Measuring the acoustical properties of mallets

Applied Acoustics 30 (1990) 207-218

Measuring the Acoustical Properties of Mallets

Ingolf Bork Physikalisch-Technische Bundesanstalt, D-3300 Braunschweig, Bundesallee 100, FRG

A BSTRA CT Measuring the shock spectrum of a mallet when it strikes a stationary force transducer is a convenient way to determine its effectiveness in exciting vibrations in a musical instrument. The maximum value of the residual shock spectrum, which indicates the frequency range over which a mallet is most effective, may vary with the strength o f the blow. Experimental measurements show how this maximum frequency varies with blow strength for a number of mallets.

1 INTRODUCTION For striking the various percussion instruments, the percussion player has at his disposal a great variety of mallets, which differ in shape, weight and surface area. By selecting the appropriate mallet, the player can influence the tone colour of a given instrument, using accurate knowledge about the mallet's qualities. How are these qualities to be quantified in physical terms'?. A simple method based on a procedure for shock testing which may help the manufacturer test and quantify the mallet head quality will be described. This quantification should also be helpful to the musician in the selection of the mallet to be used for obtaining a desired sound.

2 SOUND SPECTRUM OF PERCUSSION INSTRUMENTS The sound of percussion instruments is governed largely by the resonances of the vibrating parts which are excited by the blows of" the mallet. The 207 Applied Acoustics 0003-682X/90/$03.50 © 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain

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Sound spectrum of a xylophone bar struck by a hard (upper) and soft mallet (lower curve).

dependence of the spectrum of a xylophone bar on the hardness of a mallet head is shown in Fig. 1. The upper curve shows the sound spectrum radiated when the bar is struck by a wooden mallet, while the lower curve shows the spectrum of the same bar struck by a soft felt mallet. In the latter case, only the fundamental o f 110 Hz is heard, while in the case of the hard wooden mallet, the spectrum is dominated by the higher partials. It is evident that the kinetic energy o f the moving mallet head is transferred and spectrally distributed in different ways. The force transmission during the contact is responsible for the energy

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Measuring the acoustical properties of mallets

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transfer between mallet and percussion instrument. The force exerted by the mallet can be determined by measurement when the mallet head strikes upon a fixed force transducer. The output signal of the force transducer will then give the time dependence of the applied force. It should be borne in mind that when freely vibratory structures are excited, the force pulse is modulated due to the feedback of the excited vibration, in particular when the duration of the contact becomes greater than the period of the excited vibration, t This feedback will be disregarded in the following, as here only the properties of the mallet are to be investigated. A force signal from a soft felt mallet measured in this way is shown in Fig. 2. The shape of this curve, which approximately corresponds to a sin 2 function, allows only the duration of contact to be read. The analysis using an F F T (fast Fourier transform) analyser additionally provides information on the spectral distribution of the force F(t) transmitted:

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Figure 3 shows the result obtained when a soft mallet is used. From this statements on the sound quality of a struck instrument can be inferred on the basis of the following considerations. As is well known, when a vibratory system with the pulse response ,~(t) is excited by a signal x(t), the reaction of the system is calculated by convolution: 2 y(t)

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pulse response is defined as the response of the system to an ideal shockf(t) with the characteristics

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The sound of the instrument depends decisively on the amplitudes of the individual partial components or their ratios which are primarily determined by the point of excitation. This relationship results from the shapes of the individual modes which can be determined by modal analysis? The greater the vibration amplitude of a mode at the striking point, the greater its excitation. At the vibration nodes, therefore, excitation is theoretically impossible. The amplitude of vibration after excitation by a real force pulse F(t) is now given by eqn (2). For the assessment of the mallet quality, the effects of the excitation function x(t)= F(t) on the output signal y(t) are therefore to be investigated. For the sake of clarity, a simple vibration system will first be analysed in the following discussion.

3 SHOCK S P E C T R U M For vibration testing, shock tests are carried out by measuring the vibration amplitudes during and after the impact, the object measured being subjected to a well-defined pulse of acceleration a(t) applied to its base (Fig. 4). The time function of the shock can be optimized so that the critical resonances are excited very strongly. The so-called shock spectrum serves to describe the effect of the shock.-' It indicates which vibration amplitudes a mass-spring system with a resonance frequency shows during (initial shock spectrum) and after (residual shock spectrum) the impact, the attenuation being disregarded here. When the excitation (in this case, the acceleration of the base a(t)) is known, the time behaviour of the vibration can be calculated by solving the convolution integral (eqn (2)). The appropriate values of the initial and residual shock spectra can be calculated from the vibration amplitudes during and after the impact; for this purpose, these amplitudes

Measuring the acoustical properties of mallets

211

o

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Response of a single-degree-of-freedom system to shock excitation (from Broch~).

have to be plotted as a function of the resonant frequency of the mass-spring system. According to Ayre, 2 the curves represented in Fig. 5 are obtained for a sin-' pulse, the product of natural frequencyf by pulse duration r being plotted as the independent variable. In the range fr = 1, both curves assume their maximum value; that is, when the period of the natural vibration T = l/f is equal to the pulse duration, a maximum acceleration amplitude of the massspring system is attained for the blow strength given. There are important differences for large fr values; if the period of the natural vibration ts appreciably smaller than the shock duration, the amplitude during the blow will be substantially greater than after it (cf. Fig. 4, right-hand side). In this connection, it is a remarkable feature that the values of the shock spectra and, as a result the vibration amplitudes, are 1.6 1.4

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independent of the mass of the vibration system. ~ This applies, however, only to the case considered here in which the excitation is produced by a pulse applied to the base. It can be inferred that the residual shock spectrum results from the Fourier spectrum of the excitation pulse: '~ R(oJ) = ~olf(j~o)l

(4)

The vibration amplitude of a non-attenuated mass-spring system with resonance frequency ~ = 2rcf after impact is thus calculated by multiplying the angular frequency by the value of the Fourier spectrum at this frequency. With the aid of an F F T analyser (in which differentiation with time is realized by spectral multiplication by ~o), the residual spectrum can be easily determined by measurement of the excitation pulse. If the system to be investigated has several resonances, the appropriate vibration amplitudes can be directly read at the resonance frequencies. To what extent can these results be applied to the case where the system is excited by a force pulse? In case the excitation is produced as for a percussion instrument by a force pulse from outside, the equation of motion ofthe mass-spring system F(t) = -M~nZx(t) + Kx(t) is valid. The calculation in the Appendix shows that the acceleration amplitude of the mass-spring system with resonance frequency ~o after the action of force is R,(~o) = (~/ M)IF(je~)I

(5)

Though in this case the magnitude of the residual shock spectrum still depends on the vibrating mass, this relation clearly describes the quality of the excitation pulse. The residual shock spectrum ~lF(j~)l can therefore also be considered to be a measure of the acceleration amplitude. Here the selection of acceleration as the variable of motion appears to be appropriate, since at a given surface velocity the sound pressure radiated by a spherical source also increases with frequency. The acceleration a(t) measured on a xylophone bar in the bass register with a fundamental frequency of 110 Hz is shown in Fig. 6. The second and third modes have frequencies of 440 and l l 0 0 H z , respectively. For the measurement, an acceleration transducer was fixed to the middle of the bar's upper side while the excitation was produced by a mallet of average hardness (Jma,, = 800 HZ) in the area of a node in the third mode of vibration. 3 The selection of these two points allowed the second and third vibration modes to be excited only weakly. It can clearly be seen how the development of the vibration with time is decomposed into two parts; during contact with the mallet, a substantially higher value (Fig. 5) is obtained than during the decay, which is essentially characterized by the superposition of fundamental and third partial. This can be attributed to the fact that for the fundamental the

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productfr is much greater than unity, and thus strong differences between the initial and residual shock spectra result (Fig. 5). For the pulse spectrum of a soft mallet represented in Fig. 3, the residual shock spectrum shown in Fig. 7 is obtained by multiplication by joy. The frequency of the maximumfm~x now indicates at which resonant frequency a mass-spring system can be maximally excited with this mallet. Conversely, in the case of a complex vibration system, this curve allows the ratio in which the individual resonances are excited to be stated. The curve of measured values shown here applies to a blow of constant strength. With increasing mallet velocity, both the overall value and the shape of the residual shock spectrum change; the frequency of the maximum rises. For describing the dynamic mallet qualities, it has thus been found to be suitable to represent this frequency as a characteristic quantity which is a function of the blow strength. 50

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4 M E A S U R E M E N T ON MALLETS For the measurement of this dependence, an electromagnetic device was used to strike the mallets to be investigated onto a fixed force transducer. The strength of the blow can be varied by adjustment of the supplied voltage, the velocity of the mallet head prior to striking being measured with a photo cell. For calculation of the residual shock spectrum, the output signal of the force transducer is processed in an F F T analyser. For a number of different mallets of two manufacturers, the dependences of the frequencies of maximum excitationfm,x on the strength of the blow were determined in this way. In Figs 8 and 9 the mallets with soft surfaces are found in the lower portions and the hard ones in the upper portions of the diagrams. The degree of rise in each curve gives the range of variation with the change in the strength of the blow: a horizontal curve indicates that the pulse duration (and with it the optimum tonal range) is independent of the strength of the blow. A strong rise in the curve indicates that increasing the 5k-

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Properties of different mallets (Sonor).

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Measuring the acoustical properties of mallets

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Properties of different mallets (Studio 49).

blow strength will shift the peak to a higher spectral range; that is to say, with such a mallet the player has better opportunities to influence the sound character. On the basis of these diagrams, a suitable mallet can be selected for each frequency range. For playing over several octave ranges of a xylophone, a mallet is selected whose curve rises steeply. The selection of the corresponding tonal range can be controlled by the blow strength. From these curves the characteristic dynamic qualities can be deduced for each mallet. It is clear in each curve that the spectral maximum value changes to a greater extent in the region of weak blows than in the region of strong blows. This is especially clear for Sonor mallet $9, a wooden mallet with a rubber ring primarily intended for glockenspiels. With weak blows, the deformation of the rubber ring depends on the force resulting from it. With increasing blow strength, a saturation effect is reached, where further broadening is hindered by the wooden core. In the case of this mallet, whose rubber layer is only 2 m m thick, the elastic region in which the curve has a large slope is especially small. In principle, all mallets with a hard core and soft surface show this relationship; the thickness and hardness of the striking

216

lngolf Bork

surface determine the size and condition of this domain, in which the maximum e x c i t a t i o n ) ~ strongly depends on the blow strength. In the case of the S10 mallet, the wooden core is bigger than that of $9, and the curve is shifted to lower frequencies because of the increased mass. $11, which has the same core size as S10, is surrounded by a felt ring 5ram thick. Mallet S1 serves as an example of a mallet with a slight dependence on the blow strength. The curve of this all-rubber mallet (without a core) is nearly horizontal. S 100, $8 and $60 are very soft felt mallets of different sizes, which all have wooden cores. They are used for kettledrums and bass xylophones. Mallet G t 1 in Fig. 9 (Studio 49), a mallet having a hard core wrapped with yarn, shows an especially great variation region of 2'5 octaves. Because of its hard surface, it is especially suited for instruments in the high range (vibraphone, xylophone and glockenspiel).

5 OTHER APPLICATIONS Measurements of the dependence of the maximum excitation frequency on the stroke strength leads to important conclusions regarding the dynamic qualities of mallet heads. Such measurement procedures are, therefore, also of special interest to piano builders and especially to piano tuners. The sound of a piano is largely determined by the quality of the hammer head, and this raises the question of how the thickness and hardness of the felt, as well as the form and material ofthe core, affect the sound quality. The relationship between the blow strength and the frequency range of maximum excitation, which appears as dependence of the sound character on the playing dynamics, is of particular importance. On the basis of a series of measurements, a statement can thus be made on how the overtone content increases with the stroke strength; here not only the frequency of maximum excitation is of interest but also the total shock spectrum in the range of string and soundboard resonances. With it a checking of the optimum adaptation of hammer head and string is possible before the hammer is mounted into the piano action. This procedure offers advantages to the piano builder; unlike the measurement of effective compliancefl it is directly related to the sound spectrum produced.

6 CONCLUSIONS The procedure presented here for the measurement of the dynamic qualities of mallet heads has proved to be closely related to practice. The statement of

Measuring the acoustical properties of mallets

217

a frequency range of optimum excitation of vibrations is easy for both the manufacturer and the musician to interpret. Owing to the use of modern F F T analysers, the measurement is relatively easy to carry out.

REFERENCES 1. Hall, D., Piano string excitation 1II: General solution for a soft narrow hammer. J. Acoust. Soc. Anler.. 81 (19871 547-55. 2. Ayre, R. S., Transient response to step and pulse functions. In Shock and Vibration Handbook, 3rd edn, ed. C. M. Harris. McGraw-Hill, New York, 1988. 3. Bork, I., Schwingungsformen und Klangbewertung bei Xylophonen und Trommeln. In Qualitdtsaspekte bei Musikinstrumenten, ed. J. Meyer. Moeck Verlag, Celle, t988. 4. Broch, J. T., Mechanical Vibration and Shock Measurements, 2nd edn. Bri,iel & Kjaer, 1980. 5. Hall, D. & Askenfelt, A., Piano string excitation V: Spectra for real hammers and strings. J. Acoust. Soc. Arner., 83 (1988) 1627-38.

APPENDIX A mass-spring system consisting of a mass M and a spring with the spring constant Kis excited from rest by a force F(t) during the time 0 < t < 7". The equation of motion is F(t) = Ma(t) + Kx(t)

(A1)

The attenuation can be disregarded here. In the formulation x(t) = x e ~', with the complex variable p and with a ( t ) = d Z x / d t 2 = p Z x ( t ) , the Laplace transformations of force and displacements are F(p) = x(p)(pZM + K)

1 x(p) = F(p) M(p2 + 0)2)

(A2)

with 0,02= K/M. The initial conditions x = 0 and d x / d t = 0 at t - 0 are assumed. By inverse Laplace transformation, the right-hand part of eqn (A2) is transformed in the time domain into a convolution integral. The inverse transform of the right factor yields ~-~{1}lsin(ot). =_ (p2 + o2) co F r o m this the following relationship results: x(t) -

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218

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When calculating the residual shock spectrum, we are interested only' in the vibration after the impact. As the shock is restricted to the time 0 < t < 7-, the integration limits can be extended to + z¢ without the value of the integral changing. The transformation of the sine function according to the formula sin (z~- fl) = sin (:0 cos (/3) - cos (zO sin (ill yields x(t) = - -

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sin(cot)

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:t

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1

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and

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the following is valid: x(t) = ~

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1 1

IF(jco)l {sin (cot) cos (¢,(jco)) - cos (cot) sin (¢~(jco))} (A3) ]F(J'CO)I sin (cot - (b(jco))

For the acceleration a(t), the following is then valid: a(t) =

-

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=

-

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(A4)

whence it follows that the amplitude and thus the maximum value of the acceleration is proportional to colF(l'co)t,M after the action of the force. The a m o u n t of the Fourier transform ]F(jco)[ represents the spectral force density with the dimension N/Hz = Ns.