Waves and vibrations in curved ultrasonic transmission lines

Waves and vibrations in curved ultrasonic transmission lines K. F. GRAFF Curved transmission lines are used in several areas of ultrasonic and power sonic applications such as delay lines and sonic wave drawing apparatus. The optimum design for such applications requires knowledge of the vibration and wave propagation characteristics of such devices, but theoretical developments in this area have been generally lacking. A review of some recent developments is given, as well as brief remarks on desired future work.

Introduction The transfer of ultrasonic energy from a transducer to a receiver or load frequently involves a solid transmission line having the form of a straight or curved wire, rod or plate. Ultrasonic delay lines, in wire or strip form, 1 are examples of such transmission devices for high frequency applications. In power sonics applications, half-wavelength or multiple half-wavelength rods are often attached to the horns of sonic transducers in order to transfer energy to a distant load or to increase the energy storage characteristics of the system. While many applications use straight transmission lines, it is necessary or desirable in many instances to use curved devices. For example, in the case of delay lines, a coiled configuration enables a long line to be placed in a compact space. A typical configuration of a coiled wire delay line is illustrated in Fig. 1. Other illustrations of such devices have been given by May,2 Palfreeman 3 and Davidson (reference 1, p 141). The nature of the transmitted signals may be longitudinal or torsional pulses in curved wires, or shear waves in curved strips. In the area of power sonics, curved transmission lines, or 'bent waveguides', have found several actual or suggested applications. Rozenberg et al 4 have pointed out that early ultrasonic dental drills employed curved waveguides. The use of curved transmission lines allows a single transducer to drive several tools in certain applications, as pointed out by Frederick 5 and by Rozenberg et al (reference 4, pp 7 7 78). Apparatus used in ultrasonic wire drawing may also employ curved transmission lines. Illustrations of such apparatus have been given by Balamuth 6 and recently Dr Karl F. Graft is Professor of Engineering Mechanics at The Ohio State University, Columbus, Ohio, USA. Paper received 10 November 1970,

ULTRASONICS. MARCH 1972

Output J

t FGQsdUC~F

Fig.1 Coiled wire delay line

ssion line

Transducer

Fig.2 Curved transmission line used in wire drawing apparatus

77

reported by Buckley and Freeman. 7 The sonic wire drawing apparatus developed by Funk 8 also employs a curved transmission line and is illustrated in Fig.2. The rational design and optimum use of transmission lines, be they straight or curved, requires an accurate governing theory for describing their vibration and wave propagation characteristics. When transmission lines are straight, either in the form of rods or plates, there is no particular problem in this respect. At low frequencies, when the vibrational wavelength is long compared to the transmission line thickness, simple longitudinal rod or plate theory in the form of the wave equation describes the motion. At higher operating frequencies, where the simple theory no longer applies due to the influence of lateral inertia and shearing effects, other approximate theories are available. Thus, various higher order theories have been presented by Love, 9 Mindlin and Herrmann, 10 Medick 11 and others, that include mechanical effects important at high frequencies. On the other hand, the vibration and wave propagation characteristics of curved transmission lines are not as well known as for straight lines. The complicating effect of curvature alters the response of such transmission lines from that of their straight counterparts. Thus, in the case of delay lines, wavelength-thickness designs that would be nondispersive in straight form would show pulse dispersion when coiled. In the case of power sonics, it is found that curved lines, equal in curvilinear length to resonant straight lines, have shifted resonant frequencies. The physical basis for the more complicated behaviour of curved lines lies mainly in the coupling between flexural and longitudinal motions of the line. In the case of straight lines, there is no tendency for such coupling; in curved ones, it is unavoidable. While the physical basis for the behaviour of curved lines is sufficiently clear, the mathematical description of their vibration and wave characteristics has been only partially established. Although the vibrations of circular rings have been considered by many investigators, their attention has been mainly focused on the inextensional, flexural behaviour of complete or incomplete rings with applications to the low-frequency behaviour of arches and reinforcing rings. In ultrasonic and power sonic applications, however, frequencies are high, forcing functions are longitudinal and the primary mode of energy transfer is by extensional action. Therefore, theoretical analysis of vibration and propagation phenomena under such circumstances must include such effects. Several efforts have been made to include longitudinal , extensibility effects in the theory. Philipson L2 used Love s original development to include extentional effects in the equations for lines of arbitrary curvature. Buckens 13 has included extensional effects in determining the natural frequencies of rings. However, wave propagation was not considered in these developments. In addition to centre line extension, rotatory inertia and shear effects may be important in high frequency sonics applications. These effects should also be assessed using a higher order theory analogous to the Timoshenko development for straight rods. Morley, 14 in one of the few treatments of wave propagation in rings, has derived governing equations with these effects included. However, wave propagation characteristics,in terms of dispersion curves and frequency spectra were not presented. Volterra 15,16 and Lincoln and Volterra 17 have presented the results for the natural frequencies of

78

complete rings using higher order theories. Wittrick 18 has reported on waves in helical springs, and recently the results of some experimental studies on stress waves in rings have been reported by Britton and Langley. l 9 In the following matter, some additional developments in the formulation of governing theories for vibrations and waves in circular rings will be reviewed. The wave propagation characteristics of curved lines, as indicated by their dispersion curves, will be presented, and some results obtained on the natural frequencies of ring segments will be given. Finally, a short discussion of desired future developments in precision analysis and approximate theories is given.

Governingtheory The usual approach to formualting dynamical approximate theories for rods, plates or sheels begins with the statement of the conservation of energy,

f

( T + X})dv=O

1

v where T and V are the kinetic and strain energy densities, given by P T = - ( u + w) 2 V = V(eoo,erO ) The tangential and radial displacement components for a circular ring are shown in Fig.3. Also in Equations 2, p is the material density and e00, er0 are the tangential and shearing strain components in polar coordinates, defined by

e0o -

r 30 1 3~,

er0 -

+ r 3fi

÷ .... r 30 3r

fi

r A

A-A

i Fig.3 Circular ring section

U L T R A S O N I C S . MARCH 1972

involved and are also omitted here for brevity. The results are given by

Noting that OV tO0-

OV

8e00

EAk2(u ' + w - Re')

EA(u' + w) N _

, frO-

aer 0

the following results are obtained for I", V:

R

R3

Q = G'A(1 + k2/R 2)

(w' - u + Re)

ll

R

EAk2(u ' + w - R~b') M= _ - -

--

+

+

frO

aO

Or

for the ring stresses. The equations of motion are 20 (k2~)

+-----

r

R2

5

( oo,oo) r

r

pRA

O0

EA(u"+w')

~+--.

-

R

The most important stage in the development of any approximate theory now arises. Certain assumptions must be made on the kinematics of deformation. Thus, in the development of classical Bernoulli-Euler beam theory, this takes the form of assuming that plane cross-sections remain plane and at right angles to the neutral surface. In the present development, the deformations are presumed to be given by

E A k 2 ( u " + w ' - R ~ ") +

R3

R

+ G'A

(

k~2) (w' - u - R~)

1+

R

EAk2(u " + w' - RqS") pAk2(~ + R'q~)= -

~i(r, O, t) = u(R, O, t) + zO(O, t)

R2

-G'A

1+

(w'-u+R¢)

12

W(r, O, t) =w(R, O, t) The next step is to substitute Equation 5 in Equation 1 and perform the integration to obtain the equations of motion. While the details of this process will not be presented here, the results are given by

p R A ~ =G'A

k 2 \ (w" - u' + Re') 1 +~-~

)

R

EA(u' + w)

E A k 2 ( u ' + w - R~')

R

R3

p R A ( ~ + k2~b'/R) = N' + Q

In the above equations, E is Young's modulus and G' is a modified shear modulus in the sense of Timoshenko beam theory, defined by G' = KC,. Here G is the usual shear modulus and g is the Timoshenko-type of shear coefficient.

pAk2(;J + R'~6)= M' - RQ pRA~ = Q'-N

where N, Q and M are the ring stresses (normal force, shear force and bending moment, respectively) defined by

N=f 'oodA, O=f','rOdA,M=f oozdA 8 A

A

A

Also in Equations 7, A and k are the cross-section area and the radius of gyration of the ring, defined respectively by f dz=A, fz2dA=Ak 2 9 A A In addition to the equations of motion, the boundary conditions for finite rings accrue as a natural by-product of the integration and are given by 6N, ~M, ~,Q = 0, 0 = 0 1 , 0 2 Thus, boundary conditions corresponding to various types of constraint are obtained by specifying one each of each of the three pairs of quantities. The steps necessary to express the ring stresses, N, Q, M and equations of motion (Equations 7 7 strictly in terms of the displacements and rotations u, w, ~bare slightly

U L T R A S O N I C S . MARCH 1972

10

The resulting equations of motion contain the effects of centre-line extensibility, shearing deformation and rotational inertia. In addition, a type of Winkler-Bach effect associated with the shift in neutral axis location due to ring curvature is indicated by the EAk2/R 3 term in the defining relation for N. Shear effects are carried by the G' terms of Equations 12 while rotational inertia effects arise from the left-hand side of the second of Equations 12. It is possible to develop more simple governing theories for the motion of rings by neglecting the effects associated with shearing deformation, rotational inertia or the shift in the neutral axis location. For example, when shearing effects and rotational inertia are ignored, the resulting equations are given by 21 pRAii -

EA(u" + w') R

pRAY, =

EA(u' + w) R

EAk2(w TM + 2w" + w)

13

R3

79

Waves in curved lines The wave propagation characteristics of curved lines are of interest in delay line applications or in determining the response of sonic transmission lines to impact. The usual approach in determining these characteristics is to establish the relationship between wave velocity and wavelength, or between frequency and wavelength, for propagating harmonic waves. For the case of the governing Equations 12, this is done by considering solutions of the form u = A t ei~( RO - ct) w = A2ei~(R0 - ct)

14

~b = A 3 ei~( R0 - ct) Where A 1 , A 2, A 3 are frequency-dependent wave amplitudes, c is the wave velocity and ~ is the wave number. This latter quantity is related to the wavelength k by the relationship ~ = 27r/k. The dispersive characteristics of the curved line will be indicated by the relationship between c and. ~. The relationship between frequency and wavenumber is obtained from the dispersion relations by using the basic relationship co = ~c or, alternatively, by first considering solutions of the form u = A 1 exp [i(~R0 - cot)], and so forth. Substitution of Equations 14 in the governing equations (Equations 12) yields a rather complicated relationship between the wave number and wave velocity which can be expressed in the form of a three-by-three determinant, given by [aij [ =~0 , ( i , j = 1 , 2 , 3 )

It is possible to assess the influence of, say, shear or rotary inertia by appropriately modifying the dispersion relations and recomputing and comparing the curves to the general case of Fig.4. The results from the simple theory, governed by Equation 19, where rotational inertia and shear effects are neglected, are presented in Fig.5. Since shearing deformation is absent in such a development, there are only two modes of propagation, the flexural and the longitudinal and, consequently, only two sets of branches to the dispersion curves. It should be noted that the results for zero curvature (K 2 = 0) agree with those of classical straight rod and beam theory. Thus, the zero curvature branches are given by ~ = 0, ~'. The former is the nondispersive longitudinal mode of classical rod theory, the latter is the result from Bernoulli-Euler beam theory. Similarly, the zero curvature results for the general theory shown in Fig.4 agree with those obtained from Timoshenko beam theory.

Vibrations of curved lines

15

A typical coefficient of the determinant has the form at3 = K[~ 2 + G ( I + K 2) +~2~2]

from solving Equations 15 from the general theory are shown in Fig.4 for the case of a steel rod (G = 0.44). As can be seen, for a given value of K 2, there are three branches to the dispersion curves, corresponding to various modes of propagation. The lowest set of curves corresponds to the flexural mode, the middle set of branches to the longitudinal mode, and the highest set represent the radial shear mode.

16

In power sonics applications, it is the steady vibration characteristic of curved ring segments that is of interest. Specifically, if proper matching of the curved transmission line segments to transducers is to be done, the line resonances must be known. To illustrate developments in this respect, the simplest form of governing equations, previously given by Equations 13, will be used.

3.5(

where ~-, e, K, G are dimensionless quantities defined by 30( g = k/~, -d= cx/(p/E)

17

K= k/R, G = G'/E

18

As previously mentioned, simpler theories are obtainable by neglecting various effects due to shear or rotational inertia. In each case, dispersion relationships between the wave velocity and wave number may be obtained. In the case of governing equations as given by Equations 13, the resulting dispersion relation is sufficiently simple to display in complete form as 23

2.0

I-5(

FO(

~4 _ [1 - 2K 2 + K 2 ( I + K2)/{ 2 + ~21 e-2 + ('~2

25~

2K 2 + K4/~2) O.~

=0 The actual numerical relations between wave velocity and wave number are determined by solving the dispersion relations (Equation 15 or Equation 19) with the use of a digital computer for various values of the curvature parameter K 2. The dispersion curves that result, for example,

80

19 c 0

050

I O0

1"50

2,00

250

Fig.4 Dispersion curves for the general theory: curves are labelled w i t h values of K 2

U L T R A S O N I C S . MARCH 1972

Considering U and W to be of the form

2.50 --

- K 2 = O.15 Kz =

0.10

K 2=

005

U = Ae s0,

200

W = Be s0

25

substitution in Equations 23 leads to two homogeneous equations in A and B. The characteristic equation results from equating the determinant of coefficients to zero and is given by

1-50

Iu s6 + (2 + ~k4)s4 + [1 + X4(2 -- K'2)] s2 I.OO +X 4 [ l + ( 1 - ; k 4 ) K "2] K z = 002

-

= 0

K , = o.,

/ / / / / /

26

050 I I

I

2.OO

250

O u"

©

0.50

I.OO

1.50

The roots of this equation may be real, imaginary, complex or some combination of these types. Ignoring for the moment the nature of the roots, we realize there will be a total of six, s = s I , s 2 , . . . , s 6. Then U and W will be of the form

Fig.5 Dispersion curves for the simplified theory: curves are again labelled with K 2 values

U = A1 esl0 + . . . + A6 es60,

W=B1 esl0 + . . . + B 6 es60 Since the ring will be of finite arc length, boundary conditions must be applied at the ring ends. These have been previously given for the general theory. However, for the simpler theory of relevance here, the boundary conditions have a somewhat different form, being given by

(:)

fi N +

,wM,wM =0,0=01,02

20

A total of eight different boundary conditions are possible at each end of the ring, representing various combinations of the pairs of quantities of Equations 20. They may be described as clamped, pinned, pinned-radially, guided, and so forth. Many of the combinations represent mathematically acceptable conditions that are of little practical interest. Others, such as the clamoed or free end are physically realizable and thus of practical value. In determining the resonances of sonic lines, the free end conditions are of greatest interest. These are given by M N + -- = M = M ' = 0 ,

0=01,02

21

R

The eigenvalue problem which must be solved to determine the ring natural frequencies is formulated by considering the ring to be undergoing harmonic vibrations, described by u = U(0)e iwt, w = W(0)e iwt

22

Substitution of the above in the equations of motion (Equations 12) gives U" + W' + ~4U = 0 23 U ' + W + K 2 ( W IV + 2 W " + W ) - X 4 W = 0 . where X4 =6o2/a 2,

a 2 = E / p R 2,

K2 = k 2 / R 2

ULTRASONICS. MARCH 1972

24

27

For any root s = si, the constants Ai, Bi will be related via the homogeneous equations previously mentioned. Thus, it is found that Ai Bi

-s i

28

s 2 + X4

The solutions (Equations 27) are now substituted in the boundary conditions (Equation 21), where 01 = 0, 02 ---00. Six homogeneous equations in A 1 , . . . A 6 result and the requirement that the determinant of coefficients vanish yields the frequency equation for the ring. The determinant is a six by six array, and a typical coefficient aij is a rather complicated function of the frequency 6o and ttie angle 00. Determination of the roots of the frequency equation must, of course, be done numerically. Results are shown 22 in Fig.6 for the variation in the natural frequency parameter (defined as ~ = X2K) as a function of the subtended angle of the ring, for a specific value of the ratio k/L. Various vibrational modes are plotted and identified as F1 ('first flexural'), L1 ('first longitudinal'), F2, L2 and so forth. The identification of the modes as longitudinal or flexural is based on the vibrational mode from which they originate when the bar is straight. Of course, longitudinal and flexural actions are coupled in the curved configuration, so that the ring modes are 'mainly' longitudinal or flexural.

Desirable future work Although the results of previous investigators, as well as the results reviewed in this report, allow some aspects of vibrations and waves in curved lines to be determined, certain additional developments are highly desirable. Of particular value would be the exact frequency spectrum for waves in a curved ring or strip, under plane strain or plane stress conditions. Such a result would be the analogue of the

81

References

L4

0"25

F7 L3

0"20

F6 v 0"15 ~,~

jL2

7

i_.

8 F5

0 ¢7

~- o . l o LI

c

9 10

Is.

11 0 05

i

1F3

12 13

-

0 0

1__ I __ SO IO 0 150 Deflection angle e[degrees]

F2 FI

200

14 15 16

Fig.6 Natural frequencies for a curved ring segment ( k / L = 0.02)

Rayleigh-Lamb spectrum for a plate, or the P o c h h a m m e r Chree spectrum.for a rod. Similar to the case for plates and rods, a major use of such a result w o u l d be to provide a basis o f comparison for various a p p r o x i m a t e theories for waves in curved transmission lines. A logical n e x t step in the d e v e l o p m e n t o f an a p p r o x i m a t e theory for waves in rings w o u l d be to include higher longitudinal effects. In fact, this w o u l d probably be o f greater use in sonic and ultrasonic applications than some of the d e v e l o p m e n t s reviewed here, since higher order longitudinal effects are likely to be o f greater i m p o r t a n c e than transverse shear effects.

82

17 18 19 20 21 22

DAVIDSON, S., Wire and strip delay lines, ULTRASONICS (July 1965)pp 136-146 MAY, J. F., Jr, Chapter 6 of Guided wave ultrasonic delay lines, Physical Acoustics Vol 1- A. W. P. Mason (Ed) Academic Press (1964) pp 435,443,444 PALFREEMAN, J. S., Acoustic delay lines - a survey of types and uses, ULTRASONICS (January 1965) p 4 ROZENBERG, L. D., MAZANTSEV, V. F., MAKAROV, L. O., YAKHIMOVICH, D. F., Ultrasonic Cutting, Consultants Bureau, New York (1964) pp 77 FREDERICK, J. R., Ultrasonic Engineering, John Wiley and Sons Inc (1965) p 183 BALAMUTH, L., Progress in ultrasonic metal forming presented at the International Automotive Engineerilag Congress, Detroit Michigan (January 1965) p 8 BUCKLEY, J. T., FREEMAN, M. K., Ultrasonic tube drawing, ULTRASONICS (July 1970) p 157 FUNK, E. R., Research on wire drawing, Department of Welding Engineering Ohio State University (1970) LOVE, A. E. H., The mathematical theory of elasticity, 4th Edition Dover Publications New York p 428 MINDLIN, R. D., HERRMANN, G., A one-dimensional theory of compressional waves in an elastic rod, Proc First US Nat Congr Appl Mech ASME New York (1952) p 187 MEDICK, M. A., One-dimensional theories of wave propagation and vibrations in elastic bars of rectangular cross section, JAppl Mech (September 1966)p 489 PHILIPSON, L. L., On the role of extension in the flexural vibrations of rings, JAp'plMech Vol 23 (1956) p 364 BUCKENS, F., Influence of the relative radial thickness of a ring on its natural frequencies, JAcoust Soc Amer Vol 22 No 4 (1950) p 437 MORLEY, L. S. D., Elastic waves in a naturally curved rod, Quart JMech and Appl Math Vol 14 (1961)p 155 VOLTERRA, E., The equations of motion for curved elastic bars deduced by the use of the 'Method of Internal Constraints', lngArch Vol 23 (1955)p 402 VOLTERRA, E., A one-dimensional theory of wave propagation in elastic rods based on the 'method of internal constraints', IngArch Vol 23 (1955) p 410 LINCOLN, J. W., VOLTERRA, E., Experimental and theoretical determination of frequencies of elastic toroids, ExperMech (May 1967) p 211 WITTRICK, W. H., On elastic wave propagation in helical springs, Int J ofMech Sci Vol 8 (1966) p 25 BRITTON, W. G. B., LANGLEY, G. O., Stress pulse dispersion in curved mechanical waveguides, J Sound Vib Vol 7 (1968) p 417 GRAFF, K. F., Elastic wave propagation in a curved sonic transmission line, IEEE Sonics and Ultrasonics (January 1970) p 3 GRAFF, K. F., A note on dispersion of elastic waves in rings, lnt J Mech Sci (to appear) DETTLOFF, B. B., GRAFF, K. F., The vibrations of a curved sonic transmission line, Engineering Experiment Station Report 2 2 0 - Q 2 - 6 9 The Ohio State University (1969)

U L T R A S O N I C S . MARCH 1972