dynamic properties of self-similar structures in mechanics

ISSN 1052�6188, Journal of Machinery Manufacture and Reliability, 2013, Vol. 42, No. 2, pp. 89–94. © Allerton Press, Inc., 2013.Original Russian Text © L.Ya. Banakh, O.V. Barmina, 2013, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2013, No. 2, pp. 3–9.

89

SELF�SIMILAR STRUCTURES IN MECHANICS

Regular structures with translation symmetry make up a rather limited class and do not describe mostnatural and technical systems. However, it can be significantly enlarged by adding a class of self�similarstructures:

Structures, wherein each cell copies the structure of the previous one at a certain scale (here, the trans�lation symmetry is accompanied by the transformation of similarity (scaling) of neighboring cells; it is oneof the fractals);

Structures with helical symmetry, wherein the cells are placed along a helical line with a uniform step,including systems with reflection symmetry); and

Self�similar structures, for which the helical symmetry is accompanied by the transformation of simi�larity of neighboring cells.

It is precisely these structures which prevail in biology, polymer chemistry, and physics. The growth andformation of polymer molecules and crystals are accompanied by the formation of fractals. Many scien�tists study fractals primarily as a way of shaping different structures [1, 2].

Such structures created by man are simpler in comparison with natural formations, though the studyof them is of specific character. Dynamic and vibration properties are of first importance in this research.The present paper is aimed at the investigation of the dynamic properties of self�similar structures, whichare poorly known.

The most widespread mechanical self�structures are the bar with the stepped section, a shaft with disks,whose parameters change by segments, the ribbed conical shell, the bent shaft (helical symmetry), conicalsprings, and so on [3].

Let us consider the principal dynamic properties of such structures and the methods for investigatingthem using fairly simple systems.

VIBRATIONS OF SELF�SIMILAR STRUCTURES.DISPERSION EQUATION AND VIBRATION MODES

Mathematical investigation of the dynamic processes in self�similar structures can be carried out onthe basis of the unified approach [4]. We study the longitudinal vibrations of a bar of round section withfixed localized masses. Assume that the geometric parameters of the bar (l is the length and r is the radiusof cross section) vary with the uniform scale γ; the values of masses in Fig. 1a also change. Then, the rigid�ity and the mass for the s + 1 elements of system are the following:

(1)ks 1+EFs 1+

ls 1+

��Eπγ2

rs2

γls

�� γks, ms γms 1+ .= = = =

MECHANICSOF MACHINES

Dynamic Properties of Self�Similar Structures in MechanicsL. Ya. Banakh and O. V. Barmina

Moscow, RussiaReceived December 3, 2012

Abstract—The structures, in which each cell is similar in structure to the previous one at a certainscale, are called self�similar. Their mechanical vibrations are considered in the present paper. Thevibrations of the systems with the parameters changing in length with some scale, namely, longitudinalvibrations of a bar with localized masses, a beam with a stepped section, and the two�dimensional self�similar grating, are studied as examples. The features of such systems during vibrations are identified:self�similar systems consist of a mechanical band�pass filter; and the capability of amplification(deamplification) of a transmitted signal along the length.

DOI: 10.3103/S1052618813020039

90

JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY Vol. 42 No. 2 2013

BANAKH, BARMINA

Thus, the partial frequencies for each mass are identical, being equal to = = … = ks/ms. Con�dition (1) can be considered as the condition of self�similarity.

The equation of self�similar structure (Fig. 1a) in matrix form is

(2)

where XT = [x1, x2, x3…xn]; ω is the eigenfrequency.

These equations can be considered as equations in finite differences with variable coefficients depend�ing on the number of the node s, i.e., on x. Then, the equation for the s node is

(3)

Let us perform a change of variables in (2):

(4)

This means the multiplication of the matrix [K – ω2M] by the matrices N and NT on the right and leftrespectively. As a result,

. (5)

Equation (5) obviously describes the regular structure of translation symmetry with identical massesm1, the rigidity between which is constant and equal to k1γ–1/2. Here is the additional fixing of massesk1(1 + γ)γ–1 – 2k1γ–1/2 = k1(1 – γ1/2)2γ–1 = k* (Fig. 1b).

ν12 ν2

2 νn2

–m1ω2k1+ k1–

k1– –m2ω2k1 1 γ+( )+ γk1–

γk1– –m3ω2k1γ 1 γ+( )+ γ2

k–

γ2k–

X 0,=

…

k1γs 1–xs 1–– m1γsω2

– k1γs 1–1 γ+( )+[ ]xs k1γs

xs 1+–+ 0.=

X* NX, N2

diag 1/γ 1/γ2 … 1/γn[ ].= =

K ω2M–[ ]* N

TK ω2

M–[ ]N,=

K ω2M–( )*

m1ω2–

k1

γ��+

k1

γ��–

k1

γ��– m1ω2

–k1 1 γ+( )

γ��+

k1

γ��–

k1

γ��– m1ω2

–k1 1 γ+( )

γ��+

=

…

m1 m2 m3 mn...

(a)

(b)m1 m1 m1 m1

Fig. 1. A self�similar bar structure with localized masses (a) and the equivalent periodic structure with the same eigenfre�quencies (b).


DYNAMIC PROPERTIES OF SELF�SIMILAR STRUCTURES IN MECHANICS 91

Equation (5) can be written in finite differences,but with constant coefficients. The equation for thes mass is

(6)

A particular solution of Eq. (6) has the form [5]:

(7)

The solution can be interpreted as a wave, whereω is the eigenfrequency and μ is the wavelength

k1

γ��xs 1–– m1ω2

–k1 1 γ+( )

γ��+ xs

k1

γ��xs 1+–+ 0.=

xs Cei ω t μs–( )

.=

constant characterizing a change in phase at the transition from the s element to the s + 1 element anddefining the wavelength, which is Lw = 2π/aμ, where a is the distance between the neighboring masses inthe initial system. Substituting (7) into (6), we find the dispersion equation relating the vibration fre�quency with the wave number. The dispersion equation for the system (Fig. 1b) takes the form:

(8)

for real μ; and

for pure imaginary μ = iμ'.The real values of μ are determined from Eq. (8) at |cosμ| ≤ 1. Hence it follows that the given system is

the mechanical band�pass frequency filter; the band of the harmonic signal transmission is ω0 < ω < ω*(Fig. 2), where

Consequently, the transmission band is defined as

(9)

Thus, the transmission band is in inverse proportion to the similarity parameter γ. The larger γ, thesmaller the transmission band. For mechanical systems, this parameter is small, i.e., of the order of 1.5–3.It can be larger for biological systems. If γ is very large, the transmission band in the limit becomes verynarrow, and the system is configured on a certain frequency.

Outside the transmission band (Fig. 2), the vibrations are exponentially damped (or they increase), andthe solution takes the form:

The infinite increase in the amplitude of eigenfrequencies is obviously impossible in the real systemowing to the absence of the energy source.

Since the linear transformation of coordinates (4) does not change the frequency properties, the struc�tures (Fig. 1) have the same frequencies. Therefore, the self�similar system (Fig. 1a) is the mechanicalband�pass filter of frequencies.

The vibration modes are easily derived from the corresponding mode of vibrations of the regular struc�ture (Fig. 1b) by proportionally varying the amplitude of vibrations of each segment by γ times, accordingto the coordinate transformation (4). Consider the construction of the higher mode of vibrations as anexample for the self�similar system (Fig. 1a) at fixed ends. For the regular structure (Fig. 1b), it is a sine

m1ω2–

k1 1 γ+( )γ

�� 2k1

γ�� μcos–+ 0=

m1ω2–

k1 1 γ+( )γ

�� 2k1

γ��coshμ'–+ 0=

ω02

k*/m k1 1 γ–( )2/mγ at μ 0,= = =

ω*2 k1 1 γ+( )2/mγ at μ π.= =

Δω2 ω*2 ω02

– 4kγ 1/2–.= =

Xi* A 1–( )i

eηi

B 1–( )ie

ηi–, ηcosh+ 1 ω2

2νi2

��– k*

2k1/ γ��+

⎝ ⎠⎜ ⎟⎛ ⎞

1 ω2

2νi2

��– 1 γ–( )2

γ��+

⎝ ⎠⎜ ⎟⎛ ⎞

.= = =

ω

ω*

ωn

π/ γ– π/ γ

Fig. 2. A dispersion curve for the real wave values.

92


BANAKH, BARMINA

curve, and the neighboring masses are in antiphase. Therefore, the corresponding mode of vibration of theinitial self�similar structure (Fig. 1a) is obtained by increasing the amplitude of each mass by γ times.Thus, we find the vibration mode with constantly increasing amplitudes (Fig. 3). The envelope of thismode is the right line with the slope angle γ, while the points with the maximal amplitude shift along theaxis x every time.

Let us consider a further example of the mechanical system, namely, the longitudinal vibrations of abeam with a stepped section. Such beams are applied often in engineering constructions. Specifically, theyare used in ultrasonic vibration systems to increase the amplitudes of vibrations transmitted from thevibration exciter to a tool [6, 7]. In the self�similar beam with the stepped section, the cross�sectional areaof each s segment varies by an exponential law (Fig. 4):

(10)

The mass of the s segment of the beam is Ms = ρlFs. We consider this mass to be evenly divided betweenthe segment ends, equal to Ms/2, and localized at each end of the s element. Then, the mass ms in the snode is equal to the half�sum of the localized masses of the neighboring elements of the beam, i.e.,

ms = .

The rigidity of each segment with longitudinal vibrations along the horizontal axis is ks = EFs/l. Thepartial frequencies νs = ks/ms for each mass are the same at l = const:

i.e., the self�similarity conditions (1) are satisfied for such a beam.

The equations of the longitudinal vibrations in finite differences for the s node are the following:

(11)

Taking correlations (10) into account, we can write (11) in the following form:

Hence it follows that the structure of Eqs. (11) is absolutely similar to (3). Thus, all the results are true

for the given beam: it is the band�pass filter with the transmission band Δω2 = ω*2 – = 4k1/ (9).The vibration modes are characterized by increasing amplitude at the thin end, as in Fig. 3.

For the given beam of stepped section in the limit at n → ∞ (n is the number of segments), we obtainthe beam with the cross�section changing in length by an exponential law: F(s) = Fexp(ρF0(s)/P).

Let us show that Eq. (11) in the limit for the continuous system is the Klein�Gordon equation. Thewave equation in new variables with the exponential law of changing parameters can be reduced to thisequation. Let us write the wave equation for the longitudinal vibrations:

(12)

Let the cross�sectional area and the linear mass change according to F(x) = , m = m0ebx.

F s( ) γse

bs.= =

12��ρl Fs 1– Fs+( )

νs ks 1– ks+( )/ms 2E/ρl2

const,= = =

ks 1– xs 1– ks 1– ks msω2

–+( )xs ksxs 1++ + 0, s 1 … n., ,= =

k1γs 1–xs 1– k0a

s 1–k1γs

smsω

2–+( )xs k1γs

xs 1++ + 0, s 1 … n., ,= =

ω02 γ

∂∂x�� F x( )∂u

∂x��⎝ ⎠

⎛ ⎞ m∂2u

∂t2

��.=

F̃0ebx

x

Fig. 3. Vibration modes of self�similar structure.

l

Fig. 4. A beam of stepped section.


DYNAMIC PROPERTIES OF SELF�SIMILAR STRUCTURES IN MECHANICS 93

Substitute the variables u(x) = (x)eαx. Then, from (12) it follows that

(13)

Let us put α = –b/2. Then, u(x) = (x) , which is in accordance with the substitution of variables (3)for the discrete case. From (13) follows the Klein�Gordon equation

Note that the beam of uniform resistance [7] also has the exponential law of cross�section variation.It is known that such a vertical beam under gravity has uniform static loading in each section. It is easy tounderstand, when taking into account the change in load (force of gravity) along the length, it obeys anexponential law.

In particular, the well�known “golden section,” which has been used in architecture from ancienttimes, satisfies the exponential law of changing parameters. This is a special case of self�similar variation

in parameters of segments by the law T(s) = as, a = Φ = .

BRANCHED STRUCTURES (TWO�DIMENSIONAL GRATING)

Let us consider the vertical vibrations of a self�similar two�dimensional grating shown in Fig. 5. Suchstructures are found in many biological systems, specifically, in identification of the genetic code [9]. Eachelement of the grating has the double index sj, where s is the number of the row (along the axis x), and j isthe number of the mass in the s row (along the axis y).

Let us find the conditions of grating self�similarity. For each s element, we assume that the elastic iner�tia elements vary according to the similarity coefficient γ, i.e., kxs = kx0/γs, ms = m0/γs is the first self�similar

condition of = kx0/m0 = const along x.

All the elements in each row are supposed to be identical along y. Then, the second condition is

kys/ms = = const, kysj = const.

The equation of vibrations in the vertical plane for the sj element of grating has the form:

(14)

Assume that

(15)

We search for the solution of the dispersion equation in the form Zsj = XsYj, and the resultant vibrationsare the multiplication of waves along the axes x and y. Due to (15), equations (14) are independent alongthe axes x and y. Vibrations of standard periodic structure occur along the axis у; the dispersion equationof the structure is (–mω2 + ky) = ky(e

iν + e–iν), ys0 = ysn = 0. Hence, Ysj = Cssinνj at ω < ωy at fixed ends.

u

Fx'∂u∂x�� F∂2

u

∂x2

��+ F0eα b+( )x ∂2

u

∂x2

�� b 2α+( )∂u∂x�� ba α2

+( )u+ + m0eα b+( )x∂2

u

∂t2

��.= =

u e

b2��x–

F0∂2

u

∂x2

�� α2u–⎝ ⎠

⎛ ⎞ m0∂2

u

∂t2

��.=

5

ωxs2

ωy2

Mλ2K+( )Zsj– Kx Zs 1– j, Zs 1 j,++( )– Ky Zs j 1–, Zs j 1+,+( )– 0.=

K Kx Ky, Kx+ kx 0

0 0, Ky

0 0

0 ky

.= = =

s − 1

s

s + 1

Fig. 5. A two�dimensional self�similar grating.

94


BANAKH, BARMINA

The dispersion equation along the axis x (ν = 0) is found from (14), assuming that xs + 1 = xs/ .In this case, the dispersion equations coincide with the equations for the self�similar structure shown in

Fig. 1a. The transmission band is similarly determined to correlations (9): Δω2 = .

Thus, the self�similar grating is also the band�pass filter and has the property of amplifying (deampli�fying) the signal.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (project no. 11�08�90434Ukr_f_a).

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Algebras of Genetic Code, Noise Immunity), Moscow, 2008.

γ/2

2aωx2/γ

dynamic properties of self-similar structures in mechanics

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