Dispersion of elastic waves in a triangular bar

Journal of Sound and Vibration (1971) 18 (2), 261-269

DISPERSION

OF ELASTIC WAVES IN A TRIANGULAR R.E.

BAR

BOOKER?

Physics and Engineering Laboratory,

Lower Hutt, New Zealand

AND W. B. FRASER Department

of Applied Mathematics,

University of Sydney, Australia

(Received 14 April 1971)

It is shown that in a bar with an equilateral triangular cross-section longitudinal, torsional, and bending modes of wave propagation are possible. The first few branches of the dispersion curves for each of these modes have been calculated using the collocation method. The first branch of the longitudinal mode shows excellent agreement with experiment. The existence of an “end-resonance” is inferred from the experimental results. PART I : THEORETICAL

INVESTIGATION

W. B. FRASER 1. INTRODUCTION

In this part, the method of collocation is used to investigate the modes of vibration of an infinite elastic bar of equilateral triangular cross-section. It is shown that there are three modes of harmonic wave propagation in such a bar. The first three real branches of the dispersion curves for longitudinal and bending waves and the first two for torsional waves are obtained. Briefly, the method of collocation consists of forcing exact solutions of the elastic wave equations in cylindrical polar coordinates to satisfy stress-free boundary conditions at a discrete set of points on the surface of the bar. This gives rise to a determinant, the zeros of which give points on the dispersion curves. The method has been used previously to investigate elastic wave dispersion in rectangular bars [ 11, and elliptical bars [2], and full details can be found in these papers. Here, we describe the collocation method only in sufficient detail to derive the modes of wave propagation in the equilateral triangular bar. 2. THE

METHOD

OF COLLOCATION

oriented as in Figure 1 so that the z-axis Let (r,B,z) be cylindrical polar coordinates coincides with the intersection of the three planes of symmetry of the bar. The normal and tangential components of stress at the surface of an infinite elastic bar of arbitrary cross-section along which harmonic waves are propagating have been given in reference 2. They are 0” =

{A,[-J,+@R)

cos (no + 2~) + J,_@R)

cos (no - 2y)] +

II=0 + B,[Jn+&3R)

cos

(n0 + 2~) + J,_#?)

+ C,[J,+z(ctR) cos (no + 27) + J,_,(c&) +"go

cos (no - 2~) - 2J,@R) cos no] + cos (no - 2~) - 2(2K - l)J,(aR)

cos n@J> +

{R,G,F,,sinl,

f Present address: Department Illinois 61801, U.S.A.

of Metallurgy and Mining Engineering, 261

University of Illinois, Urbana,

262

R. E. BOOKER

AND W. B. FRASER

rf = $LO(--A,[J,+2@R) sin (no + 2~) + J,_#R)

sin (no - 2y)] +

+ B,[J,+dj3R) sin (d + 27) - J&W)

sin (120- 2y)] +

+ C,[J,+,(crR) sin tne

sin cne

+ B,[.&+#R)

+

27) - J,_,(aR)

COS(d + Y) - &#R)

+ C,[--J,+~(~R)COS(~~

-

2y)]} -

COS(no - r)] i@’ - k’)/pk

+ y) + .T_,(~)cos(~~

- ~)]2ikja)

+

+

+ 2 ID,, En, F,, sin),

where (R,8) are the plane polar coordinates of the cross-section boundary, y is the angle between the normal to the boundary and the radial direction, and a factor exp [i(kz - wr)] has been omitted.? This notation, {D,,&,F”,sin}, is used to indicate that this expression is the same as that in the preceding braces with D,, E,,, F,, and “sin”, replacing A,,, B,,, C,, and “~0s” respectively.

k

30

Figure 1. Equilateral triangular cross-section

showing notation.

The boundary conditions U, = TV= T, = 0

at

I+= R,

all 8,

(2)

are to be satisfied approximately at a discrete number of points on the boundary using suitable truncations of the series solutions (1). Since this cross-section is symmetric with respect to the axis 0 = 0, the boundary conditions can be satisfied by series solutions involving either terms with arbitrary constants A,, B,, C,, or those with arbitrary constants D,, E,, F,,. Further, if either of these series is made to satisfy boundary conditions on sector ABD of the boundary (Figure 1) then by symmetry they also satisfy the boundary conditions on the rest of the boundary DFA. Suppose we retain Nterms, involving 3Narbitrary constants, in one of these series solutions and set u,, 7t, T,, equal to zero at each ofNcollocation points; then we obtain 3Nhomogeneous linear algebraic equations for the 3N unknown constants. This set of equations has a nontrivial solution if and only if its determinant (which we will call the collocation determinant) is equal to zero. For a particular value of Poisson’s ratio, and configuration of collocation points, the elements of the collocation determinant depend only on the dimensionless t Definitions of notation are given in the Appendix.

WAVE DISPERSION

IN A TRIANGULAR

BAR

263

frequency parameter? G? = wa/cz and wavenumber ka, where 3a is the height of the triangular section, and c2 is the velocity of shear waves in the infinite elastic medium. The real values of Sz and ka for which the determinant is zero give the frequencies and wavenumbers of waves that propagate along the bar without attenuation. In the next two sections we show, for the equilateral triangular bar, that the series solutions (1) can be separated into four sets of infinite series each of which is capable of satisfying the boundary conditions. However, two of these solutions lead to identical collocation determinants and thus there are only three modes of wave propagation in such a bar. They are a longitudinal mode, torsional mode and bending or flexural mode which we will designate by L(m), T(m) and B(m), respectively, m being an integer indicating the branch of the dispersion curve in the particular mode.

3. LONGITUDINAL

AND

TORSIONAL

MODES

If we retain only the terms involving A, B, C, with n = 0,3,6, . . ., in (I), we obtain a solution that is symmetric with respect to all three axes of symmetry of the section. Enforcing the boundary conditions on one-sixth of the boundary, say on section AB (Figure 1), ensures their satisfaction on the rest of the boundary due to this symmetry. The solution represents longitudinal waves in the bar. Similarly, retaining only the terms involving D, EnF, with n = 0, 3,6, . . ., in (l), we obtain a solution that is antisymmetric with respect to the axes of symmetry of the section, and this solution represents torsional waves in the bar.

4. BENDING

MODE

Consider now the solution involving the terms A,, B,, C,, with n = 1, 2,4, .5,7, 8, . . . . Let the surface stress components given by this solution be designated by o;, T:, T,C,the superscripts c and s being used to indicate that a series involves either cosine or sine functions. Examination of the terms in these series leads to the following relations between the stress components at the symmetrically arranged boundary points P, Q and S, (Figure 1): G(P) =

-4(Q)

- 4(S),

T;(P) = +3(Q) - T;(s), T;(P)

=

--7;(Q)

-

(3)

T;(S).

Since the series are also symmetric with respect to 19= 0 (AD Figure l), if the boundary conditions are satisfied at all points P, Q lying on the boundary sector ABC, then relations (3) insure their satisfaction on the rest of the boundary. We will refer to this solution as the bending mode of wave propagation. Similarly, if boundary stress components corresponding to series involving terms D,, E,,, F,,, with n = 1, 2, 4, 5, 7, 8, . . ., are designated by aS,, T:, T:, then the relations between them at points P, Q and S, are u;(P) = u;(Q) - a;(S), T:(P) = -T;(Q) T;‘(P)

=

T;(Q)

- T;(S), -

(4)

T;(s).

t In references [I] and i.21the collocation determinants were taken to be functions of c/c2 and ka which is seen to be equivalent to the above dependence by noting that k = w/c and thus $2 = @c/c,). The advantage of showing Q VS. ka dispersion curves is that (a) the cut-off frequencies can be shown and (b) the slope of the curves gives the group velocity.

264

R. E. BOOKER AND W. B. FRASER

Further, if D,, E. and F. in this solution are replaced by A,, B, and C,, respectively, then the relation between the individual terms in these two series solutions for the bending modes is

(5)

where6,=2sin(2n7r/3);n=1,2,4,5,7,8 ,.... Relations (3), (4) and (5) can be used to show that the second bending mode collocation determinant can be obtained from the first by simple row transformations of the determinants. Thus there is only one independent bending mode for an equilateral triangular bar. Any other bending mode can be obtained from that represented in expression (3) by a suitable rotation of this result about the z-axis. 5. RESULTS The numerical procedure is the same as that described in papers [I] and [2] and will not be discussed here. The first three real branches of the dispersion curves for the longitudinal and bending modes, and the first two branches for the torsional mode have been constructed in the range 0 G ka G 3*5,0 G Sz =G4.0, and these are shown in Figure 2. The points from which the L(1) branch was plotted are tabulated in Table 1, and Table 2 gives the dimensionless cut-off frequencies of the higher order branches shown in Figure 2. TABLE

1

Frequency parameter Q and phase velocity ratio c/c2 as a function of ka for branch L(1) Q

ka 0 0.2 0.4 0.6 0.8 1-o 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

0 0.327 0649 0.959 1.248 1.499 1.703 1.868 2.013 2,155 2.300 2449 2603 2.762 2.926 3.093 3.264 3.436

CICZ 1.638 1.634 1.622 I .599 1.559 I *499 1.419 1.334 1.258 1.197 I.150 1.113 1.085 1.063 1.045 1.031 1.020 1.011

Poisson’s ratio = 0.3424.

WAVE DISPERSION IN A TRIANGULAR BAR

265

For the branch L(l), six equally spaced collocation points on the boundary sector AB (Figure 1) gave Q accurate to three decimal places (the value of ku being specified), and the residual stresses midway between the collocation points were less than a,,, x 10P4, where TABLE

2

Cutof frequencies for some dispersion curve branches

Branch Q cutoff L(2) L(3) T(2) B(2) B(3) Poisson’s ratio =

Figure 2. Frequency vs. wavenumber dispersion longitudinal mode; ---, (T) torsional mode; ---,

2.094 2*30(2) 2.07(4) 1.209 1.30(l) 0.3424;ka

= 0.0.

curves for the equilateral triangular bar. -, (B) bending mode. Poisson’s ratio = O-3424.

(L)

urnaxis the maximum stress amplitude in the bar. Similar results were obtained for the other branches, using six collocation points for the longitudinal and torsional mode branches and twelve for the bending mode branches. In the second part of this article it is shown that the calculated L(1) branch, as plotted on a graph of dimensionless phase velocity C/Q vs. ku, is in excellent agreement with experiment.

266

R. E. BOOKER AND W. B. FRASER

We also note that the fundamental torsional wave-branch T(1) has a phase velocity ratio that is very nearly constant, and is in close agreement with the result of the simple onedimensional theory of dynamical torsion. It varies from c/c2 = 0.775 at ka = 0.2 to c/c2 = O-764 at ka = 3.4. Simple theory gives the result c/c2 = J/I = d(3/5) t 0.775, independent of ka, where J is Saint Venant’s torsional constant and Z is the polar second moment of area of the cross-section. PART II: EXPERIMENTAL INVESTIGATION OF THE LONGITUDINAL MODE, AND COMPARISON WITH THE THEORY R. E.

BOCIKER

To be able to compare theory and experiment an isotropic bar of triangular cross-section and known elastic moduli was required. The material chosen for this rod was “24 ST” aluminum. This material has been used in previous investigations [3,4,5], possesses very low mechanica damping, and has known structure, composition, and elastic moduli. It is assumed that the shear wave velocity and the elastic moduli of the triangular bar, the square bar of references [3] and [5], and the “24 ST” cylindrical bar of references [3] and [4] are the same, as the bars were machined from adjacent lengths of the same original specimen. Young’s modulus is 7.370 x IO’ON/m2 [4]. Poisson’s ratio is 0*342(4) [4], and the shear wave velocity is 3.134 x lo3 m/s [3]. The slightly preferred grain-orientation of the aluminum [4] has negligible effect on the isotropy of the bulk material, as the elastic moduli of aluminum exhibit only small orientation-dependence. The length of the bar is 8.821 cm, and it possesses an equilateral triangle cross-section of average height 0.6080 cm. The height varies between a maximum of 0.6109 cm and a minimum of 0.6045 cm, the mean deviation from the average being 0.0011 cm. The standard deviation in the average height is about 0.05 %. The bar was electrically earthed and supported horizontally by two 0.0005-in (I.27 x lo-’ cm) diameter tungsten wires in a vacuum chamber. It was excited into vibration by the application of an alternating voltage to a small electrode facing one bar endface. A second electrode at a d.c. potential of 600 V faced the other bar endface. Longitudinal vibration of the bar causes a small change in capacitance between the latter electrode and the bar endface. This generates a small alternating voltage with an amplitude directly proportional to the vibration amplitude of the bar. Because of the low damping (Q-i + 3 x 10m6),the resonant frequencies can be determined to 1 part in 106. They are unaffected by the electrode spacings and the tautness of the supporting wires, but a small temperature drift of 1.0 “C during the experiment reduced the significance of the frequency readings to I.5 parts in 104. The wave-velocity at each harmonic was calculated from the relation c = 2Lf(n)/n, wheref(n) is the resonance frequency of the nth harmonic, and L is the bar-length. A graph of dimensionless velocity, c/c2, vs. ka is plotted in Figure 3, together with the theoretical values of Table 1. It is clear that theory and experiment are in excellent agreement. Table 3 shows the experimental and theoretical values of c/c2 at each harmonic, the difference d(c/c2), as well as Sz. On the whole, the difference between the experimental and theoretical values of c/c2 is only about 0.002 (Table 3). The theoretical values of c/c2 at each harmonic were calculated from the values of Table 1 by a non-linear interpolation method. The experimental error in the determination of the values of c/c2 due to frequency determination errors and systematic errors in c2 and a does not exceed O+)Ol . The maximum harmonic number that can be excited is the 27th, with ka = 1.95 and IR = 2.25. Higher harmonics cannot be excited because at higher values of fi the L(3) branch

WAVE DISPERSION IN A TRIANGULAR

267

BAR

becomes real (Table 2 and Figure 2). The wave-motions associated with the real L(l), L(2), and L(3) branches are coupled at the bar endfaces, so that the only resonances that can be excited with an 8 value above 2.30 are a composite of the three real longitudinal branches. As for the cylindrical bars [4], it is possible to excite the L(1) resonances in the Q-region TABLE 3

Values at each harmonic of the theoretical and experimental phase velocity ratio c/cl, the dijkrence A(c/c,), and thefrequency parameter Sz

cIc2 Harmonic , number Experimental Theoretical d(c/c*) x lo3 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1.641 1.638 1.635 1.631 1.626 1.619 1.612 1603 1.591 1.578 1.561 1.542 1.520 1.521 1.495 1.467 1.437 1406 1.375 1.344 1.344 1.311 1.315 1.287 11.291 1.261 1.236 1.237 1.214 1.193

1

1.172 1.152

1.637 1.636 1.633 1.630 1.625 1.619 1.612 1602 1.591 1.577 1.561 1.541 1.520 1,495 1.467 1.438 1408 1.377 1.346 1.316 1.289 1.262 1.238 1.216 1.196 1.178 1.161

+4 -1-2 +2 +I +l 0 0 +I 0 +1 0 +l 0 -tl 0 0 -1 -2 -2 -2 -2 -5 -1 -2 i-2 -1 -2 -1 -2 -3 -6 -9

Q O-1184 0.2365 0.3541 0.4708 0.5869 0.7013 0.8145 0.9253 1.033 1.139 1.239 1.335 1.426 1.427 1.510 1.588 1.659 1.726 1.786 1.843 1.843 1.893 1.898 I .950 1.957 2.002 2.052 2.053 2.103 2.152 2.199 2.246

between the cut-off frequencies of L(2) and L(3). The resonances of the L(1) branch are slightly affected by the real L(2) branch, however, which accounts for the slight difference between experiment and theory for the 26th and 27th harmonics. The L(2) branch in Figure 2 exhibits a minimum at Q = 2.05. In cylindrical and square bars such a minimum is associated with the generation of doublets, [4,5] : i.e. two resonances very close together in frequency. This phenomenon also occurs for the triangular bar, as the 23rd harmonic is split into a doublet of resonances of 505.016 and 505.292 kHz respectively, at an J2 value of 2.05.

268

R.E.BOOKERANDW.B.FRASER

Cylindrical and square bars show either a displaced harmonic, a doublet, or a triplet at the “end resonance” frequency [4, 51. A similar type of end-resonance exists for the triangular bar, as evidenced by the splitting of the 13th harmonic into a doublet (ka = 0.938, Q = 1.43). The reason for the doublet-splitting of the 19th, 20th, and 21st harmonics is not yet understood.

2.1

ku

Figure 3. Graph of dimensionless phase velocity vs. wavenumber comparison of experimental (+) and theoretical ( 0) results. d = doublet.

for the first longitudinal

branch;

Theory and experiment differ more for the fundamental than for subsequent harmonics. This may be caused by the slight non-uniformity of height along the bar, which would have its greatest effect when adjacent nodes and anti-nodes are far apart. In a previous paper [3] it was found that the velocity dispersion curves of the first longitudinal mode of a square bar and a cylindrical bar could be made to coincide by assuming an equivalent diameter for the square bar such that the geometric moments of inertia of the bars were equal. It was found that no such relationship exists between the first longitudinal branches of the triangular bar and its “equivalent” cylindrical rod, however.

CONCLUSION It has been shown that the L(1) branch calculated by the collocation method agrees extremely well with experiment. The minimum of the L(2) branch and the cut-off frequency of the L(3) branch also agree with experiment. Finally, the existence of an end-resonance was inferred. REFERENCES 1. W. B. FRASER1969 International Journal of SoIids and Structures 5,379-397.

Stress wave propaga-

tion in rectangular bars. 2. W. B. FRASER 1969 Journal of Sound and Vibration 10,247-260.

Dispersion of elastic waves in elliptical bars. 3. R. E. BOOKER 1969 Journal of the Acoustical Society of America 45,12&I-1286. Velocity dispersion of isotropic rods of square cross section vibrating in the lowest-order longitudinal mode. 4. R. E. BOOKERand F. H. SAGAR1971 Journal qfthe Acoustical Society of America 49,1491-1498. Velocity dispersion of the lowest-order longitudinal mode in finite rods of circular cross section. 5. R. E. BOOKER 1971Journal of the Acoustical Society of America 49,1671-1672. Velocity dispersion

of the lowest-order longitudinal mode in finite rods of square cross sections. 6. R. E. BOOKERand F. H. SAGAR 1963 Ultrasonics 1,223-231. An experimental system for ultrasonic attenuation measurements in solids in the upper kilocycle range.

WAVE DISPERSION

IN A TRIANGULAR

BAR

APPENDIX NOTATION

height of the triangular section phase velocity ‘12,velocity of dilatational waves in an infinite elastic medium 1 w~~)/Pl velocity of shear waves in an infinite elastic medium constant loefficients in the series solutions n frequency f = v’_1 Bessel’ s function of order n J,(s) dJn J.‘(s) ds k w/c, wavenumber K = $(/3’- k2)/a2 L rod length N number of collocation points R radial coordinate of a point of the cross-section boundary time cylindrical polar coordinates = /+2/c: - 1)“2 = /@/c; - 1)“2 angle between the normal to the cross-section boundary and the radius vector Lam& elastic constants Poisson’s ratio density component of stress normal to the boundary component of shear stress on the boundary, perpendicular to the z-direction component of shear stress on the boundary parallel to the z-direction maximum stress (amplitude) in the bar = 24 angular frequency = wa/cz, dimensionless frequency 3a

A PI,...,

2

269