Development of a diagnosis technique for failures of V-belts by a cross-spectrum method and a discriminant function approach

Transcription

1 Journal of Intelligent Manufacturing (1996) 7, Development of a diagnosis technique for failures of V-belts by a cross-spectrum method and a discriminant function approach HAJIME YAMASHINA, 1 SUSUMU OKUMURA 2 and ISAO KAWAI 3 1 Department of Precision Mechanics, Faculty of Engineering, Kyoto University, Kyoto 66-1, Japan 2 Department of Mechanical Systems Engineering, University of Shiga Prefecture, Hikone, Shiga 522, Japan 3 Japan Tobacco Inc., Kyoto 61, Japan Received January 1995 and accepted July 1995 This paper describes the development of a failure diagnosis technique for V-belts through vibration monitoring. The V-belt vibration is monitored at a driven bearing body attached to a power transmission device. Seven basic causes of belt failure and their combinations are considered. Power spectra of the vibration data are calculated through noise reduction by a cross-spectrum method. Six parameters characterizing the vibration data are extracted, and 16 typical combinations of the basic causes and a normal belt state are diagnosed successfully by a Bayes discriminant function approach. Two types of incorrect diagnosis are examined: Type I leaves a failed belt not repaired, and type II causes overmaintenance. A risk ratio for the Bayes discriminant function is determined to minimize the two types of incorrect diagnosis. Moreover, the risk ratio is determined to minimize type I error. Keywords: V-belt, failure diagnosis, vibration monitoring, cross-spectrum method, Bayes discriminant function 1. Introduction In order to put condition-based maintenance into practice it is necessary to have an effective technique for monitoring and diagnosing each machine or component. Power transmission devices with belt drives are widely utilized in industry because of the following advantages (Palmgren, 1986): (1) Speed ratio and shaft centre distance are optionally selected; (2) The elasticity of belts reduces shock and protects equipment against sudden overloads; (3) Acquisition cost of belt drives is low; (4) Belts and pulleys are well standardized and easily available. The power transmission capacity of a V-belt is superior to that of a flat belt, since the wedge effect arising from the shape of the V-belt helps the transmission of power. Therefore, power transmission devices with V-belts are # 1996 Chapman & Hall commonly utilized in various industries. Disturbances in a V-belt drive can induce errors in power transmission. The development of a technique for diagnosing failures for V-belts would be an important contribution to protecting V-belts against unexpected failures. The vibration characteristics of a V-belt have not been fully analysed for a number of reasons: (1) the elasticity of a V-belt is nonlinear; (2) the distribution of elastic modulus and linear density is not uniform along a V-belt; (3) vibration due to the collision of a V-belt with a pulley causes changes in centrifugal tensile stress; and (4) boundary conditions at the ends of a pulley change with time. The conventional maintenance method for V-belts is based on regular inspection by maintenance staff (Palmgren, 1986; Erickson, 1987), while only a few automated failure diagnosis methods have been developed. To develop an automated technique for diagnosing failures of V-belts we need to solve several problems: the method for monitoring conditions, the method for Article number = ji718

2 86 Yamashina et al. processing signals, and the diagnosis method itself. When a power transmission device utilizing V-belts is in operation, sound and vibration are produced. One may assume that these sounds and vibrations change in accordance with the conditions of the V-belts. However, the method utilizing sound levels is impractical in most workshops because the method is apt to be interfered with and influenced by environmental noises. To overcome these difficulties, vibration is therefore monitored by an accelerometer for the purpose of this paper. When vibration data are analysed a noise reduction is important in the frequency domain, as vibration data include random noise. There are three general noise reduction methods: (1) the average of vibration data additions in the time domain; (2) the average of power spectra additions in the frequency domain; (3) the average of cross-spectra additions in the frequency domain (Ono et al., 1988). Trigger signals are indispensable for method (1). V-belts are operated with a certain degree of slippage: therefore the trigger point varies during the operation of V-belts, and vibration data that are expected to be effective for failure diagnosis cannot easily be obtained. Methods (2) and (3) are suitable for noise reduction because these methods do not need trigger signals, and method (3) is more effective than method (2) as method (3) needs a smaller sample size than method (2). Method (3) is, therefore, selected as a noise reduction method. A Bayes discriminant function approach (Chi-hau, 1973) is one of the statistical pattern recognition methods for failure diagnosis. This approach decides the exact class to which an unknown pattern belongs after the a priori probability and the probability density function for each class are known. The minimization of the expected risk when incorrect diagnosis occurs is assured theoretically by this pattern recognition method. From the point of view described above, this paper addresses itself to establishing a diagnosis technique for failures of V-belts, which consists of collecting vibration data from a power transmission device that utilizes V-belts, analysing vibration data by the cross-spectrum method, and diagnosing by using discriminant functions. 2. Basic causes and failure modes Typical failure modes of V-belts can be classified into the following: (1) Total power transmission failure due to complete fracture of V-belts or the separation of V-belts from pulley grooves: this type of failure is caused by such circumstances as (a) foreign material embedment in V- belts; (b) transverse cracks at the bottom of V-belts; or (c) misalignment of pulleys. (2) Power transmission degradation due to excessive slippage or vibration of V-belts: this type of failure is caused by such conditions as (a) resonance due to decrease of V-belt tension; (b) excessive slippage due to the presence of oil and/or water on V-belts; (c) twisted V- belts; or (d) bent V-belts. The above seven basic causes of failure will now be considered. 3. Experimental apparatus and method A power transmission device utilizing a V-belt is configured with two parallel shafts, each with a pulley, and the pulleys are located in the same plane as shown in Fig. 1. A drive motor runs at 5 rev min ÿ1. The drive motor and a drive pulley can be moved perpendicular to the drive shaft. The distance between the two pulleys and the belt tension can therefore be adjusted. Back electromotive force is applied to a load motor. A driven shaft, driven bearings, a driven pulley, a coupling and a driven Bearing Pulley Drive motor V-belt Axis Pulley Coupling Load motor (a) (b) Fig. 1. Outline of experimental apparatus: (a) front view of drive and driven sides; (b) side view of driven side.

3 A diagnosis technique for failures of V-belts 87 (c) (b) (d) (f) Drive axis θ (a) (e) Driven axis Fig. 4. Misalignment of pulleys. Fig. 2. Vibration measurement points on a driven bearing. motor can be moved parallel to the driven shaft. This is how alignment of the driven pulley is achieved. Three types of classical V-belts are discussed in this paper: A-45, A-55 and A-65 (Japan JIS standards). Tension of a normal V-belt is 32 N. An accelerometer is attached vertically to one of six vibration monitoring points (a) (f), as shown in Fig. 2, which are the most realistic vibration monitoring positions for practical usage. Figure 3 shows a vibrationmonitoring system schematically. The vibration signal from an accelerometer is first amplified, then passes through a low-pass filter (2 Hz cut-off frequency), and is finally fed into a signal analyser (1.22 ms sampling interval, 5 s measurement time for obtaining one vibration datum) connected with a 16-bit personal computer and an X Y plotter. The seven types of basic failure causes described in section 2 are simulated in the experiment in the following ways. (1) Foreign material embedment in V-belts. Small objects such as metal pins seem to be the most probable foreign material in production systems. A machine screw of small diameter is therefore fastened at the bottom of V-belts or the side of V-belts while the degree of embedment is adjusted by changing the exposed length of the screw; (2) Transverse cracks at the bottom of V-belts. If such cracks are distributed along a V-belt at a distance of 5 6 cm, the V-belt is usually replaced. Cracks are therefore Amplifier Accelerometer Low-pass filter CPU Signal analyzer Interface X-Y plotter Fig. 3. Block diagram of the measurement and data processing system. made artificially with the interval and the depth of each crack varied; 1 4, 1 2 and 3 4 of the height of a V-belt; (3) Misalignment of pulleys. The driven shaft of the experimental apparatus is shifted, as shown in Fig. 4, with an angle ; (4) Resonance. Assume that the revolutions per minute of a drive motor are fixed; resonance arises owing to the decrease of the tension of a V-belt, which is caused by the expansion and/or wear of the belt. Tension of the V- belt is therefore decreased until resonance emerges; (5) Excessive slippage. This is simulated by placing lubricating oil on the surface of the V-belt and subjecting the belt to heavy loads. Slippage rate is defined as being when 1 (actual moving distance of V-belt per unit time)/(moving distance of a V-belt per unit time in cases where there is no slippage) is above 3%; (6) Twisted and bent belts. These kinds of V-belt are simulated by applying external force to the dismounted V-belts over a long period of time. 4. Vibration characteristics of V-belts 4.1. Vibration analysis by cross-spectrum method Power spectra of the vibration data from an accelerometer are calculated through the average of added cross-spectra in the frequency domain. A cross-spectrum is calculated by inputting two vibration data into a signal analyser with a 5 s lag, which is the time required for getting one vibration datum. It is confirmed experimentally that the effect of a noise reduction by the cross-spectrum method is remarkably high. However, the noise reduction effect is not substantially improved with more than 1 additions. Therefore, the number of additions adopted is Vibration characteristics of normal V-belt in operation The equation of motion of transverse vibrations of a V-belt under off-operation is written as follows: 2 y 4 y ˆ (1)

4 88 Yamashina et al. where x is the coordinate to specify the position of the V- belt, y is the transverse displacement, t is time, r is the volume density of V-belt material, EI is the modulus of flexible rigidity, and T is the tension of the V-belt. Assume that a V-belt is simply supported at both ends. The natural frequency of the kth mode is as follows: f k ˆ k 2l s T 1 k2 π 2 EI r l 2 T where l is a span. The natural frequency of the first mode of a V-belt, for example, an A-65 belt, is calculated as 4.5 Hz from Equation 2. Using u as the velocity of a moving V-belt, the equation of motion of transverse vibrations of a V-belt in operation is obtained by the (2) 2 y ÿ ˆ (3) Boundary conditions are assumed as 2 y y 2 ˆÿ1 (xˆ, l) (4) R where R is the radius of a pulley. The solution of Equation 3 is obtained as the sum of the spatial solution that satisfies Equation 4 and the solution that satisfies the following boundary 2 y y ˆ, ˆ (x ˆ, l) The spatial solution is written as follows: y ˆÿ l2 δ 2 cosh δx ÿ 1 ÿ tanh δ δx sinh (6) R l 2 l where s r T δ ˆ l EI r ÿ u2 Assuming the general solution to be we can obtain the following equation: 4 (u2 ÿ p 2 ) q 2 y(x, t) ˆ e x=l e t (7) 2 ÿ 2ul3 q 2 r s! T EI p ˆ, q ˆ R r 2 l 4 q 2 ˆ (8) If 1, 2, 3 and 4 are the solutions of Equation 8, the general solution can be rewritten as y(x, t) ˆ [C 1 e 1x=l C2 e 2x=l C3 e 3x=l C4 e 4x=l ]e t (9) where C 1, C 2, C 3, and C 4 are constants. If C C2 2 + C2 3 + C2 4, the following equation is obtained: e 1 e 2 e 3 e 4 ˆ (1) 2 1 e e e e 4 The unknown value in Equation 8 is determined from the above equation. The natural frequency is consequently obtained (Cyubachi, 1958; Sekiguchi and Sugawara, 1964). The natural frequency of the first mode of a V- belt, for example an A-65 belt, is 4.2 Hz from the calculation based on the above equations. Figure 5 shows an example of the power spectrum of a normal V-belt through the cross-spectrum method at the point (b) shown in Fig. 2. It is observed that the natural frequency of the first mode and its higher modes (integral multiple of 3.36 Hz) appear, and that rotating frequency and its higher modes appear. Tension force differs between the tight side and slack side in operation. Therefore, their natural frequencies are different. The spectrum shown in Fig. 5 is the result of overlapping natural frequencies in both the tight side and the slack side of the belt Vibration characteristics and method for detecting belt failures An impact is loaded when failures such as embedment of foreign material, transverse cracks at the bottom, twisted Amplitude db Natural frequency of first mode Fig. 5. An example of a spectrum of a V-belt in operation (belt: A-65).

5 A diagnosis technique for failures of V-belts 89 belts or bent belts occur. These failures cause collision of the belt with the pulleys. As the impact energizes free vibration in the belt, each amplitude of higher modes of rotating frequency increases. The amplitude increase of belts compared with that of normal belts is therefore proposed for detecting these four kinds of causes of belt failure. The amplitude increase is also considered for identifying resonance of V-belts. Rotating frequency of rolling elements and amplitude of vibrations of driven bearings are considered for identifying misalignment of pulleys and excessive slippage of V-belts Foreign material embedment in belts Figure 6 shows an example of the amplitude increase of higher modes of the rotating frequency of belts. The increase differs in accordance with the location, such as the side or the bottom of the belts, and/or the degree of embedment. The amplitude increase is most notably observed in the case of foreign material embedment, Amplitude increase db 1 2 Fig. 6. An example of the amplitude increase of higher rotating harmonics for a V-belt with foreign material embedment (belt: A-65). Amplitude increase db Fig. 7. An example of the amplitude increase of higher rotating harmonics for a V-belt with transverse cracks at the bottom (belt: A-65). compared with other failure causes such as transverse cracks at the bottom, resonance, twisted belts, and bent belts Cracks at the bottom of belts Figure 7 shows an example of the amplitude increase of higher rotating harmonics. As the degree of crack depth grows, the amplitude increase from 12 Hz to 2 Hz is extremely high because intervals between cracks are distributed from 5 cm to 6 cm Misalignment of pulleys The relationship between the misaligned angle and the pass amplitude of driven bearing balls is considered here. The stiffness between two rings of a bearing has a characteristic of nonlinearity, and is generally approximated by the following equation: K() ˆ k k 3 3 (11) where K() is the stiffness between two rings; is the displacement of a driven bearing in the direction of a drive shaft; and k, k 3 represent stiffness. Let β be the displacement in the direction of a driven shaft, and P be the force in the direction of a driven shaft. The following equation is obtained: P ˆ k β k 3 β 3 (12) Assume that the generation of the forced vibration is in the neighbourhood of = β, and the displacement in the direction of a driven shaft is very small. The stiffness is estimated as a linear characteristic. The equation of motion for the forced vibration is consequently given by m c _ k ˆ f sin!t (13) where m is the mass of the outer ring; K is the stiffness at = β; c is the damping coefficient; and f is a positive constant. The above equation is a linear, nonhomogeneous, second-order differential equation with constant coefficients. Assuming that the solution of Equation 14 is given in the form X sin (!t + ), the following equation is obtained: f X ˆ p (k ÿ m!2 ) 2 (c!) 2 (14) The relation between force P in the direction of a driven shaft and belt tension T is P ˆ 2T sin (15) From Equations 12, 14 and 15, the relation between the misalignment angle and the pass amplitude X of a driven bearing is obtained.

6 9 Yamashina et al. Figure 8 shows the experimental results superimposed on the results of numerical calculation between and X. It is observed that as angle increases the pass amplitude X of bearing balls decreases. It therefore seems reasonable that misalignment of a pulley can be detected by monitoring the change of the pass amplitude of bearing balls Resonance Resonance arises when the natural frequency of a belt in operation corresponds to an integer multiple of the rotating frequency of the belt. It is considered that the amplitude of the higher rotating harmonics that correspond to resonance frequency increases. The amplitude of resonance frequency and its double frequency increased sharply in the results of experiments (Fig. 9) Excessive slippage If the ratio of the rotating frequency to the rotating frequency of a drive shaft is considered, excessive slippage can be detected since the pass frequency of the driven bearing balls is proportional to the rotating frequency of the driven shaft Twisted belts and bent belts Figure 1 shows an example of the amplitude increase of the higher rotating harmonics concerning a twisted belt. The amplitude is not extreme because the impact caused by twisted belts is weaker than that caused by foreign material embedment. Similar experimental results are obtained for bent belts. The magnitude of the impact of twisted belts is, therefore, considered approximately as the Amplitude ratio X/X Misaligned angle : Actual : Computed deg. Fig. 8. Relationship between the misaligned angle of pulleys and the pass amplitude of bearing balls (belt: A-65). X = amplitude for a normal belt; X = amplitude for a misaligned belt. Amplitude increase db Resonance frequency 2xResonance frequency 1 same as that of bent belts. Similar experimental results from other kinds of V-belts (A-45 and A-55) are obtained. Rotational harmonics of belts were monitored at point (b) shown in Fig. 2. The pass amplitude of driven bearing balls was monitored at point (f). It became clear that random noise is effectively reduced and amplitude is maximized as a result of the extra experiments on vibration monitoring points. 5. Diagnosis by Bayes discriminant function 2 Fig. 9. An example of the amplitude increase of higher rotating harmonics for a V-belt with resonance (belt: A-65). Each cause of failure was treated separately in the previous section. In a real situation a V-belt failure can occur from a combination of causes. It is therefore important to consider a diagnosis method that can treat a combination of failure causes. As there are seven kinds of failure cause, the number of combinations including the normal state is 2 7 (= 128). On the other hand, maintenance action is conducted when V-belts fail. V-belts are replaced Amplitude increase db Fig. 1. An example of the amplitude increase of higher rotating harmonics for a twisted V-belt (belt: A-65).

7 A diagnosis technique for failures of V-belts 91 when failure causes such as foreign material embedment, transverse cracks at the bottom, twisted belts or bent belts occur. Belt tension is adjusted when resonance occurs. Pulleys are aligned properly when misalignment occurs. Causes of excessive slippage are eliminated when they occur. Combinations of failure causes that do not matter in practical applications are not considered: (1) Combinations of more than or equal to three failure causes: these types of combination are considered rare; (2) Combinations of failure causes, such as a combination of excessive slippage and resonance, that do not occur simultaneously; (3) Combinations that induce the following vibration characteristics: vibration characteristics due to one failure cause dominant vibration characteristics due to other failure causes when a failure due to multiple causes occurs. It is sufficient to detect the dominant failure cause for such a combination when maintenance actions are considered. For example, when a combination of foreign material embedment and cracks at the bottom occur simultaneously, the effect of foreign material embedment appears strongly on the power spectrum. From the point of maintenance carried out it is sufficient to detect foreign material embedment as if a single failure cause has occurred; (4) Combinations of failure causes that induce the following occasions: while one failure is being repaired the other failures can be detected. For example, while maintenance action for recovery of excessive slippage is conducted, failure causes such as foreign material embedment that needs belt replacement is easily detected. For the reasons mentioned above, the following 17 kinds of failure, including normal state of belts, are identified: (1) Normal V-belts; (2) Foreign material embedment in V-belts; (3) Cracks at the bottom of V-belts; (4) Misalignment of pulleys; (5) Resonance; (6) Excessive slippage; (7) Twisted belts; (8) Bent belts; (9) A combination of cracks at the bottom and resonance; (1) A combination of twisted belts and resonance; (11) A combination of bent belts and resonance; (12) A combination of foreign material embedment in V-belts and misalignment of pulleys; (13) A combination of cracks at the bottom and misalignment of pulleys; (14) A combination of resonance and misalignment of pulleys; (15) A combination of excessive slippage and misalignment of pulleys; (16) A combination of twisted belts and misalignment of pulleys; (17) A combination of bent V-belts and misalignment of pulleys. There are two maintenance actions: (1) V-belt failures are not repaired if belt tension is adjusted when belt replacement is the appropriate maintenance action; (2) V- belt failures are overmaintained if belt replacement is conducted when an adjustment of tension is the appropriate maintenance action. Therefore, two kinds of risk are considered. Type I risk, denoted by R 1, occurs when failures are not repaired; type II risk, denoted by R 2, occurs when failures are overmaintained. The risk ratio R 1 /R 2 is determined to minimize type I risk. The following risk ratios are chosen: 1, 5, 1, 1, 1, 5 The following six characteristic parameters are calculated: (1) The sum of the amplitude increase of higher rotating harmonics of belts. This parameter is effective for identifying foreign material embedment in V-belts, twisted belts and bent belts; (2) The sum of the amplitude increase of higher rotating harmonics of belts from 12 Hz to 2 Hz. This parameter is effective for identifying cracks at the bottom; (3) The pass amplitude of driven bearing balls. This parameter is effective for identifying misalignment of pulleys; (4) The sum of the amplitude increase at the frequencies f m,2f m, 1 2 f m, where f m is the frequency that maximizes the amplitude increase of higher rotating harmonics. This parameter is effective for identifying resonance; (5) The ratio of the rotating frequency of a drive shaft to the pass frequency of driven bearing balls. This parameter is effective for identifying excessive slippage; (6) Area of power spectrum at the point (b) shown in Fig. 2. This parameter takes into account vibration characteristics other than higher rotating harmonics. In total, eight instances are considered for each type of V-belt. The V-belt failures are simulated in the way described in Section 3. A combination of the parameter values for each instance after the standardization is called an input pattern. We thus have a total of eight input patterns for each type of belt. Four patterns are used for the learning phase of the Bayes discriminant function, and the remaining four patterns for the succeeding checking phase. Consequently, we have a total of 4 17 = 68 learning patterns and a total of 68 checking

8 92 Yamashina et al. patterns. Assuming that the a priori probabilities for each of the 17 kinds of V-belt condition are identically and normally distributed, the results of discrimination by Bayes approach are shown in Fig. 11. It is obvious that characteristic parameters considered in this paper are proper on the grounds that the rate of incorrect diagnosis to each V-belt is low when the risk ratio is 1. The low rate of incorrect diagnosis (I) is equivalent to the small number of failed belts not repaired. We need to increase the risk ratio to decrease type I failures. However, too great an increase of the risk ratio corresponds to an extreme increase of type II incorrect diagnosis: in other words, the cases when normal belts are diagnosed as failed belts. Therefore, when the minimum risk ratio that makes the ratio of type I incorrect diagnosis in the neighbourhood of is introduced, most failures can be repaired and needless maintenance actions are eliminated. From Fig. 11, if the risk ratio is taken as in the neighbourhood of 5 for V-belt A-45, 5 for V-belt A- 55, and 1 for V-belt A-65, we can make the ratio of type I incorrect diagnosis almost with the minimum ratio of type II incorrect diagnosis. 6. Conclusions A failure diagnosis technique for V-belts has been developed by introducing a combination of cross-spectrum method and a discriminant function approach to the vibration data monitored at the driven bearing body attached to a power transmission device. The following are considered as failure causes of V-belts: foreign material embedment in V-belts, transverse cracks at the bottom of V-belts, misalignment of pulleys, resonance of V-belts, excessive slippage of V-belts, twisted V-belts, bent V-belts, and combinations of the above. The following results have been obtained: (1) The cross-spectrum method does not require trigger signals, and does not need large numbers of average times for noise reduction. Therefore it is suitable for vibration analysis for V-belts. (2) When failures due to foreign material embedment in V-belts, cracks at the bottom, resonance, twisted belts or bent belts occur, the amplitude of higher rotating harmonics of V-belts increases compared with that of normal V-belts. Moreover, the degree of increase varies according to different failure causes. (3) As a result of investigating the relationships between misalignment of pulleys and pass amplitude of driven bearing balls, misaligned pulleys are detected as a decrease of the pass amplitude occurs. (4) Excessive slippage can be detected by the ratio change of the rotating frequency of a drive shaft to the pass frequency of driven bearing balls. The ratio change Ratio of incorrect diagnosis % Ratio of incorrect diagnosis % Ratio of incorrect diagnosis % Logarithm of loss ratio log (R 1 /R 2 ) (a) (I) (II) (I) (II) (I) (II) : Failed belt not repaired : Overmaintenance : Failed belt not repaired : Overmaintenance : Failed belt not repaired : Overmaintenance Logarithm of loss ratio log (R 1 /R 2 ) (b) Logarithm of loss ratio log (R 1 /R 2 ) (c) Fig. 11. Diagnosis performance profile with varying loss ratio: (a) A-45; (b) A-55; (c) A-65. is detected by monitoring the ratio of the rotating frequency of a drive shaft to the pass frequency of bearing balls. (5) Risks when incorrect diagnosis occurs are divided into two types: the risk when failures are not repaired, and the risk when failures are overmaintained. Under this

9 A diagnosis technique for failures of V-belts 93 classification of risks, failure diagnosis is conducted by a Bayes discriminant function approach with six types of parameter that characterize vibration data. The appropriateness of the characteristic parameters and the effectiveness of a discriminant function approach are clarified. It is verified that the ratio of two types of risk is determined to repair most failures of V-belts and eliminate needless maintenance actions. References Chi-hau, C. (1973) Statistical Pattern Recognition, Hayden Book Co., Rochelle Park, NJ. Cyubachi, T. (1958) Lateral vibration of an axially moving wire or belt form material (4th Report on Vibration Characteristics of Less Elastic Materials). Transactions of the Japan Society of Mechanical Engineering (Part 1), 24, Erickson, W. D. (1987) Belt Selection and Application for Engineers, Marcel Dekker Inc., New York. Ono, T., Katayama, H. and Suzuki, H. (1988) Detection of frequencies and power spectra of a periodic signal buried in noise by a cross-spectrum method. Journal of Acoustical Society of Japan, 44, Palmgren, H. (1986) The V-belt Handbook, Hans Palmgren and Studentlitteratur, Lund. Sekiguchi, H. and Sugawara, K. (1964) Vibration of belt (1st Report, Approximation of Frequency). Transactions of the Japan Society of Mechanical Engineering, 3,