The sound generated by a transverse impact of a ball on a circular

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1 J. Acoust. Soc. Jpn. (E) 1, 2 (1980) The sound generated by a transverse impact of a ball on a circular plate Toshio Takahagi*, Masayuki Yokoi*, and Mikio Nakai** *Junior College of Osaka Industrial University, Daito, Osaka, 574 Japan **Department of Precision Mechanics, Kyoto University, Kyoto, 606 Japan (Received 19 June 1979) The purpose of this investigation is to get a fundamental theory for the reduction of impact noise. The sound caused by the impact of a ball on a relatively thin circular plate clamped at its edge was investigated experimentally and theoretically and the theoretical results almost agreed with the experimental results. The impact sound in this paper is due to the forced vibration of the plate. The forced vibration problem was solved with consideration of the damping of the plate and Hertz's contact theory was also introduced to analyze the impact phenomenon. The impact force was calculated by the small increment method and the sound pressure was calculated by assuming that the point sources were distributed all over the plate surface. The influence of both impact duration and maximum impact force on the impact sound were investigated and it was found that the impact sound pressure level depends on the amplitude of the impact force histories and the frequency components of the impact sound pressure depend on the impact duration. PACS number: r 1. INTRODUCTION There are many cases in which impact phenomena cause machine noise. We can easily find examples of machine noise such as when the parts of a machine structure collide with each other in the process of movement. The noises from machines which utilize impact phenomena such as pile drivers, forming machines and punch presses cause serious problems. To control these noises, noise insulation methods are usually taken which enclose the source of the noise and reduce the radiation of the sound. But the method of controlling noise at its source, which is most effective for reducing noise, is not usually taken. This is because the generating mechanism of impact noise has not been precisely analyzed, and effective methods for reducing noise have not been found. Several studies on impact vibration have been made. Timoshenko1) solved the problem of the transverse impact of a mass on a beam strictly, using the Hertz law of contact and considering the vibration of the beam. The small increment method which he developed to solve the problem is reliable but tedious. Lee2) developed an approximate method that is reliable for problems in which impact energy is absorbed mainly in the fundamental mode of vibration of the beam. Tobe and Kato3) also studied the transverse impact of a mass on a beam by Timoshenko's small increment method using a computer, and he investigated the influence of the mass ratio and the rigidity between the mass and the beam, and also the effect of the size of the beam on the impact. As to the fundamental study of impact noise, 121

2 J. Acoust. Soc. Jpn. (E) 1, 2 (1980) some researchers analyzed the rigid body sound that was caused by the collision of two bodies which are supposed to be nearly rigid, and others analyzed the sound of impacted bodies which were flexible like plates. In the former case, Nishimura and Takahashi4) investigated the collision of two steel balls experimentally and theoretically and showed that the peak sound pressure level was proportional to the impact velocity raised to the 1.2 power and the impact sound was not caused by free vibration, but by the acceleration of the center of gravity of the ball. Koss and Alfredson5) made similar investigations and corroborated the results of Nishimura and Takahashi. Endo et al.6,7) studied the impact noise of a ball on a cylinder and on a freely suspended plate and suggested that the evaluation of the energies in the separate modes of vibration made it possible to decide which of the free vibration sounds and the rigid body sounds are dominant. Akay8) indicated two types of impact sound generation mechanisms for impact on a plate. One is a radiation due to rapid surface deformation, that is, an initial sound pressure pulse due to rapid local deformation of a plate was generated before the radiation from natural modes of the plate occurred. The other is a pseudo-steady-state radiation due to impulsively excited steady state acoustic radiation. And we will call this radiation free vibration sound. In the investigations of free vibration sound, Tokita9,10) conducted a precise experiment on the impact of a sphere on freely suspended plates, and his careful considerations were full of suggestions, but unfortunately no theoretical analysis has been obtained yet. In this paper, a plate was regarded as the most simple element of a machine structure, and to investigate the impact noise generating mechanism precisely, the impact of balls on a plate was analyzed experimentally and theoretically. Usually plates of machine structures are not freely supported but clamped, so we clamped the plate at its edge, and we used a circular plate in order to make vibrational mode analysis easier. The plate which we used was thin in comparison with its radius and modal vibration cannot be ignored when considering impact vibration. The sound from the plate belongs to the free vibration sound. 2. THEORETICAL ANALYSIS Substituting Eq. (7) into Eq. (1), multiplying both

3 T. TAKAHAGI et al.: THE SOUND GENERATED BY A TRANSVERSE IMPACT (8) where Fig. 1 Geometrical impact condition. The impact force applied as a concentrated load is expressed by Dirac delta function. (9) Substituting Eq. (9) into Eq. (8) and solving the resulting equation, we have where (10) (11) (15) 2.2 Numerical Solution of Impact Force The geometrical impact condition when a ball contacts a plate is shown in Fig. 1. The relative (12) Denoting the mass of the ball by m and the initial 123

4 where e is the value for judging convergence. 2.3 Sound Field The sound pressure at any point in the field can be obtained, assuming that the sound source consists of many simple sources and the sound pressure at time t is the sum of the symmetric spherical waves radiated from each simple source at time (t-r/c), where R is a distance from the source and c is the sound speed. When the function S gives the instantaneous value of the total flow of air away from the center of the source, the sound pressure p from the simple source is12) where Substituting Eq. (11) into Eq. (18), we have Substituting the obtained impact force into Eq. (19) and integrating the equation by numerical methods, the impact sound pressure can be calculated. 3. EXPERIMENTAL METHOD Fig. 2 Coordinate system. The arrangement of measuring instruments is illustrated schematically in Fig. 3. The clamped circular plate that was struck by the ball has a diameter of 350mm, and a thickness of 1.95mm and is made of steel. It was clamped with twelve screws by a rigid circular frame to a rigid steel cylinder (25mm in thickness and 400mm in depth) so as to be supported rigidly. Absorption material is

5 T. TAKAHAGI et al.: THE SOUND GENERATED BY A TRANSVERSE IMPACT Fig. 3 Experimental apparatus. packed inside the cylinder and its bottom is covered with a thick steel plate (4mm in thickness) in order to avoid the influence of sound reflection. Therefore, the sound which radiates from the plate corresponds to that of a baffle. The plane of the circular plate is vertical. Two kinds of steel balls (12.7mm, 9.5mm in diameter) and rubber balls (26mm, 22mm in diameter) were used. The ball was hung by a string 1m long and struck the central point of the circular plate like a pendulum. Initial velocity can be changed by adjusting the initial position of the ball. The impact sound was measured at a distance of a 1/2 inch condenser microphone (B& K 4133). Vibrational acceleration was measured with a lightweight vibration pick-up (B& K 4335) attached to the central point of the rear of the plate. The measured signals were transmitted to a magnetic tape recorder (TEAC R500) and were recorded at a tape speed of 60 i. p. s. which could measure frequencies up to 20kHz. The waveforms of signals were reproduced by a photo recorder. The signals are digitalized by an A-D converter, and the frequency spectra of the signals are obtained by using a Fourier transform computer program. The damping coefficients K at each natural frequency of the plate were obtained by observing the variation of the level of the peak frequency at timed intervals in the spectrum of vibrational accelaration. An electrical circuit is completed when the ball contacts the plate; this is used to measure the contact duration. This method can only be applied to the steel ball's impact. 4. RESULTS AND CONSIDERATIONS 4.1 Experimental Results In order to investigate impact sound as a kind of noise, the frequency range of both the vibrational acceleration and the sound pressure is taken up to 20kHz. In the experiments, only circular nodal line modes of the plate could be generated because the impact point was at the center of the plate. According to the calculation of the natural frequencies of the plate, eleven circular nodal line modes are included in the frequency range up to 20kHz. Figure 4 shows waveforms of the sound pressure and vibrational acceleration when a steel ball (12.7mm in diameter) and a rubber ball (22mm in diameter) struck the center of the plate at a velocity of 120cm/s. In this figure the frequency spectra of vibrational acceleration are also shown in the frequency range up to 20kHz ((n, s): n shows the number of nodal diameters and s shows the number of circular nodal lines). There is little difference in mass between the two balls. When the steel ball strikes the plate, the frequency components contain the natural frequencies of all modes in the frequency range up to 20kHz, and the sound pressure level is much higher than that of the rubber ball. When the rubber ball strikes the plate, the frequency components consist of almost solely (0, 1) and (0, 2) mode. Figure 5 represents the relation between the peak sound pressure level and the momentum when different kinds of balls strike the plate. The peak sound pressure level is measured from the waveform reproduced by the photo recorder, and that level of the steel ball is a little lower than the level measured by an amplifier's (B& K 2606) peak hold meter. This phenomenon suggests that the frequency components contain frequencies higher than 20 khz in fact. But in this case we consider the frequency range of impact sound only up to 20kHz. Consequently, it is reasonable that we use the level measured from the waveform. From Fig. 5, it is found that even if balls have the same momentum, peak sound pressure levels differ according to materials, and for balls of the same material the less the mass of the ball, the higher the levels become. The peak sound pressure level is proportional to the momentum of the ball when the same ball in both size and material is used to strike the plate.

6 J. Acoust. Soc. Jpn. (E) 1, 2 (1980) Fig. 4 Waveforms and frequency spectra (experimental results). Fig. 5 Relation between peak sound pressure level and momentum. In general, the vibrational acceleration damps exponentially if the vibrational acceleration level decreases linearly with time. From the frequency spectra of the vibrational acceleration at timed Fig. 6 Relation between damping coefficient and frequency. intervals, it is found that the amplitude of the peak frequency decreases almost linearly with time. Thus, the damping coefficient at each natural frequency can be calculated. These results are plotted in Fig. 6. The relation between the damping 126

7 T. TAKAHAGI et al.: THE SOUND GENERATED BY A TRANSVERSE IMPACT coefficient and frequency can be approximated by the curve shown in Fig. 6 and we calculated the impact force and the sound pressure in the next section by using the damping coefficients obtained from this curve. 4.2 Calculation Results and Considerations The impact force histories Let us calculate the impact force histories by the small increment method under the same conditions as shown in Fig. 4. That is, a steel ball (12.7 mm in diameter) and a rubber ball (22mm in diameter) strike the plate at a speed of 120cm/s. In calculation ƒã is 10-3 and the small time increment t is ~T1 for a steel ball and ~T1 for a rubber ball. Here, T1 is the period of (0,1) mode of the plate, 6.23ms, and no nodal diameter modes are considered because the impact point is at the center of the circular plate. The decision whether or not it is permissible to neglect the effect of the higher modes of the vibration of the plate is important. Figure 7 shows the impact force histories of the steel ball calculated under consideration of several modes. It is found that the more the influence of high frequency vibrational modes is considered, the smaller the impact force becomes. But it may be said that the maximum value of impact force Pmax, the duration of the impact T, and its shape are scarcely effected by the higher modes over the 20th. For the transverse impact of a mass on a beam, Tobe and Kato3) indicated that the influence of high Figure 8 shows the calculated impact force for the rubber ball's impact. Here, impact force which was calculated above the 11th mode is not shown because there is no change in the shape of the impact force beyond the 11th mode. There is little difference in mass between two balls, but ƒ of the rubber ball is much lower than that of the steel ball. In comparison with Fig. 7, first, the steel ball's impact force history which is calculated by considering the vibrational modes up to the 20th shows a characteristic figure; it rises sharply and decreases gradually, contrary to the symmetric shape of the rubber ball's impact force. Secondly for the rubber ball, T is much longer, Pmax is much smaller, and the influence of the vibrational modes on T and Pmax seems to be minimal. Therefore we may say as follows. The impact force history of the rubber ball seems to be determined by contact deformation and to be hardly influenced by the modal vibration of the plate. Also, the damping coefficients as shown in Fig. 7 Impact force histories of the steel ball by the small increment method. Fig. 8 Impact force histories of the rubber ball by the small increment method.

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9 T. TAKAHAGI et al.: THE SOUND GENERATED BY A TRANSVERSE IMPACT lated by this procedure are almost similar to those calculated by considering the vibrational modes up to the 11th mode. Therefore, the vibrational modes up to the 11th circular nodal line mode and damping coefficients as shown in Fig. 6, are considered in actual calculations. In the case of the steel ball, theoretical waveforms almost agree with experimental waveforms. For the rubber ball, the initial peak rises higher than that of the experimental waveform and the waveform has a different tendency from the experimental waveform. Modes higher than the third circular nodal line mode are not generated in the frequency spectrum of Fig. 4. The waveform considered up to the third circular nodal line mode is shown in Fig. 10, and this almost agrees with the experimental waveform. From these results, it can be predicted that another mechanism restrains the modal vibration of high frequencies besides the impact duration and damping coefficients of the plate. This may be due to the damping of the ball itself. Figure 11 shows the calculated results of sound pressure waveforms without damping. For the steel ball, the sound pressure level becomes higher and the shape of the waveform changes if damping coefficients are not considered. However, for the rubber ball, the waveform which includes the modes to the third mode without damping has little difference in shape and magnitude from those with damping because the damping coefficients are small in low modes The effect of the initial velocity on both the impact force and the peak sound pressure level The next calculation was attempted to investigate the effect of the initial velocity v0 on the impact force history. For the steel ball, Pmax, T and rise time for reaching Pmax are shown for various initial velocities in Fig. 12 considering the vibrational modes up to the 20th circular nodal line mode and the damping coefficients. T measured by experiments are also shown and agree with theoretical results. From this figure, the following relations are shown approximately. For the rubber ball, Fig. 13 shows the calculated results of Pmax and T. But is not plotted because it has the same tendency as T. From this figure, we obtain the following results For purpose of comparison, the results of Pmax and T calculated by the Hertz law of contact are 129

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12 J. Acoust. Soc. Jpn. (E) 1, 2 (1980) 4) G. Nishimura and K. Takahashi, "Impact sound of steel ball collison," J. Soc. Precision Mech. Jpn. 28, (1962) (in Japanese). 5) L. L. Koss and R. J. Alfredson, "Transient sound radiated by spheres undergoing an ellastic collision," J. Sound Vib. 27, (1973). 6) M. Endo, N. Nishi, M. Nakagawa, and M. Sakata, "Impact sound of a ball striking a cylinder," Trans. Jpn. Soc. Mech. Eng. 44, (1978) (in Japanese). 7) M. Sakata, M. Nakagawa, and M. Endo, "Impact sound of a ball striking a circular plate," Trans. Jpn. Soc. Mech. Eng. 45, (1979) (in Japanese). 8) A. Akay "A Review of impact noise," J. Acoust. Soc. Am. 64, (1978). 9) Y. Tokita, "Vibration and sound radiation of a plate generated by an impulsive force (I)," J. Acoust. Soc. Jpn. 16, (1960) (in Japanese). 10) Y. Tokita, "Vibration and sound radiation of a plate generated by an impulsive force (II)," J. Acoust. Soc. Jpn. 17, (1961) (in Japanese). 11) A. E. H. Love, A Treaties on the Mathematical Theory of Elasticity, 4th ed. (Dover, New York, 1944), p ) M. Morse and U. Ingard, Theoretical Acoustics (McGraw Hill, New York, 1968), p.310.