There are many different methods that can be applied to calculate damping. They can be divided into three groups:
• Vibration decay measurements (time domain)
• Bandwidth determination of measured modal resonances (frequency domain)
This study does not deal with the steady-state technique which is based on the energy balance of vibrationally excited structures.
The complete dynamic behaviour of a structure can be described by its individual modes (harmonics) of vibration. The so-called modal parameters characterize the modes in a given frequency range (Brüel & Kjær, 1994). The modal parameters are:
• The resonance or modal frequency
• The damping of the resonance – the modal damping
• The mode shape
Damping can be described by many factors which are all interrelated; for example, loss factor, damping ratio, quality factor, reverberation time and logarithmic decrement.
Quality Factor method
The quality factor or Q factor is an indicator of the "quality" of a resonant system. Resonant systems respond to frequencies which are close to their natural frequency much more strongly than they respond to other frequencies. The Q factor measures the sharpness of the resonance peak, as displayed in a frequency response function (FRF) spectrum or by fast fourier transform (FFT), and therefore this method belongs to the frequency domain category. Systems with a high Q factor resonate with greater amplitude at the resonant frequency compared with systems with a low Q factor. Damping within the system decreases the Q factor.
The Q factor is defined as the resonant frequency (frequency of the peak) f0 or fc
divided by the bandwidth between the half power point values. When the Y axis is converted to a decibel scale, the half power points are located on either side of the peak where the energy is lower by 3 dB (Figure 2-4). Decibel is the standard unit in acoustics for measuring the level or level difference. The decibel scale is based on the
ratio 10 which takes account of the fact that the perception of loudness of the human ear is roughly logarithmic.
The Q factor can be calculated by the following equation:
f f f f f Q c c ∆ = − = 1 2 where;
Q = Q factor, f2 = the upper, f1 = lower half-power frequency, ∆f = f2 - f1 and fc =
frequency of the peak
The Q factor is related to the damping ratio by the following equation: Q 2 1 = ξ where;
Q = Q factor and ξ= damping ratio
Figure 2-4: Bandwidth of a resonant peak.
The accuracy of the frequency measurements is determined by the frequency resolution of the measurement. The accuracy can be increased by reducing the
Frequency (Hz) Ac ce ler at ion (m /s 2 ) dB sc al e Half power points
frequency range of the baseband, or making a zoom measurement around the frequency range of interest.
Whenever the Q factor was used in this study, it was given by the auto damping function of the “PULSE” unit which is based on the fundamental frequency. This automated function is very useful especially in terms of time efficiency. However, it is also possible to calculate the Q factor on subsequent harmonics. In this study calculating the damping based on the fundamental frequency was chosen. The reason for this is twofold. Firstly the fundamental frequency is usually the strongest mode in the FRF spectrum and secondly, higher frequencies receive higher damping which might affect the accuracy of the frequency measurement of subsequent harmonics.
Logarithmic Decrement
The logarithmic decrement provides a variable which tells how quickly the motion decays. It therefore belongs to the time domain measurements. It is important to point out that the logarithmic decay curve displays the whole frequency spectrum; therefore logarithmic decrement measurements give accurate results when only one resonance is present (Brüel & Kjær, 1994). Figure 2-5 shows a typical decay curve (of a homogeneous cylinder of plastic) where the time is plotted against acceleration. Logarithmic decrement can be calculated by the following equation:
n e
X
X
n
log
11
=
δ
where;δ = logarithmic decrement and X1 and Xn are two amplitudes n cycles apart.
Logarithmic decrement is related to the damping ratio (ξ) by the following relationship:
π
δ
Figure 2-5: The logarithmic decay in amplitude (of accelereation) of the stress wave with time of a plastic cylinder as measured by “PULSE”.
Impulse Response method
The impulse response method in its simplest form is used by woodmen to judge the quality of a tree trunk by ear. When the trunk is hit with a hammer (impulse) the vibrational response is detectable by the human ear owing to the low resonance frequencies of a log (< 20 kHz).
For accurate calculations, the specimen is excited by a short pulse (for example a hammer blow) and the impulse response is detected by means of a vibration sensor, most often by an accelerometer as in this study. Essentially, this method calculates the logarithmic decrement by isolating the signal to the fundamental frequency (or subsequent harmonics) and converting the signal to a logarithmic dB scale where the decay can be observed as a straight line (Figure 2-6). The damping can then be extracted from the measured slope of the decay curve.
Figure 2-6 Impulse response of an isolated mode displayed on a logarithmic scale.
The decay rate σ for the isolated mode is related to the time constant τ by
(Brüel&Kjær, 1999):
σ τ = 1
The decay corresponding to time constant τis given by the factor e–1, or in dB:
-20loge= –8.7 dB.
The damping ratio ξ is related to the decay rate by: f f τ π π σ ξ 2 1 2 = =
Reverberation Time method
Reverberation time measurements are usually used to assess the amount of sound absorption in a room. By using Schroeder’s elegant “Method of Integrated Impulse Response” the impulse response is first squared and then integrated backwards to obtain the energy decay curve from which the reverberation time can then be calculated. However, Ouis (2000 and 2002) applied this method successfully to detect
decay and knots in logs and timber. The reverberation time T60 is the time that is required for the energy of the system to drop to one millionth of its initial value, or alternatively, the time within 60dB decay in energy or amplitude can be observed. The following equation can be used to calculate the loss factor based on reverberation time (Cremer and Heckl, 1988):
60 60 6 2.2 2 10 ln fT fT ≈ = π η