuous derivatives with respect to time, t. These are d dt respectively. The prime denotes di!erentiation with respect to the whole argument of the saw-tooth sine, ,d /d(2 t/ # ). In the above expressions, one should take into account that the -type singularities of second derivative, , are`locateda at those points +t,, where "$1. As a result, the underlined above term is zero.
Thus the temporal mode shapes of the particles with no constraints are smooth functions of time, although they are expressed through the saw-tooth sine.

Conclusion
A class of strongly non-linear many degrees of freedom vibrating systems including bilateral rigid barriers was considered. Exact analytical periodic solutions were expressed through the saw-tooth temporal argument. The conditions of existence and non-existence for impact mode regimes were formulated based on the exact unit-form solutions. More precisely, Proposition 1 gives a su$cient condition of existence of impact modes, whereas Proposition 2 gives a su$cient condition of their non-existence. These two propositions do not compose any necessary and su$cient condition of existence for the impact modes. Namely, one cannot indicate all the cases when impact modes exist, and also all the cases when they do not. A certain exemption is given only by the high frequency vibration ' , , when the impact modes become localized spatially and always exist. The examples clearly showed that a complete investigation presents a global geometrical problem. This, however, can be done at least numerically, since the exact expressions have been obtained in terms of trigonometric sums. The problem of stability can be considered by evaluating the eigenvalues of the Jacobian matrix for known periodic solutions as it was done, for example, in [17]. The systems above considered are strongly non-linear due to barriers only. In some cases, the system displacements must be signi"cant in order to provide an interaction with barriers, and hence, linearization of the equations of motion between the barriers may become doubtful. In general terms, however, the proposed formulation remains applicable. For example, Eq. (2) can be generalized as provided that the function f (x, x , t) is periodic (¹"4/ ) with respect to time, and the condition "x") holds. In this case, the saw-tooth temporal argument can be introduced by using the twocomponent representation for periodic solutions [14,18].
This results into the coupled non-homogeneous boundary-value problem with respect to unknown functions X( ) and >( ), X( )#R D (X, >, X, >, )"0, where functions R D and I D are obtained by substitution of the above representation for periodic solutions into the equation of motion. Such a kind of boundary value problems admit approximate analytical solutions in the power series form with respect to the new temporal argument, , since it automatically accounts for the periodicity of the motion. Unfortunately, the related solutions may appear to be so complicated that it will be either di$cult or even impossible to separate real and phantoma solutions. To this end, the above manipulations with the sawtooth temporal variable, should not be confused with the method of non-smooth transformation of the con"guration space coordinates for the systems with rigid barriers [10]. These two methods are mutually antithetic in its physical meaning as well as in its mathematical implementation. For example, being applied to system (1), the method of work [10] would give a non-linear oscillator (compare with (5)), sK # (s) (s)"0 where s(t) is a new positional variable, such that x" (s) ; the result is written in the notations of present work for the reason of comparison. The later transformation does not require the motions to be periodic. Unfortunately, this advantage brings essential non-linearity of the transformed di!erential equations. Further comparison of these two techniques as well as their composition can be found in [19]. phantoma solutions associated with large magnitudes of this parameter.
In Fig. 7, the`phantoma solutions correspond to those \ at which solutions, shown by the meshed surface, intersect the horizontal (non-meshed) planes, x ? / ? "$1. These planes represent the two barriers. As follows from expressions (A.6) and (20), the parameter determines a contribution of the ith (resonating) mode into the solutions and thus is an important characteristic of the system. Namely, the ith mode is responsible for violation of the condition "x ? / ? ")1. Fig. 7 also shows that increasing the frequency ratio / and thedistance from the basic frequency will expand the \-region of non-existing`phantoma solutions.