Engineering Acoustics. Malcolm J. Crocker

H₁₂(f) is a complex quantity because it will have amplitude and phase. In addition, the coherence function (also called coherency squared) between the input and output may be calculated, defined by [12, 13]

      (1.21)

      The coherence function varies between 0 and 1. If the coherence is 0 then the input and output are completely random with respect to each other. On the contrary, if the coherence is one, all the power of the output signal is due to the input signal, indicating a completely linearly dependence between the two signals. In cases where there are multiple inputs and a single output (e.g. several cylinders [inputs] on a diesel engine), but one microphone position (output), the situation becomes more complicated. However, in such cases the coherence function may be used to estimate the contribution to the output from each input. Main applications of the coherence function are in checking the validity of frequency response measurements and the calculation of the signal, S, to noise, N, ratio as a function of frequency [13]

      (1.22)

      Measurement of the coherence function, transfer function, cross‐ and auto‐power spectral densities have been successfully used to identify sources and predict noise levels in machinery such as diesel engines, punch presses, and other noise problems. The necessary theory is dealt with in detail in references [1, 2, 12] and in the application manuals supplied by the instrument manufacturers.

The FFT analyzer can also be used as a real‐time analyzer up to a certain frequency limit (which depends on the accuracy desired). CPB (such as one-octave and one‐third octave band) analysis can also be approximately performed from FFT spectra, as illustrated in Figure 1.13, where each decade is converted separately. The bandwidth of the individual lines in the original FFT spectra must be less than the percentage bandwidth being converted to at the lowest frequency in the FFT band. The conversion is achieved by calculating the lower and upper cutoff frequencies of each constant percentage band, and then integrating up the power in the FFT lines (and parts of lines) between the limits [21]. Care must be taken at low frequencies since just a few frequency samples may be available to calculate the band value.

Figure 1.13 Conversion from FFT spectra to a constant percentage bandwidth (CPB) spectrum [21].

With the development of and decreasing prices of computer technologies, traditional FFT‐based instruments are becoming replaced by computer‐based instruments because of the processing, interfacing, and networking power of computers. Thus, digital signal processors can be replaced by specific software running on a personal computer. Computation speed reduction is not a problem because normally, calculations are made faster than they can be displayed. For data acquisition, since computers have hard drives of huge data storage capabilities, computers have advantages over traditional data logging equipment [20]. In addition, users can now develop their own programs to design instruments for noise and vibration signal analysis.

References

      1 1 Piersol, A.G. and Bendat, J.S. (2010). Random Data: Analysis and Measurement Procedures, 4e. Hoboken, NJ: Wiley.

      2 2 Piersol, A.G. and Bendat, J.S. (1993). Engineering Applications of Correlation and Spectral Analysis, 2e. New York: Wiley.

      3 3 Oppenheim, A.V., Willsky, A.S., and Hamid, S. (1997). Signals and Systems, 2e. Harlow: Pearson.

      4 4 Bracewell, R.N. (1999). The Fourier Transform and Its Applications, 3e. New York: McGraw‐Hill.

      5 5 Tohyama, M. and Koike, T. (1998). Fundamentals of Acoustic Signal Processing. London: Academic Press.

      6 6 Hansen, E.W. (2014). Fourier Transforms: Principles and Applications. New York: Wiley.

      7 7 Newland, D.E. (2005). An Introduction to Random Vibrations, Spectral & Wavelet Analysis, 3e. Mineola, NY: Dover.

      8 8 Lathi, B.P. and Ding, Z. (2009). Modern Digital and Analog Communication Systems, 4e. Oxford: Oxford University Press.

      9 9 Magrab, E.B. and Blomquist, D.S. (1971). Measurement of Time‐Varying Phenomena: Fundamentals and Applications. New York: Interscience.

      10 10 Herlufsen, H., Gade, S., and Zaveri, H.K. (2007). Analyzers and signal generators. In: Handbook of Noise and Vibration Control (ed. M.J. Crocker), 470–485. New York: Wiley.

      11 11 Fourier, J. (1822). Théorie analytique de la chaleur. Paris: Firmin Didot Père et Fils.

      12 12 Randall, R.B. (1987). Frequency Analysis, 3e. Naerum, Denmark: Bruel & Kjaer.

      13 13 Piersol, A.G. (2007). Signal Analysis. In: Handbook of Noise and Vibration Control (ed. M.J. Crocker), 493–500. New York: Wiley.

      14 14 Oppenheim, A.V. and Schafer, R.W. (2009). Discrete‐Time Signal Processing, 3e. Upper Saddle River, NJ: Prentice‐Hall.

      15 15 ISO R 266:1997 (1997) Acoustics – Preferred Frequencies. Geneva: International Standards Organization.

      16 16 IEC 1260:1995‐07 (1995) Electroacoustics – Octave‐band and Fractional‐octave‐band Filters, Class 1. Geneva: International Electrotechnical Commission.

      17 17 ANSI S1.11‐2004 (2004) Specification for Octave‐band and Fractional‐octave‐band Analog and Digital Filters, Class 1. New York: American National Standards Institute.

      18 18 Cooley, J.W. and Tukey, J.W. (1965). An algorithm for the machine computation of the complex Fourier series. Math. Comput. 19 (90): 297–301.

      19 19 Duhamel, P. and Vetterli, M. (1990). Fast Fourier Transforms: a tutorial review and a state of the art. Signal Process. 19: 259–299.

      20 20 Li, Z. and Crocker, M.J. (2007). Equipment for data acquisition. In: Handbook of Noise and Vibration Control (ed. M.J. Crocker), 486–492. New York: Wiley.

      21 21 Randall, R.B. (2007). Noise and vibration data analysis. In: Handbook of Noise and Vibration Control (ed. M.J. Crocker), 549–564. New York: Wiley.

2 Vibration of Simple and Continuous Systems

      2.1 Introduction

      The vibrations in machines and structures result in oscillatory motion that propagates in air and/or water and that is known as sound. The simplest type of oscillation in vibration and sound phenomena is known as simple harmonic motion, which can be shown to be sinusoidal in time. Simple harmonic motion is of academic interest because it is easy to treat and manipulate mathematically; but it is also of practical interest. Most musical instruments make tones that are approximately periodic and simple harmonic in nature. Some machines (such as electric motors, fans, gears, etc.) vibrate and make sounds that have pure tone components. Musical instruments and machines normally produce several pure tones simultaneously. The simplest vibration to analyze is that of a mass–spring–damper system. This elementary system is a useful model for the study of many simple vibration problems. In this chapter we will discuss some simple theory that is useful in the control of noise and vibration. For more extensive discussions on sound and vibration fundamentals, the reader is referred to more detailed treatments available in several books [1–7]. We start off by discussing simple harmonic motion. This is because very often oscillatory motion, whether it be the vibration of a body or the propagation of a sound wave, is like this idealized case. Next, we introduce the ideas of period, frequency, phase, displacement, velocity, and acceleration. Then we study free and forced vibration of a simple mass–spring system and the influence of damping forces on the system. In