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Dynamics of a weakly non-linear periodic chain
International Journal of Non-Linear Mechanics 36 (2001) 375}389
Dynamics of a weakly non-linear periodic chain G. Chakraborty, A.K. Mallik* Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, UP - 208016, India Received 1 December 1999
Abstract Harmonic wave propagation in an in"nite, non-linear periodic chain is investigated. Both hardening and softening types of non-linearity are considered. A perturbation approach is used to obtain both propagation and attenuation constants which are amplitude dependent for such a non-linear system. Only the "rst-order non-linear e!ect is retained in the analysis. Special attention is given to the bounding frequencies of the propagation zone. Propagation constants are used to obtain the non-linear natural frequencies and the associated non-linear modes of both "nite chains with homogeneous boundary conditions and endless cyclic chains. The computational e!ort is shown to be independent of the number of elements present in the chain. The interaction of two opposite-going primary waves in semi-in"nite or "nite chains are seen to generate secondary waves. The non-linear normal modes are found to consist of atmost two linear modes and for some boundary conditions exhibit restricted orthogonality properties. Some explicit numerical results are included to validate the wave-propagation approach for studying free vibration of such non-linear periodic chains. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Non-linear periodic chain; Wave propagation; Free vibration
1. Introduction A periodic structure is constituted by repeating identical systems which are called elements. Such a discrete structure without ends is called a cyclic periodic structure. A number of real life systems can be modelled as either "nite or cyclic periodic structures, e.g., a tall apartment block having identical stories, an aeroplane fuselage structure consisting of a shell reinforced at regular intervals by an orthogonal set of identical sti!eners, a pipe having equidistant identical supports, a ring with identical
* Corresponding author. Tel.: #91-512-590-7098; fax: #91512-597-995. E-mail address: [email protected] (A.K. Mallik).
supports at equal angular interval, etc. Harmonic waves can propagate in a linear in"nite periodic structure provided the frequency lies in certain &propagation zones'. Otherwise, the wave attenuates (even in the absence of damping) with a uniform attenuation constant. Utilizing the space periodicity of the system, the dynamics of such structures can be studied by analysing only a single element. Two methodologies, namely, wave-propagation approach and transfer matrix approach are generally used. An outstanding review of various vibrational aspects of one- and two-dimensional linear periodic structures has been published by Mead [1]. Analysis of the free and forced vibration of a "nite periodic structure becomes particularly simple by the wave-propagation approach. During
0020-7462/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 0 2 4 - X
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a free modal oscillation, the phases of the travelling waves change by an integer multiple of 2p as they traverse to and fro once across the total span. This is known as the &phase-closure principle'. Determination of the natural frequencies and the normal modes is easy for a periodic structure. Both analytical [2] and graphical [3] methods are available. If the structure is cyclic, the natural frequencies may be degenerate, i.e., two di!erent modes may exist at a single frequency [4,5]. Enormous analytical di$culties are encountered if the elements of the above periodic structure are non-linear. The wave propagation then becomes considerably complicated. Various studies have con"rmed that not only the waves may get modulated in a non-linear medium [6] but solitary wave solutions (solitons) are also possible [7}9]. In most of the studies [10}13], a one-dimensional chain consisting of non-linear elements is approximated as a non-linear continuum. This assumption is valid only for waves having wavelengths longer than the length of any element. Although this may be a reasonable approximation for studying lattice vibration and related topics, the mechanical structures are, on the other hand, too large to be assumed as a single continuous body. Moreover, these are often subjected to external convective loads with wavelength much smaller than the element length. The continuum approach has another disadvantage. Even if the formulation of the nonlinear partial di!erential equation remains easy for a periodic chain with single-degree-of-freedom elements, the complexity of the problem increases with increasing degree-of-freedom of the element. The modal oscillation is, in general, not possible in a non-linear, multi-degree-of-freedom system. However, periodic motions are still possible when all the coordinates cross their equilibrium positions simultaneously and reach the extremum positions similtaneously. Consequently, a concept of nonlinear normal modes, similar to the linear modes, has been proposed [14]. The associated frequencies are known as the non-linear natural frequencies. Both the non-linear normal modes and natural frequencies are not system properties but also depend on the extent of motion. Various methods have been suggested to obtain the non-linear normal modes and the associated natural frequencies
[15}18]. It has also been shown that the phaseclosure principle is satis"ed during a non-linear modal oscillation [19]. In this paper, the vibration of a one-dimensional periodic chain consisting of identical masses and weakly non-linear springs has been considered. A perturbation approach retaining only the "rstorder non-linear e!ect is used. Amplitude-dependent propagation and attenuation constants are derived for harmonic waves in an in"nite chain. For a "nite chain, the phase-closure principle is used to obtain the non-linear normal modes and the associated natural frequencies. Results are obtained for di!erent types of homogeneous boundary conditions and also for an endless cyclic chain. Results for forced vibration dealing specially with the non-linear e!ects will be presented in a subsequent paper.
2. Harmonic wave propagation in an in5nite periodic non-linear chain Consider an in"nite periodic chain consisting of identical springs and masses as shown in Fig. 1. The springs are assumed to be non-linear with the force}deformation characteristics given by F"k d#ek d, where d is the deformation due to a force F with e as a small parameter. The equation of motion of any mass (say the nth) is dxH L #k (xH!xH )#k (xH!xH ) m L L\ L L> dt #ek (xH!xH )#ek (xH!xH )"0, (1) L L\ L L> where xH is the displacement of the nth mass. Using L the following non-dimensional scheme q"t
k k , x "xH , L L k m
Fig. 1. Non-linear periodic chain.
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377
the following equations are obtained: dx L #(2x !x !x )#e(x !x ) L L> L\ L L\ dq #e(x !x )"0. (2) L L> To study the propagation of a harmonic wave with non-dimensional frequency u (non-dimensionalised by dividing the actual frequency by (k /m), the displacements are assumed in the form q q x " e SO# e\ SO, L 2 2 q q x " eIe SO# eI e\ SO, L> 2 2 q q x " e\Ie SO# e\I e\ SO, L\ 2 2
(3)
where the complex number k is called the propagation constant, q is the complex amplitude and the overbar indicates complex conjugate. Substituting Eq. (3) into Eq. (2) and equating the coe$cients of e SO from both sides, one gets !uq#(2!eI!e\I)q#eqq ;+(1!e\I)(1!e\I )#(1!eI)(1!eI ),"0. (4) Eq. (4) is solved to get the relationship between u and k. For a linear chain (e"0) the roots of k can have either purely imaginary (0("k"(p) or a complex value with imaginary part as p [20]. The real and imaginary parts of k are shown in Figs. 2a and b, respectively. The frequency regime with imaginary value of k is known as the propagation zone and the frequency regime having the real value of k is called the attenuating zone. For the non-linear case, the value of k depends not only upon u but also upon q, the complex amplitude of the nth mass. An inspection reveals that both k and !k are the roots of Eq. (4). As in the linear case, the negative and positive values of k denote the right- and left-going waves, respectively. This implies that the wave-propagation characteristics are equal in both directions. This is expected from the symmetry of the structure.
Fig. 2. Variation of propagation ("k ") and attenuation ("k ") ' 0 constant with frequency. e"0: (a) propagation constant; (b) attenuation constant.
For small non-linearity, it seems encouraging, at least at a "rst glance, to use a perturbation technique for "nding k for a given value of u and vice versa. For a given u, assume k in the following form: k"k #ek #2
(5)
and substitute this into Eq. (4). Equating the coe$cients of the like powers of e from both sides, one gets e: u"(2!eI !e\I ),
(6)
e: !k (eI !e\I )#qq +(1!e\I )(1!e\I ) #(1!eI )(1!eI ),"0.
(7)
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The value of k corresponds to that obtained from the linear theory. Eq. (7) is solved to obtain
0)u)4#16Z.
k "qq +(1!e\I )(1!e\I ) (8)
One can easily verify that the real and imaginary natures of k remains the same as that of k . Thus, the propagation characteristics of the non-linear system, predicted by the above analysis, follow that of the linear system. But this simple perturbation technique fails in some frequency ranges where the correct analysis can be carried out as discussed below. Let us now consider only the propagation zone for the non-linear system, where the exact value of the propagation constant (k purely imaginary) can be obtained in closed form by assuming
where both a and b are real with a#b"1. (9) Substituting Eq. (9) into Eq. (4) and equating the real and the imaginary parts, the following equations are obtained: (10)
and (11)
where Z"(3/4)eqq . Eq. (10) is seen to be trivially satis"ed. From Eq. (11) the roots of a are obtained as a"1!+!2$2(1#4Zu,/8Z.
(12)
For Z'0 (e.g., e'0, i.e., a hard system) the negative sign before the square root yields a'1 and can be discarded. Hence, the only possible root is a"1!+!2#2(1#4Zu,/8Z. The restriction "a")1 implies 0)+!2#2(1#4Zu,/8Z)2
a"1#+!2$2(1!4"Z"u,/8"Z".
(15)
It can be seen that again the negative sign before the square root cannot give a feasible solution of a since this solution does not converge as "Z"P0. Thus, the only possible root is a"1#+!2#2(1!4"Z"u,/8"Z".
(16)
The condition "a")1 yields 0)u)4!16"Z".
(17)
Thus, for both hard and soft oscillators the bounding frequencies for the propagation zone are given by
eI"a#ib
u"2(1!a)#4(1!a)Z,
(14)
Now, for Z(0 (if e.g., e(0, i.e., a soft system),
#(1!eI )(1!eI ),/(eI !e\I ).
b"b/(a#b)
or
(13)
0)u)4#16;eqq .
(18)
It can be veri"ed from Eq. (4) that the bounding frequencies predicted by the equality signs of expression (18) correspond to eI"1 or eI"!1. The variation of the upper bounding frequency (u ) with the amplitude of the nth mass (i.e., "q") is shown in Figs. 3a and b for e'0 and e(0, respectively. The lower bounding frequency (u ) is zero (Eq. (18)) for both hard and soft systems under consideration. There exist two types of regions in each "gure. In region of type-I, the wave propagations of the linear and non-linear chains have similar characteristics. In such a region, the propagation and attenuation constants can be obtained by perturbation analysis (i.e., from Eq. (8)). In region of type II, however, the wave propagation in the non-linear structure shows marked di!erence from that of the linear one. The perturbation analysis fails in this region and the propagation constants can be determined as follows. For e'0, the value of k is purely imaginary in region II and can be obtained from Eq. (13). For e(0, the value of eI is real in region II which is an attenuation zone. To determine the corresponding attenuation constant (k), eI"a (a real) is
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379
with the value of a written as a"!1#ec.
(21)
Substituting Eqs. (20) and (21) into Eq. (19) and neglecting the terms of order higher than e, one gets ec "eu 2 or ec"$(2(u!4!16;eqq ).
Fig. 3. (a) Variation of upper bounding frequency (u ) of the propagation zone with amplitude; e'0. (b) Variation of upper bounding frequency (u ) of the propagation zone with ampli tude; e(0.
substituted into Eq. (4) and after simpli"cation one gets the following algebraic equation:
1 3 !u# 2!a! # eqq +(1!a) a 4
1 # 1! "0. a
(19)
Since eI"!1 at the upper bounding frequency and any nearby frequency in regoin II can be expressed as (see Eq. (18)) u"4#16;eqq #eu ,
(20)
(22)
The negative value of c corresponds to Re(k)'0 and hence to the left attenuating wave. The other root corresponds to the right attenuating wave. For the non-linear periodic chain, the attenuation or propagation constant depends on the amplitude ("q") of the nth mass. Figs. 4a and b show the variation of attenuation and propagation constants for linear and non-linear systems (both hard and soft). For a propagating wave without attenuation, the phases rather than the amplitudes of the masses vary and the propagation constant is "xed for a given amplitude. But when the wave attenuates, the amplitudes of the masses change progressively. Consequently, there is no "xed attenuation constant for a non-linear structure. The change of the wave characteristics with amplitude (see Figs. 3a and b) accounts for a special phenomenon observed in a non-linear chain. For a soft system (e(0), if the initial amplitude and frequency lie in region II, the wave attenuates and the amplitude starts decaying. After attenuating over a length spanned by few periodic elements, the value of the amplitude enters region I. The wave then starts propagating without any further decay. A typical amplitude envelope looks like that in Fig. 5 where the masses are indicated by the dots. In this "gure the longitudinal displacement amplitudes of various masses are plotted in the transverse direction. The dots are not coincident with the crest or trough of the diplacement wave to indicate the phase di!erence (propagation constant) between successive elements.
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Fig. 6. An element of the non-linear periodic chain.
carried out considering only a single element [2]. The boundary frequencies of the propagating zones are related to the natural frequencies of the element. In this section, such an analysis is carried out for the non-linear periodic structure. Consider the element shown in Fig. 6, where Q refers to the non-dimensional exterior coordinate with the corresponding non-dimensional force F (non-dimensioanlised by dividing the actual force by (k /k ) and the subscript L and R refer, respec tively, to the left and right ends of the element. The harmonic responses are assumed as q q Q " e SO# e\ SO * 2 2
(23)
and Fig. 4. E!ect of non-linearity on the propagation and attenuation constant. **: e"0; * ) *: "q"("e""0.3, e'0; } } }: "q"("e""0.3, e(0: (a) propagation constant; (b) attenuation constant.
q q Q " eIe SO# eI e\ SO. 0 2 2
(24)
The corresponding forces can be written as f fM F " e SO# e\ SO * 2 2
(25)
and
fM f F "! eIe SO# eI e\ SO . 0 2 2
(26)
Fig. 5. Longitudinal displacement amplitudes of a soft nonlinear periodic chain (e(0) with the amplitude of the centremass (marked 0) and frequency in region-II of Fig. 3b.
The negative sign in F is necessary for the condi0 tion of force equilibrium. The complex force amplitude is denoted by f. The non-dimensionalised equations of motion for the end masses are
3. Vibration of a single element and the wave propagation
1 dQ * #(Q !Q )#e(Q !Q )"F * 0 * 0 * 2 dq
One of the most important advantages rendered by the space periodicity of a linear repetitive structure is that the wave-propagation analysis can be
(27)
and 1 dQ 0 #(Q !Q )#e(Q !Q )"F . 0 * 0 * 0 2 dq
(28)
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
Substituting Eqs. (23)}(26) into Eqs. (27) and (28) and equating the coe$cients of e SO from both sides of the resulting equations one gets the following relations:
q q 3 1 !u # (1!eI)# eqq (1!eI)(1!eI ) 2 2 2 8 f " 2
(29)
and
eI q q 3 !u ! (1!eI)! eqq (1!eI)(1!eI ) 2 2 2 8 f "!eI . 2
(30)
381
vibratory mode which can be obtained as discussed below. The antisymmetric shape is assumed as (see Eqs. (23) and (24) with eI"!1) q q Q " e SO# e\ SO, * 2 2
q q Q "! e SO# e\ SO 0 2 2
and F "F "0. Substituting these in Eqs. (27) or * 0 (28) and applying the harmonic balance method, one gets u"4#16;eqq ,
(33)
which is the non-linear natural frequency of the periodic element and identical to u given by Eq. (32).
Elimination of f from Eqs. (29) and (30) yields !uq#(2!eI!e\I)q 4. Interaction of harmonic waves in a semi-in5nite periodic chain
3 # eqq +eI>I (1!e\I)(1!e\I ) 4 #(1!eI)(1!eI ),"0.
(31)
Eqs. (31) and (8) are seen to be identical if k"!k, i.e., if k is purely imaginary. Thus, the element can still be used to study the harmonic wave propagation. During attenuation of the waves, the elemental approach does not give correct result. This is expected since there is no uniform attenuation constant and each element (having di!erent amplitudes) behaves di!erently from the other. The lower (u ) and upper (u ) bounding frequencies of the propagation zones are obtained by substituting, respectively, eI"1 and eI"!1 into Eq. (31) as u "0 and u "(4#16;eqq .
(32)
It can be veri"ed that u and u are the two non-linear natural frequencies of the element. The zero frequency corresponds to the rigid-body mode and the other frequency is the only frequency of the
Standing modes are possible for "nite, linear periodic structures. The knowledge of the wave propagation in an in"nite structure can then be of great use in obtaining the normal modes and the corresponding natural frequencies [2]. During such a modal oscillation, two opposite-going waves with same phase velocities travel in such a way that the total phase di!erence is an integral multiple of 2p after traversing to and fro once across the span. The characteristics of a wave in a linear chain are not a!ected by the oppositely travelling wave. Consequently, the phase change of each wave is considered separately to get the natural frequencies. In a non-linear chain, however, the two opposite-going waves undergo complicated interaction rather than just getting linearly superimposed and the wave characteristics are altered. In what follows, the interaction of the waves in a semi-in"nite nonlinear chain is studied by considering a perturbation of the linear waves. The boundary is assumed to be linear and non-dissipative. The phase of the re#ected wave, generated at the boundary, di!ers from that of the incident wave by a certain amount (k , say).
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The total harmonic response of the nth mass can now be written as a x " [(e\ LI#e I e LI)#eq ]e SO#c.c., L 2 L
(34)
where ik and !ik are, respectively, the propagation constants of the left and right propagating waves, both of frequency u. The term (e\ LI# e I e LI) indicates the linear superposition of the two opposite-going waves. The e!ect of the non-linearity is taken into account by introducing a small perturbation in the form of eq which also is a comL plex quantity. For the same value of the propagation constants, let the waves in the corresponding linear system (i.e., e"0) have a frequency u . Then one can write u"u #eu . (35) Substituting Eq. (34) into equation of motion (Eq. (2)) and collecting the coe$cients of e SO from both sides one gets a a !u ( #eq )# (2!e I!e\ I)
L L 2 L 2
Eq. (38), as expected is identical to Eq. (6) which holds good for the linear in"nite chain. Using the relation given by Eq. (37) and expressing
and L\
in terms of one can, after algebraic manipL> L ulation, write the right-hand side of Eq. (39) (in terms of only) as given below: L a 3 u ! aa [3(1!e I)(1!e\ I) 2 L 8 3 #(1!e\ I)(1!e I)] ! aa e\ I [(1!e I) L 8 #(1!e\ I)](e\ LI#e I e LI). The presence of the term in the above expression L yields an unbounded value of q . So the solvability L condition requires vanishing of the coe$cient of
yielding L u "aa [(1!e I)(1!e\ I) #(1!e\ I)(1!e I)].
(40)
a #e (2q !q !q ) L> L\ 2 L
Combining Eqs. (38) and (40) the frequency relation is obtained as
3 )( M ! M ) # eaa [( !
L L\ L L\ 8
u"(2!e I!e\ I)#eaa [(1!e I)(1!e\ I)
#( !
)( M ! M )]"0, L L> L L> where
#(1!e\ I)(1!e I)]. From Eq. (39), the response q is found to be L
"e\ LI#e I e LI. (37) L Eq. (35) is now substituted into Eq. (37) and the coe$cients of the like powers of e are collected to get e: [!u #(2!e I!e\ I)] "0 L and
3 aa e\ I q "! L 4 [(e I#e\ I)!(e I#e\ I)] ;[(1!e I)#(1!e\ I)](e\ LI#e I e LI). (42)
(38)
a a e: !u q # (2q !q !q ) 2 L 2 L L> L\ a 3 "u # aa [(
2 L L 8 )( M ! M ) L\ L L\ #( !
)( M ! M )]. L L> L L>
(41)
(36)
!
(39)
Thus, it should be noted that due to the interaction of two opposite-going harmonic waves of propagation constant $ik, another set of opposite-going waves of propagation constant $3ik are generated. The phases of the latter pair of waves di!er by 3k . These new waves are designated as the second ary waves (with propagation constants $3ik) in the rest of the paper whereas the original waves are called the primary waves (with propagation constants $ik).
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
Attention must be drawn at this stage to some special situations where q may vanish implying L non-interacting waves. This situation occurs when e I "$1
(43)
and k satis"es the following equation: e\ LI#e I e LI"j(e\ LI#e I e LI)"j . (44) L It may be mentioned that e I "1 is satis"ed by a free boundary and e\ I "!1 is satis"ed by a "xed boundary. When Eqs. (43) and (44) are satis"ed the solution for u is obtained (by following the general method described above) as u"(2!e I!e\ I) # eaa [3(1!e I)(1!e\ I) #(1!e\ I)(1!e I)]#eaa je\ I ;[(1!e I)#(1!e\ I)]
(45)
In the presence of non-linearity secondary waves are generated. Further, when the primary waves have propagation constants $ik, the secondary waves have propagation constants $3ik. The phase changes of the secondary waves at the boundaries are 3k and 3k after re#ection. Thus, it * 0 is evident that the secondary waves always satisfy the phase-closure principle if the primary waves do so. Hence, for the modal oscillations of a "nite non-linear chain, the allowed propagation constants are still given by Eq. (46). Knowing the value of k, the non-linear natural frequencies are obtained from Eq. (41) where the amplitude a is the strength of the incident wave. However, in a "nite structure the non-linear mode shapes and corresponding natural frequencies are usually obtained as a function of the amplitude of a speci"c mass particle (say the rth mass). If b be the amplitude of the rth mass, then b"(e\ PI#e I* e PI)a
and q "0. L
383
(48)
or
5. Normal modes and natural frequencies of a 5nite non-linear chain As discussed in Section 1, the non-linear normal modes of a multi-degree-of-freedom system correspond to periodic motions when all the coordinates cross their equilibrium positions simultaneously and also reach their extremum positions simultaneously. The associated frequencies are called the non-linear natural frequencies [15]. In a "nite linear periodic chain, the harmonic waves get re#ected at the boundaries and form a standing mode only when the phase-closure principle is satis"ed. If the phase changes at the left and right boundaries are k and k , respectively, then * 0 the phase-closure principle suggests that the modal oscillation is possible if the propagation constant ik bears the following relationship [2]:
aa "bbM /(e\ PI#e I* e PI)(e PI#e\ I* e\ PI).
(49)
Hence, the natural frequencies are calculated from Eq. (41) after replacing aa by bbM using Eq. (49). Similarly, the non-linear normal modes can be expressed as t " #eq , (50) L L L where and q are given by Eqs. (47) and (42), L L respectively, after replacing aa by bbM using Eq. (49). In what follows some explicit results for speci"c boundary conditions are included. 5.1. Non-linear chain with both ends xxed Consider a non-linear chain of N masses having both ends "xed as shown in Fig. 7.
2Nk#k #k "2mp, m"0, 1, 2,2, (46) * 0 where N is the number of elements in the chain. Furthermore, in a linear chain the modal response of the nth mass is given by
"e\ LI#e I* e LI. L
(47)
Fig. 7. Non-linear periodic chain with "xed}"xed boundary conditions.
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In this case, k "k "p. The phase-closure * 0 principle yields
p #(1!e\ IH )(1!e IH ),! +(1!e IH ) "p"
jp k" , j"1, 2, 3,2 . H N#1
#(1!e\ IH ),],
(51)
Thus for the jth non-linear mode the natural frequency in terms of the amplitude of the "rst mass (r"1) is u"(2!e IH !e\ IH ) H 9 bbM #e [(1!e IH )(1!e\ IH ) 4 4 sin k H #(1!e\ IH )(1!e IH )]
(52)
and the jth non-linear normal mode is 3 bbM tH" H#e L L 4 4 sink H (1!e IH )#(1!e\ IH ) ; [(e IH #e\ IH )!(e IH #e\ IH )] ;(e\ LIH !e LIH ),
(53)
with
H"e\ LIH !e LIH . L For any given j it can be veri"ed that
(54)
(55) e\ LIH !e LIH " H if 3j(N#1, L p " N with p"3j!2(N#1) "p" L
(59)
and tH" H. L L Thus, the non-linear natural frequencies can be easily obtained either from Eq. (52) or graphically from the plots of k vs. u for a given "b" using Eq. (52). So the computation e!ort is independent of the number of periodic elements (N), a feature clearly established in the literature for linear periodic structures [2]. From Eqs. (53) and (54) it is obvious that any non-linear mode consists of either two linear normal modes when conditions (55) and (56) apply. On the other hand, if condition (57) or (58) holds good then the non-linear normal mode is identical to the linear mode of the same order. Further, a careful study of conditions (55) and (56) reveal that any odd-order non-linear mode consists of the corresponding linear mode and another odd-order linear mode. Exactly similar conclusion can be drawn for even-order modes. The orthogonality of linear modes immediately imply that any odd (even)-order non-linear mode is orthogonal to all even (odd)-order non-linear modes. 5.1.1. Numerical results for N"3 In this section the "rst non-linear natural frequency with N"3 is obtained directly considering a three-degree-of-freedom system and veri"ed against that obtained from Eqs. (51) and (52) with j"1 and N"3. The equations of motions of the system can be written in the following non-dimensional form:
if 3j'N#1 and "p"Oj,
(56)
"0 if 3j"N#1.
(57)
dX M #KX#eN"0, dq
(58)
where X"+x , x , x ,2, M"diag+1, 1, 1,2, 2 !1 0
A special situation occurs when p"3j!2(N#1), 3j'N#1 and "p""j.
(60)
In this case, u is given by Eq. (45) with aa replaced by bbM as given below:
K" !1
u"(2!e I\H!e\ IH ) H 3 bbM # e [+3(1!e IH )(1!e\ IH ) 4 4 sin k H
x #(x !x ) N" (x !x )#(x !x ) . (x !x )#x
0
2 !1
!1 , 2
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
The linear mode shapes are ' "+1, (2, 1,2, ' "+1, 0, 1,2 and ' "+1, !(2, 1,2. Assume the responses as X"b( cos u q, (61) H H where ( "+t , t , t ,2 is the jth non-linear H H H H normal mode and u is the corresponding nonH linear natural frequency. Substituting Eq. (61) into Eq. (60) and applying the method of harmonic balance, one gets !uMX#KX#eN "0, H H where
(62)
t #(t !t ) H H H 3 N " b (t !t )#(t !t ) . H H H H H 4 t #(t !t ) H H H For small non-linearity one can assume ( "' #e* #2 H H H and
(63)
(64)
u"(uJ )#ed #2, (65) H H H where uJ is the jth linear natural frequency. SubstiH tuting Eqs. (64) and (65) into Eq. (62) and collecting the coe$cients of like powers of e from both sides, one obtains e: !(uJ )M' #K' "0, (66) H H H e: !(uJ )M* #K* "d M' !N , (67) H H H H H ( where N is the vector obtained by replacing t 's ( IH (k"1, 2, 3) by 's in N (see Eq. (63)). Eq. (66) is IH H trivially satis"ed. To solve Eq. (67), N can be ( written as
385
and solution of Eq. (67) yields c M' N N . * " H [(uJ )!(uJ )]('2M' ) N H N N N$H For the chosen example, if j"1, c "b(9!6(2), c "0, c "b((2!1). Thus, the "rst natural frequency is u "(uJ )#eb(9!6(2). The same value of u can be obtained from Eq. (52) by substituting k "p/4. H Furthermore, c "0 implies that the "rst non linear normal mode has contribution from only the "rst and the third linear normal modes but not from the second one. It can be veri"ed that similar is the case for the third mode. The second nonlinear mode is identical to the second linear mode. 5.2. Non-linear chain with both ends free Consider the non-linear chain as shown in Fig. 8. In this case k "k "0 and the phase-closure * 0 principle yields jp k" , H N
j"0, 1, 2, 3,2, N.
(68)
The value k "0 corresponds to the rigid-body H mode. The jth non-linear natural frequency in
N " c M' , ( N N N where '2 N c " N ( . N '2 M' N N The solvability condition yields d "c , H H
Fig. 8. (a) Non-linear periodic chain with free}free boundary conditions, (b) A typical element.
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terms of the amplitude of the "rst mass is obtained as follows: u"(2!e IH !e\ IH ) H 9 bbM #e [(1!e IH )(1!e\ IH ) 4 4 #(1!e\ IH )(1!e IH )] if jON,
5.3. Non-linear chain with one end xxed and the other end free (69) Consider a non-linear chain which is "xed at one end and free at the other end as shown in Fig. 9. In this case k "p and k "0, thereby yielding * 0 1 p k " j! , j"1, 2, 3,2, N. (73) H 2 N
or u"(2!e IH !e\ IH ) H 3 bbM #e (2!e IH !e\ IH ) if j"N, 4 4 3 "4# ebbM ;16. 4
(70)
The non-linear normal modes are given by
The jth non-linear natural frequency and the corresponding normal modes are given, respectively, by Eqs. (52) and (53). One can also verify that e\ LIH !e LIH " H\ if 3j(N#1, L (2p!3) "
N\ "(2p!3)" L
3 bbM tH" H!e L L 4 4 (1!e IH )#(1!e\ IH ) ; [(e IH #e\ IH )!(e IH #e\ IH )] ;(e\ LIH #e LIH ) if jON
The observations made in Section 5.1 regarding restricted orthogonality of various non-linear modes for the "xed}"xed chain remain una!ected for the free}free chain.
with p"3j!2N if 3j'N#1. (71)
" H if j"N. (72) L Comparing Eq. (70) with Eq. (34), it is easily seen that the highest non-linear natural frequency of the chain (for any value of N) is identical to that of the periodic element. It may be further noted from Eq. (72) that at this frequency, the non-linear normal mode is identical to the linear mode of the chain when each element oscillates independently. As in Section 5.1, for the present system it can be shown that e\ LIH #e LIH " H if 3j(N#1, L " N with p"3j!2(N#1) L if 3j'N#1 and "p"Oj.
It should be noted that in this case the even- or odd-order non-linear normal modes have contribution from both even- and odd-order linear normal modes. Another interesting feature of the present system needs to be pointed out. An inspection of Eqs. (51) and (73) suggests that the propagation constant for the jth mode of the chain under discussion is identical to the (2j!1)th mode of a "xed}"xed chain with 2N!1 masses. Consequently, the non-linear natural frequencies and normal modes of the former are given by those of the latter. Similarly, comparing Eqs. (68) and (73) it is found that the propagation constant of the jth mode of the "xed}free chain with N masses is identical with that of the (2j!1)th mode of a free}free chain
A special case occurs if "p""j, then u is given by H 4 bbM u"(2!e IH !e\ IH )#e (2!e IH !e\ IH ) H 44 and tH" H. L L
Fig. 9. Non-linear periodic chain with "xed}free boundary conditions.
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387
having (2N#1) masses. However, the natural frequency and the non-linear mode shape of a "xed}free chain given by Eqs. (52) and (53) are di!erent from those of a free}free chain. The natural frequencies and the associated normal modes of a free}free chain are functions of the amplitude of the end mass, whereas the corresponding quantities of the "xed}free chain are calculated with respect to the "rst mass from the "xed end or the Nth mass from the free end. Thus, when the frequencies and the normal modes of the free}free chain are calculated as a function of the amplitude of the same mass the following results are yielded: u"(2!e IH !e\ IH ) H 9 bbM #e [(1!e IH )(1!e\ IH ) 4 4 cos(N!1)k H #(1!e\ IH )(1!e IH )] if jON,
(74)
and 3 bbM tH" H!e L L 4 4 cos(N!1)k
H
(1!e IH )#(1!e\ IH ) ; [(e IH #e\ IH )!(e IH #e\ IH )] ;(e\ LIH #e LIH ) if jON.
(75)
As k "(2j!1)p/2N, it can be veri"ed easily that H cos(N!1)k "sin k . Thus, Eq. (74) becomes H H identical to Eq. (52). However, the mode shapes do not match. This is due to di!erent ways of numbering the elements of the chain in these two cases. By changing the numbering suitably (replacing n by N!n) it can be shown that Eqs. (75) and (53) generate the same modes. 5.4. Cyclic one-dimensional non-linear chain Consider a cyclic chain made of N masses as shown in Fig. 10. For studying the free vibration of such a structure one needs to take a reference mass (say the mass numbered 1). In the absence of any boundary, the waves travelling in a cyclic structure undergo no re#ection. However, two oppositegoing waves still exist. The phase di!erence
Fig. 10. Cyclic one-dimensional non-linear chain.
between these two waves depend upon the way in which the system is disturbed. Depending on the initial disturbance, the system may vibrate symmetrically or antisymmetrically about the reference mass. In the symmetric mode, the reference mass undergoes "nite oscillation and in the other case this mass remains stationary. The phase di!erence between the opposite-going waves is k "0 for the symmetric mode and k "p for the antisymmetric mode. Since there is no re#ection in this case, both the clockwise and anticlockwise waves must satisfy the phase-closure principle independently. The phase constant at the jth non-linear natural frequency must satisfy the following relation: 2jp k" H N
where j"1, 2,2, N .
(76)
The value of N depends upon various condi tions. If N is even then the maximum number j can take is N/2 when the excitation is symmetric and (N/2!1) when the excitation is antisymmetric. Similarly, if N is odd then the maximum value of j is (N!1)/2 for both symmetric and antisymmetric excitations. In all cases the system has atmost N!1 natural frequencies. The values of j"0 and N correspond to no motion for an antisymmetric excitation and a rigid-body motion for symmetric excitation.
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For a value of k given by Eq. (76), the non-linear H natural frequencies and the corresponding mode shapes are obtained from Section 5.1 (see Eqs. (52) and (53)) for antisymmetric excitation and from Section 5.2 (see Eqs. (69)}(72)) for symmetric excitation. These quantities are calculated as a function of the amplitude of the mass adjascent to the reference mass for the former case and as a function of the amplitude of the reference mass for the latter. However, when the natural frequencies of both the cases are calculated with respect to the amplitude of the adjacent mass, the term bbM in Eqs. (69)}(72) are replaced by bbM /cos k . For a linear system with H odd number of elements, both symmetric and antisymmetric modes exist at all natural frequencies. However, for a non-linear chain the degenaracy is absent. In a chain with even number of elements, symmetric and antisymmetric modes at the same natural frequency may not be distinguished except for a &rotation' of the mode shape through an integral number of elements.
latter pair of waves are three times that of the former pair. (vi) The normal modal oscillation of a "nite chain is possible at certain frequencies where the associated propagation constants satisfy the phase-closure principle. The corresponding nonlinear natural frequencies can be obtained either graphically or analytically. In either case, the computational e!ort is independent of the number of elements in the chain. (vii) For a chain having both ends "xed or free, the non-linear normal modes are either identical to the linear modes or comprise of only two linear modes. In the latter case, the odd-order non-linear modes are orthogonal to the even-order non-linear modes. For a chain with one end "xed and the other end free, such restricted orthogonality condition does not hold good. (viii) For a cyclic one-dimensional chain, the non-linear normal modes are either symmetric or antisymmetric. The degeneracy, present in such a linear chain with odd number of elments, vanishes because of the non-linear e!ects.
6. Conclusions The major conclusions of the paper are listed below: (i) The value of the propagation constant depends on the amplitude of the harmonic wave. (ii) In some frequency range, a simple perturbation of a linear periodic chain fails to predict the wave-propagation characteristics of the weakly non-linear periodic chain. (iii) The propagation (and not the attenuation) constant of an in"nite chain can be obtained by considering only a single element. The bounding frequencies of the propagation zone are identical with the non-linear natural frequencies of a single element. (iv) The amplitude dependence of the propagation characteristics may result in transforming an attenuating wave into a propagating wave for softening type non-linearity. (v) In a semi-in"nite non-linear periodic chain, two opposite-going harmonic (primary) waves interact to form another pair of opposite-going (secondary) waves. The propagation constants of the
References [1] D.J. Mead, Wave propagation in continuous periodic structures: research contributions from Southampton, 1964}1995, J. Sound Vib. 190 (1996) 495}524. [2] D.J. Mead, Wave propagation and natural modes in periodic systems: I: monocoupled systems, J. Sound Vib. 40 (1975) 1}18. [3] G. Sen Gupta, Natural #exural waves and the normal modes of periodically supported beams and plates, J. Sound Vib. 13 (1970) 89}101. [4] A.K. Mallik, D.J. Mead, Free vibration of thin circular rings on periodic radial supports, J. Sound Vib. 54 (1977) 13}27. [5] A.K. Mallik, G.V.R. Murthy, Free out-of-plane vibration of circular rings on identical equi-spaced elastic supports, J. Sound Vib. 122 (1988) 433}439. [6] A.H. Nayfeh, D. Mook, Nonlinear Oscillations, Wiley, New York, 1979. [7] M. Toda, Theory of Nonlinear Lattices, Springer, Berlin, 1981. [8] M. Peyrard, M.D. Kruskal, Kink dynamics in the highly discrete Sine-Gordon system, Physica D 14 (1984) 88}102. [9] R. Hirota, K. Suzuki, Theoretical and experimental studies of lattice solitons in nonlinear lumped networks, Proc. IEEE 61 (1973) 1483}1491. [10] P. Rosenau, Quasicontinuous spatial motion of a massspring chain, Physica D 27 (1987) 224}234.
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389 [11] A.M. Kosevich, A.S. Kovalev, Self localization of vibrations in a one-dimensional anharmonic chain, Sov. Phy. } JETP 40 (1975) 891}896. [12] A.F. Vakakis, M.E. King, A.J. Pearlstein, Forced localization in a periodic chain of non-linear oscillators, Int. J. Non-Linear Mech. 29 (1994) 429}447. [13] A.F. Vakakis, M.E. King, Non-linear wave transmission in a monocoupled elastic periodic system, J. Acoust. Soc. Am. 98 (1995) 1534}1546. [14] R.M. Rosenberg, On non-linear vibration of systems with many degrees of freedom, Adv. Appl. Mech. 9 (1966) 155}243. [15] W. Szemplinska-Stupnicka, The Behavior of Nonlinear Vibrating Systems, Vol. II, Kluwer Academic Publishers, London, 1990.
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[16] S.W. Shaw, C. Pierre, Normal modes of vibration for non-linear continuous systems, J. Sound Vib. 169 (1994) 319}347. [17] A.H. Nayfeh, S.A. Nayfeh, On nonlinear modes of continuous systems, J. Vib. Acoust. 116 (1994) 129}136. [18] G. Chakraborty, A.K. Mallik, H. Hatwal, Normal modes and near-resonance response of beams with non-linear e!ects, J. Sound Vib. 210 (1998) 19}36. [19] G. Chakraborty, A.K. Mallik, Wave propagation in and vibration of a travelling beam with and without nonlinear e!ects: Part I: free vibration, J. Sound Vib. (2000) in press. [20] L. Cremer, M. Heckl, Structure-Borne Sound, Springer, Berlin, 1971.
Dynamics of a weakly non-linear periodic chain G. Chakraborty, A.K. Mallik* Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, UP - 208016, India Received 1 December 1999
Abstract Harmonic wave propagation in an in"nite, non-linear periodic chain is investigated. Both hardening and softening types of non-linearity are considered. A perturbation approach is used to obtain both propagation and attenuation constants which are amplitude dependent for such a non-linear system. Only the "rst-order non-linear e!ect is retained in the analysis. Special attention is given to the bounding frequencies of the propagation zone. Propagation constants are used to obtain the non-linear natural frequencies and the associated non-linear modes of both "nite chains with homogeneous boundary conditions and endless cyclic chains. The computational e!ort is shown to be independent of the number of elements present in the chain. The interaction of two opposite-going primary waves in semi-in"nite or "nite chains are seen to generate secondary waves. The non-linear normal modes are found to consist of atmost two linear modes and for some boundary conditions exhibit restricted orthogonality properties. Some explicit numerical results are included to validate the wave-propagation approach for studying free vibration of such non-linear periodic chains. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Non-linear periodic chain; Wave propagation; Free vibration
1. Introduction A periodic structure is constituted by repeating identical systems which are called elements. Such a discrete structure without ends is called a cyclic periodic structure. A number of real life systems can be modelled as either "nite or cyclic periodic structures, e.g., a tall apartment block having identical stories, an aeroplane fuselage structure consisting of a shell reinforced at regular intervals by an orthogonal set of identical sti!eners, a pipe having equidistant identical supports, a ring with identical
* Corresponding author. Tel.: #91-512-590-7098; fax: #91512-597-995. E-mail address: [email protected] (A.K. Mallik).
supports at equal angular interval, etc. Harmonic waves can propagate in a linear in"nite periodic structure provided the frequency lies in certain &propagation zones'. Otherwise, the wave attenuates (even in the absence of damping) with a uniform attenuation constant. Utilizing the space periodicity of the system, the dynamics of such structures can be studied by analysing only a single element. Two methodologies, namely, wave-propagation approach and transfer matrix approach are generally used. An outstanding review of various vibrational aspects of one- and two-dimensional linear periodic structures has been published by Mead [1]. Analysis of the free and forced vibration of a "nite periodic structure becomes particularly simple by the wave-propagation approach. During
0020-7462/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 0 2 4 - X
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a free modal oscillation, the phases of the travelling waves change by an integer multiple of 2p as they traverse to and fro once across the total span. This is known as the &phase-closure principle'. Determination of the natural frequencies and the normal modes is easy for a periodic structure. Both analytical [2] and graphical [3] methods are available. If the structure is cyclic, the natural frequencies may be degenerate, i.e., two di!erent modes may exist at a single frequency [4,5]. Enormous analytical di$culties are encountered if the elements of the above periodic structure are non-linear. The wave propagation then becomes considerably complicated. Various studies have con"rmed that not only the waves may get modulated in a non-linear medium [6] but solitary wave solutions (solitons) are also possible [7}9]. In most of the studies [10}13], a one-dimensional chain consisting of non-linear elements is approximated as a non-linear continuum. This assumption is valid only for waves having wavelengths longer than the length of any element. Although this may be a reasonable approximation for studying lattice vibration and related topics, the mechanical structures are, on the other hand, too large to be assumed as a single continuous body. Moreover, these are often subjected to external convective loads with wavelength much smaller than the element length. The continuum approach has another disadvantage. Even if the formulation of the nonlinear partial di!erential equation remains easy for a periodic chain with single-degree-of-freedom elements, the complexity of the problem increases with increasing degree-of-freedom of the element. The modal oscillation is, in general, not possible in a non-linear, multi-degree-of-freedom system. However, periodic motions are still possible when all the coordinates cross their equilibrium positions simultaneously and reach the extremum positions similtaneously. Consequently, a concept of nonlinear normal modes, similar to the linear modes, has been proposed [14]. The associated frequencies are known as the non-linear natural frequencies. Both the non-linear normal modes and natural frequencies are not system properties but also depend on the extent of motion. Various methods have been suggested to obtain the non-linear normal modes and the associated natural frequencies
[15}18]. It has also been shown that the phaseclosure principle is satis"ed during a non-linear modal oscillation [19]. In this paper, the vibration of a one-dimensional periodic chain consisting of identical masses and weakly non-linear springs has been considered. A perturbation approach retaining only the "rstorder non-linear e!ect is used. Amplitude-dependent propagation and attenuation constants are derived for harmonic waves in an in"nite chain. For a "nite chain, the phase-closure principle is used to obtain the non-linear normal modes and the associated natural frequencies. Results are obtained for di!erent types of homogeneous boundary conditions and also for an endless cyclic chain. Results for forced vibration dealing specially with the non-linear e!ects will be presented in a subsequent paper.
2. Harmonic wave propagation in an in5nite periodic non-linear chain Consider an in"nite periodic chain consisting of identical springs and masses as shown in Fig. 1. The springs are assumed to be non-linear with the force}deformation characteristics given by F"k d#ek d, where d is the deformation due to a force F with e as a small parameter. The equation of motion of any mass (say the nth) is dxH L #k (xH!xH )#k (xH!xH ) m L L\ L L> dt #ek (xH!xH )#ek (xH!xH )"0, (1) L L\ L L> where xH is the displacement of the nth mass. Using L the following non-dimensional scheme q"t
k k , x "xH , L L k m
Fig. 1. Non-linear periodic chain.
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377
the following equations are obtained: dx L #(2x !x !x )#e(x !x ) L L> L\ L L\ dq #e(x !x )"0. (2) L L> To study the propagation of a harmonic wave with non-dimensional frequency u (non-dimensionalised by dividing the actual frequency by (k /m), the displacements are assumed in the form q q x " e SO# e\ SO, L 2 2 q q x " eIe SO# eI e\ SO, L> 2 2 q q x " e\Ie SO# e\I e\ SO, L\ 2 2
(3)
where the complex number k is called the propagation constant, q is the complex amplitude and the overbar indicates complex conjugate. Substituting Eq. (3) into Eq. (2) and equating the coe$cients of e SO from both sides, one gets !uq#(2!eI!e\I)q#eqq ;+(1!e\I)(1!e\I )#(1!eI)(1!eI ),"0. (4) Eq. (4) is solved to get the relationship between u and k. For a linear chain (e"0) the roots of k can have either purely imaginary (0("k"(p) or a complex value with imaginary part as p [20]. The real and imaginary parts of k are shown in Figs. 2a and b, respectively. The frequency regime with imaginary value of k is known as the propagation zone and the frequency regime having the real value of k is called the attenuating zone. For the non-linear case, the value of k depends not only upon u but also upon q, the complex amplitude of the nth mass. An inspection reveals that both k and !k are the roots of Eq. (4). As in the linear case, the negative and positive values of k denote the right- and left-going waves, respectively. This implies that the wave-propagation characteristics are equal in both directions. This is expected from the symmetry of the structure.
Fig. 2. Variation of propagation ("k ") and attenuation ("k ") ' 0 constant with frequency. e"0: (a) propagation constant; (b) attenuation constant.
For small non-linearity, it seems encouraging, at least at a "rst glance, to use a perturbation technique for "nding k for a given value of u and vice versa. For a given u, assume k in the following form: k"k #ek #2
(5)
and substitute this into Eq. (4). Equating the coe$cients of the like powers of e from both sides, one gets e: u"(2!eI !e\I ),
(6)
e: !k (eI !e\I )#qq +(1!e\I )(1!e\I ) #(1!eI )(1!eI ),"0.
(7)
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The value of k corresponds to that obtained from the linear theory. Eq. (7) is solved to obtain
0)u)4#16Z.
k "qq +(1!e\I )(1!e\I ) (8)
One can easily verify that the real and imaginary natures of k remains the same as that of k . Thus, the propagation characteristics of the non-linear system, predicted by the above analysis, follow that of the linear system. But this simple perturbation technique fails in some frequency ranges where the correct analysis can be carried out as discussed below. Let us now consider only the propagation zone for the non-linear system, where the exact value of the propagation constant (k purely imaginary) can be obtained in closed form by assuming
where both a and b are real with a#b"1. (9) Substituting Eq. (9) into Eq. (4) and equating the real and the imaginary parts, the following equations are obtained: (10)
and (11)
where Z"(3/4)eqq . Eq. (10) is seen to be trivially satis"ed. From Eq. (11) the roots of a are obtained as a"1!+!2$2(1#4Zu,/8Z.
(12)
For Z'0 (e.g., e'0, i.e., a hard system) the negative sign before the square root yields a'1 and can be discarded. Hence, the only possible root is a"1!+!2#2(1#4Zu,/8Z. The restriction "a")1 implies 0)+!2#2(1#4Zu,/8Z)2
a"1#+!2$2(1!4"Z"u,/8"Z".
(15)
It can be seen that again the negative sign before the square root cannot give a feasible solution of a since this solution does not converge as "Z"P0. Thus, the only possible root is a"1#+!2#2(1!4"Z"u,/8"Z".
(16)
The condition "a")1 yields 0)u)4!16"Z".
(17)
Thus, for both hard and soft oscillators the bounding frequencies for the propagation zone are given by
eI"a#ib
u"2(1!a)#4(1!a)Z,
(14)
Now, for Z(0 (if e.g., e(0, i.e., a soft system),
#(1!eI )(1!eI ),/(eI !e\I ).
b"b/(a#b)
or
(13)
0)u)4#16;eqq .
(18)
It can be veri"ed from Eq. (4) that the bounding frequencies predicted by the equality signs of expression (18) correspond to eI"1 or eI"!1. The variation of the upper bounding frequency (u ) with the amplitude of the nth mass (i.e., "q") is shown in Figs. 3a and b for e'0 and e(0, respectively. The lower bounding frequency (u ) is zero (Eq. (18)) for both hard and soft systems under consideration. There exist two types of regions in each "gure. In region of type-I, the wave propagations of the linear and non-linear chains have similar characteristics. In such a region, the propagation and attenuation constants can be obtained by perturbation analysis (i.e., from Eq. (8)). In region of type II, however, the wave propagation in the non-linear structure shows marked di!erence from that of the linear one. The perturbation analysis fails in this region and the propagation constants can be determined as follows. For e'0, the value of k is purely imaginary in region II and can be obtained from Eq. (13). For e(0, the value of eI is real in region II which is an attenuation zone. To determine the corresponding attenuation constant (k), eI"a (a real) is
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379
with the value of a written as a"!1#ec.
(21)
Substituting Eqs. (20) and (21) into Eq. (19) and neglecting the terms of order higher than e, one gets ec "eu 2 or ec"$(2(u!4!16;eqq ).
Fig. 3. (a) Variation of upper bounding frequency (u ) of the propagation zone with amplitude; e'0. (b) Variation of upper bounding frequency (u ) of the propagation zone with ampli tude; e(0.
substituted into Eq. (4) and after simpli"cation one gets the following algebraic equation:
1 3 !u# 2!a! # eqq +(1!a) a 4
1 # 1! "0. a
(19)
Since eI"!1 at the upper bounding frequency and any nearby frequency in regoin II can be expressed as (see Eq. (18)) u"4#16;eqq #eu ,
(20)
(22)
The negative value of c corresponds to Re(k)'0 and hence to the left attenuating wave. The other root corresponds to the right attenuating wave. For the non-linear periodic chain, the attenuation or propagation constant depends on the amplitude ("q") of the nth mass. Figs. 4a and b show the variation of attenuation and propagation constants for linear and non-linear systems (both hard and soft). For a propagating wave without attenuation, the phases rather than the amplitudes of the masses vary and the propagation constant is "xed for a given amplitude. But when the wave attenuates, the amplitudes of the masses change progressively. Consequently, there is no "xed attenuation constant for a non-linear structure. The change of the wave characteristics with amplitude (see Figs. 3a and b) accounts for a special phenomenon observed in a non-linear chain. For a soft system (e(0), if the initial amplitude and frequency lie in region II, the wave attenuates and the amplitude starts decaying. After attenuating over a length spanned by few periodic elements, the value of the amplitude enters region I. The wave then starts propagating without any further decay. A typical amplitude envelope looks like that in Fig. 5 where the masses are indicated by the dots. In this "gure the longitudinal displacement amplitudes of various masses are plotted in the transverse direction. The dots are not coincident with the crest or trough of the diplacement wave to indicate the phase di!erence (propagation constant) between successive elements.
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Fig. 6. An element of the non-linear periodic chain.
carried out considering only a single element [2]. The boundary frequencies of the propagating zones are related to the natural frequencies of the element. In this section, such an analysis is carried out for the non-linear periodic structure. Consider the element shown in Fig. 6, where Q refers to the non-dimensional exterior coordinate with the corresponding non-dimensional force F (non-dimensioanlised by dividing the actual force by (k /k ) and the subscript L and R refer, respec tively, to the left and right ends of the element. The harmonic responses are assumed as q q Q " e SO# e\ SO * 2 2
(23)
and Fig. 4. E!ect of non-linearity on the propagation and attenuation constant. **: e"0; * ) *: "q"("e""0.3, e'0; } } }: "q"("e""0.3, e(0: (a) propagation constant; (b) attenuation constant.
q q Q " eIe SO# eI e\ SO. 0 2 2
(24)
The corresponding forces can be written as f fM F " e SO# e\ SO * 2 2
(25)
and
fM f F "! eIe SO# eI e\ SO . 0 2 2
(26)
Fig. 5. Longitudinal displacement amplitudes of a soft nonlinear periodic chain (e(0) with the amplitude of the centremass (marked 0) and frequency in region-II of Fig. 3b.
The negative sign in F is necessary for the condi0 tion of force equilibrium. The complex force amplitude is denoted by f. The non-dimensionalised equations of motion for the end masses are
3. Vibration of a single element and the wave propagation
1 dQ * #(Q !Q )#e(Q !Q )"F * 0 * 0 * 2 dq
One of the most important advantages rendered by the space periodicity of a linear repetitive structure is that the wave-propagation analysis can be
(27)
and 1 dQ 0 #(Q !Q )#e(Q !Q )"F . 0 * 0 * 0 2 dq
(28)
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
Substituting Eqs. (23)}(26) into Eqs. (27) and (28) and equating the coe$cients of e SO from both sides of the resulting equations one gets the following relations:
q q 3 1 !u # (1!eI)# eqq (1!eI)(1!eI ) 2 2 2 8 f " 2
(29)
and
eI q q 3 !u ! (1!eI)! eqq (1!eI)(1!eI ) 2 2 2 8 f "!eI . 2
(30)
381
vibratory mode which can be obtained as discussed below. The antisymmetric shape is assumed as (see Eqs. (23) and (24) with eI"!1) q q Q " e SO# e\ SO, * 2 2
q q Q "! e SO# e\ SO 0 2 2
and F "F "0. Substituting these in Eqs. (27) or * 0 (28) and applying the harmonic balance method, one gets u"4#16;eqq ,
(33)
which is the non-linear natural frequency of the periodic element and identical to u given by Eq. (32).
Elimination of f from Eqs. (29) and (30) yields !uq#(2!eI!e\I)q 4. Interaction of harmonic waves in a semi-in5nite periodic chain
3 # eqq +eI>I (1!e\I)(1!e\I ) 4 #(1!eI)(1!eI ),"0.
(31)
Eqs. (31) and (8) are seen to be identical if k"!k, i.e., if k is purely imaginary. Thus, the element can still be used to study the harmonic wave propagation. During attenuation of the waves, the elemental approach does not give correct result. This is expected since there is no uniform attenuation constant and each element (having di!erent amplitudes) behaves di!erently from the other. The lower (u ) and upper (u ) bounding frequencies of the propagation zones are obtained by substituting, respectively, eI"1 and eI"!1 into Eq. (31) as u "0 and u "(4#16;eqq .
(32)
It can be veri"ed that u and u are the two non-linear natural frequencies of the element. The zero frequency corresponds to the rigid-body mode and the other frequency is the only frequency of the
Standing modes are possible for "nite, linear periodic structures. The knowledge of the wave propagation in an in"nite structure can then be of great use in obtaining the normal modes and the corresponding natural frequencies [2]. During such a modal oscillation, two opposite-going waves with same phase velocities travel in such a way that the total phase di!erence is an integral multiple of 2p after traversing to and fro once across the span. The characteristics of a wave in a linear chain are not a!ected by the oppositely travelling wave. Consequently, the phase change of each wave is considered separately to get the natural frequencies. In a non-linear chain, however, the two opposite-going waves undergo complicated interaction rather than just getting linearly superimposed and the wave characteristics are altered. In what follows, the interaction of the waves in a semi-in"nite nonlinear chain is studied by considering a perturbation of the linear waves. The boundary is assumed to be linear and non-dissipative. The phase of the re#ected wave, generated at the boundary, di!ers from that of the incident wave by a certain amount (k , say).
382
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
The total harmonic response of the nth mass can now be written as a x " [(e\ LI#e I e LI)#eq ]e SO#c.c., L 2 L
(34)
where ik and !ik are, respectively, the propagation constants of the left and right propagating waves, both of frequency u. The term (e\ LI# e I e LI) indicates the linear superposition of the two opposite-going waves. The e!ect of the non-linearity is taken into account by introducing a small perturbation in the form of eq which also is a comL plex quantity. For the same value of the propagation constants, let the waves in the corresponding linear system (i.e., e"0) have a frequency u . Then one can write u"u #eu . (35) Substituting Eq. (34) into equation of motion (Eq. (2)) and collecting the coe$cients of e SO from both sides one gets a a !u ( #eq )# (2!e I!e\ I)
L L 2 L 2
Eq. (38), as expected is identical to Eq. (6) which holds good for the linear in"nite chain. Using the relation given by Eq. (37) and expressing
and L\
in terms of one can, after algebraic manipL> L ulation, write the right-hand side of Eq. (39) (in terms of only) as given below: L a 3 u ! aa [3(1!e I)(1!e\ I) 2 L 8 3 #(1!e\ I)(1!e I)] ! aa e\ I [(1!e I) L 8 #(1!e\ I)](e\ LI#e I e LI). The presence of the term in the above expression L yields an unbounded value of q . So the solvability L condition requires vanishing of the coe$cient of
yielding L u "aa [(1!e I)(1!e\ I) #(1!e\ I)(1!e I)].
(40)
a #e (2q !q !q ) L> L\ 2 L
Combining Eqs. (38) and (40) the frequency relation is obtained as
3 )( M ! M ) # eaa [( !
L L\ L L\ 8
u"(2!e I!e\ I)#eaa [(1!e I)(1!e\ I)
#( !
)( M ! M )]"0, L L> L L> where
#(1!e\ I)(1!e I)]. From Eq. (39), the response q is found to be L
"e\ LI#e I e LI. (37) L Eq. (35) is now substituted into Eq. (37) and the coe$cients of the like powers of e are collected to get e: [!u #(2!e I!e\ I)] "0 L and
3 aa e\ I q "! L 4 [(e I#e\ I)!(e I#e\ I)] ;[(1!e I)#(1!e\ I)](e\ LI#e I e LI). (42)
(38)
a a e: !u q # (2q !q !q ) 2 L 2 L L> L\ a 3 "u # aa [(
2 L L 8 )( M ! M ) L\ L L\ #( !
)( M ! M )]. L L> L L>
(41)
(36)
!
(39)
Thus, it should be noted that due to the interaction of two opposite-going harmonic waves of propagation constant $ik, another set of opposite-going waves of propagation constant $3ik are generated. The phases of the latter pair of waves di!er by 3k . These new waves are designated as the second ary waves (with propagation constants $3ik) in the rest of the paper whereas the original waves are called the primary waves (with propagation constants $ik).
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
Attention must be drawn at this stage to some special situations where q may vanish implying L non-interacting waves. This situation occurs when e I "$1
(43)
and k satis"es the following equation: e\ LI#e I e LI"j(e\ LI#e I e LI)"j . (44) L It may be mentioned that e I "1 is satis"ed by a free boundary and e\ I "!1 is satis"ed by a "xed boundary. When Eqs. (43) and (44) are satis"ed the solution for u is obtained (by following the general method described above) as u"(2!e I!e\ I) # eaa [3(1!e I)(1!e\ I) #(1!e\ I)(1!e I)]#eaa je\ I ;[(1!e I)#(1!e\ I)]
(45)
In the presence of non-linearity secondary waves are generated. Further, when the primary waves have propagation constants $ik, the secondary waves have propagation constants $3ik. The phase changes of the secondary waves at the boundaries are 3k and 3k after re#ection. Thus, it * 0 is evident that the secondary waves always satisfy the phase-closure principle if the primary waves do so. Hence, for the modal oscillations of a "nite non-linear chain, the allowed propagation constants are still given by Eq. (46). Knowing the value of k, the non-linear natural frequencies are obtained from Eq. (41) where the amplitude a is the strength of the incident wave. However, in a "nite structure the non-linear mode shapes and corresponding natural frequencies are usually obtained as a function of the amplitude of a speci"c mass particle (say the rth mass). If b be the amplitude of the rth mass, then b"(e\ PI#e I* e PI)a
and q "0. L
383
(48)
or
5. Normal modes and natural frequencies of a 5nite non-linear chain As discussed in Section 1, the non-linear normal modes of a multi-degree-of-freedom system correspond to periodic motions when all the coordinates cross their equilibrium positions simultaneously and also reach their extremum positions simultaneously. The associated frequencies are called the non-linear natural frequencies [15]. In a "nite linear periodic chain, the harmonic waves get re#ected at the boundaries and form a standing mode only when the phase-closure principle is satis"ed. If the phase changes at the left and right boundaries are k and k , respectively, then * 0 the phase-closure principle suggests that the modal oscillation is possible if the propagation constant ik bears the following relationship [2]:
aa "bbM /(e\ PI#e I* e PI)(e PI#e\ I* e\ PI).
(49)
Hence, the natural frequencies are calculated from Eq. (41) after replacing aa by bbM using Eq. (49). Similarly, the non-linear normal modes can be expressed as t " #eq , (50) L L L where and q are given by Eqs. (47) and (42), L L respectively, after replacing aa by bbM using Eq. (49). In what follows some explicit results for speci"c boundary conditions are included. 5.1. Non-linear chain with both ends xxed Consider a non-linear chain of N masses having both ends "xed as shown in Fig. 7.
2Nk#k #k "2mp, m"0, 1, 2,2, (46) * 0 where N is the number of elements in the chain. Furthermore, in a linear chain the modal response of the nth mass is given by
"e\ LI#e I* e LI. L
(47)
Fig. 7. Non-linear periodic chain with "xed}"xed boundary conditions.
384
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
In this case, k "k "p. The phase-closure * 0 principle yields
p #(1!e\ IH )(1!e IH ),! +(1!e IH ) "p"
jp k" , j"1, 2, 3,2 . H N#1
#(1!e\ IH ),],
(51)
Thus for the jth non-linear mode the natural frequency in terms of the amplitude of the "rst mass (r"1) is u"(2!e IH !e\ IH ) H 9 bbM #e [(1!e IH )(1!e\ IH ) 4 4 sin k H #(1!e\ IH )(1!e IH )]
(52)
and the jth non-linear normal mode is 3 bbM tH" H#e L L 4 4 sink H (1!e IH )#(1!e\ IH ) ; [(e IH #e\ IH )!(e IH #e\ IH )] ;(e\ LIH !e LIH ),
(53)
with
H"e\ LIH !e LIH . L For any given j it can be veri"ed that
(54)
(55) e\ LIH !e LIH " H if 3j(N#1, L p " N with p"3j!2(N#1) "p" L
(59)
and tH" H. L L Thus, the non-linear natural frequencies can be easily obtained either from Eq. (52) or graphically from the plots of k vs. u for a given "b" using Eq. (52). So the computation e!ort is independent of the number of periodic elements (N), a feature clearly established in the literature for linear periodic structures [2]. From Eqs. (53) and (54) it is obvious that any non-linear mode consists of either two linear normal modes when conditions (55) and (56) apply. On the other hand, if condition (57) or (58) holds good then the non-linear normal mode is identical to the linear mode of the same order. Further, a careful study of conditions (55) and (56) reveal that any odd-order non-linear mode consists of the corresponding linear mode and another odd-order linear mode. Exactly similar conclusion can be drawn for even-order modes. The orthogonality of linear modes immediately imply that any odd (even)-order non-linear mode is orthogonal to all even (odd)-order non-linear modes. 5.1.1. Numerical results for N"3 In this section the "rst non-linear natural frequency with N"3 is obtained directly considering a three-degree-of-freedom system and veri"ed against that obtained from Eqs. (51) and (52) with j"1 and N"3. The equations of motions of the system can be written in the following non-dimensional form:
if 3j'N#1 and "p"Oj,
(56)
"0 if 3j"N#1.
(57)
dX M #KX#eN"0, dq
(58)
where X"+x , x , x ,2, M"diag+1, 1, 1,2, 2 !1 0
A special situation occurs when p"3j!2(N#1), 3j'N#1 and "p""j.
(60)
In this case, u is given by Eq. (45) with aa replaced by bbM as given below:
K" !1
u"(2!e I\H!e\ IH ) H 3 bbM # e [+3(1!e IH )(1!e\ IH ) 4 4 sin k H
x #(x !x ) N" (x !x )#(x !x ) . (x !x )#x
0
2 !1
!1 , 2
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
The linear mode shapes are ' "+1, (2, 1,2, ' "+1, 0, 1,2 and ' "+1, !(2, 1,2. Assume the responses as X"b( cos u q, (61) H H where ( "+t , t , t ,2 is the jth non-linear H H H H normal mode and u is the corresponding nonH linear natural frequency. Substituting Eq. (61) into Eq. (60) and applying the method of harmonic balance, one gets !uMX#KX#eN "0, H H where
(62)
t #(t !t ) H H H 3 N " b (t !t )#(t !t ) . H H H H H 4 t #(t !t ) H H H For small non-linearity one can assume ( "' #e* #2 H H H and
(63)
(64)
u"(uJ )#ed #2, (65) H H H where uJ is the jth linear natural frequency. SubstiH tuting Eqs. (64) and (65) into Eq. (62) and collecting the coe$cients of like powers of e from both sides, one obtains e: !(uJ )M' #K' "0, (66) H H H e: !(uJ )M* #K* "d M' !N , (67) H H H H H ( where N is the vector obtained by replacing t 's ( IH (k"1, 2, 3) by 's in N (see Eq. (63)). Eq. (66) is IH H trivially satis"ed. To solve Eq. (67), N can be ( written as
385
and solution of Eq. (67) yields c M' N N . * " H [(uJ )!(uJ )]('2M' ) N H N N N$H For the chosen example, if j"1, c "b(9!6(2), c "0, c "b((2!1). Thus, the "rst natural frequency is u "(uJ )#eb(9!6(2). The same value of u can be obtained from Eq. (52) by substituting k "p/4. H Furthermore, c "0 implies that the "rst non linear normal mode has contribution from only the "rst and the third linear normal modes but not from the second one. It can be veri"ed that similar is the case for the third mode. The second nonlinear mode is identical to the second linear mode. 5.2. Non-linear chain with both ends free Consider the non-linear chain as shown in Fig. 8. In this case k "k "0 and the phase-closure * 0 principle yields jp k" , H N
j"0, 1, 2, 3,2, N.
(68)
The value k "0 corresponds to the rigid-body H mode. The jth non-linear natural frequency in
N " c M' , ( N N N where '2 N c " N ( . N '2 M' N N The solvability condition yields d "c , H H
Fig. 8. (a) Non-linear periodic chain with free}free boundary conditions, (b) A typical element.
386
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
terms of the amplitude of the "rst mass is obtained as follows: u"(2!e IH !e\ IH ) H 9 bbM #e [(1!e IH )(1!e\ IH ) 4 4 #(1!e\ IH )(1!e IH )] if jON,
5.3. Non-linear chain with one end xxed and the other end free (69) Consider a non-linear chain which is "xed at one end and free at the other end as shown in Fig. 9. In this case k "p and k "0, thereby yielding * 0 1 p k " j! , j"1, 2, 3,2, N. (73) H 2 N
or u"(2!e IH !e\ IH ) H 3 bbM #e (2!e IH !e\ IH ) if j"N, 4 4 3 "4# ebbM ;16. 4
(70)
The non-linear normal modes are given by
The jth non-linear natural frequency and the corresponding normal modes are given, respectively, by Eqs. (52) and (53). One can also verify that e\ LIH !e LIH " H\ if 3j(N#1, L (2p!3) "
N\ "(2p!3)" L
3 bbM tH" H!e L L 4 4 (1!e IH )#(1!e\ IH ) ; [(e IH #e\ IH )!(e IH #e\ IH )] ;(e\ LIH #e LIH ) if jON
The observations made in Section 5.1 regarding restricted orthogonality of various non-linear modes for the "xed}"xed chain remain una!ected for the free}free chain.
with p"3j!2N if 3j'N#1. (71)
" H if j"N. (72) L Comparing Eq. (70) with Eq. (34), it is easily seen that the highest non-linear natural frequency of the chain (for any value of N) is identical to that of the periodic element. It may be further noted from Eq. (72) that at this frequency, the non-linear normal mode is identical to the linear mode of the chain when each element oscillates independently. As in Section 5.1, for the present system it can be shown that e\ LIH #e LIH " H if 3j(N#1, L " N with p"3j!2(N#1) L if 3j'N#1 and "p"Oj.
It should be noted that in this case the even- or odd-order non-linear normal modes have contribution from both even- and odd-order linear normal modes. Another interesting feature of the present system needs to be pointed out. An inspection of Eqs. (51) and (73) suggests that the propagation constant for the jth mode of the chain under discussion is identical to the (2j!1)th mode of a "xed}"xed chain with 2N!1 masses. Consequently, the non-linear natural frequencies and normal modes of the former are given by those of the latter. Similarly, comparing Eqs. (68) and (73) it is found that the propagation constant of the jth mode of the "xed}free chain with N masses is identical with that of the (2j!1)th mode of a free}free chain
A special case occurs if "p""j, then u is given by H 4 bbM u"(2!e IH !e\ IH )#e (2!e IH !e\ IH ) H 44 and tH" H. L L
Fig. 9. Non-linear periodic chain with "xed}free boundary conditions.
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
387
having (2N#1) masses. However, the natural frequency and the non-linear mode shape of a "xed}free chain given by Eqs. (52) and (53) are di!erent from those of a free}free chain. The natural frequencies and the associated normal modes of a free}free chain are functions of the amplitude of the end mass, whereas the corresponding quantities of the "xed}free chain are calculated with respect to the "rst mass from the "xed end or the Nth mass from the free end. Thus, when the frequencies and the normal modes of the free}free chain are calculated as a function of the amplitude of the same mass the following results are yielded: u"(2!e IH !e\ IH ) H 9 bbM #e [(1!e IH )(1!e\ IH ) 4 4 cos(N!1)k H #(1!e\ IH )(1!e IH )] if jON,
(74)
and 3 bbM tH" H!e L L 4 4 cos(N!1)k
H
(1!e IH )#(1!e\ IH ) ; [(e IH #e\ IH )!(e IH #e\ IH )] ;(e\ LIH #e LIH ) if jON.
(75)
As k "(2j!1)p/2N, it can be veri"ed easily that H cos(N!1)k "sin k . Thus, Eq. (74) becomes H H identical to Eq. (52). However, the mode shapes do not match. This is due to di!erent ways of numbering the elements of the chain in these two cases. By changing the numbering suitably (replacing n by N!n) it can be shown that Eqs. (75) and (53) generate the same modes. 5.4. Cyclic one-dimensional non-linear chain Consider a cyclic chain made of N masses as shown in Fig. 10. For studying the free vibration of such a structure one needs to take a reference mass (say the mass numbered 1). In the absence of any boundary, the waves travelling in a cyclic structure undergo no re#ection. However, two oppositegoing waves still exist. The phase di!erence
Fig. 10. Cyclic one-dimensional non-linear chain.
between these two waves depend upon the way in which the system is disturbed. Depending on the initial disturbance, the system may vibrate symmetrically or antisymmetrically about the reference mass. In the symmetric mode, the reference mass undergoes "nite oscillation and in the other case this mass remains stationary. The phase di!erence between the opposite-going waves is k "0 for the symmetric mode and k "p for the antisymmetric mode. Since there is no re#ection in this case, both the clockwise and anticlockwise waves must satisfy the phase-closure principle independently. The phase constant at the jth non-linear natural frequency must satisfy the following relation: 2jp k" H N
where j"1, 2,2, N .
(76)
The value of N depends upon various condi tions. If N is even then the maximum number j can take is N/2 when the excitation is symmetric and (N/2!1) when the excitation is antisymmetric. Similarly, if N is odd then the maximum value of j is (N!1)/2 for both symmetric and antisymmetric excitations. In all cases the system has atmost N!1 natural frequencies. The values of j"0 and N correspond to no motion for an antisymmetric excitation and a rigid-body motion for symmetric excitation.
388
G. Chakraborty, A.K. Mallik / International Journal of Non-Linear Mechanics 36 (2001) 375}389
For a value of k given by Eq. (76), the non-linear H natural frequencies and the corresponding mode shapes are obtained from Section 5.1 (see Eqs. (52) and (53)) for antisymmetric excitation and from Section 5.2 (see Eqs. (69)}(72)) for symmetric excitation. These quantities are calculated as a function of the amplitude of the mass adjascent to the reference mass for the former case and as a function of the amplitude of the reference mass for the latter. However, when the natural frequencies of both the cases are calculated with respect to the amplitude of the adjacent mass, the term bbM in Eqs. (69)}(72) are replaced by bbM /cos k . For a linear system with H odd number of elements, both symmetric and antisymmetric modes exist at all natural frequencies. However, for a non-linear chain the degenaracy is absent. In a chain with even number of elements, symmetric and antisymmetric modes at the same natural frequency may not be distinguished except for a &rotation' of the mode shape through an integral number of elements.
latter pair of waves are three times that of the former pair. (vi) The normal modal oscillation of a "nite chain is possible at certain frequencies where the associated propagation constants satisfy the phase-closure principle. The corresponding nonlinear natural frequencies can be obtained either graphically or analytically. In either case, the computational e!ort is independent of the number of elements in the chain. (vii) For a chain having both ends "xed or free, the non-linear normal modes are either identical to the linear modes or comprise of only two linear modes. In the latter case, the odd-order non-linear modes are orthogonal to the even-order non-linear modes. For a chain with one end "xed and the other end free, such restricted orthogonality condition does not hold good. (viii) For a cyclic one-dimensional chain, the non-linear normal modes are either symmetric or antisymmetric. The degeneracy, present in such a linear chain with odd number of elments, vanishes because of the non-linear e!ects.
6. Conclusions The major conclusions of the paper are listed below: (i) The value of the propagation constant depends on the amplitude of the harmonic wave. (ii) In some frequency range, a simple perturbation of a linear periodic chain fails to predict the wave-propagation characteristics of the weakly non-linear periodic chain. (iii) The propagation (and not the attenuation) constant of an in"nite chain can be obtained by considering only a single element. The bounding frequencies of the propagation zone are identical with the non-linear natural frequencies of a single element. (iv) The amplitude dependence of the propagation characteristics may result in transforming an attenuating wave into a propagating wave for softening type non-linearity. (v) In a semi-in"nite non-linear periodic chain, two opposite-going harmonic (primary) waves interact to form another pair of opposite-going (secondary) waves. The propagation constants of the
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