Solitary transversal waves and vibro-impact motions in infinite chains and rods

This paper deals with traveling waves in non-linear infinite elastic systems (chains and rods). A passage to a long wavelength approximation is realized. Conditions of the solitary waves existence are analyzed. The waves with regard to elastic impacts have been investigated.

. Numerical and experimental investigations of some elastic and rotor-dynamic systems with impact non-linearities were carried out in Emaci et al. (1997). Chaotic motions of vibro-impact systems were studied in works by Shaw and Holmes (1983), Moon et al. (1991), Ivanov (1994), Valkering (1994), Farahanchi and Shaw (1994), Han and Luo (1995). Methods of nonlinear and normal modes theory (Vakakis et al., 1996;Mikhlin, 1996) were used in vibro-impact problems in a work by Mikhlin et al., 1998. Studies of traveling waves in non-linear elastic systems were carried out by many researchers. Solitary traveling longitudinal waves were analyzed by Ostrovsky and Sutin (1977); the re¯ection of the soliton at almost ®xed and almost free ends were detected in a numerical simulation by Soerensen et al. (1984Soerensen et al. ( , 1987 and Clarkson et al. (1986) studied longitudinal wave propagation in a non-linear elastic rod modeled by the non-linear hyperbolic equation, containing odd nonlinearities (u 3 and u 5 ) and only one dispersive term with mixed fourth-order derivatives. Samsonov (1995) reduced the initial highly non-linear elastic wave problem governed by coupled PDE to the only one`double dispersion' equation, describing longitudinal strain waves in a rod; before, he obtained some results in the framework of the KdV approach (Samsonov, 1984). Solitary traveling waves were analyzed by Toda (1981) in chain systems with special potential. Potapov (1985) determined a principal possibility of the solitary transversal waves existence in essentially non-linear in®nite rods.
Although the stability problem is not considered here, some works on the subject will be referred. Zhuravlev (1977) and Vedenova et al. (1985) studied a stability of some vibro-impact motions by employing averaged equations and eliminating a discontinuity in the impact points. Nagaev (1985) considered a stability of vibro-impact motion with a number of inelastic impacts equal to in®nity. The author analyzed a one-dimensional system of n solids with masses change by a special exponential rule. Local and global bifurcations of periodic orbits in systems with vibroimpacts and an appearance of chaotic motions were examined in some recent works by Shaw and Holmes (1983), Shaw (1985), Ivanov (1994), Valkering (1994), Farahanchi and Shaw (1994) and Han and Luo (1995).
We analyze here traveling transversal waves in non-linear in®nite chains and rods with regards to elastic impacts. The work includes: a passage to a continual, long wavelength approximation in nonlinear in®nite chain systems (Section 2), one uses here the translation operator approach; a passage to the long wavelength approximation in non-linear in®nite rod system in the framework of Kirchho hypothesis (Section 3); an analysis of the solitary waves equations obtained previously (Section 4); an analysis of vibro-impact motions: the solitary waves with impacts in chains (Section 5) and periodical vibro-impact motions (Section 6) are considered.

Non-linear transverse waves in chains
Consider transverse waves in a symmetric in®nite chain consisting of particles (masses of the particles are equal to m ) connected by linear springs (Fig. 1). Stinesses of the springs are equal to g. The chain is found between two rigid barriers (boundaries) with gaps equal to L; a is the distance between undisplaced particles.
The renewal coecients are equal to 1. The impact is absolutely elastic. The solitary waves exist only in this case because if the energy fails at the time of impact, the solitary waves collapse.
The displacement vector of the particle with the number n is presented as: S n W n V n n 0, 21, 22, . . . : 1 where V n =an+U n . Here W n is a relative transverse displacement, U n is a relative longitudinal displacement and V n is an absolute (from zero) longitudinal displacement.
One assumes that the functions continuously depend on n: W n Wn, t, U n Un, t: Let a n W n À W nÀ1 2 a U n À U nÀ1 2 1=2 j S n À S nÀ1 j : The equations governing the transverse motions of this system are expressed as: mW HH n g1 À a=a n W n À W nÀ1 À g1 À a=a n1 W n1 À W n PW 0 mU HH n g1 À a=a n a U n À U nÀ1 À g1 À a=a n1 a U n1 À U n 0 n 0, 21, 22, . . . : 2 P(W) is the elastic impact function. It is impossible to utilize dierent forms of the function, for example, power form Manevitch et al., 1989): PX CX=L 2nÀ1 C const:, n 4 I: It will be restricted to low-frequency (long wavelength) approximation and will be realized in a passage to a continual system in place of system (2). Consider the function F(n ) de®ned on the integer number set. One writes the Taylor formula for F(n ), assuming the function is continuous and in®nitely dierent: Stinesses of the springs are equal to g. The chain is found between two rigid boundaries with gaps equal to L; a is the distance between undisplaced particles.

Fn21 Fn
utilising an operator form of the notation. The operator (3) is used by  and Manevitch et al. (1989), and called`a translation operator'. There are obvious combinations: Introduce the operators: A 1 À e À@ =@ n ; B e @ =@ n À 1: 5 One passes to a continual system replacing the discrete variable n by a continuous variable x.
Consider traveling stationary waves. Corresponding solutions are described by functions of a single argument named`phase', where: x is the space coordinate; k is the wavenumber, k = 2p/l; l is the length of the wave; o is the frequency of the traveling wave.
Let W=W(F ), U=U(F ), V=V(F ), respectively, VF na UF and @ V=@ n a @ U=@ n, etc: 6 Using the formulas, one obtains new representations for the operators A, B from (5): r! ak r @ r @ F r ka ( 1, this is a long wavelength approximation: 7 Note that a passage to a continual system of high-frequency approximation is possible using a continualization of an envelope (Kosevich and Kovalev, 1975;Manevitch et al., 1989): Taking into account (6) and (7) we obtain a system of ODEs characterizing the traveling stationary waves, in place of system (2): where n 2 =g/m; here the prime means a derivation by F.
One introduces a small parameter m=k 2 , also let y=a 2 . One expands the components of system (8) in the Taylor series in the vicinity of m=0 and retains the leading terms. Note that Finally, making a transformation, , one obtains after some transformations of the following system: The impact function is not presented in (10) and will be used later. In new variables the impact happens if v W v =L/E. One obtains from the ®rst equation of (10) the estimate: o 2 =O(m 2 ). Let o 2 =o 2 0 +0(m 3 ). From the second equation of (10), retaining the leading terms of O(m ), we have the following equation: Integrating by F, one has dU '+E 2 W ' 2 /2=D (D being arbitrary constant). Write out the ®rst equation of (10) taking into account (11) and the terms of O(m 2 ) in the second equation in (10): In the case W0=0 one has, with regard to impacts (in a position v W v =L/E ), a non-smooth`saw-tooth' solution . Consider here other solutions. Let o 2 0 Àyn 2 D=0. It is clear that Dr0. The second equation of (12) gives us the following: The analysis of solitary waves in elastic chains will be carried out in Section 4; the impact function will be considered in Section 5.

Non-linear transverse waves in rods
Consider the in®nite elastic rod being between two ideal and absolutely ®xed catches (barriers) with gaps equal to L.
The impact process is not considered in this section and will be considered later.
One describes the plane rod dynamics using equations of coupled longitudinal-bending (transverse) vibrations of rods, obtained by Potapov (1985). Note that the following analysis will be made for nonlinear rods within the framework of the Kirchho hypothesis (Kauderer, 1958).
The equations may be deduced from expressions of potential and kinetic energies containing quadratic, cubic terms and principal terms of the fourth degree.
Corresponding equations of motion are expressed as (Potapov, 1985): Here: U(X, t ) characterizes a longitudinal displacement; W(X, t ) characterizes a transverse displacement of a middle curve of the rod; the coecients a 1 E=2 3 2 l 1 6 À nn 1 1 À 2nn 2 4 3 n 3 and a 2 E=2 1 2 n 1 3n 2 4n 3 determine geometrical and physical non-linearities; C S E=r 0 p is the velocity of longitudinal waves propagation in the rod; C t M=r o p is the velocity of translation waves propagation in the rod; C M l=r 0 p ; r 0 is the density per unit length; E is the elasticity modulus of the material; n is Poisson's coecient; l and M are LameÂ constants of the second-order; n 1 , n 2 , n 3 are one's of the third-order; I r =ff F (Y 2 +Z 2 )dF is the polar moment of inertia; I Y =ff F Y 2 dF, I Z =ff F Z 2 dF are the axial moments of inertia; I 1 =ff F Z 4 dF, I 2 =ff F Z 2 (Y 2 +Z 2 /2)dF are the geometrical moments of inertia of the fourth-order; R r I r =F p is the polar radius of inertia; and R Y, Z I Y, Z =F p are the axial radii of inertia.
One transforms to dimensionless variables: in order that initial values of the new variables j U H 0 j 1, We are restricted to a long wavelength approximation corresponding to the Kirchho hypothesis. Orders of the principal parameters of wave processes are as follows: Eqs. (14) can be assumed to be of the form of ODEs: Then one can reduce the order of the system integrating this. Utilize the expansion Preserving the leading terms by the small parameter m, one obtains from the ®rst equation of (19) after integration, the following: D 1 is an arbitrary constant. By substituting the expression (19) to the second equation of (17) after integration, one has D 2 is an arbitrary constant.
In the case of the solitary transversal waves D 2 =0. Note that the solitary transversal waves exist if p 1 < 0, that is (C 2 S +2C 2 M )D 1 > o 2 0 . The last equation of (20) is similar to Eq. (13). But the phase places of Eqs. (13) and (20) are dierent. The point is that a shear is absent in the chain system under consideration.

22
The corresponding phase portraits are presented in Fig. 2. Every diagram has its own scale by axes. Curves on the places conform to the ®rst integral of Eqs. (20) or (13): (H is an arbitrary constant, this is an energy of the system).

Consider Case (c) appropriate to the case of the solitary waves in a non-linear rod system
It is possible to write out the well-known solution corresponding to the separatrix: where q=b 3 /(2b 1 ), C A2 b 1 p F, and the arbitrary parameter A (this is equal to the initial phase by a traveling variable) may be eliminated without loss of generality by imposing the initial condition.
Integrating the relation (24) one obtains B being an arbitrary constant. In addition, one writes: The solution W(F ) just W '(F ) and W0(F ) are bounded functions. Taking advantage of Eq. (19) one has the function U(F ): Here C is an arbitrary constant. The constant D has the meaning of the preliminary deformation of the rod. The function U(F ) is unbounded in this case. The constant D is not equal to zero because in the opposite case (D=0) one has o 2 0 < 0.

Consider Case (d) appropriate to the non-linear chain system
It is possible to write out the well-known solution corresponding to the separatrix: (the initial phase A may be eliminated without loss of generality by imposing the initial condition); one obtains, integrating (27): where B is an arbitrary constant. The transformation to new variables (9) was introduced provided that One has from here that This is a relation between the wave frequency o and the wave amplitude E. Using Eq. (20) one has: As a consequence one has the equality: U '=1ÀW ' 2 .
Here the constant B de®nes some initial value of the transverse displacement, the constant C de®nes some initial value of the longitudinal displacement.
The expressions (34) and (35) represent a one-parameter family of solutions. The amplitude E is a free parameter.

Vibro-impact motions in chains
The solution W(F ) in a form of the solitary wave (34) is unbounded at in®nity. In the presence of barriers the impacts take place necessarily.
Construct the phase place (W, W ') of the obtained solution. At the time of impact a sign of the velocity changes and the following principal cases are possible: 1. The impacts are possible and there is a`jumping' to the same phase trajectory. 2. The impacts are possible and there is a`jumping' to another phase trajectory. 3. The impacts are impossible, that is the solution does not exist in the presence of barriers.
In the new coordinates [see the transformation (9)] the distance between the barriers equal to 2L/E. Using the expression (34), Case (d), one obtains the relationship W '(W): where V 2 =b 1 /(2E 2 ). Branch`+' (branch a ) of the solution corresponds to W > B, and one`À' (branch b ) corresponds to W < B. The phase place (W, W ') is depicted in Fig. 4. The solutions increase or decrease inde®nitely. In Fig. 4 the constant B=À1; À1 < W < 4 (branch a ), À4 < W < À1 (branch b ).
If barriers exist, three variants are possible:

Consider the variant 1
The corresponding phase portrait is presented in Fig. 5. Here B=À1, L/E=1.7, the range of values of the variable W is the same as in Fig. 4.
There are two separated modes of vibrations, each of them strikes its own barrier. It reaches à jumping' to the same vibration mode (Fig. 5).
Evaluate the period of the vibro-impact modes. One uses the representation (34): The impact takes place if W=L/E (branch a ). One has from here L=E B 1 VE ln chVEF: In Fig. 5 a point A 1 corresponds to the sign`À' in relation (37) (here F=F 1 ), a point A 2 corresponds to the sign`+' (here F=F 2 ), moreover F 1 =ÀF 2 . One assumes that the impact is stereomechanical and takes place instantly. The stereomechanical impact theory was established by Newton (Goldsmith, 1960;Nagaev, 1985). A period of the vibro-impact motion is a time of the passage from A 1 to A 2 along the phase trajectory. We consider a period of the vibro-impact motion by branch a, T a 1 VE ln e LÀBEV e 2LÀBEV À 1 p e LÀBEV À e 2LÀBEV À 1 p or T a 2 VE lne LÀBEV e 2LÀBEV À 1 p : 38 One obtains from (38) Consider the branch b. One has: A period of the motion by branch b is a time of the passage from B 2 to B 1 : Consequently, the minimal and maximal times of the motion between barriers are not dependent on the vibration modes, the times are the functions of the system parameters only.

Consider the variant 2
The corresponding phase portrait is presented in Fig. 6. Here B=À2, L/E=1.7, À2 < W < 6. It is only presented in branch a.
T 1 is a time of the motion from A 1 to B 1 ; T 2 is the same from B 2 to A 2 . Let B=ÀL/EÀG(G > 0).
One obtains from (41) T 1 1 VE ln 1 À 1 À e À2GEV p e 2LV À e 4LV À e À2GEV p ; 1 À e À2GEV p : It is possible to make sure of T 1 =T 2 . Consequently, By analogy with the results one obtains (B=L/E+G; G > 0) It is possible to make sure that T (a ) =T (b ) (if the constants G are coinciding).

Consider the variant 3
The corresponding situation is depicted in Fig. 7. Here B=À1, L/E=1, À1 < W < 4 (branch a ), The variant can be represented as a special case of the variant 1. In this case: T (a ) =T max , T (b ) =T min =0. Simultaneously the variant 3 can be represented as a special case of the variant 2.
In this case branch b is absent and T (b ) =0, T (a ) =T max .

Construction of periodical vibro-impact solutions
As is obvious from the preceding, there are jumps of the function W '(F ) at the point of impact. The function W0(F ) is singular. Rejected inertia terms containing U0 are not small. Therefore, the examined vibro-impact approximation circumscribes suciently well only for the transverse displacements W(F ).
Consider, in the ®rst place, the variant 1 (Section 5.1). There are impacts to single barriers in this case. The phase values corresponding to impact times are the next: F=F 1 and F=F 2 . In the chosen coordinate system F 1 =ÀF 2 and T=F 1 ÀF 2 is a time of the motion between impacts [see (38); F 1 < 0, F 2 > 0]. Fig. 7. Phase portrait of the solution (34) with impact, the variant 3. The constant B=À1, L/E=1, À1 < W < 4 (branch a ), À4 < W < À1 (branch b ). Point A 1 corresponds to the sign`À' in relation (37), here F=F 1 ; point A 2 corresponds to the sign`+' in (37), here F=F 2 ; F 1 =ÀF 2 .
The vibro-impact discontinuous solution under consideration can be expressed as a continuous periodical function. Namely, a special non-smooth transformation can be introduced: xF T=2 fF=T gT 44 where curly brackets mean a selecting of the fractional component. The transformation (44) is like the non-smooth saw-tooth time transformation by  for essentially non-linear systems close to the vibro-impact ones and to the non-smooth transformation by Zhuravlev (1977). One has ÀT=2ExFET=2 if ÀI < F < +I. Impact points are the following: F T=2 Tk; k 0, 21, 22, . . . : A solution of the problem is a function W(x(F )) de®ned on ÀI < F < +I. The functions W(x(F )) and W '(x(F )) are presented in Fig. 8 on the interval [ÀT/2, 3 T/2]. Consider present the variant 2 (Section 5.2). In this case there are impacts to both barriers. The impacts in the vicinity of the origin takes place in the following points ( Fig. 6): A 1 : here F 1 < 0, coordinates of the barriers: F=L/E (branch a ), and F=ÀL/E (branch b ); B 1 : here F 2 < 0, coordinates of the barriers: F=ÀL/E (branch a ), and F=L/E (branch b ); B 2 : here F 3 > 0, coordinates of the barriers: F=ÀL/E (branch a ), and F=L/E (branch b ); A 2 : here F 4 > 0, coordinates of the barriers: F=L/E (branch a ), and F=ÀL/E (branch b ).
Here the variable x(F ) is given by formula (44).

Conclusions
We analyzed transversal traveling waves in one-dimensional essentially non-linear in®nite elastic systems: chains and beams. In the framework of a long wavelength approximation the result was that the non-linear systems with regard to impact (only the case of absolutely elastic impact is regarded) sometimes assume the solitary traveling waves. An analysis of the phase planes of a long wavelength traveling wave gives us conditions of the solitary waves existence. A special non-smooth transformation which gives us a periodical representation of the vibro-impact motion was also introduced. The presented analysis can be extended to other non-linear elastic systems including two-dimensional, such as plates or membranes.