VIBRATING-STRING ELECTROMECHANICAL FILTERS

.... ~ . .. ' .. ~

by STANLEY H. LOGUE

stJBKI'l'TED IX PARTIAL FULFII,IJIENT OF THE REQ.UIREURTS FOR THE DEGREE OF KASTER OF SCIENCE at the KASSACHUSl'l'TS D'S'fI'l'UTE OF TECHNOLOGY (1952)

Signature Redacted Signature of A11thor ••••.••.•••••• ·~· •.•••• •()I• •••••••• Dept. of Electrical ~ineering Signature Redacted Kay 1 ' 1952 Certified bY•••••••••••••••r·~···,·•••••••••••••••••• /? . /'I Thesis SUpenieor Signature Redacted ·ch~~~·~p~;t~--;;i~·c~~iii~;·;~·G;'aci~~i;·si~d;~i; J . \ ~' . VIBRATING-STRING ELECTROMECHANICAL FILTERS (FOR HARMONICS) by STANLEY H. LOGUE SUbmitted to the Depa~tment of Electrical Engineering on May 16, 1952 1n partial fulfillment of the requirements for the degree of Master of Science. The purpose of thie thesis is to investigate the overtone properties of an electromechanical filter made up of a vibrating wire or "string" tightly stretched between two supports. Special transducers are coupled to the wire to excite and detect mechanical vibrations. The investigation considers the properties of the overtone or possible modes

of oscillation and their relationship to the harmonic series of overtones for a theoretically perfect string. Techniques -are discussed for oontroling the filter bandwidth, the relative amplitude of transmission for each overtone, and . . for adjusting the phase·response. An outline is given of the theory of vibrating-string filters; a.nd characteristics are given for several different types of string filters, including photogr~phs of the input and output waveforms. Thesis supervisor: Prlf. Fay Title: Asl!ooiate Prot•ssor of Electrical Communications CONTENTS Section Page I Introduction 1 1. Statement of the Problem 1

2. Importance of the Problem 1

!1 History of the Problem 2

III Consideration of Theory 6 1. Reeonant Frequencies 6 2. Amplitude of Vibration 7 3. Phase Propert1ee 10 4. Some Electrical Analogs 12 5. Damping 15 6. Frequency Difference between overtones and 16 Harmonics 7. Transducers 20 8. Amplitude Correction 21 IV Observations and Resulte 26 1. Equipment 26 2. Response Measurements 30 ( a) Undamped Strings 30 (b) A Bar Magnet Transducer 33 (c) Bridge Filter 34 (d) Motion of the End Supports 34 (e) Damping 36 CONTENTS Section Page 3. The Frequency Difference between overtones 38 and Harmonics 4. Some Special Transducer Arrangements ;2 5. String Filters 42

V Conclusions 48 1. The Amplitude Differences between the 48 Harmonics 2. Phase Response of the String Filter 48 3. Ratio of On-resonance to Between-resonance 49 Amplitudes 4. Properties of Damping Materials 49 Table of Wire Properties 50 I Introduction l. Statement of the Problem The purpose of this thesis is to investigate the overtone properties of an electromechanical filter made up of a. vibrating wire or "string" tightly stretched between two supports. Special transducers are coupled to the wire to excite and detect mechanical vibrations. The investigation considers the proper- ties of the overtones or possible modes of oscillation and their relationship to the harmonic series of overtones for a theoreti• cally perfect string. Techniques are discussed for controlling the filter bandwidth, the relative amplitude of transmission for each overtone, and for adjusting the phas~response. Character­ istics are given for several different types of filters. 2. Importance of the Problem While the vibrating string tilter is a rather novel idea, it is also of considerable practical value. The selective properties are excellent. · One can readily obtain ratios of resonant frequency to bandwidth as high as 2,000 ( up to 7000 with special care i which is much larger than can be obtained with filters constructed with inductors and condensers. Vibrating string :filters are easily tuned by adjusting the string tension.:, . , • They are inexpensive, the major cost being the construction of a mount to hold the strings. Since a mount can be designed to hold more than one string, the cost for each ad­ ditional filter is quite small. Also, while one string and its mount may seem bulky, a large number of strings may be paralled in close proximity, otten all utilizing the same two transducers, 2

resulting in a relatively compact array of tilters. In some ap­ plications, when vibrating string tilters are used instead ot the more conventional inductor-capacitor filters, a number of

I otherwise necessary isolating &m9litiers are eliminated, re- sulting in additional reductions ot cost and space. One of the ways in which vibrating string tilters are most different from other tilters is that one string is capable ot transmitting specific harmonics of some frequency. For example, it is possible to transmit the even harmonics and· reject the odd, transmit only the third, fourth, and fifth harmonics, separate two different wave forms even though their fundamental harmonics differ by only a few cycles, and other similar combinations of the above. Thus vibrating string filters can not only perform the functions of conventional audio filters, they can also be used to construct filtering systems difficult to obtain with inductance-capacitance tilters. This thesis attempts to pre- sent a basis for the construction of such systems, and provides data on the harmonic properties of string filters. II. History of the Problem A search of the literature on electromechanical filters has uncovered almost no information on vibrating string filters except the fact that very few people have investigated their properties. Most of the information is the result of re- , search conducted by Dr. Vilbig of the Air Force Research Center.

His results are presented in a collection of seven monographs 1 briefly summarized here. I - "Construction of a String Filter" discusses the basic principles of operation, suppression of undesired harmonics and cross-talk, and the damping of the strings. II - "The String Filter and its Application as a Readily variable Filter" discusses the phase and amplitude transmission curves obtainable with a multi-string filter.

III - "The String Filter *s a Phase Shifter" shows how string filters can be used to correct amplitude distortion in phase-shifting networks. IV - "Frequency-band Multiplication and Time Expansion by means of a"String Filter" shows how vibrating strings may be

used to double all frequencies of an audio spectrum. An example is given that demonstrates how much simpler vibrating strings are for this application than the conventional electronic circuits. v. - "Frequency-band Division through Excitation of Subharmonics of the Strings of a String Filter. Application of

the same Principle to Frequency-Band Multiplication" presents, a

1 Dr. Friedrich Vilbig, A String Filter and its Utilization Possibilities~ Air Force Cambridge Research Center, Report No. E5071, 1951 method for frequency-dividing spectrums (and thus reversing the process of multiplication given in M:mogra.ph IV). 'I.sclmiciues are given that permit multiplication and division by fractional factors. VI Discusses "The String Filter as an Amnl1tude Limiter and as Multiple

Frequency Audio Oscillator." VII discusses the "Behavior of the String Filter During the Application of Impulses, Noise Signals, and Wobbled Frequencies." It is the purpose of this thesis to extend the above work, which gene­ rally considers only one frequency per string, and consider filtering many frequencies with only a few strings.

The United States Navy Electronics Laboratory in San Diego, California has published a report on "Vibrating Strings as Band-Pass Filters• which considers using the fundamental mode of vibration as a filter. The strings used varied in length from 2 to 8 cm. and were completely enclosed in a strong magnetic field, excitation being produced by passing a current through the string. Attempts at driving strings with crystals proved unsatisfactory due to undesired resonances. Several string materials were used, tungsten, beryllium:- copper, and dural, with the first two giving best results. The output was obtained from the following balanced bridge arrangement:

reeietor

String

The string 1s suspended 1n a magnetic field. S'

Undesired harmonics were suppressed by using tuned circuits and damping was adjusted by changing the magnetic field intensity about each wire. The results are empirical and apply' to strings arranged as indicated in the report.' Vibrating strings form the heart or a control and measurement device invented by the Rieber Research Laboratory (New York City).il The string is used as the frequency-determining element of an oscillator; thus, any changes in the tension of the vil:rating string will be converted into changes in oscillator frequency. The _tension may be made a function or a rather wide range of physical quantities, such as temperature, pressure, flow, potential displacement, etc. The utility of the device for telemetering purposes lies in the fact that the oscillator frequency may be readily transmitted and measured with a high degree of accuracy.

I' R.E. Johnson, J.W. Sampsell, K.K. Wyckoff, Vibrating Strine;s as Bandpass F1lters(1950), u.s. Na.vy nectronics Lab., San Diego, Calif., NE 050928 Report No. 196

2. "The Vibratron", Electronic Industries, IV (April 1949), 79._ '

III Consideration of Theory

The vibrating string filter is a special application of vibrating systems; therefore., to gain an understanding of how it functions and to point out its basic properties., this section outlines some of the more pertinent parts of the general theory of vibrating strings. l. The Resonant Frequencies

One of the limitations of using vibrating strings for filters is the difficulty of obtaining high values.for the fun­ damental frequency. From page 8., the resonant frequencies are given by

( I ) where n is ti+~ harmonic number, J is the length (cm.), T the tension (dynes)., and/1 the mass per·:unit length (gms per cm.) It is instructive to change the form of this equation by noting that ~ = ~ , where Sis the stress in the string (dynes per cm. ) and,o is the density of the string, as well as to convert to more convenient units. The result is

f,. • lSl,t ~ $ 1 (2.) where J is in cm • ., Sin lb./in~, and Pis the specific weight of the string. Since for any given string materialp is con­ stant, and Scan only be so large before the string breaks; 7

for any fixed length of wire there is a limit to how high one can make the fundamental frequency. This limit is independent of' the size or cross-section of' the wire and depends only upon

its material and length. Figure If , page 9'4'. gives a listing of wire properties including the limiting frequency.

2. Amplitude of Vibration

Most text-book treatments of this topic either neglect the losses in the string or give them superficial treatment. 'Although the losses are very small in most instances,· at fre­ quencies close to resonance they are the controlling factor. In the following solution losses will be assumed to be a fune­ tion of frequency and of a visgous nature (proportional to string velocity). The mass per unit length will be assumed constant. Neglecting the restoring force due to wire stiffness for the time being, for the small displacements encountered in this . work the differential equation of motion is:

e>a., ~ a '1 .1!l /'I ~ta =- T ii'- - 8 ) t (3) M mass per unit length {gm/cm) T string tension (dynes) 8 viscous resistance per unit length

Solutions to this equation may take various forms.

Assuming a sinusoidal driving force of frequency w • J.trf located a distance • from one end of a string of length I , ,

Driving force= F £jut . t (where the imaginary part of e'w is ignored when checking physical measurements), a solution can be obtained in closed form for the vibration at a distance ,c from one end1:

Si1t ~ = F t'i,.,t "'~ "I w St~{': (J- "~ s;,, -r-1

s,n[~(I-•)] = F£jwt ~ Sin wx "I WT Sl11 wl ~ ( 0 < >C < ct ) (If. .., -ir- where c is the wave velocity (cm/sec} and

ca= T' • L (5') M -J ..I.cJ M unless damping is great. Resonance occurs whenever is of minimum magnitude:

.,... a (2rrf.) .. = ( "J t; -('J. ~ r (6) Unless damping is large,1-·RzJ .q·· The magnitude of vibration is of chief importance since the phase relationship can be pre­ dicted from a knowledge of the amplitude. Expanding c as a series and introducing hyperbolic functions gives: (7J

"1 _ F c ~jwt Sin(a «) coala(e1.S)-Jc.. {oll)$ill•(ca4)] [s,,,{cbJ c.,,-,Ut) -Jco.S,W•J1111"8•J] WIT [ Sin(dl) coaltle/)-J ceaJJ 4iltlt(4JJ]

1 P. M. Morse, Vibration;~ Sound (New York, 1948), p. 96. the magnitude of which is

Sin~d.• + s1111t1~.. sin1tth + sin h 2,8 C V -J o' (BJ 1"11 =F 4.IT ,/ 5;,.1..,1..J + Sir,J,"/J./ 1 where a (ta<:ic<.I) ~= b - { ';" 2.,}MT ( 0 < '('< .. )

For most all applications, because damping is small,~ is much larger than,8 , so the expression can be simplified by assuming

~ = o in the numerator. The following definitions are introduced:

w,.M B

The Q used here is analogous to the one used in electric tuned­ circuits, a measure of the bandwidth:

( 8. w.],, - -1!!.. Q,. ('I CJ

The resulting expression for amplitude is

F 1"11 - (,oJ 2.11' f ,J T l'1 and shows various factors affect the amplitude of string vi-· bration. The ampli tude"1, at & from one end, is proportional to the exciting force, F , applied a distance a from one end. The sine terms in the numerator describe the effect of the ex­ citing force and the shape of the string, while the numerator controls the properties near resonance, increasing the amplitude /0

and decreasing the bandwidth as :- increaseso Figure I gives plots of amplitude vs. frequency for various va1 ues of-;;-.e..

At resonance (when f• 11 t, ) the amplitude of vibration of each anti­ node, when the exciting force is also applied at an antinode, is

(,, 1 tr2. n a f, ~ T M

This demonstrates the effects of varying the tension in the string and its mass per unit length. As tne harmonics increase in frequency, the amplitude decrease as the square of the har­ monic number,n. At frequencies between the harmonics the am­ plitude of vibration is only slightly affected by the Q of the string and is approximately

F ~~ 211' I ,J T M This is referred to as the "minimum amplitude"o As Q, is decreased the amplitude at resonance approaches this minimum amplitude, the ratio ~etween the output at resonance and the output midway between resonances being approximately

This result can be seen in figure 17 on page 'II •

3. The Phase Properties

Returning to equation 7, the chief factor influencing phase is the term Sln d./ C•slt/J.J -J COJ rl./ Slllh,&..I II

I 8 ' IC>

I

t

! i i I I

I ...; I I t ' I i I l= I i -z : iii I "I 0 I ; - I I 1111 I ~~..; I J I I 8 i V N J:... 1 I >­ z () iii I z I I ! "'

<{ I s I f ~ I II I i 0 I 2 I I I ~LL I I i 0 JI IC> ! I

\ i

/ f 0 I' N I I ~ I I I i ~ i 0 I I ! ~c.-. ~ y I 0 ~ 2 ~ ~ ~ ~I .. - N IO iij Si _-::; Ii"" / II N I IQ- --- i7 -- I ... t'! / I I I ----- I 0 ,._.0 0 ~ 0 0 0 • i.t

which results in a phase shift hetween the input and output s_1gnals . Reference to figure 3 on page. 14 shows that near resonance each mode of oec111at1on can be considered as a separate. tuned c1rcui t. Realizing that "1 and ~cl't differ in phase by fo·, then in the vicinity of fn:

f'- fn A :a \X + 1l- ,.. fJ/411 5 e z. fn '2.

where 6 is the phase aruzle between F and ll/ • The rema.1n1ng factors in equation 7 that affect the phase are those 1n the numerator and indicate that for Q> 10, moving the points of excitation and pick-up shifts the phase only a few degrees unless the transducer passes through a node point, at which time the phase very rapidly shifts through 180. ~hat this should be so .can be seen by considering the geometry of the vibrating string.

4. Some Electrical Analogs Since persons interested in electromechanical filters are usually very familiar with electrical networks, it is in­ structive to point out that the different,ial eaue.tion for a vibrating string ( see equation~) has the ~a.m.e. form as the equation for electrical transmissinn lines. For a line having series resistance, inductance, and aaunt capacitance (L,R, and C per unit length) ''

The following analogies occur between the electrical and mech- anical systems 2 :

Mechanical Electrical y i F d.E/dt

M L

B R

T 1/c

Thus many of the widely published concepts of transmission lines

can be directly applied to vibrating strings. Figure '2. on page '""·shows the two systems.

Another useful solution to the differential equation of motion, equation 3, can be obtained by assuming a solution of the form • Jwt "1 = ,..,~ A.. ~ si11(1r,pc/.1)

Evaluating the coefficient A.. by the conventional Fourier tech­ niqu&s gives

• Sc1t 1111 f Sin fl"~ ; "1 = J r-_...,...... _ __.;;;; __ (IS") ,, ., Jr.J M -i- + a A. + ..+- T ..I.. [ n-,q '- • z. Jc.., z. TJ

where driving force F is applied at a distance a from one end3 o

2 M. J. Gardner, J. To Barnes, Transients in Linear Systems (N'ew York, 1951), p. 64. 3 Morse, .2E!,_ cito, p. 105. I It

Drb1111, F.,-ce

Cu,rut VS~~:?: / I11,llcf•11ce ------E>>------.L-----'lff"'---- /77777777777777 7 7 G,...,,.,,77 7777 7///777/?//777/777/7

Fig. '2. The Analogy between Vibrating Strings and Electric Tranemieeion Linee

Cur,.• 11 t' - Ra----~------R,

S, Sa. S1t La. .__ ___...... ___l. I ___., ______r_:.r ~,,. ______

- C,c./ B L C[,.. J M 2. Ir --2.

Volt•,• source - o,.,~;"' Ferce ( ~ ,,..,.,, o"• ••• J C &I,.,.. •t "" Ve.le cit'j ( ..!. , ,.,,.. •,r edf e e1tfl J

Fig. 3 A wmned-parameter Analog for Vibrating Strings IS'

Applying the above analogs to this equation results in an elec­ trical network that clearly shows the resonant nature of a vi­ brating string, each section of the network representing one of the natural modes of vibration. Since the parameters in this electrical analog are of a lumped rather than a distributed nature, the network lends itself more readily to thinking in terms of conventional network theory. Figure 1 on page,~ shows a diagram of the network and gives th~.electrical-mech­ anical analog relationships.

5. Damping

If the difrerential equation of motion (equation 1 ) is to be linear and result in a velocity of wave propagation that is independent of the damping factor 8, the restoring force due to damping must be proportional to the.velocity of the string. This avoids such undesirable effects as having input and output amplitudes not proportional, the resonant frequencies dependent upon the amplitude of vibration, and fr·equency shifts due to damping. Fluids and magnetic damping devices provide viscous damping, while damping by touching the string with fine fibers gives the above difficulties. Investigations have shown that wool damping results in pronounced shifts in the resonant frequencies of the string as its amplitude of vibration is changed4.

4 Vilbig, .2£!_ cit., P• 29. 16

The minimum amount of damping is obtained by placing the string in a vacuum, and Q's on the order of 7000 have been ob­ tained in this manner5. Damping due to air results in Q's of . about 2000 or 3000 f'or wires of ro·und cross-section. When the

string is round and damped by a rluid, the Q is given approxi­ mately by6: '"'

where r is string radius, p is fluid viscosity, ,.01 and p are respectively fluid and string densities, and JR is the resonant

frequency of the string. If the fluid is air the resulting Q is too high by a factor of from 2 to 3. This is due to dam.ping ••d within the metalAat the supports which the equation neglects. Agreement is better f'or liquid damping, but in any case the equa­ tion serves to indicate some factors that affect damping,includ­ ing frequency.

6. The Frequency Difference between Overtones and Harmonics

Various factors prevent the fundamental string resonance and overtones from forming a harmonic series of frequencies. When the string is uniform along its length and undamped this shifting of harmonics is largely due to the stiffness of the string. Solving the differential equation of motion considering

5 B. H. Geyer, Multi-element Electromechanical Wave Filters, MIT E. E. Seminar, Jan. 1949. ~ 6 Ao B. Wood, A Textbook of Sound (New York, 1930), p. 105. 11

stiffness, the f~actional shift in frequency is7

(17) where A is constant for a given string: f Ys1t" A = TV~ _...;;...... r __

Yis Young's modulus ands the cross-sectional area for the string, If is the radius of gyration for the cross-section, T and .I are string tension and length. f" is the frequency of the nth harmonic and 4 f the frequency shift due to stiffness.

Figure If on page la shows the frequency shifts for the three types of strings used in_the experiments. Note that the bronze ribbon ha~ very little shift and that all harmonics are shifted by nearly the same percentage. A filter that shifts the fre­ quencies of all the harmonics by the same per cent does not pro­ duce phase distortion.

As the amount of damping applied to a string is increased, the frequency shifts due to stiffness become less important. Unfortunately the damping medium shifts the frequencies. That portion_of the medium (air, liquids, etc.) in contact with the string moves with the string and increases its ef'fective mass per unit length. The amount of shift changes with frequency, being greatest at the lower harmonics. One advantage of magnetic

7 'M orse, 2.E.!_ £1:....!,.,"t Po· 166 a 18

I I I i I ' I I I ! i 4 I I ! I i i I I 3 I I I ; I i I - 2 I I I I I i I i [~I : ! i I ··"' .7 ~ ~

! '1'. 1· I •• ~~ i l.°-9!- .51-+i~~I~~+-1, ~--11~--+-L-----"~--+-L-,,---"'------=~1...-=:::;.;....----+~----+-~--+--~--+------I

.3~,~~:~==~==r...--·__J_: ---+-· ----l-~----i.1_ __j_ _ __j_ _ ___j_ _ ___j, _ ___J

Ir 1· , I ! .2 t-t-/. ---,------+i---~ I

1 ! : i I ! !~ .I ....+---~,--...-,---+---+--,---+----+-----+---+--...... ,--...... ----t z I ! w:::, , I l 1' ! ,Q7..+--+-!--+-,I --+----+I;--+ 1. --+----+--+----+---1---~

I&. .05 ..+--+------.---+-----1---+--_....--+---+---l-----+---II I i .001' • .013" .ROMZI RIBBOIN I ' ! ! .03~----+ll__ +-----1----+----<.----+---+------+--+---+--~

i I ' 01 i I ; . ------,--- ,----+-----1---+---f----,f----+---+-----+-----t ! i I I

I ; 1 l 3 7 9 II 13 15 17 19 21 HARMONIC NUMBER II

Fip:. '+ Fre1uency Shift due to Stiffnees /ff

damping is that it does not produce this effect.

Several methods can be used to intentionally shift the resonant frequencies. One way is to make the string's mass per unit length non-uniform by attaching a small weight to the string. Referring to the electrical analog on page/¥ shows that this is equivalent to inserting an inductance in series with a transmission line. If the weight is placed near one end of the string, the lower harmonics will be shifted less than the higher onesa .

An easy way to compute the amount each harmonic is shifted is to use a Smith chart8• Suppose a small weight of mass mis attached at a distance~ from the end of the string. The chart gives the normalized mechanical reactance for any length of string (normalized by dividing by the characteristic impedance of the string, 'Zc•~TM). If the reactance for a string ,c units long is Xs and the reactance of the weight is w,.•, the total reactance of the combination is Xs•c.l,.111. The weight and " uni ts of string are now replaced by a new string x' units long and having a rea.ctance x; =r x,.+w.."'. The Smith chart is used to find the length ,c'. (,c' - x) is the effective change in the length of the string for the harmonic of frequency""-, thus the new fre- quency will be~,c ' times the old one. This calculation is re- peated for each harmonic and is accurate as long as the fre­ quency change is not great., Figure 18 on page Ill compares some 8 ' M. I. T. Radiation Lab, Staff, Principles of Radar (New York, 1946), PP• 8 - 66., 2o

calculated and measured frequency shifts.

Another way to shift harmonics is to attach small springs

to the string, such as fine rubber fibers (see page 36 ). · If the fibers are uniformly distributed along the length of the string

( ,4F"e.,ce ) . and have a spring factor of K per unit length K• 4 c,.. ,.,,.,. , solv- ing the resulting differential equation of motion gives.the res­ onant frequencies 1 a (18) w,. = [ ~tr ] :

Thus the lower harmonic frequencies are increased more than the higher ones. This frequency shift is evident in the curves for

fiber-damping on page 3 7 •

7. Transducers

The relationship between current in the string and the force produced by the bar magnets is simplyF"•fJI/, where,1:1 is magnetic flux,.,/ is length of string in the magnetic field, and i is string current. This type of transducer produces no phase shift.

The capacitance pickup is best analyzed by considering the circuit in the.following figure. c. '~* ~ tR>~ f,,.c,, 2. I

C,is the capacitance between the string and the pickup electrode

and is considered to be a constant capacitance plus a time-vary­ ing capacitance. C. - c .... c.. f(tJ

Caand R represent the input impedance of the cathode followero In general i = d."' 't = 'h-(ce)

Exchanging the battery, E, for a current source gives9 I .l tctJ - I = (c,E:) = £:.. _!£! = C c,., ... -k dt ., t" ea. C, R

Another source exchange gives

Ca.

e,ac • :I (,,, 1+Ca.I Jr -'t ~ +~-& f(tJ C1 R

1,,. (c} This circuit clearly shows how the parameters in figure(•) affect the performance of this transducer. Sensitivity is proportional to E: , and the values must be chosen to give little phase shift at the lowest frequency used.

80 Amplitude Correction

Since the amplitude of string vibration for each harmonic decreases with increasing frequency, it is necessary to equalize

9 Gardner and Barnes, .£E!_ cit., p. 43. the amplitudes if the relationships between the harmonics are to remain unchanged by the filter.

A good way to correct amplitudes is to place the trans­ ducers near the ends of the string. Here the shape of string vibration varies as Y sen ~~ , where Y is the amplitude at an antinode. Then the slope near the end is Y 7 . If the maximum amplitude of vibration, Y, is inversely proportional to frequen­ cy, the slope a~ the ends of the string will be the same for all harmonics. Thus placing a transducer near the string supports makes the effect of the transducer inversely proportional to fre­ quencyo The nearer the transducer is to the support, the greater the number of corrected harmonics and the smaller the transducer sensitivity. This correction system was used in all measurements.

Another way to correct the harmonic amplitudes is to place a properly shaped pickup electrode along the entire length of the string. Consider small elements of the string and elec­ trode:

///4////,,ij//af~ehho////h//"/u//f".1 --,c .. 1 J.- "1 "1 d'IC' Assume the voltage induced in the small pickup electrode element is given by lac~J ·•~-<:,cf where h(,rJ relates amplitude of vibration to induced voltage. Now let hC~) be expressed as a function of x by a sum of sines: N ltC,cJ = £ H,. sm(1tttfJ II• t (2o) 2.3.

The shape of the string is also a sum of sines:

At(:IC') = £- Y,. Sitt(ntr f) cos(llc.,t) I II•.

The total output voltage is th~n

• - ~ E,. c.-.,(nwt) ••• or E, c.Gwt + Ca c:..& Z.t.>t + E ~ c.o 1wt • .. • • •• • . =

I " = T ....Z H11 y,. Ceca (nwtJ (2.1)

,I thus E:. I - - H, 'ri I I I • I • I J. I I

Since the values of Y. can be measured for each harmonic, and

the values of £ft are given, the values of Hn are known. Thus hc,c}is known as a function of distance, • Equation '' on page or.1tp 11t voltage 'to 1l relatesAstring-to-electrode capacitance change. The follow- ing gives an approximate relationship between amplitude of vi- bration and capacitance change 10 •

lO J. Eo Term.an, Radio Engineer's Handbook (New York, 1943), p. 113. ~ C •--~-~- .)/)I,,./. h los,. [ i-] / LIIC'

i• r:, • 0,'13 ~ • .. AC ... 6 « M - h c:::::J]. ,.,;. [-t] ~ ~eo•o.,1 ~- o.a, A point by point plot can be made of the shape of the pickup electrode. To allow hC~)to be positive as well as negative, a straight electrode can be placed on the opposite side of the string as shown below.

Unless c :-..>Aca in the above equations., a slight amount of ampli­ tude distortion will be produced. This is true of all such capacitance pickups. 2.5 IV. Observations ~ !ES Results 1. Fquipnent. The vibrating string system was mounted on a brass plate clamped to

a section or a steel beam for increased. rigidity. The entire assembly' was shock-mounted on large pieces or sponge rubber to isolate the string from externaJ. vibrations. Since it was often necessary to place the . string at some potential above gro,mJ in order to use a capacitance pick­ up system, the end sup~rts were made or fibre blocks to which were attached screw assemblies for anchoring the wire and adjusting the

tension. See the photographs on pages I.~ ,r.

A schematic for the electrical equipment used is given in figure ' ,

page z. 7 • Since some of the measurements were made with strings damped

in air, the resulting bandwidths were quite small. This required that the signal source be.quite stable. After testing a number or signal generators, the Hewlett-Packard model 200c employing a Wien bridge

oscillator was found to be quite satisfactory. It was modified by altering one of the resistance arms in the Wien bridge to provide a fine frequency

control. (see figure 7 ) • The frequency of oscillation is given i,,- I w • iic where ft is the same for both arms. Then when only' one resistance arm is changed:

Thus, by incre8J3ing R some accurately known percentage, the frequency

will decrease by half that percentage,· so changes in frequency can be

measured by measuring resistance. A motor.:..driven potentiometer was used to sweep through the string resonances at a linear rate.

The first string used was piano wire having a circular Ol'O?S section 0.016 inches in diameter. The first exciting transducer consisteci of an '2. 7

Fig. G Block Diagram of Equipment

1-,. .._.w, String -aso. velts d.c. Exciting Pickup •• ~ Tran~ducer Tran~ducer

Cathotle S;,,,,.I Fol/owe,- - Oscilloscope "4- Genef"at•r

Vo/tmete,. Vo/f,nete,.

Headphonee P•p• ,- Record•r

,' ' I ' R' ' '

R

Original Altered Circuit Circuit Fig. ·7 Changine: a 'Hien Bridge Oec1llator to obt81n a fine frequency adjuetment. R'alters the frequency range of the oscillator. R, and R1 are fine frequency controls. )8

electromagnet held close to the steel wire. Magnetic bias was supplied by' a bar magnet. A cross section or the arrangement is shown in figure 8 . To determine the transfer characteristics or this type of filter, it is . . necessary to lmow the relationship between exciting coil current or voltage which m•'I · · aiad, the air gap flux density"'\Q be less than proportional to coil current. The following scheme was used to investigate this relationship. A small coil or wire was mounted in the magnetic field of the exciting coil. If: -' e1 : ,CI Co-ca. tvt' l = I ....;... "'t" e. N. Then:

e& • N1. il., 't • le N, Nz I w c.,. wt' • "wlJrNa. ~I It the circuit tor l is mostly inductive, L. :

... c~ If N, N 1.. -=E', ...

Measurement showed ;~ to be independent or frequency for frequencies at least as high as 5000 cycles, so no correction was needed for ampli­ tudes when using this transducer. This is not true of' phase, but a better transducer was available by the time phase became important.

The string vibrations were detected by making the wire 850 volts negative with respect to. ground (the brass base) and placing a small electrode near the wire. Since excess sh'tmt capacitance or a low resis­ tance between electrode and ground decrease the sensitivity, a cathode follower was used to provide high input and low output impedances.

Details of this pickup system are given in figure 10 , and an analysis or such transducers is given on page 2. o • 2.,

Fig. 8 F1g. 8 Electromagnetic Bar Magnet Transducer Tranl!"ducer

Bar Magnet -----frtr1ng

String N S • N S .C., N

Ber Magnet!

Fig. 10 Capec1tan~e Pickup Traneducer and Cathode Follower

..l_--~1-2-A-T-7-_1_---~~--• 175.volte cW1re Electr e

20. output JI)''"· ....._ __...... ______....,._--0

The entire cathode­ The lower half-power follower circuit 1e point 1e 66. cyclee. housed in a copper box. 30

The cathode follower was .connected directly to a Ballintine volt­ meter (model 300) having a maximum sensitivity of 0.001 volt. Since these instruments also have output Jacks,they ll8re used to amplify the

. ' signal from the string and drive headphones or additional. measuring equipant. Their accuracy is 2 per cent and their frequency response is entirely flat over the range of frequencies encountered in any of the experiments. 2. Response Measurements. a. Undamped Strings

The first measurements were ma.de on strings without any external ·.l damping, the only damping being due to air and the resistance of the wire. This was done to determine the properties ot the s:,stem independent of the type of damping that might be applied and al.so because-one of the valuable characteristics of string filters is the~ high undamped

40· em. and the fundamental. frequency was .500 cycles. To equalize ampli­ tudes as frequency increased, as discussed earlier, both transducers were placed near the ends of the string.· The motor drivea potentiometer slowly moved the signal generator frequency through string resonances,while a Sanborn recorder ma.de paper recordings. of the amplitude response for each harmonic. In this way, it was possible to obtain the bandwidth at the half power points and thus the • and bandwidth for each harmonic.

As a check, curves were run of the exponential. decq of the harmonics by driving the string at resonance, then removing the signal source. The rate of decay also allows the calculation of ca as f'ollows a 31

6000 FREQ. ~ CHANGE

/ ~ 0.2 Q V ~ • CH~ ~ / 0.1•1. .--- ~

3000 ~I 0

• 0 500 1000 1&00 2000 2500 3SOO.· FREQENCY F1g. // .Q and Freauency Change trom Harmonic 5er1ea tor 0.008 Diameter steel Wire

100 FREQ. CHANGE Q ~ Q 0.f% ~ - ~ CHANGE 0 500 / V !/

300

400 800 1200 l800 2000 2400 .2800 FREQUENCY

Fig. 12. Q · and Fre~uency Change trom }:la.rmonic series for O.OOl'x 0.013" Phosphor-bronze Ribbon Since from the analog given in figure 3 each mode of vibration is equivalent to one series-resonant R J.. c circuit, the damping factor

- tl "t e is ~ • « is in the electrical circuit or 2M for mecha- nical circuits. ~" is """8 /II\ so Gl11• icr-"'" and. oc. can be obtained from the recordings. Once the values of Ci,, are determined for one .frequency, Q fo .. otlt•,. ,,.~,11.enc/•$ l,'1 lflt•fuilft it is possible to evaluateAthe relative amplitudes of vibration at the antinodes for these frequencies. {See equation • The results are tabulated on page '3. J No attempt was ma.de to relate the actual amplitude of vibration to exciting current or output voltage since these would depend upon the particular transducers one might build as well as the very small distances between the transducers and the wire; these distances could not be easi1y­ measured or held constant as the transducers were moved and the strings changed.

Equation II indicates that by decreasing the wire size, the sensitivity (amplitude of vibration for a given force) can be increased. This indicates the use of smaller wire. Also decreasing wire size reduces the effects of wire stiffness and reduces the strain upon the supports, which is desirable if maey strings are to be used. Piano wire 0.008 inches· in diameter was obtained (Central Scientific Co.) and substituted. for the 0.016 inch wire. The.same measurements were repeated giving substan­ tially the same results. Since the smaller wire has less volume, the pull from the exciting coil was reduced and there was no marked increase in sensitivity. Also,using an electromagnet to excite the string limits the type of string material to steel (iron and nickel break too easily to give a sufficiently high fundamental frequency-). b. A Bar-Magnet Transducer.

It was decided to excite vibrations by placing part of the wire in a strong magnetic field and passing the signal current through the wire. This change resulted in greater excitation for the same amount of driving power (by a factor of five for the transducers used) and had the advantage that the bar magnets used could be spaced quite close to the wire and not interfere with large amplitudes of vibration since the string moved parallel to the face of the magnets. Al.so, this transducer is free from the amplitude distortion produced by the previous transducer due to the string's moving too far in and out of the restricted field near the pole pieces of the electromagnet. An output transformer was used so that the string could be placed at the high potential necessary- to use the electro­ static pickup and to match the string resistance to the signal generator.

A simple bridge arrang~ment balances out any pickup due to the signal current in the string. This balance proved to be independent of frequency­ and so complete that removing the magnetic field applied to the wire . ,_) decreased the output to less than one tenth the output normally obtained at a frequency halfway- between two reso~ces. This new excitation transducer was used in all further work. A phosphor-bronze ribbon of the type used in galvanometer suspensions was substituted for the steel wire. Except for increased air damping due to the shape of the ribbon (0.013 in. lC 0.001 in. ), the·amplitude response was much the same as for the steel strings. c. Bridge Filter. A balanced bridge string filter was built using the same arrangement

employed by the Navy Electronics Laboratory (as discussed on page '+ ) ,

but using two strings instead of one (see page '3 s- ) • Bronze ribbons were

used to form two bridge arms, both excited by placing one point of each string in a magnetic field. The balanced nature of the bridge eliminated output due to the signal current in the bridge arms. When the strings

vibrated, the back emf'. induced by the magnets appeared at the output terminals. A low-impedance-to-grid transformer was used to raise the output voltage level. The bridge arrangement did not prove very success­

ful for the following reasons: The balance was critical and any- slight change in the resistance of the bridge arms resulted in undesired output. Slight temperature changes would alter the string resistance and unbalance

the bridge. Thus, all four bridge arms should be identical even though

only two of them vibrate, and they should be symmetrica,lly arranged to

prevent t/J cycle hum. Even with a stepup transformer, the output voltage was smaller than for the capacitance pickup, so the bridge filter system · was rejected. d. M:>tion of the End Supports. The strings were mounted on fiber and supports in order to insulate

them from the frame. To determine if there was any- motion of the fiber

supports, and if metal supports would be more satisfactory, the string was supported on brass blocks. Tests were made of the amplitude response,'

· considering both the Gl and the relative values of amplitude between harmonics. Also, the amount that the overtone frequencies differed from

being exact harmonics was measured. No difference could be detected between brass and fiber supports. 3S e; Damping. When a string is air damped, its bandwidth is too narrow for most

a;:plications; while damping in oil gives bandwidths as wide as conventional L - C filters. In an attempt to cover the range of bandwidths between these two extremes, several different damping techniques were studied. Any form of damping where the string rubqed or slid against the damping material, as would be the case if textile fibers were held against the wire,

resulted in amplitude distortion and frequency dependence upon amplitude. Since the latter effect is prohibitive in a filter for harmonics, such damping schemes were rejected. Small amounts of damping were obtained in the following manner. Two copper wires were run parallel to the string and very fine rubber fibers spun between these wires and the string.

The fibers were made from a mixture of rubber cement and plastic thinned

to give fibers of desired thickness. The advantages of this system are

that the damping can be readily adjusted by adding or removing fibers, and the distribution along the length of the wire can be as desired.

Results are given on page 37 for one application of uniformly spaced

fibers. A disadvantage is that the small restoring force, due to the spring like nature of the fibers, slightly detunes the harmonics (as discussed

on page '2.o ) • No amplitude distortion or dependence of frequency upon amplitude was detected. Another damping scheme tried was to place small pieces of sponge rubber against the string. The portion of the rubber nearest the string · was cut wedge-shaped to reduce the damping and effective mass added to the

string. This type of damping seemed linear in that it produced no ampli­ tude distortion or frequency dependence upon amplitude; however, the spring action of the sponge rubber shifted the frequencies. The results are given on page 37 • 37

-+-~~~~-~~~~---+-~----,.__~,% FREQ. Q CHANGE

0

0 ~ _ __.______....., ___.....i.. ______..;.,.._,,_ ___ ...... , 500 1500 2~ 3500 FREQUENCY Fig./¥ Q and Fre~uency Change from Harmonic series tor 0.0081 W1re Damped with Fine .Rubber Fibere

i I

; I I 60 I I I Q CHANGE '.I\. I 1% 40 ' --~ I - FREQ. Q k I ~ Ir CHANGE - ill - 30 1i"' I It ..______0 20 r--~! . . - - I D I I I I I ' I I -- 10 ! -r-I I

I I 0 360 720 1080 1440 1800 2160 2~20 FREQENCY Fig. 1s Q and Frequency Change from Harmonic Serie~ for Phosphor-bronze Ribbon Damped ~1th Sponge Rubber at 3.8 cm. from .One End 38

Dam.ping was also produced by placing the entire string in oil.

'.rhe oil used was Dow Corning Silicone Oil D C 200, having a viscosity ~t about one centistoke. Its damping n frequency characteristics differ

somewhat from those of an air-damped string. The resulting Q was quite

low, and the difference between maximum (on resonance) and minimum values of output voltage as frequency was changed was not very great. Due to . viscous nature of the damping, there was no dependence of resonant .fre­ qU4!lcies upon amplitude or amplitude distortion. Al.so, much higher c~ents could be passed through the string because oil dissipates heat more readil.7 than air. The results are given on page 3, . 3. The .Frequency Difference Between Overtones and Harmonics. The overtones are usually not a harmonic series because of string stiffness or the effects of damping. Thus, when.the signal applied to such a filter has the same .frequency as the fundamental string resonance,

the filter attenuates the higher harmonics. If the frequencies of al1 overtones are shifted from the natural resonant frequencies of an ideal

string by the same percentage, the result would still be a harmonic series

of frequencies. Then, all harmonics woul~ pass through such a filter. It is not important how much the overtones shift. The ratio between the shifted frequencies is what matters. The best way to relate the overtones is to

measure their frequencies and divide by the harmonic nUlllber. If the quotient is constant, the overtones form a harmonic series. Even if

all ovei~tones are shifted in frequency, usually only the first few are not part of a set of harmonics. To measure resonant frequencies, a Stroboconn audio frequency meter .was used. It gives readings to five significant .figures and can still more accurately determine if frequenc;ef!are harmonically- related. :,,

Q FREQ. t---+---~~--+----:.,o~=------+----!-----L__J 1. CHANGE

0

0 _____._ ____._ ___ ...... _.,._ __...._ ____.. ______.. ____.___. 360 720 . 1080 1440 1800 2160 . 2520 FREQUENCY

Fig. 16 Q and Frequency Change from Harmonic Seriee tor a 0.008" Diameter Wire Damped with 011 ( 1.0 Cent1etokes at 25 C) lfO

This method was used on all types of strings and for each damping system to determine how closely the overtones approximated harmonics. The results were nor~ized by dividing ~" by f. •

Since the fundamental and first .few overtones were usually lower in frequency than their corresponding harmonics, the fundamental being off the largest amount, methods were investigated for increasing these frequencies in order to make them harmonics. One way was to place a mass load on the string near one support, the size and position of the small mass being adjusted in accordance with the system outlined. on pages ,,.zoto give best results. The weights were originally pieces of copper wire, but wax was used later because it was easier to apply' and adjust. The result was to increase the frequencies of the higher over­ tones more than the lower overtones, which is about the same thing as lowering the fundamental and first few overtone frequencies. This correction method was reasonably satisfactory and could usually shift the .first .two resonances into harmonics. Figure 18 on page,,., gives. examples of how mass loading can shift frequencies. The fine rubber fibers, in ad~tion to providing damping, al.so shift the .frequencies due to their·a.cting as small springs. The effect is somewhat the inverse of the mass load scheme. in that the .fundamental. and lower overtone frequencies are increased.while the higher overtones are less affected. Equation 18 on page zo indicates how much the over­ tones will be shifted. Figure l't on page :, , gives an example or frequency correction with rubber fibers.

As can be seen from the curves of ,. ~, for the various types of damping, the lower resonant frequencies are usually the ones that must be corrected to produce a harmonic series. One of the two methods described above can IOOOr------+------±------.1~---~

FILTER OUTPUT (MV.)

IOOr------ttt-----+ff------++1----~~---~

.AL.L Q.: 30. F R DAMPED CURVE.

380 760 1140 1520 FREQUENCY, CPS Fig. 17 Comparison between Maximum and Minimum Amplitudes for a Damped and Undamped string

~

'- ::::--- ~ ~ .... \. 4 - MEASURED v, LUES FREQUENCY --d 40l FOR 20 MG. CHANGE % ~ 3 - /// 20.1 t '/ 2 52 I l&J 10 ~

WEIGHT 2CM. FROM END. 0 380 760 1140 1520 FREQUENCY CPS Fig. '8 Measured and Calculated Values of Frequency Change when a Weight is Added to the String . 11'2.

be used in such cases, the .amount of success obtained will largely depend upon the nature of the uncorrected set of frequencies. 4. Some Special 'l'ransducer Arrangc:1ents. In an attempt to simplify the problem of equalizing the amplitude

and phase responses for the string filter harmonics, it was decided to

sort the harmonics by passing the even ones through one filter, the odd through another, and combining the outputs. With only half the resonances on each string, ~he equaJ.izing problem should be halt as difficult as for a single string. Two magnets were placed at alternate ends of the strings (a total of four magnets). For one string both magnets applied a driving

force in the same direction, and by making these forces have equal affects, only the odd harmonics were excited. The other string's magnets applied forces in opposite directions and thus excited only even harmonics. Measurements were made to determine the extent to which the even or odd

harmonics were rejected. With care, the magnets could be adjusted so

the undesired harmonics were below the "minimum·signa]. n level and usually undetectable.

Even harmonics can also be eliminated by placing either the pickup

or excitation transducer in the exact center o:f the string •..This method

also reduced the even harmonics below the "minimum S1gna1 • level. Odd

harmonics can be e]jmjnated by tuning the :fundamental string frequency to the second harmonic. 5. String Filters. A string filter was constructed using two strings, one for even the other for odd harmonics •. B>th strings were made of phosphor-bronze ribbon

40 cm. long and were excited by magnets placed near the ends of each

string. Damping was provided by touching the strings :with small pieces of sponge rubber, the point of contact so chosen that each string passed two frequencies with equal amplitudes. Since, to maintain a constant

Q and amplitude for each harmonic requires the effects of both the

be excitation and pickup transducers .to proportional to frequency,. the trans4ucers were placed close to the ends of the strings. ·lbth pickup electrodes were connected to a single cathode follower and amplifier.

The frequencies were corrected by placing a small piece of wax on the strings. At first, the -amplitude response was measured with a voltmeter while a phase angle meter was used to determine the phase response for • each harmonic. While accurate, this method proved to be too slow.so the following system was used to adjust the .filter. Wave f'orms of the f'llter input' and output voltages were pl.aced on a dual trace os.cilloscope (both traces having a common sweep circuit), one wave form superimposed upon the other.

The signal generator was accurately adjusted to each harmonic and the two traces were compared. The signal generator provided a constant amplitude signal, so aJlY' relative difference among the harmonic amplitudes in the f'ilter output was readily noticed. A phase shift through the filter will cause the 01iltput trace to shift to the right of the input trace. If the oscilloscope sweep rate is held constant, then as long as the traces are displaced by the same distance for each harmonic, the :filter output wave

.form will not be phase distorted. Br using this system, the two strings were adjusted to pass a total of four harmonics with little distortion.

Some of the higher harmonics also passed through the f'ilter but with decreased amplitude. Photographs o.f the results obtained with this filter are given on page "Is- •

A filter was built using only one string, 40 cm long of phosphor­ bronze ribbon • .Amplitude equalization was obtained in part by placing the transducers near the ends of the string and by attaching a sponge rubber damping block to one end of the string. A small wax weight 2 cm. from one end of the string helped correct the frequencies. While the amplitudes and phase shifts among the harmonics were not completely corrected, the results were good as can be seen from the photographs on page 't.S- • The

input wave form, a trapezoid-like pattern, was obtained by overloading the output from the signal generator, causing it to distort. The photographs compare the filter input and output wave forms.

A square wave contains only odd harmonics. By placing a small sponge rubber damping block at the exact center of the string, the odd harmonics were suppressed and only the even harmonics passed tbrought the filter. The result was that the trapezoidal input produced a triangular output of· twice the input frequency (see page 1/7 ). Information for Photographs on Pages 46-47

All waveforms are tor a tilter made of phosphor-bronze ribbon 0.001 x 0.013 inches in cro1a section and 40 om. long. The tundamental frequency tor each filter is 360. cps. Waveforms (a) through ·cg) apply to a filter consisting of two strings, one passing the odd harmonics, the other pa11eing the even harmonics. The signal was generated by two oscillators connected in series thus producing two harmonica. (See p.42) Waveforms (h) through (m) use a half-wave rectified sine wave for the signal, and the filter is as above. The filter for waveforms (n) through (p) consists of a single string adjusted to pass as many harmonics as possible.

( See p. 43 ) • (a), (b), and (c) are second and fourth harmonics. (c) is detuned from resonance by 20 cps. (d), (e), and (f) are for the first and third harmonics with (f) detuned 20 cps. (g) shows all four harmonics. The input amplitude was held constant. (h) superimposes input and output for half-wave. (1) is detuned 30 cps. (k) and (1) are the odd and even parts of (j). The half-wave source for (m) was distorted with an inductor. (n) shows response of second filter to near-trapezoid wave, and (o) is (n) detuned 30 cps. (p) is as in (n) and Co) except only the even harmonics are passed.

FOR EACH SET OF WAVEFORMS THE TOP PATTERN IS THE INPUT, THE BOTTOM PATTERN THE OUTPUT. TIME PROGRESSES FROM RIGHT TO LEFT. ( cl J

(b) (e)

( (') V (f\

I~ • '+7

(h) {j J

(k) (i)

(l)

Ut1)

(o)

(p) Cn) V Oonolusions

1. The .A.mplltude Differences between the Harmonics The normalized response curves on page 11 give the shape of the response patterns for the various harmonics.provided the fl 'a are known tor each harmonic. This determinee the bandwidth at the half-power points. If the type of damping employed does not have a very great ·effect upon the mass·· per unit length of the string, the relative amplitudes allow a computation of the relative bandwidths (and thus relative Q's) of each harmonic. The position of the transducers can be arranged so as to provide a considerable amount of amplitude equalization. If the Q's are all equal for each harmonic, the losses increasing in proportion to\frequency, the amplitude of vibration at the antinodes tends to decrease with the square of the harmonic number, n. As long as this ie true, placing the pick-up and and excitation transducers near the supports equalizes the output harmonic amplitudes. Should·it be desired to eliminate one of the harmonica or all of the even or odd harmonics, this can be done quite satisfactorily by proper poeitioning of the transducers.

2. Phase Response of the String Filter In order to pass the harmonics of some complex waveform without distortion, all of the harmonics must be shifted by an amount proportional to their harmonic number.

This can only be true if the Q 'e for each harmonic are equal ( see equation 14 on page 12 ). Even if the Q's for each harmonic are different, it the relative amplitudes of all the harmonics are equalized, there will still be no distortion if all of the waveform harmonics coincide with the string resonances. In this case the phase shift for each harmonic is the same ( t-5n = 0, see equation 14).

3. The Ratio of On-resonance to :Between-resonance ( "min­ imum amplitude") Amplitudes. As damping is increased, the amplitude at resonant frequencies approaches the amplitude at frequencies half way between resonances. In some applications, such as where bandwidth must be large, this effect may be objectionable. Were one to construct a filter of electrical elements, such as the one on page 14, it would also possess~hie amplitude effect, so the effect is not due to having chosen a-·:etr1ng as the filtering medium.

~. Properties of Damping Materials Unfortionately the amount of vibrational resistance for any particular type of damping material can not be determined with accuracy without resorting to actually measuring the damping and frequency- shift properties of the material. The curves given 1n the main body of this report allow some generalizations to be made concerning several different types of damping. The problem of correcting the amplitude response for the various hannonice is much less of a problem than is adjusting the phase response. More work 1s needed on this problem. !i I

Some Physical Properties of Wires 1. The following information was obtained from several handbooks, the most typical values being used: Metal :Maximum Density Young's :U:aximum Tensile Kodulus Frequency streng~h (cpa. tor (lb/in) (gm/cm3) (lb/in2) 10 om.of wire) Piano 250-300xl03 7.8 28x106 2480. wire

Phosphor 110-140 tt 8.8 15 .. 1600. bronze (hard drawn) Tungsten 590" 51 II 2320.

ti Aluminum 30-40 II 10 1480. Beryllium 180-200" 19 n 2000. copper

2. The wiree ueed in thie theeie had the following properties: Bronze ribbon: 0.001 in. thick, 0.013 in. wide Weight per cm. o.69 mg. Electrical resistance 17.3 ohme per meter Maximum frequency ( wire 10 cm. long.) 1690. cpe. Piano wire; Diameter 0.008 in. 0.016 in. Weight per cm. 2.47 mg. 1.05 mg. Resistance 6.6 ohms per 1.56 ohms per meter meter Maximum frequency 2400. cpe. ( wire 10 cm. long)