Receptance methods and the dynamics of disordered one-dimensional lattices

Journal of Sound and Vibration (1984) 92(3), 427-445

RECERANCE METHODS AND THE DYNAMICS OF DISORDERED ONE-DIMENSIONAL LATTICES D. J. MEAD Department of Aeronautics and Astronautics, University of Southampton, Southampton SO9 5NH. England AND

S. M. LEE Keeweenaw

Research

Center, Michigan

Technological U.S.A.

University, Houghton,

Michigan

4993 1.

(Received 20 January 1983)

The method of receptance analysis is used to set up a frequency equation for the free vibration modes of a one-dimensional periodic lattice (mass-spring system) containing a disorder which is itself a one-dimensional periodic lattice. The concepts of the propagation constant and wave-receptance function are used to determine the receptances of the component systems, and these are used to set up a simple frequency equation. An accurate root-searching computer programme has been used to find the natural frequencies and

corresponding modes of particle displacement. Some computed results are shown to demonstrate the capability of the method and programme. Special attention is given to modes which occur in the frequency “forbidden” zone, and receptance methods are used to derive formulae for the frequencies of systems with a single-mass disorder. The wider usefulness of the method is briefly discussed.

1. INTRODUCTION

bulk properties of crystalline solids that arise from the microscopic behaviour of constituent atoms can be studied from the vibrational analysis of idealized simple models. Elastic properties, thermodynamic properties (particularly, the specific heat), and the optical properties of ionic crystals are good examples of those which can be studied by treating the solids as one-dimensional arrays of point-masses coupled together by massless Hooke’s law-type springs. Thus, the vibrational analysis of finite, free-ended periodic systems in the harmonic approximation has been a subject of active investigation among physicists. The simplest array of such a periodic system, consisting of identical point-masses coupled together by identical springs (the monatomic linear lattice), exhibits normal modes whose frequencies fall within a band of continuum referred to as “the frequency band”. This band has an upper limit of frequency that separates the frequency band from the “forbidden zone”. For a more complex linear system, such as a diatomic linear lattice, the frequency zone is divided into more than one branch separated by “gap zones”. These branches are characterized by the modes of oscillation of the masses in the system. Real crystals are seldom perfectly ordered. Local defects, such as substitutional impurities, destroy local periodicity. These defects (to be referred to in this paper as disorder) not only present an interesting problem in vibrational analysis but also play an

Certain

427 0022-460X/84/030427

+ 19 $03.00/O

@ 1984 Academic

Press Inc. (London)

Limited

428

D. J.

MEAD

AND

S. M. LEE

important role in affecting the physical properties of solids. For example, the defects cause discrete frequency modes to appear in the forbidden zone or in the gap. These impurity (defect) levels provide semiconductors with the seat for electrons from which they can easily make transitions into conduction bands. Transport processes such as heat conduction are often controlled by the scattering of electrons or phonons by defects. A particularly interesting phenomenon is the case of a defect occurring on a lattice surface. The local surface mode may have a vibrational frequency nearly equal to some molecular vibrational frequency, which then will find it easy to adsorb itself onto the crystal surface. From the above discussion, the importance of the study of disordered lattices is easy to realize. Many investigators have treated the problem from the phenomenological approach dealing with real crystals and with their behaviour as determined experimentally. Some of the attempts to solve the problem in closed form have met with a limited degree of usefulness. It is now generally accepted that the cyclic boundary condition [l], which was successful in simplifying the vibrational analysis of a free-ended system, is either not applicable or leads to wrong results when applied to most vibrating systems that are not perfectly periodic [2-51, particularly the disordered systems. Other attempts by various investigators to analyze vibrations of disordered lattices have often been complicated and, in the end, have their applicability limited to some simplified systems. In addition, in most of these works only the frequency spectrum, without the associated amplitude distribution, has been considered. Mazur et al. [6] reported a study of localized vibrations for a linear diatomic chain with fixed boundary conditions and having a defect at the center of the chain. This work was extended by Bjork [7] who allowed different nearest-neighbour force constants for the defect atom. Lee [8] calculated the surface vibrations of a diatomic linear chain by application of the extra force method. This study was extended by Malik and Ghatak [9] by allowing the nearest-neighbour force constant for the surface atoms to be different from those for the interior atoms. Studies of disordered systems by Montroll and Potts [lo], Bacon et al. [ll], and Dean [12] are frequently referred to by other investigators. Bala et al. applied the method of extra forces to some real crystals with impurities [ 131. A study of surface modes for free-ended diatomic chains was published by Wallis [ 141. One important characteristic of the defect modes is known to be that the amplitudes of displacements for the defect modes are highly localized. Taking advantage of this, Rosenstock and McGill [15,16] have developed a method of calculation in which the localized modes of a disordered system are analyzed by solving the problem of the vibrations of a small system in the neighbourhood of the defect atom whose ends are regarded as fixed several neighbours away. Even though this is a good technique, it cannot be applied to the study of surface modes nor if the defect occurs very near the surface. The purpose of the present study is to devise an alternative method of vibrational analysis that can be applied to simple disordered periodic systems. The aim is to calculate both the frequencies and the amplitudes of the modes associated with disorder. The method to be used is to set up the frequency equation of the whole finite, one-dimensional system by using the receptances of the two ordered parts and of the one disordered part of the whole system. These receptances are found by using the well-known concepts of harmonic wave propagation in periodic systems [ 171. The number of roots of the frequency equation is one less than the number of masses in the system, so there may be many. They are found by using a computer-based root-searching technique which can be very rapid. Once the roots are known, the receptance approach quickly leads to the accurate calculation of the particle displacements in the corresponding modes, which may be in the frequency band or in the forbidden zone.

DISORDERED

ONE-DIMENSIONAL

429

LATTICES

This paper describes the method on which a computer programme has been based and presents some sample results. The receptance analysis has also been used to derive formulae for the frequencies of very simple disordered systems which occur in the forbidden zone. Although these problems can be solved by other computational methods, the purpose of this paper is to show how receptance methods can be used to find the natural frequencies of disordered periodic systems. Very large systems can be studied in no longer computer time than small systems. The physical model on which the present work is based is the simple one-dimensional lattice of point masses joined by linear elastic springs. The analytical approach which is developed can readily be applied to one-dimensional engineering periodic structures consisting of, say, beams and shafts. An example of one of these would be a finite beam on more than one set of regularly spaced supports: e.g., over one set of supports the beam has a uniform section, and over another set of supports the beam has a different uniform section; alternatively, the support spacing could be different in the second region. In analyzing the beam structure, the receptance functions for the basic periodic unit would be quite different from those of the unit of the mass-spring lattice, but the general approach of this paper can be applied equally well to disordered systems of either type. 2. THE RECEPTANCE ANALYSIS FOR THE NATURAL FREQUENCIES AND MODES 2.1.

THE

SYSTEM

TO BE CONSIDERED

Natural frequencies and modes are to be found for a one-dimensional periodic system which is disordered by a defect cluster located arbitrarily within the total system. The defect cluster itself is a periodic system, and is connected to the adjacent periodic systems through elastic springs which may differ from the others of the whole system.

Figure

1. (a) The whole disordered

periodic

system;

(b) the constituent

periodic

elements.

The disordered system is shown diagrammatically in Figure l(a). Each sub-system consists of N, (r = 1,2,3) complete periodic elements. The elements of each system are shown in Figure l(b). Each system has N,+ 1 masses of magnitude M,, and N, springs of rate K,. The additional springs KC,, KC2 connect system II to systems I and III, respectively. The masses and stiffnesses in the three different systems are allowed at this stage to be different so that any one of the systems may subsequently be regarded as the defect cluster. 2.2. DYNAMIC ANALYSIS OF THE WHOLE SYSTEM To study the dynamics of the combined system, consider it sub-divided into the three sub-systems as shown in Figure 2, with the inter-connecting forces FA and FB between the sub-systems. It is to be understood that all forces and displacements considered are harmonically varying, and are positive when in the direction from left to right. The

430

D. J. MEAD

C

E

Figure

AND

S. M. LEE

A

R

2. Displacements

and forces within the disordered

D

F

system

displacements u at different points of the system are related to the forces F through the frequency-dependent receptance functions (Yas follows: uAL-

-F AaAAt

uAH

=

uBL=F~ff21+F~(y22,

-F~'y,,+FBff,2,

(132)

u BR=--FL@BIP

(3*4)

In these, (YAAand (YBBare the direct receptances of the left- and right-hand systems, respectively, at points A and B. (~ii and (y22 are the direct receptances of the middle system at its left and right-hand ends, and oi2 (= (Ye,) is the transfer receptance between the two ends of the middle system. For continuity of displacement at A and B, uAL= uAR and uBL= uBR. Substituting into these the above expressions for the U’S, and re-arranging, one obtains FA(~AA

+a,,)-F+rlz=O,

FA~I~-FB(QBB+~~~)=~~

(%6)

For non-trivial solutions of these equations for the Fs,the determinant of the coefficients must vanish, so (BAA

+(y11)(~BB+(y22)-(y:2

=ft")=O.

(7)

The frequencies at which this equation is satisfied are the frequencies of free vibration of the whole system. Since the system is semi-definite, and has (N, + N2 + N3 + 3) masses, it has (N, + N2 + N3 f 2) non-zero natural frequencies. There are some systems and frequencies for which the solution FA = FB = 0 is not a trivial case, but corresponds to an actual natural frequency and mode. This has to be considered separately, but is such a special case that it is not pursued further in this paper. Equation (7) is the “frequency-receptance equation” for the combined system, the roots of which are to be found by a computer root-searching technique. This is described in a later section. RECEPTANCES 2.3. EXPRESSIONS FOR THE SUB-SYSTEM Expressions for the sub-system receptances, (Y,will now be obtained in terms of the sub-system parameters. The method presented by Mead [17] for mono-coupled periodic systems will be used. The direct and transfer receptances of a finite periodic system of N, + 1 masses and N, interconnecting springs will first be investigated. The receptances of this system when excited at the left-hand end, P (Figure 3(a)) are unaffected when another massless spring K, is added at the right-hand end, Q (Figure 3(b). Since the right-hand end is free, no force acts on the spring, so the displacement of the right-hand mass is the same as the displacement of the right-hand end of the additional spring. This system (Figure 3(b)) now consists of N,+ 1 complete periodic elements, and its receptances are readily found, as follows. When such a finite, mono-coupled periodic system is excited by a force at one of its ends, a “characteristic wave” is propagated away from that end, and an opposite-going

DISORDERED K, M,

K< M,

ONE-DIMENSIONAL

K, M,

K,

K,

M,

,+f,

4

4 M,

431

LATTICES 6

Mr

M,

P

u K

M, (b)

K,

K,

M,

)

z

n elements

N,

P

X

+

N, t

I complete

elements,

I masses 0

K

,Q

(c) *

Figure 3. Mass-spring models used in the analysis. with an extra spring added; (c) the periodic element.

b

(a) The N,+ 1 mass model;

(b) the N,+ 1 mass model

characteristic wave is reflected back from the other end. The total motion of the periodic system consists of the sum of these two waves. Corresponding to each of these waves is a “characteristic receptance”. aW+ or (Ye_. (The + or - sign depends upon whether the wave propagates to the right or to the left.) This characteristic receptance is the quotient (displacement/force) at a point within an infinite periodic system when one wave alone is present. (The presence of just one wave is normally only possible in an infinite system with no disorders.) This concept can now be used to find, firstly, the direct receptance of the system of Figure 3(b) at P. Suppose the component of displacement at P due to the positive-going characteristic wave is u+, and due to the reflected, negative-going wave is u_. The forces at P corresponding to the two waves are Fp+ and Fp_ and are related to u+ and u- through the characteristic receptances by u+ = Fp+aw+ and u_ = Fp_ aw_. The total displacement at P is up = u+ + u_ = Fp+~w+ + Fp_aw_

(8)

and the total force at P is Fp = Fp+ + Fp-. The direct receptance at P of the system is then (~pp=

UPIFP=(FP+~~W++FP-(YW-)/(FP++FP-).

(9)

The ratio Fp+/Fp- and the characteristic receptances (~w+, czw- must now be determined. As the right-hand end of the system is unrestrained, no force acts upon it. This implies that the sum of the forces at that end from the two characteristic waves must vanish. When the positive-going wave propagates away from P with the propagation constant (i.e., phase constant) E, per periodic element, the force and displacement at a point N elements to the right of P are Fp+ exp (-iN&,) and u+ exp (-iN&,), respectively. Likewise, when the negative-going wave is present with force F_ and displacement u- at P, its force and displacement at a location N elements to the right of P are Fp- exp (+iNc,) and u_ exp (+iN&,), respectively. At the free end R, where N = N,+ 1, the total force must vanish so FP+exp(-i(N,+l)&,)+Fp-exp(+i(N,+l)&,)=O. At the left-hand end P, the total force is equal to the applied force Fp, so Fp++Fp_=Fp.

432

D. J. MEAD

AND

S. M. LEE

Solving this pair of equations, one finds FP+=FPexp(i(N,+1).5,)/2isin(N,+1)E,, Fp- = -Fp exp (-i(N, + l)s,)/2i

(10)

sin (NT+ l)~,,

(11)

so that Fp+/Fp_ = - exp (i2(N,+ 1)~~).

The direct receptance

(12)

app (from equation (9)) can now be expressed in the form

~PP={~YW_-~W+exp(i2(N,+1)~,)}/{1-exp(i2(N,+1)~,)}.

(13)

The characteristic wave receptances, (Ye+ and czw_, are readily obtained from the receptances (Y,,, ffy,bof a single periodic element (Figure 3(c) and reference [17]). It can be shown that %+ = %I -

aab

%Ll= - ~/MP~, Furthermore,

exp

(Yw-

(-id =

%b

=

-1/Mf12,

ff,,

-

aab

exp

(14715)

(+k),

Q=(l/KI)-(l/M&.

(16-18)

it can be shown [17] that cos E, = (L& + (Ybb)/2a,,, = 1- w2MJ2K,.

(19)

After substituting these expressions for the (Y’Sinto equation (13) and manipulating the equation, one obtains (Ypp=-

1

sin (N, + 1) E, - sin N& sin (N,+l)E, M&J2

1’

(20)

This is the direct receptance of the system when excited at the left-hand mass. It is independent of the extra spring added at Q (Figure 3(b)) and is the same as the direct receptance czoo of the system of Figure 3(a) when excited at the right-hand mass. To find the transfer receptance LY op( = ape) it is convenient to find firstly the transfer receptance anP, where n locates the point X, n elements to the right of I? The displacement at this point is U, = u+ exp (-ins,)+

u_ exp (+ine,) = Fp+aW+ exp (-in&,) +Fp_aw_ exp (+inc,).

(21)

The transfer receptance (Y,~= un/Fp = un/(Fp++Fp_). Substituting into this for Fp+, Fp(equations (10) and (11)) and u, (equation (21)), one obtains

I.

sin(N,+l-n)&,--sin(N,-n)c, sin (N,+ l)~,

1 (ynp=-M#2

(22)

The transfer receptance (YRpis given by this expression with n = N, + 1, but this is also the same as the transfer receptance sop since the spring QR does not deflect. Hence, 1 aRP=QOP=-MrW2

sin E, sin(N,+ljE,=QPO~

The frequency-receptance equation (7) requires the receptances ffAA and ffBB, which are the receptances of the left- and right-hand periodic systems with the springs KC, and KC2 interposed between the driving points A, B and the periodic systems (see Figure 2). These are given by aAA

=

%TC

+

1/KCl~

ffBB

=

aDD

+

1/Kc,.

(24,25)

DISORDERED

ONE-DIMENSIONAL

433

LAlTICES

(YDDand cycc are given by equation (20) with M, = M3, M,, respectively and N, = N3, N,, respectively. The value of E, appropriate to the sub-system must be used. crll and a22 are given by equation (20) with M,=M,, N,=N2, E, = e2. From the symmetry of the system, it is evident that LYE, = aZ2. (Y~*is given by equation (22) with M, = M,, N, = N2, E, = Ed. 2.4. DEVELOPMENT OF THE FREQUENCY-RECEPTANCE EQUATION To find the roots of equation (7), computerized numerical techniques are required. It is convenient, firstly, to introduce certain non-dimensional parameters into the equation and to rearrange it slightly to avoid the possibility of ill-conditioning. A non-dimensional frequency, n,, will be used which is defined by (26)

0, = WI%,

in which w,i is the cut-off frequency of free wave propagation in sub-system 1. The cut-off frequency is that frequency above which free waves cannot propagate and for which &1becomes complex. It is found by setting cos &I= -1. Equation (19) then shows that %I = 2JK,/M,,

(27)

0, = (w/2)dM,/K,.

(28)

so that

It follows that cosEt=1-2~:=Xi,

COSF~=~-~LIT(M~/M~)(K~/K~)=X~,

COSF~=~-~~:(M~/M~)(K,/K~)=X~.

(29a,b) (29c)

When any one of these is less than - 1, the corresponding value of E, is of the form 7r + i6, Instead of computing and using this in the complex form, it is more convenient in this case to use cash 8, = -X, and to find 8, by using real arithmetic from the derived equation S,=log(-x,+Jx;

-1).

(30)

Furthermore, when E, = n + ia, the complex sine terms in the receptance functions can be replaced by real sinh terms, as follows: sin E, by -i sinh- 6,, sin N,E, by i(-l)r sinh N& and sin (N,+ l)~, by i(-1) Nr+lsinh (N, + l)S, The i( =J- 1) terms in these all cancel in the receptance functions which are therefore still entirely real. In this way, the computer programme to determine the natural frequencies can be run throughout with real arithmetic routines. Complete non-dimensionalization of the frequency-receptance equation (7) may be accomplished by multiplying each of the five receptances involved by M, w$, These non-dimensional receptances then take the form ~*A=M1Wcl~*A= 2

-- 1

sin(N,+l)E1-sinNi&i sin (N1 + 1)~~

0: GBB= MIo:,aBB 5 II

= --

1

sin (N,+l)Eg-sin sin (N3+1)+ n: [

MI =

%&~~I

=-M

2

1 yp

1

1 KI’ 1 1= I 4K1

N3E3 ( 4K1

sin (N2+ l)e,-sin N2~2 sin (N2+ 1)~~

(31)

(32)

K2’

f522,

(33)

D. J.

434 L?12 =

2

MEAD

AND

S. M. LEE

-__-

M,uc1(~,2

In terms of these non-dimensional becomes (after expansion)

(34)

receptances,

the frequency receptance

equation (7)

‘YAA~~B+(Y1,((YAA+(YBB)+(Y:l -cl;:, =o.

(35)

At the natural frequencies of sub-system II on its own, the terms Grr and al2 both approach infinity. This leads to problems of ill-conditioning when the left-hand side of equation (35) is evaluated when seeking its roots. The problem is circumvented by writing -2 -2 -(Gl,+(Y12)((Yl,a,- 2), substituting into this from equations (33) and (34)) and a11 --a12 putting A, = sin Nc2/sin (N2+ 1)~~ and A2 = sin &,/sin (N,+ 1)~~. This’yields (Y:, - (Y:? = (l-Al--A2)(1-A1+A2)(M1/M2LR:) so the finally developed frequency-receptance equation is f(G)

)+(l-A,-A,)(l-A,+A,)(M,/M,R;)=O.

= &4&3+~11(~/%4+&3

(36)

OF THE WHOLE SYSTEM 2.5. THE MODES OF VIBRATION Once the natural frequencies have been accurately determined, the modes of vibration (i.e., the particle displacements) can be found by using some of the receptance functions. First of all, the particle displacements at each end of the three sub-systems will be found, and then the intermediate particle displacements in terms of those end displacements will be determined. The displacements at the ends of the three sub-systems are (see Figure 2) uE = FAaYAE, uc=FA~YCC=FA((YAA-l/K=l), UAR=--FA(Y~~+F&Y~~, &L=--&(Y,2+F@z2, UD= -FBaDD = -FB( aBB - l/K,,) and uF = -FBcxBF In these

(37) F,=F~~12I(+x+ff22) (from equation (6)). Expressions for all of the above receptances, except aAE and aBF have already been quoted. (YAEand aysF are of the same form as (Y,~and are given by aAE

=-sin

aBF =-sin

el/(w2Ml sin (N, + 1)~~).

(38)

EJ(w’M~

(39)

sin (N,+ 1)~~).

For convenience, the modal displacement amplitudes can be expressed in terms of Us Hence, one puts uE = 1, and divides all other displacements by UE = FA(YA~ Then i&=1, UAR=- ~II/%E

+(FB/FA)(~I~/~AE),

~D=-(FB/FA)((YBS-~/KC~)/(YAE,

~~=UCIUE=((YAA-~IK,*)I(YAE, &L

=

-@I~/@AE

&

(4Oa, b)

+(FB/FA)(Q~~/~AE)Y

=-(FB/FA)~QBF/~AE),

(41a,

b)

(4’k

b)

where FB/FA=cxI~/((YBB+cY~~). The intermediate amplitudes are found by considering the system of Figure 3(a), when the two end amplitudes and the frequency are known. As in equation (8), the total motion at any point consists of the sum of equal but opposite going wave motions such that i&=ii++L and iio = ti+ exp (-iN,E,) + ZXexp (+iN,e,). Solving for zi, and L, one finds ii+ = -(ii,

- UPexp (+iN,e,))/2i

U- = ( ilo - tip exp (-iN,E,))/2i

sin Nr~r, sin NJ~,

(434 (43b)

DISORDERED

ONE-DIMENSIONAL

435

LATTICES

The total displacement at X, n elements from P, is U, = ii, exp (-ins,) + U_ exp (fins,) =(r?,sinnE,+&.sin(N,-n)&,)sinN,E,.

(44)

This expression may be used to find the displacement of any particle in each of the three subsystems, once the end-displacements (Uo, UP) of the system (given by equations (37), (38) or (39)) are known. 2.6. COMPUTATIONAL PROCEDURE FOR FINDING NATURAL FREQUENCIES Values of R, are to be found which cause f(0,) of equation (36) to vanish (i.e., f(0) of equation (7)). Figure 4 shows a typical plot of f(0,) versus 0,. The zero crossings may be located approximately by computing the values of f(n,) at regular (small) intervals of a,, and identifying adjacent frequencies which cause the sign of F(R,) to change. The sign-changes caused by singularities (f(0,) + 00) are ignored. Linear interpolation and iteration are then used to converge on the accurate values of 0, which cause f(0,) to vanish. -

2000

c: ;I s

1000

; ..z 5 5

0

& F x : ?r ? L

-1000

-2000

0

O,P

0.4

0.6

0.8 Non-dlmenslonol

I.0

I2 frequency,

Figure 4. Typical variation with frequency of the frequency-receptance 1.0, M,=0.06125, N, =iV,=5, N,= 7; K’s= 1.0.

I.4

2-o

fi,

function

of equation

(7). M, = M, =

The total frequency range to be scanned can be found by using approximate formulae for the lowest and highest natural frequencies expected. The lowest frequency can be found approximately by using Rayleigh’s principle and Rayleigh’s quotient for the fundamental frequency. The lowest frequency for the scan must therefore be set below the Rayleigh approximation. The highest frequency to be considered need not exceed the highest of the following: (a) the highest natural frequency of system I with its right-hand end blocked rigidly by systems II and III; (b) the highest natural frequency of system III with its left-hand end blocked rigidly by systems II and I; (c) the highest natural frequency of system II with both of its ends (i.e., points C and D) blocked rigidly. The frequencies of (a) and (b) cannot exceed the cut-off frequencies of systems I and III, respectively, and so are given by 0, Y 1 and fl, >d(K3/KI)/(M3/M,), respectively. If system II has several masses and springs, the frequency (c) will not exceed the cut-off frequency of system II: i.e., 0, 7d(KZ/KI)/(M2/M1). If system II consists of a single mass, M2, connected to systems I and III through the coupling springs Kc1 and KC2, the frequency given by (c) will not exceed O-5 d[(K,, +KC2)/Kl]/(M2/M1).

436

D. J. MEAD

AND

S. M. LEE

The highest of these may safely be used as the upper limit of the scanning range. Between this frequency and 0 = 0, the total number of natural frequencies which exist is (Nl + N2 + N3 + 2). A limited frequency range within this total range may be scanned in order to find just those modes and frequencies within that range. It is not necessary first to find all the natural frequencies with lower (or higher) values. 3. NATURAL FREQUENCIES OF SOME SIMPLE DISORDERED SYSTEMS 3.1. THE EFFECTIVE MASS OF A FINITE PERIODIC SYSTEM Consider now a uniform periodic system with a single disorder created by just one mass and its adjacent springs being different from the regular values. Without the disorder, all the natural frequencies of the periodic system would be below the cut-off frequency, wc. With the disorder, it is possible for a natural frequency to exist in the forbidden zone above wcl. The conditions for this to occur, and the value of the natural frequency will now be investigated for the systems shown in Figures 5(a) and 5(c).

System I Cd)

System

System

II

IU

v+-+j-+T E

c

A

I

28

D

F

Figure 5. Particular simple disordered systems. (a) Single mass disorder at left-hand end only; (b) block diagram of (a); (c) single mass disorder at centre of a symmetrical system; (d) block diagram of (c).

It is helpful, firstly, to consider the nature of the periodic sub-system DF above the cut-off frequency. The receptance (YDD is given by equation (20) with E, = r +i& (see section 2.4). Replacing the sine terms by the corresponding sinh terms, one obtains, for w ’ wc*(G’ 1) (YD,=-(1/M,w2){1+sinhN2S2/sinh(N,+1)6,}.

(45)

The bracketted expression is always positive and less than 2.0. (Ye,, is the receptance of an equivalent mass, Me, of magnitude M,=M,/(l+sinh

N,S,/sinh(N,+l)&).

(46)

At the cut-off frequency, &=O and M,=M,(N,+1)/(2N,+l). At a very high frequency (w >>w,), S, is large and the sinh quotient becomes very small. Me is then very nearly equal to M2. Above the cut-off frequency, the whole periodic system DF can be regarded (as far as point D is concerned) as a frequency-dependent mass, Me, having a magnitude between MJ2 and M2.

DISORDERED

ONE-DIMENSIONAL

437

LAlTICES

If system 2 is semi-infinite (N2 + 00) its effective mass is given by M,,,=~M*/2)~1+~1-1/~n)““~, where LIZ is the non-dimensional system II.

frequency,

(47)

w/o C2and oC2 is the cut-off frequency

of

Non-d~mens~onol frequency, .Q2

Figure 6. Variation with frequency, &, frequency. ---, N2 = 1; -, N2 = 00.

of the effective

mass, M., of a periodic

system

above

its cut-off

Figure 6 shows Me/ M2 for systems with N2 = 1 and N2 = CO.When N2 has an intermediate value, the corresponding curve falls between the two curves shown. The values of Me/M, when f12= 1-O are indicated for N2= 2, 3, 5 and 9. It can be seen that the semi-infinite system quite accurately represents all finite systems with N2a 1 when R,> 1.50. Even at 0, = 1.05, the effective mass of the system with N2 = 1 is only 8.3% greater than that of the semi-infinite system. 3.2. THE SYSTEM WITH AN END DISORDER This is the system of Figures 5(a) and (b), and can be regarded as two masses, MI and Me, joined by the spring Kc. Natural frequencies occur when (Y**+(Y DD- -0 (YDD=-~/M,o~

These two receptances dimensional forms 6DD

= -MM,IW~:,

and

P

(484

022 = l/K, - 1M1w2.

(48b c)

and equation (48a) can be expressed in the corresponding

42

= 4W,lKJ

-

1/hW’Wfi:,

622+cDD=o.

non-

(48d)

Figure 7 shows the variation with frequency of GDD(with N2 = co) and -&, for different values of K2/ Kc and M1/M2. Since natural frequencies exist when (Y22+&‘DD= 0, they occur at the intersections of the curves. For the natural frequency to exist in the forbidden zone of system II (at, or above a2 = l), the mass MI must not exceed a critical value MI, which makes the natural frequency fi2= 1. One then has -l/(M,/M*) = 4(K,/K,)-l/(M,,/M,). Now, at f12=1, M,=M,/2. Hence MlJM2=0*5/(2K2/Kc-1).

(49)

Note that when Kc = 2K2, MI, + 00. Hence when Kc 3 2K2, any mass MI, however large, will lead to a resonant frequency in the forbidden zone.

438

D. J. MEAD

AND

Non-dimensional

Figure 7. Variation with frequency, + = 8; curve (b), p = 8, K, = K,/1.20;

S. M. LEE

frequency,

0_2

&, of the receptances a,, and -a,,; curve (c), y = 2; curve (d) p = 1.

-,

-G,,;

---,

a,,;

curve (a),

When the right-hand system II is finite, a natural frequency in the forbidden zone can be found accurately only by the iterative process described for the more general system. However, when the system II is large enough to be regarded as semi-infinite, a formula can be derived for the natural frequency, as follows. The effective mass of a semi-infinite system is given by equation (47). Using this in equation (48d) one obtains the following equation for the normalized natural frequency, 0,: 4(K,/K,)-l/(M,/M*)R~-2/~n:[l+(l-l/n~)”’]=o. This equation of which are

can be manipulated

A=I-KJK,,

(50)

to form a quadratic equation

B= (K,/2KJ[(K,/2Kd-

(MdMd(l AR;+BR;+C=O.

C = (K,M,14K,MJ2,

in a:,

the coefficients

-K,/2KJl,

(51a, b) (5 lc, d)

Of the two roots of equation (51d), only the lower positive value is valid. A higher positive root may occur if K, < KZ, and corresponds to the intersection “A” on Figure 7. The branch of the GDD curve on which it occurs is a fictitious branch corresponding (47). This is mathematically, but not to a negative value of (1 - 1/L?2)“2 in equation physically, possible. As the branch is fictitious, so also is the intersection on it, and the corresponding root and apparent natural frequency should be rejected. When K, = KZ, the coefficient A vanishes and the quadratic equation reduces to 0; which agrees

= -C/B

with the result

3.3. THE SYMMETRICAL

= 1/4[(M,lM,)

of Bacon

SYSTEM

WITH

-

W,/Md21,

(52)

et al. [l 11. AN INTERNAL

DISORDER

This is illustrated in Figures 5(c) and (d). To find the natural frequency the complete frequency-receptance equation can be used. Systems I and II are identical, so aAA = LYBB= I/K, - 1/M,02, where Me is the effective mass of each of the systems CE, DF at C and 0, respectively. System II consists of a pure mass, M2, which is the disorder mass. Then

DISORDERED

ONE-DIMENSIONAL

439

LATTICES

ff,I=(Y2*=cy12= -l/M+‘. Equation (7) simplifies to (~~~(0~~ + 2all) = 0. Hence, either IyAA=0 or (Y.,+*=-~LY1,. The first of these implies that MZ is motionless, and systems 1 and II are resonating with their ends at A and B held rigidly. The second involves motion of M2 and leads to l/K - l/Mew2 = 2/M2u2, where Me is the effective mass of system I or III. Expressed in the non-dimensional form, this equation becomes

4WJK)

- I/(MJM,W:

(53)

= 2/(M,/M,W:,

where R, = o/wC1. The critical value of M2 (Mzc) required to make 0, = 1 is found by putting Me = M,/2 and fl, = 1. Hence M2c

=

M,IPW,IK)

-

(54)

11,

which, as expected, is twice the magnitude of the disorder-mass a single semi-infinite system (equation (49)).

4. EXAMPLES

4.1. NATURAL

required at the end of

OF THE USE OF THE COMPUTER

FREQUENCIES

AND MODES

OF SYSTEMS

WITH

PROGRAMME THE

DEFECT

NEAR

ONE

END

The computer programme for finding the frequency roots of equation (7) has been used to study a number of problems related to natural frequencies in the forbidden zone when the periodic system is disordered by a defect cluster of light masses. The diagrams above Figures 8(a) and (b) show the systems considered. The effect on these natural frequencies of the depth to which the defect is “buried” has been investigated. This has been done simply by varying the value of (N3+ 1) which denotes the number of masses between the defect and the extreme right-hand end. A range of mass ratios P ( =M~/Mdisorder ) has been considered. All the spring constants have been assigned the same value of 1 *O. Initial calculations showed that with p > 2.0, the forbidden zone frequencies did not change significantly when the number of masses in the left-hand system exceeded 5. The

ii= I-----(b)

-32--i-

1 i

1

0 No of regular

Figure 8. Frequencies light masses.

in the forbidden

mosses

I

to the right

zone of a disordered

2

3

4

5

6

7

of the dmrder

system with (a) a single light mass and (bl two

440

D. J. MEAD

AND

S. M. LEE

value of 15 assigned in the present investigation ensures that when 0, > 1 the left-hand system behaves virtually as though it were infinite. Figure 8(a) shows the natural frequencies in the forbidden zone when the disorder consists of a single mass. When N3 + 1 = 0 and &I3 is light, the disorder is at the extreme end of the whole system. Notice that the natural frequencies for the p values considered scarcely change when N3 > 1. The disorder mass is therefore effectively bounded on each side by a semi-infinite system provided N3 3 1. This is also seen in Figure 8(b), which shows the frequencies of a disorder consisting of two equal light masses. In this case, there can be two natural frequencies in the forbidden zone for each system, and these are separately identified. Figure 9 presents these results (and those for disorders with three and four equal light masses) with p as the abscissa. The number of separate curves for a given system is equal to the number of light disorder masses in the system. The lowest curve for each system rises above fi, = 1 at a value of p which increases with the number of disorder masses. The programme is readily adapted to locating these + values precisely.

(15,3,0)-(15,3,5)Highest (15, I ,O)- (15, -_(15,3,0)-(15.3

I

branch branch ,5) Hqhest branch .5) Upper mlddle

/(15,0,0)-(15.0,5)M,dd,ebranch ‘(15,2,0)-(15.2,5) -_(15,3,0)-(15,3,5)Lowermiddle --115,

0

4

I ,5) Lowest branch

-_(15.2,0)-(15,2,5)Lowest

branch

-_(15.3,0)-(15,3,5)Lowesl

branch

1

1

0

I ,O)-(15,

8 Moss

12

ratio (= regular

I6

20

mass/dtsorder

24

28

32

mass)

Figure 9. Variation with mass ratio of the forbidden zone frequencies (15,3,0)-(l&3,5); ---, (15,2,0)-(15,2,5); -. -, (l&1,0)-(15,175);

of various

disordered

systems.

-

-,

-, (15,0,0)-(15,075).

For clarity, Figure 9 does not show frequencies for systems with the disorder at the extreme end. The curves on Figure 9 for a given disorder but for different values of N3 (=l, 2, 3, etc.) are, for the most part, quite indistinguishable. The modes of displacement at some of these natural frequencies are indicated in Figure 10. For the modes in the forbidden zone, rapid decay of displacement with distance from the disorder is evident, and is associated with the left- and right-hand systems behaving almost as though they were infinite. The modes of displacement within the disorder are those which would be expected for a simple equi-mass, equi-spring system with almost fixed ends. The last displacement-mode shown occurs at a frequency just below the forbidden zone. Large displacements are seen in the regular system, and much smaller displacements in the disorder.

DISORDERED

ONE-DIMENSIONAL

441

LA-KTICES

Q,

4 4 NJ P

I-804 15 3 2 32 3.345

I5 3 2 32

I.155 I5

I 2 I 2

I.003

15

3

2

I.25

I.084

I5

3

2

I.25

I5

2

3

I.111

0.983

Figure 10. Particle are plotted vertically.

displacements

for some of the modes of Figures

TABLE

4

I.760 I5

4

8 and 9. N.B.: longitudinal

1

Particle displacements of the highest modes of two systems with M1=M3=1, M,=O-03125; 0+4*0160; (a) N,=12, N2=2, N,=30; (b) N,=5, N2=2, N,=5 Particle

number

System (a)

System

(b)

Main system masses 1.15 1.12 1.5 1.4 1.3 1.2 1.1 1.0 Defect

+1.8178-29 -4.507E-24 +1.678E-11 -l.O49E-09 +6.555E-08 -4.097E-06 +2.56OE-04 - 1.600E-02

+1.6.528-11 - l.O49E-09 +6.5558-08 -4.0978-06 +2.56OE-04 - 1.600E-02

+ 1 .OOOE+OO 0.0 - 1 .OOOE+OO

+ l.OOOE+OO 0.0 -l.OOOE+OO

+1.600E-02 -2.560E-04 +4*097E-06 -6.555E-08 +l.O49E-09 -1.678E-11 +4*507E-24 +7.565E-35 +1.27OE-45 +2.097E-56

+1.600E-02 -2.560E-04 +4*097E-06 -6.555E-08 +l.O49E-09 -1.652E-11

masses 2.0 2.1 2.2

Main system masses 3.0 3.11 3.12 3,3 3.4 3.5 3.12 3.18 3.24 3.30

displacements

442 4.2.

D. J. MEAD THE

FORBIDDEN

ZONE

MODES

AND

S. M. LEE

OF LARGE

SYSTEMS

WITH

INTERNAL

DEFECTS

As an example of the power of the method to identify particular natural frequencies and to determine the particle displacements, Table 1 gives values of the particle displacements, p, for a system with M, = 1, MS = 0.03125, M3 = 1, and N, = 15, N2 = 2, N3 = 30. All K’s were assigned unit values. The accurate frequency of the mode is R = 4.0159691393. When N, and Ni are each reduced to 5, the accurate natural frequency is the same to at least 11 significant figures; the corresponding particle displacements are also shown on Table 1. The calculation of the modes of the 15-2-30 system required scarcely any more computer time than required by the 5-2-5 system. The particle displacements in the larger system remote from the defect are evidently small enough to be regarded as zero. The results are quoted to demonstrate the capability of the method for accurately determining modal displacements of such small magnitudes. 4.3.

MODAL

DENSITIES

OF LARGE

SYSTEMS

The programme has also been used for the rapid and exact determination of the number of modes per given frequency bandwidth (i.e., the “modal density”), of a large disordered system. Results are quoted for a system of 63 masses, 21 in each of three subsystems, with mass values of M, = 1 .O, Mz = 0.5, M3 = 0.25. All the spring constants were identical (K,, etc., =l). The number of zero-crossings of the frequency-receptance function of equation (7) was counted as successive frequency ranges were scanned. This can be performed very quickly, without having to identify each natural frequency accurately.

TABLE

2

Modal density in various frequency bands; comparison and approximate theory (N, = N> = N3 = 20; M, = 1.0, all K’s = 1 .O)

of computed numbers M2 = 0.5, M3 = 0.25;

Cumulative Frequency range (fl,)

o-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1.0 1.0-1.1 1.1-1.2 1.2-I .3 1.3-1.4

1.4-1.5 1.5-1.6 1.6-1.7 1.7-1.8 1.8-1.9 1.9-2.0

Number computed

of modes

in the range

3 2 3 4 3 3 4 4 5 8 2 1 3 5 2 3 0 1 2 4

Theoretical number of

modes 2.80 2.83 2.88 2.94 3.01 3.16 3.36 3.67 4.25 7.72 2.09 2.32 2.74 4.89 3.09 1.01 1.13 1.32 1.70 5.09

No.

of modes A

I

Computed

Theory

3 5 8 12 15 18 22 26 31 39 41 42 45 50 52 55 55 56 58 62

2.80 5.63 8.51 11.45 14.46 17.62 20.98 24.65 28.90 36.62 38.71 41.03 43.77 48.66 51.75 52.76 53.89 55.21 56.91 62.00

DISORDERED

ONE-DIMENSIONAL

LA-I-FICES

443

The number of modes with natural frequencies in successive frequency bands of Afl, = 0.1 is shown on Table 2. For comparison, values deduced by an approximate method outlined in the Appendix are quoted alongside. Notice that frequencies in both the forbidden zone and the continuum zone are considered in this analysis. 5. FURTHER POSSIBLE DEVELOPMENTS OF THE METHOD It is a simple matter to develop the method to find natural frequencies and modes of di-atomic systems, in which the periodic sub-systems have alternate light and heavy masses. The characteristic wave receptances of such systems are still given by equations ( 14) and ( 15), but (Y,, and (Y,bare no longer given by equations ( 17) and ( 18). Expressions for (Y,, and c&b are, however, easily derived for this case. The method can easily be applied to a system in which the defect sub-system consists of an array of unequal masses linked by unequal springs. The defect sub-system receptantes, all, a12, (y22 can be found by the usual receptance methods (not the “wave receptance” method of this paper) and can then be used directly in the frequencyreceptance equation (7). If the masses of the periodic sub-systems are linked by “next-to-neighbour” elastic spring forces, the periodic systems are no longer mono-coupled, but are bi-coupled. Two pairs of free waves can exist in the sub-system at any frequency. Each sub-system must then be characterized by 10 receptance functions, four of which are direct receptances, and six of which are transfer receptances (cf. the three receptances for the sub-systems of this paper-cull, ai2, a22 for sub-system II). If the sub-system is symmetrical, the 10 receptances consist of only six different receptances. The determination of these receptantes and their use in finding natural frequencies of the whole system must be based on the method outlined by Mead [ 181. 6. CONCLUSIONS Natural frequencies and normal modes of finite one-dimensional periodic systems with a defect can conveniently be computed by using a method based on a receptance analysis of the whole system. Advantage can be taken of the periodic nature of the sub-systems to simplify the calculation of their receptances (especially when the sub-systems are very large), and to determine accurate modal displacements of the systems at the natural frequencies. The natural frequencies and modes in any given frequency band can readily be computed to a high degree of accuracy without having to find the frequencies and modes in other bands. Calculations performed to demonstrate the use of the method have shown how easily particular natural frequencies and modes can be found, whether in the forbidden zone or in the lower frequency band. The effect on the forbidden zone frequencies of the proximity of the defect to the end of the system has been studied briefly. If the defect mass is less than one half the periodic mass, and all the spring stiffnesses are equal, the defect is effectively bounded by two semi-infinite periodic systems provided there is at least one periodic mass on either side of it. Receptance methods of analysis can be used to derive simple formulae for forbidden zone frequencies due to single-mass defects.

REFERENCES BORN and T. VON KARMAN gungen in Raumgittern.

1. M.

1912 Physikalische Zeitschriff 13, 297-309.

ijber Schwin-

444

D. J.MEAD

AND S.M. LEE

2. J. M. ZIMAN 1960 Elec#rons and Phonons. London: Oxford University Press, 1963 re-issue. See Chapter I, Section 4, pp. 16-22. 3. S. M. LEE 1975 Journal of Sound and Vibration 38, 272-274. On application of the cyclic boundary condition to mechanical vibrating systems. 4. H. B. ROSENSTOCK 1955 Journal of Chemical Physics 23, 2415-2421. On the optical properties of solids. 5. S. M. LEE 1966 Journal of Molecular Spectroscopy 21, 183-202. A new treatment of the normal vibration modes of a linear diatomic lattice. 6. P. MAZUR, E. W. MONTROLL and R. B. POTTS 1956 Journal ofthe Washington Academy of Sciencies 46, 2-l 1. Effect of defects on lattice vibrations, II: Localized vibration modes in a linear diatomic chain. 7. R. L. BJORK 1957 Physical Review 105,456-459. Impurity-induced localized modes of lattice vibration in a diatomic chain. 8. S. M. LEE 1970Physica Status Solidi 39, K79-99. On the surface vibrations of diatomic linear lattices. 9. D. P. S.MALIK and A. K. GHATAK 1971 Physica Status Solidi 45, K51-54. On thenormal modes of a diatomic linear lattice. 10. E. W. MONTROLL and R. B. POTTS 1955 Physical Review 100, 525-543. Effects of defects on lattice vibrations. 11. M. D. BACON, P. DEAN and J.L. MARTIN 1962 Proceedings of the Physical Society 80, 174-189. Defect modes in two-component chains. 12. P. DEAN 1967 Proceedings of the Physical Society 90, 479-485. Formulae for the frequencies of a defect cluster in one dimension. 13. S. BALA D. P. S.MALIK and A. K. GHATAK 1972 Journal ofthe Physics and Chemistry of Solids 33, 1885-1890. Local and gap modes due to surface or substitutional impurities. 14. R. F. WALLIS 1957 Physical Review 105, 540-545. Effect of free ends on the vibration frequencies of one-dimensional lattices. 15. H. B. ROSENSTOCK and R. E. MCGILL 1962 Journal of Mathematical Physics 3, 200-202. Vibrational modes of disordered linear chains. 16. H. B. ROSENSTOCK and R. E. MCGILL 1968 Physical Review 176, 1004-1014. Vibrations of disordered solids. and natural 17. D. J. MEAD 1975 Journal of Sound and Vibration 40, l-18. Wave propagation modes in periodic systems: I Mono-coupled systems. 18. D. J.MEAD Journal of Sound and Vibration 40, 19-39. Wave propagation and natural modes in periodic systems: II Multicoupled systems with and without damping. 19. G. SEN GUPTA 1970 Journal of Sound and Vibration 13, 89-101. Natural flexural waves and the normal modes of periodically-supported beams and plates.

APPENDIX:

THE MODAL DENSITIES SYSTEMS-ORDERED AND

OF FINITE DISORDERED

PERIODIC

The natural frequencies of finite periodic systems with no disorder can be found by using the method presented by Mead [17] and Sen Gupta [19]. This can be done through a geometrical construction on the curve of the propagation constant, E,, versus frequency, 0,. The natural frequencies so found for the truly periodic system must all lie within the frequency band below the cut-off frequency. Suppose there are N, complete periodic elements and N,+ 1 masses in the ordered system. Figure Al shows the variation with fin, of the propagation constant F, To find the natural frequencies, draw a grid of equidistant lines over the curve, parallel to the 0, axis, at intervals of r/N,. The intersections of these lines with the E, curve locate approximately the natural frequencies. The precise location is obtained by shifting the grid by an amount considered in reference [17], but if modal densities (and not precise natural frequencies) are required, this refinement can be ignored. The modal density is the number of modes occurring in a particular unit bandwidth, and can be defined by l/(difference between successive natural frequencies, AR). From Figure Al, it is evident that Aa, + AE,/(dEI/dO,) = (~/N,)/(d&,/dL&). The modal

DISORDERED

0

ONE-DIMENSIONAL

0.2

0.4

0.6

Non-dlmenslonal

Figure Al.

frequencies

Variation of the propagation constant and modal densities of finite systems.

445

LATTICES

0.8

I-O

frequency, a,

with frequency;

geometrical

construction

to find natural

density, n,, is given by n, = l/AR, = (P/,/n) d&,/da, per unit 0,. E, and R, are related through equation (29a), and from this one finds dE,/dl(l, =2/(1 -fii)1’2, and hence fli)1’2 per ni = (2N*/7r)/(l -On:)1’2. The modal density of system II is (2&/5r)/(lunit f12. Since ~2=~/~,2 and R,=co/w~~, it follows that a2 = pi201 and 0, = pl&i, where pi2 = wC1/wC2and pi3 = wC1/wC3.The total modal density niz3 of the combined system of three sub-systems is, to first order, nl + n2+ ng. Hence n

1u=iri*$(l-p:,nV ,

per unit LJ,.

This corresponds to a total number of modes equal to N, + N2 + N3. The complete system has N, + N2 + N3 + 3 masses, and N, + N2 + NJ + 2 natural modes. It can be shown by using receptance arguments that the extra modes must exist just below each of the two highest cut-off frequencies of the three systems.