A Thesis
entitled
Analysis of a Pseudo-Harmonic Tubular Bell
by
Douglas LeRoy Oliver
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Master of Science Degree in Mathematics
______Dr. Alessandro Arsie, Committee Chair
______Dr. Sonmez Sahutoglu, Committee Member
______Dr. Denis White, Committee Member
______Dr. Amanda Bryant-Friedrich, Dean College of Graduate Studies
The University of Toledo June, 2017 Copyright 2017, Douglas L Oliver
This document is copyrighted material. This material may be copied freely so long as attribution is made. An Abstract of Analysis of a Pseudo-Harmonic Tubular Bell
by
Douglas LeRoy Oliver
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Master of Science Degree in Mathematics
Tubular bells, or chimes are used for ambient sounds as well as serious music.
Unlike most wind or stringed instruments, a tubular bell does not have a harmonic set of overtones. The lack of harmonious overtones creates a problem with using tubular bells for serious music: there is not unanimity regarding the pitch, or musical note associated with a particular tubular bell.
The Euler-Bernoulli model for vibrating thin beams was used to derive a mathematical model for vibrations of a tubular bell. Using this model, an analysis of the natural frequencies of a modified tubular bell was presented. One or more ends of the tubular bell were weighted with a mass. This mass changes the boundary conditions, and hence the ratio of the natural frequencies of the tubular bell.
Values for the ratio of the mass of weight(s) to the mass of the tube were identified such that the ratio of the frequency of the first overtone to the second overtone was 2. Under these conditions, the these overtones are one octave apart.
Frequency ratios predicted by the model have been compared with experimental results of a frequency analysis of the sound produced by two physical tubes. The experimental results were in good agreement with the theoretical predictions.
iii Table of Contents
Abstract...... … iii
Table of Contents…...... iv
List of Tables...... … v
List of Figures...... … vi
Chapter 1. Introduction ...... …. 1
Chapter 2. Lateral Vibration of a Beam - Euler-Bernoulli Theory...... ……...... 4
2.1 Free-Free Boundary Condition…………………………………….. 7
2.2 Pinned Boundary Condition……………………………………….. 10
2.3 Weighting the Tube Ends………………………………………….. 11
2.4 Boundary Conditions for a Weighted End…………………………. 12
2.5 Dimensionless Form of Equations…………………………………. 13
Chapter 3. Finding The Natural Frequencies of the Chime……………………. 15
3.1 Special Cases fro Very Large Values of m0 or m1 …………………. 17
3.2 Finding Combinations the Result in f3/f2 = 2………………………. 19
Chapter 4. Experimental Verification ………………………………………….. 22
Chapter 5. Conclusion and Limitations………………………………………… 25
References ……………………………………………………………………… 26
iv List of Tables
Table 1: Harmonics with fundamental frequency of 220 Hz. …………………. 2
Table 2: Mode frequency ratios for a chime …………………..………………. 2
Table 3: Values of β and fn/f1 for free/free boundary conditions………………. 10
Table 4: Characteristic equations and values for three boundary conditions…… 10
Table 5: Conditions for f3/f2 = 2……………………………………………….. 19
Table 6: Test Case A…………………………………………………………….. 23
Table 7: Test Case B…………………………………………………………….. 23
v List of Figures
Figure 2-1: Schematic diagram of tube length……………….. …………………. 5
Figure 2-2: Case A – Two masses added. Case B – a single mass added ………... 11
Figure 3-1: f3/f2 as a function of m0 and m1/m0…………………………………….... 20
Figure 3-2: b2 as a function of m0 and m1/m0…………………………………… . ….. 21
Figure 4-1: Raven Lite 2.0 Spectral analysis for Test Case A and Test Case B... 24
vi Chapter 1
Introduction
Chimes have become a popular percussion instrument that is used both as a wind chime and for orchestral purposes. The term often used for chimes in a musical context is tubular bells. Commercial wind chimes range from inexpensive models to expensive custom-designed chimes. Tubular bells were used as early as 1853 by Giuseppe Verdi in his opera Il trovatore.1 There are many examples of tubular bells used in popular culture including: the theme song for the television series Futurama2, and the theme to the movie Exorcist3. One of the more conspicuous uses of tubular bells in popular music is performed by Mike Old…eld.4 In this paper the term chime will be used interchangeably with tubular bell. One of the problems with using tubular bells as an orchestral instrument is that it is di¢cult to precisely identify the pitch of a tubular bell. According to the Vienna Philharmonic Library, there have "been arguments over which is the ’correct’ pitch" for a tubular bell5. Most wind and stringed instruments have overtones that are harmonic. That is, the …rst overtone (or the second harmonic), has a frequency that is twice the fundamental frequency. The second overtone, (or third harmonic), has a frequency that is three times the fundamental frequency. In general, the nth overtone, (or (n + 1)th harmonic), has a frequency that is (n + 1) times the fundamental frequency. For example, if the fundamental frequency is 220 Hz, (A3);
1 Vienna Philharmonic Library at https://www.vsl.co.at/en/Tubular_bells/History. Viewed March 21, 2017. 2 Listen at https://www.youtube.com/watch?v=QRk1s5Kf3aQ. Listened on March 21, 2017. 3 Listent at https://www.youtube.com/watch?v=1hbQpjYtbps. Listented on March 21, 2017. 4 View at https://www.youtube.com/watch?v=sSRJvq4Wd48. Viewed on March 21, 2017. 5 See https://vsl.co.at/en/Tubular_bells/Notation. Viewed, March 14, 2017.
1 then a stringed or wind instrument would typically have the following set of overtones:
Table 1: Harmonics with a fundamental frequency of 220 Hz. * approximate musical note. Harmonic 1st 2nd 3nd 4th 5th 6th Overtone Fundamental 1st 2nd 3nd 4th 5th
fn - frequency (Hz) 220 440 660 880 1100 1320 fn 1 2 3 4 5 6 f1 # Musical Note A3 A4 E4 A5 C5 E5
Chimes and most other percussion instruments have overtones, but not harmonics. An aspect of the overtones in some non-harmonic instruments is the psycho-acoustical e¤ect of a "virtual pitch". When one hears a simultaneous set of harmonic frequencies, where one or more of the lowest frequencies are weak or missing, then the pitch will often be identi…ed as the missing fundamental frequency. For example, if one hears a sound composed of the following frequencies: 600, 800, 1000, and 1200 Hz, the pitch will often be identi…ed as the missing fundamental frequency of 200 Hz. That is, the virtual-pitch of that series of frequencies is 200 Hz6. The concept of a virtual pitch relates to chimes in that there is no apparent harmonic series associated with a chime. The ratio of the modal frequency to the fundamental frequency, fn ; f1 are shown in Table 2 for a long, thin chime.
Table 2: Mode frequency ratios for a chime Overtone fund. 1st 2nd 3rd 4th 5th n - (mode) 1 2 3 4 5 6 Chime: fn 1 2.76 5.40 8.94 13.34 18.64 f1
fn 2 2.99 4.17 (f4=2)
Even though a chime is not harmonic, there may be a nearly virtual pitch associated with the fourth, …fth, and sixth modes. Notice in Table 2 that the ratios for fn for modes 4 (f4=2) through 6 form a sesequence that has ratios of nearly 2:3:4. Based on this relation, Rossing
6 Thomas Rossing, Science of Percussion Instruments, Vol. 3 of Series in Popular Science, pp. 9-10, World Scienti…c, (2000)
2 hypothesized that the virtual pitch of a chime would be one octave lower than the frequency associated with the fourth mode, (e.g. f4=2): However, the "strike pitch" suggested by the Vienna Philharmonic Library is "an octave higher than the fundamental" frequency.7 That is, twice the fundamental frequency. Thus, Rossing’s virtual pitch is not the same as the "strike pitch" suggested by the Vienna Philharmonic Library. For example, consider a chime with a fundamental frequency of 220 Hz. The frequency corresponding to the fourth mode would be about 1967 Hz (8.94220 Hz). The "virtual pitch" using the method suggested by Rossing would be half of this - or 983 Hz. Contrast this with the "strike note" suggested by the Vienna Philharmonic Library of twice the fundamental frequency. By this method, the pitch would correspond to 440 Hz, (2 220Hz). With these two suggestions for the pitch of a chime, it is not surprising that there have been disagreements over what the correct pitch is! Thus, there is a motivation to adjust the frequencies of a chime to obtain a set of overtones that is more harmonic - while retaining the general timber of a chime. Rossing suggested that the ratio of the overtone frequencies could be adjusted by weighting, or loading, one end of the chime.8 However, he only presented three speci…c examples and o¤ered no clue as to any general relationship between the weighting of the tube end and the resulting ratios of the various overtones to each other and to the fundamental frequency. The remainder of this paper investigates a method to modify a chime so that it has two prominent frequencies that are one octave apart. Such a chime would have a "pseudo-harmonic" timber.
7 See https://vsl.co.at/en/Tubular_bells/Notation. Viewed March 14, 2017. 8 See Table 7.2 of Rossing.
3 Chapter 2
Lateral Vibration of a Beam - Euler-Bernoulli (Thin Beam) Theory
The lateral free vibration of a uniform thin beam has been investigated by numerous authors1. A long uniform tube is vibrating transversely in a plane with no rotational vibrations. For su¢ciently thin tubes, the Euler-Bernoulli beam model adequately models vibration2. The following analysis will follow the nomenclature and treatment is given by Rao 3. However, in subsequent sections the boundary conditions will be modi…ed to account for adding masses at the ends. Some assumptions made in this analysis are:
1. The beam is a circular tube with a uniform cross section;
2. The chime is assumed to have no external latteral forces or moments;
3. The beam has homogeneous, linearly elastic, and isotropic material properties;
4. The beam de‡ection angle is small such that ' sin ; and
1 See for example: Problem 14 of Section 19.2 of Advanced Engineering Mathematics, (2nd ed.), by Michael Greenberg, Prentice-Hall (1998). 2 Han, Seon M, Benaroya, Haym, and Wei, Timothy, "Dynamics of Tansversely Vibrating Beams Using Four Engineering Theories", Journal of sound and Vibration, 225(5), pp 935-988, (1999). 3 See Section 8.5 of "Mechanical Vibrations" (3rd ed.) by Rao, Singiresu, Addison-Wesley (1995).
4 Do 5. The diameter of the tube is much smaller than the length of the tube: L 1:
The last assumption is required for the Euler-Bernoulli analysis. Unfortunately, this as- sumption is often not fully obtained for most chimes. Han, et al. used other models including
Do 4 the Timoshenko model which better predicts chime frequencies as the ratio L increases. Consider an in…nitesimal element of a beam of length L that is x wide as is illustrated in Fig. 2-1 . The displacement of the neutral curve from the equilibrium is denoted as w(x; t): The shear, V , on the element is taken to be positive if it is upwards on the left edge, and downwards on the right edge. The moment, M, imposed on the edges is taken to be positive if it is clock-wise on the left of the element and counter-clock-wise on the right of the element. The acceleration of this element may be obtained using Newton’s Second Law:
Figure 2-1: Schematic diagram of tube length. The angles are exagerated for clarity.
acceleration mass of element net force @2w Acsx 2 = V (x; t) V (x + x; t) (2.1) z@t}|{ z }| { z }| {
Where is the density of the tube, Acs is the cross-sectional area of the tube, and V is the
4 Han, Seon M., Haym Benaroya, , and Timothy Wei; Dynamics of Transversely Vibrationg Beams Using Four Engineering Theories. J. of Sound and Vibration, 225 (5) pp. 935-988 (1999).
5 shear force. Neglecting second-order terms, Eq. 2.1 becomes:
@2w @V A = (2.2) cs @t2 @x
With no external lateral forces or moments, the angular momentum of any beam section must remain constant. Thus, the net moment about any interior beam section must be zero. Hence: M(x; t) M(x + x; t) + V (x; t)x = 0
In the limit this is equivalent to: @M V (x; t) = : (2.3) @x
Substituting this into Eq. 2.2 yields:
@2w @2M A = (2.4) cs @t2 @x2
From elementary thin beam theory5, we have the relation:
@2w M = EI (2.5) @x2 where the material property E is Young’s modulus, and I is the area moment of inertia about the neutral plane. For a tubular chime, I is given by:
I = D4 D4 64 o i where Di and Do are the inside and outside diameters of the tube. Note that for thin walled tubes - where Do Di 1: D o
D3d I ' o where d is the wall thickness. 8
Substitution of Eq. 2.5 into Eq. 2.4 yields the following linear fourth-order partial di¤eren-
5 see Rao, page 524.
6 tial equation: @2w EI @4w 2 = 4 (2.6) @t Acs @x A standard separation-of-variables analysis is employed assuming that:
w(x; t) = Wn(x)Tn(t) (2.7) n X Substitution of Eq. 2.7 into Eq. 2.6, with separation of variables yields:
• Tn 2 = !n and Tn
4 EI 1 d Wn 2 4 = !n (2.8) Acs Wn dx Setting A 4 = cs !2 ; EI n
Eq. 2.8 becomes: d4W n 4 W = 0: (2.9) dx4 n n
The general solution to Eq. 2.9 is:
Wn(x) = An cosh( nx) + Bn cos( nx) + Cn sinh( nx) + Dn sin( nx) (2.10)
2.1 Free-Free Boundary Condition
Typically, chimes are designed such that both ends are nearly free. Hence, the following boundary conditions are imposed at both ends.
7 (sheer) V (0; t) = 0 and
(moment) M(0; t) = 0; or d3W d2W n = n = 0; (2.11) dx3 x=0 dx2 x=0
and
(sheer) V (L; t) = 0 and
(moment) M(L; t) = 0; or d3W d2W n = n = 0. (2.12) dx3 x=L dx2 x=L
Applied to Eq. 2.10, the boundary conditions at x = 0 result in the following:
2 d Wn = 0 ) An = Bn (2.13) dx2 x=0
3 d Wn = 0 ) Cn = Dn (2.14) dx3 x=0
These, with the boundary conditions at x = L; result in the following:
d2W n = 0; or dx2 x=L
An [cosh( nL) cos( nL)] + Cn [sinh( nL) sin( nL)] = 0 (2.15) d3W n = 0 or dx3 x=L
An [sinh( nL) + sin( nL)] + Cn [cosh( nL) cos( nL)] = 0 (2.16)
Equations (2.15) and (2.16) in matrix form are:
cosh( nL) cos( nL) sinh( nL) sin( nL) An 0 = 2 3 0 1 0 1 sinh( nL) + sin( nL) cosh( nL) cos( nL) Cn 0 4 5 @ A @ A
8 We seek a solution where
An 0 6= : 0 1 0 1 Cn 0 @ A @ A Hence, the permissible values for n are given by:
cosh( nL) cos( nL) sinh( nL) sin( nL) = 0, or
sinh( nL) + sin( nL) cosh( nL) cos( nL)
2 2 cosh( nL) cos( nL) = 0:
th Let n be the n positive root of:
cosh( nL) cos( nL) 1 = 0:
A dimensionless n may be de…ned by:
~ n = nL
The resulting natural frequencies are given by:
2 EI ~2 EI !n = n = n 4 or sAcs sAcsL ~2 !n n EI fn = = 4 : 2 2 sAcsL
Any ratio of frequencies between modes may be found the ratio:
~2 fn !n n = = 2 : fm !m ~ m
Values for the …rst six modes are listed in Table 3:
9 ~ fn Table 3: Values of n and f1 for free/free boundary conditions. mode(n) 1 2 3 4 5 6 ~ 6 n 4.73 7.85 11.00 14.14 17.28 20.42 ~2 n 22.4 61.7 120.9 199.1 298.5 417.0
!n fn Chime: !1 or f1 1 2.76 5.40 8.93 13.34 18.64
(Harmonic Instrument) 1 2 3 4 5 6
2.2 Pinned Boundary Condition
The boundary conditions imposed on Eq. 2.9 determine the associated natural frequencies. The above analysis considered only a free-free boundary condition. Another boundary condition that may be imposed at either end is a pin (or hinge). At pinned end the displacement and the moment are zero at the boundary. That is, w = 0 and M = 0 at the boundary:
d2W W = n = 0: (2.17) n dx2
~ The characteristic equations and the …rst few values of n for a free-free, a pinned-free, and a pinned-pinned chime are given in Table 47.
Table 4: Characteristic equations and values for three boundary conditions. ~ ~ ~ f3 Boundary Conditions Characteristic Eq. 1 2 3 f2 free free cosh( ~) cos( ~) 1 = 0 4.73 7.85 11.00 1.96
pinned free tan( ~) tanh( ~) = 0 3.93 7.07 10.21 2.09
pinned pinned sin( ~) = 0 2 3 2.25
f3 Notice that the ratio for f2 for a free-free chime is 1.96, while the same ratio is 2.09 and 2.25 for the pinned-free and the pinned-pinned chimes respectively. This suggests that an intermediate boundary condition, between a free end and a pinned end might result in a ratio
6 See Rao, Fig. 8.15. 7 See for example, Fig. 8.15 of Rao.
10 f3 of f2 = 2: In this case, the …rst and second overtones would be "harmonic-like" in that they would be one octave apart.
2.3 Weighting the Tube Ends
Rossing suggested that weighting the ends of a chime would alter the ratio of overtones to the fundamental frequency.8 The intent of this paper is to present a mathematical model that
f3 predicts the e¤ect of weighting the ends of a tubular chime on the ratio f2 : The design goal is to obtain a chime where f3 is twice that of f2. Two cases will be investigated:
1. Case A: A mass of m0 is to be …xed to one end of the tube, (at x = 0), while a second
mass of mL is to be …xed to the opposite end of the tube, (at x = L). Without loss of
generality, it is assumed that m0 mL:
2. Case B: A single mass of m0 is …xed at x = 0: Notice that Case B is a special case of
Case A, where mL = 0:
Case A – Two Masses µ m µ m0, 0 L , 1
x = 0, x = L, χ = 0 χ = 1
m , µ Case B – Single Added Mass m = µ = 0 0 0 L 1
x = 0, x = L, χ = 0 χ = 1
Figure 2-2: Case A - Two masses added. Case B - a single mass added.
8 Rossing, Thomas D., Science of Percussion Instruments, Table7.2, World Scienti…ce, 2000.
11 2.4 Boundary Conditions for a Weighted End
If the ends of the tube are weighted with an additional mass, then the boundary conditions given for Eq. 2.6 will no longer correspond to Eq. 2.11 or 2.12 . The appropriate boundary conditions may be obtained using Newton’s second law and conservation of angular momentum at each end point. Applying Newton’s second law at the left end, (or x = 0) yields:
acceleration end mass net force @2w m0 = V (0; t) (2.18) @t2 z }| x=0{ z }| { z}|{ Similarly, Newton’s second law, applied to the right end, (or x = L) yields:
acceleration end mass net force @2w m = V (L; t) (2.19) L @t2 z }| x=L{ z }| { z}|{ Application of Eqs. (2.3) and (2.5) to these boundary conditions yields:
@2w @3w m0 = EI (2.20) @t2 @x3 x=0 x=0
and @2w @3w m = EI (2.21) L @t2 @x3 x=L x=L
The mass at each end is assumed to be concentrated at a point on the neutral axis. With this assumption, the moment at each end will be remain zero:
@2w M(0; t) = EI = 0 and @x2 x=0 @2w M(L; t) = EI = 0: @x2 x=L
Thus, with weighted ends, Eq. 2.6:
@2w EI @4w 2 = 4 (2.22) @t Acs @x
12 has the following boundary conditions:
@2w = 0; (2.23) @x2 x=0 @2w = 0; (2.24) @x2 x=L
@2w @3w m0 = EI ; and (2.25) @t2 @x3 x=0 x=0 @2w @3w m = EI : (2.26) L @t2 @x3 x=L x=L
2.5 Dimensionless Form of Equations
Equations (2.22) through (2.26) may be restated in a dimensionless format using the following dimensionless parameters:
w y = - dimensionless displacement, L EI = 4 t - dimensionless time, sAcsL A L4 !~ = cs ! - dimensionless frequency, EI xr = - dimensionless distance, L m0 m0 0 = = ; and AcsL mass of tube mL mL 1 = = : AcsL mass of tube
In dimensionless form (2.22) is: @2y @4y = (2.27) @ 2 @ 4
13 The associated boundary conditions are:
@2y @2y = = 0 (2.28) @ 2 @ 2 =0 =1 2 3 @ y @ y = ; and (2.29) 0 @ 2 @ 3 =0 =0 2 3 @ y @ y = : (2.30) 1 @ 2 @ 3 =1 =1
Again, separation-of-variables techniques are used. Solutions to Eq. (2.27) are sought where y( ; ) is assumed to be of the form:
y( ; ) = Yn( )n( ) (2.31) n X Substitution of Eq. 2.31 into Eq. 2.27, with separation of variables yields:
• n 2 = !~n and n 4 1 d Yn 2 4 =! ~n: (2.32) Yn d Setting 2 ~ =! ~n
Equation 2.32 becomes: 4 d Y 4 n ~ Y = 0: (2.33) d 4 n n
A general solution to Eq. (2.33) is:
~ ~ ~ ~ Yn( ) = An cosh( n ) + Bn cos( n ) + Cn sinh( n ) + Dn sin( n ) (2.34)
14 Chapter 3
Finding the Natural Frequencies of the Chime
Substitution of Eqs. (2.34) into Eq. (2.28) yields:
@2y = 0 ) A = B and @ 2 n n =0
@2y = 0 ) @ 2 =1
~ ~ ~ ~ An cosh n cos n + Cn sinh( n) Dn sin( n) = 0 (3.1) h i Substitution of Eqs. (2.34) and (2.32) into Eq. (2.29) yields:
@2y @3y = ) 2 A~ C + D = 0 (3.2) 0 @ 2 @ 3 0 n n n =0 =0
A similar substitution into Eqs. (2.30) yields:
@2y @3y = ) 1 @ 2 @ 3 =1 =1
~ ~ ~ ~ ~ An sinh( ) + sin( ) + 1 cosh( ) + cos( ) + (3.3) n h io
15 ~ ~ ~ ~ ~ ~ Cn cosh( ) + 1 sinh( ) + Dn cos( ) + 1 sin( ) = 0 h i h i Combining Eqs. (3.1) through (3.3) yields the following system of equations:
An 0 0 1 0 1 M Cn = 0 (3.4) B C B C B Dn C B 0 C B C B C @ A @ A where
~ ~ ~ ~ cosh n cos n sinh( n) sin( n) 2 ~ 3 20 1 1 M = 6 7 6 sinh( ~) + sin( ~)+ cosh( ~)+ cos( ~)+ 7 6 7 6 0 ~ ~ ~ 1 0 1 0 1 7 6 1 cosh( ) + cos( ) ~ sinh( ~) ~ sin( ~) 7 6 1 1 7 4 @ h i A @ A @ A 5 As with the free-free solution, a solution is sought where
An 0 0 1 0 1 Cn 6= 0 : B C B C B Dn C B 0 C B C B C @ A @ A Possible values for ~ may be found by setting:
jMj = 0
~ th Thus, let n be the n positive root of:
~ ~ ~ ~ ~ ~ ~ cosh cos + (0 + 1) cosh( ) sin( ) + sinh( ) cos( ) (3.5) h i ~2 ~ ~ 201 sinh( ) sin( ) 1 = 0:
The natural frequencies of vibration may be found by obtaining the positive roots of Eq. (3.5) with: ~2 n EI fn = 4 : (3.6) 2 sAcsL
16 A Newton-Raphson technique was used to …nd the roots of Eq. (3.5) . A function F ( ~) was de…ned as:
~ ~ ~ ~ ~ ~ ~ ~ F ( ; 0; 1) = cosh cos + (0 + 1) cosh( ) sin( ) + sinh( ) cos( ) ~2 ~ ~ h i 201 sinh( ) sin( ) 1
~ For small values of 0 an initial guess for 1 was used to start iterations with:
~ 1;0 = 4:73:
~ ~ Once convergence has been achieved for n, iterations are commenced to approximate n+1. The initial guess for this level of iterations was:
~ ~ n+1;1 = n + :
~ Subsequent improvements for n were obtained using standard Newton-Raphson iterations with: ~ F ( ; 0; 1) ~ = ~ n;k n;k+1 n;k ~ F ~( ; 0; 1) where
~ @F F~( ; 0; 1) = @ ~ = sinh ~ cos ~ cosh ~ sin ~ ~ ~ ~ ~ ~ ~ ~ (0 + 1) f cosh( ) sin( ) + sinh( ) cos( ) 2 sinh( ) sin( )g ~h ~ ~ ~2 ~ ~i ~ ~ 01[f4 sinh( ) sin( ) + 2 cosh( ) sin( ) + sinh( ) cos( ) : h i
3.1 Special Cases for Very Large Values of 0 or 1:
There are two special case for when the ratio of the mass at either end, to the mass of the tube, becomes large. In this case, the corresponding end will behave as a pinned end. When the mass at an end becomes large, its position will become nearly …xed - due to its large mass.
17 Yet, if the mass at the end is concentrated near the neutral plane - there will be still be no moment at the end. As such the boundary conditions with very large masses at the end will be approximately:
d2Y If 1 then Y (0) = n = 0 0 n d 2 =0 2 d Yn and if 1 then Y (1) = = 0 1 n d 2 =1
Thus, when 0 1 and 1 = 0, Eq. (3.5) approaches:
~ ~ ~ ~ ~ lim F ( ; 0; 1) = 0 cosh( ) sin( ) + sinh( ) cos( ) 0 ; 1=0 !1 h i
Hence, the limit of Eq. (3.5) as 0 increases without bound, with 1 = 0; is:
cosh( ~) sin( ~) + sinh( ~) cos( ~) = 0:
This is algebraically equivalent to the characteristic equation for a pinned-free chime given in Table 4 as: pinned-free: tan( ~) tanh( ~) = 0
Similarly, when both 0; 1 1, the roots of Eq. (3.5) are given by:
~ ~2 ~ ~ lim F ( ; 0; 1) = 01 sinh( ) sin( ): 0 ; 1 !1 !1
Thus, the limit of Eq. (3.5) as both 0 and 1 increase without bound is:
sinh( ~) sin( ~) = 0:
Since sinh( ~) 6= 0; except at ~ = 0; this is algebraically equivalent to the characteristic equation for a pinned-pinned chime listed in Table 4 as:
pinned-pinned: sin( ~) = 0:
18 f3 =2 3.2 Finding Combinations that Result in f2
A combination of 0 and 1 was sought such that the ratio of the frequencies of the second overtone to the …rst overtone is 2:
~2 f3 3 = 2 $ 2 = 2: f2 ~ 2
With this ratio, the …rst and second overtone will be "pseudo-harmonic". Recall that = 0 was arbitrarily set at the end with the largest mass, so that:
1 0 1 0 or 0 1: 0
Four values of 1 were considered: 1 2 f0; 0:1; 0:3; 1g : The values of f3 are plotted on Fig. 0 0 f2
3-1 for a broad range of values of 0:
Note that as 0 becomes small, all four curves converge to the limit of a free-free chime where f3 = 1:96: As increases, when 1 > 0 the curves for f3 converge to the pinned-pinned f2 0 0 f2 limit of f3 = 2:25: In contrast, if 1 = 0 the curve for f3 converges to the pinned-free limit of f2 0 f2 f3 f2 = 2:09:
f3 The ratio of frequencies f2 = 2 occurs in the following situations for the special cases of = 0 and 1 = 1 : 1 0
f3 Table 5: Conditions for f2 = 2: ~ 0 2 1 = 1 0.0627 7.248 0
1 = 0 0.1162 7.420
19 Figure 3-1: f3 as a function of and 1 : f2 0 0
20 Figure 3-2: ~ as a function of and 1 : 2 0 0
21 Chapter 4
Experimental Veri…cation
These mathematical predictions may be experimentally veri…ed by comparing the frequency ratios predicted with those of a physical chime. The Euler-Bernoulli beam theory requires the tube length be much larger than the diameter of the tube:
D 1: L
For this reason a narrow diameter tube was selected. A mild steel tube was used with a nominal outer diameter of 0.25 inches (6.4 mm) and a nominal wall thickness of 0.035 inches (0.9 mm). Two cases were experimentally investigated. A spectral analysis was performed for each case using the software Raven Lite 2.0 which was created by the Cornell Ornithology Lab.1
4.1 Test Case A: 0 ' 1
The length of the tube was 55.8 cm. The mass of the tube was 69.9 gr. Masses of 5.24 gr and 5.07 gr were …xed to the ends of the tube. With these values:
5:24 gr 5:07gr = = 0:07496; = = 0:07253; and 0 69:9 gr 1 69:9 gr D 0:64 cm = = 0:0115 L 55:8 cm
1 Raven Lite 2.0 is available at http://www.birds.cornell.edu/brp/raven/RavenOverview.html
22 Using 0 = 0:07496 with 1 = 0:07253; the roots of Eq. (3.5) result in the following values for ~ fn fn n and f2 . For comparison purposes, the experimentally obtained values for fn and f2 are given in the table below.
Table 6: Test Case A mode or n 1 2 3 4 ~ n 4:259 7:187 10:19 13:22 ~ 2 fn or n 0.351 1 2.009 3.384 f2 ~2 fn (Hz - experimental) 93 264 529 889
fn f2 (experimental) 0.35 1 2.00 3.37
=0 4.2 Test Case B: 1 :
The length of the tube was 46.4 cm: The mass of the tube was 58.1 gr. A single mass of 7.52 gr was …xed to one end of the tube. With these values:
7:52 gr ' = 0:1294 and 0 58:1gr D 0:64 cm = = 0:0138 L 46:4 cm
~ Using 0 = 0:1294 and 1 = 0; the roots of Eq. (3.5) result in the following values for n and fn fn f2 . For comparison purposes, the experimentally obtained values for fn and f2 are given in the table below.
Table 7: Test Case B mode or n 1 2 3 4 ~ n 4:380 7:399 10:47 13:57 ~ 2 fn or n 0.351 1 2.004 3.364 f2 ~2 fn (Hz - experimental) 143 408 814 1357
fn f2 (experimental) 0.35 1 2.00 3.33
In both Test Case A and B, the predictions were within the experimental uncertainty of the predictions.
23 Test Case A Test Case B
889 Hz 1357 Hz
529 Hz 814 Hz kHz kHz
264 Hz 408 Hz
93 Hz 143 Hz
Figure 4-1: Raven Lite 2.0 Spectral Analysis for Test Case A and Test Case B.
24 Chapter 5
Conclusion and Limitations
The Euler-Bernoulli model for beam vibration was used to predict the natural frequencies of lateral vibrations of a tubular chime with variable masses …xed at the ends. Standard separation of variables techniques were used to predict the natural frequencies of a tubular chime. The resulting equation has been used to predict which ratio of masses (end mass to the mass of the beam) will result in a ratio for the frequency of the third mode to that of the second mode of f3 f3 f2 = 2: When f2 = 2 the frequency f3 is one octave higher than is f2: The predictions were compared with a spectral analysis of two special cases of physical chimes: one where both end masses were nearly equal, and a second where only one end mass was attached. In both test cases, the predicted ratio of frequencies were within the experimental uncertainty of the experimentally obtained frequencies. The Euler-Bernoulli model is strictly valid only in the limit when the ratio of the tube D diameter to the tube length is much less than 1, L 1: Most real chimes do not fully meet this criterion. Hence, it is suggested that further investigations include an analysis for situations where this limitation is not satis…ed.1 Further, adding a …xed mass to the boundary condition is a passive means of vibration control. There are means of actively controling the frequency using transient boundary contions such as a parametric oscillator.2Introduction
1 See for example, Han, S. M., H. Benaroya, and T. Wei. Dynamics of Transfersely Vibration Beams Using Four Engineering Theories. J. of sound and Vibrations, 225(5), pp. 935-988 (1999). 2 See for example, Landau L D and Lifschitz E M, Mechanics, Addison-Wesley (1960)
25 References
Greenberg, Michael, Advanced Engineering Mathematics, (2nd ed.), Prentice-Hall (1998).
Han, Seon, Haym Benaroya, and Timothy Wei, "Dynamics of Tansversely Vibrating Beams Using Four Engineering Theories", Journal of sound and Vibration, 225(5), pp 935-988, (1999).
Landau, L. D. and Lifschitz E. M., Mechanics, Addison-Wesley (1960).
Rao, Singiresu, Mechanical Vibrations (3rd ed.), Addison-Wesley (1995).
Rossing, Thomas, Science of Percussion Instruments, Vol. 3 of Series in Popular Science, pp. 9-10, World Scientific, (2000)
Vienna Symphonic Library. See https://vsl.co.at/en/Tubular_bells/Notation. Viewed, March 14, 2017.
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