Issue 
Mechanics & Industry
Volume 21, Number 1, 2020



Article Number  103  
Number of page(s)  6  
DOI  https://doi.org/10.1051/meca/2019061  
Published online  07 January 2020 
Regular Article
An efficient method for estimating the damping ratio of a vibration isolation system
^{1}
School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu 611731, PR China
^{2}
Department of Mechanical Engineering, Tsinghua University, Beijing 100084, PR China
^{*} email: xudf@mails.tsinghua.edu.cn
Received:
17
October
2018
Accepted:
16
July
2019
As the damping ratio determines the response of a vibration isolation system at resonance, it is very necessary to estimate the damping ratio quickly and economically for an evaluation of the effectiveness to adjust the damping in practical engineering applications. An efficient method named the “ζT_{r}” method with the characteristics of simple operation and a high accuracy is introduced to estimate the damping ratio in this paper. According to the transmissibility curve, the specific mathematical relationship in which the value of the resonance peak corresponds to the value of the damping ratio is analysed theoretically. In this case, the recognition of the resonance peak can be used to directly estimate the damping ratio without an approximation or simplification. The “ζT_{r}” method is faster, more accurate and less costly than other estimation methods. Finally, the correctness of the “ζT_{r}” method is verified by a simulation and an experiment.
Key words: Vibration isolation / damping ratio / estimation / resonance peak / “ζT_{r}” method
© AFM, EDP Sciences 2019
1 Introduction
Vibration isolation systems developed to prevent the transfer of vibrations are necessary for implementation of building structures, ultraprecision manufacturing and measurement, aerospace science, vehicle engineering and other fields [1–3]. In general, the harmful vibration energy imported by the environment or ground is mainly consumed by viscous damping in these systems. To simplify the theoretical analysis and numerical calculation, the damping model can similarly be considered in linear viscous form. The damping ratio is one of the key parameters in this linear model, and the estimation of the damping ratio is very important for accurately predicting the response of a vibration isolation system. The estimation methods are commonly known as the halfpower bandwidth method and the timedomain attenuation method [4,5].
The procedure of the halfpower bandwidth method [6,7] is shown in Figure 1. The damping ratio can be estimated by locating the halfpower points and calculating the halfpower bandwidth based on the measured transmissibility curve. This method is widely used because of its simplicity, but it may be influenced by many factors, such as the sampling frequency and the frequency resolution [8–10]. Moreover, this is a simplified and approximate method, so there are some errors between the estimated and actual results.
The procedure of the timedomain attenuation method [11,12] is shown in Figure 2. The damping ratio can be estimated by calculating the average attenuation rate based on measuring the timedomain attenuation curve by adding a singlefrequency vibration excitation. The estimated results are very accurate, but the operational process is complex. In particular, the calculation of the average attenuation rate often needs to fit the envelope curve of the timedomain attenuation curve.
After measuring the transmissibility curve or timedomain attenuation curve, the above two methods still require further analysis or calculations before estimating the damping ratio. The halfpower bandwidth method must locate the halfpower points and calculate the halfpower bandwidth shown in the “Step 3” in Figure 1. The timedomain attenuation method must calculate the average attenuation rate shown in the “Step 3” in Figure 2. During the course of the product design, product development, product installation and testing of any vibration isolators or isolation system, the damping ratio needs to be estimated rapidly multiple times to verify whether the design value meets the actual requirement. Furthermore, these further operations such as “Step 3” in the above two methods require necessary professional knowledge or skills, and ordinary employees may not complete these tasks. Thus, more money must be spent in employing senior employees, and more time must be taken to finish the estimation of the damping ratio.
In this paper, an efficient method for estimating the damping ratio is proposed by analysing the relationship between the resonance peak of the transmissibility curve and the damping ratio. Compared with the above two methods, the proposed method only needs the transmissibility curve, and the damping ratio can be estimated directly depending on the value of the resonance peak without extra operations like “Step 3” in Figure 1 or 2.
Fig. 1
Procedure of the halfpower bandwidth method. 
Fig. 2
Procedure of the timedomain attenuation method. 
2 “ζ  T _{r}” method for estimating the damping ratio
Typical passive vibration isolation systems can be reduced to a massspringdamper system as depicted in Figure 3, where m, k and c are the system mass, the system stiffness and the system damping, respectively.
The transmissibility curve, which is defined as the ratio of the output and the input, is often used to measure the performance of a massspringdamper system in the frequency domain. Through dynamic analysis, the motion equation of a passive vibration isolation system is derived, and then the transmissibility curve T(g) can be expressed as [4,13](1)and(2) where ζ, ω, ω_{n} and g are the damping ratio, forcing frequency, natural frequency and frequency ratio, respectively.
The variation in the transmissibility T(g) with the frequency ratio g is shown in Figure 4. It is obvious that T(g) is a convex function. Set the first derivative of T(g) to zero:(3)
Substitute equation (1) into equation (3) and further solve equation (3). The resonance frequency ratio g_{r} shown in Figure 4 can be obtained:(4)
Substituting equation (4) into equation (1), the resonance peak T_{r} shown in Figure 4 can be obtained:(5)
As shown in equation (5), the function f expresses a onetoone relationship between the resonance peak T_{r} and the damping ratio ζ. Conversely, if we can obtain the value of the resonance peak T_{r} according to the measured transmissibility curve or the frequency response curve, the damping ratio ζ can be expressed as(6)where the function f ^{−1} is the inverse function of f. Note that T_{r} and ζ in equation (6) have no units. If T_{r} recognized from the transmissibility curve has units of “dB”, a simple transformation must be made before calculating the damping ratio ζ by substituting the resonance peak T_{r} into equation (6).
Furthermore, the variation in the damping ratio ζ with the resonance peak T_{r} is shown in Figure 5. Then, the damping ratio ζ can be estimated by a mathematical calculation method with equation (6) or diagramming method with Figure 5. These two methods are named the “ζT_{r}” method for short in the following parts. In this case, the “ζT_{r}” method can be directly used to estimate the damping ratio ζ based on the value of the resonance peak T_{r}, which can be obtained easily from the measured transmissibility curve. The procedure of the “ζT_{r}” method is shown in Figure 6, and the damping ratio can be estimated in only three steps. Both the halfpower bandwidth method and the timedomain attenuation method require four steps and further work, such as locating the halfpower points or calculating the average attenuation rate. Therefore, the “ζT_{r}” method is more efficient and has great importance in engineering practice for the estimation of the damping ratio ζ.
Fig. 3
Massspringdamper system. 
Fig. 4
Variation in T(g) with g. 
Fig. 5
Variation in ζ with T_{r}. 
Fig. 6
Procedure of the “ζT_{r}” method. 
3 Comparison with the halfpower bandwidth method
3.1 Simulation analysis
Through the above analysis, the estimation of the damping ratio using the “ζT_{r}” method is deduced directly without an approximation or a simplification and the estimation results have less error. To clarify this point, we perform further simulation analysis and a comparison with the halfpower bandwidth method, as shown below.
Without loss of generality, we assume that the natural frequency ω_{n} is 1 Hz. Then, we use MATLAB software to simulate the variation in the transmissibility T with the forcing frequency ω in the cases of a damping ratio of ζ = 0.01, 0.05, 0.1, 0.2 or 0.3, as shown in Figure 7. Because the natural frequency ω_{n} is very low, the sampling frequency is just set to 10 Hz. Finally, based on the simulation results of the transmissibility curve, different damping ratios can be calculated by the halfpower bandwidth method [8,9] and the proposed method using the “ζT_{r}” method. By comparing the calculated results with the standard values of the different damping ratios, the errors of these two different methods can be analysed. To analyse whether the frequency resolution affects the accuracy of the estimation of the damping ratio, we choose four different values of the frequency resolution: 10^{−4 }Hz, 10^{−3 }Hz, 10^{−2 }Hz, and 10^{−1 }Hz.
The calculation of the damping ratio ζ by the halfpower bandwidth method is shown in Table 1. While the frequency resolution is 10^{−4 }Hz or 10^{−3 }Hz, the calculation errors are small, within 6.83%. However, while the frequency resolution is 10^{−2 }Hz or 10^{−1 }Hz, some of the calculation errors are even 200% or 25%, which can be unacceptable. In some cases, the halfpower bandwidth may be nonexistent so that the damping ratio is incalculable. This also demonstrates that this method can be influenced by the frequency resolution and that the estimation results are incredible in some conditions.
The calculation of the damping ratio ζ by the “ζT_{r}” method is shown in Table 2. While the frequency resolution is 10^{−4 }Hz, 10^{−3 }Hz or 10^{−2 }Hz, the calculation errors are very small. While the frequency resolution is 10^{−1 }Hz, the calculation errors are still less than 2%. On one hand, the calculation errors of the proposed method are all less than those of the halfpower bandwidth method. On the other hand, regardless of how the frequency resolution or the damping ratio changes, we can also complete the calculation of the damping ratio.
The damping ratio ζ is set to 0.01, 0.05, 0.1, 0.2 and 0.3 because the halfpower bandwidth method can only be applied to the situation in which the damping ratio ζ must be less than 1/2. However, the “ζT_{r}” method has no limit to the value of the damping ratio ζ. Therefore, the “ζT_{r}” method not only reduces the estimation error of the damping ratio ζ but also has a wider range of applications.
Fig. 7
Variation in T with ω for different damping ratios ζ. 
Calculation of the damping ratio ζ by the halfpower bandwidth method.
Calculation of the damping ratio ζ by the “ζT_{r}” method.
3.2 Experimental analysis
The experimental platform shown in Figure 8 is a passive vibration isolation system consisting of six air spring isolators. The testing devices are an ultralowfrequency vibration pickup sensor 991B produced by Institute of Engineering Mechanics, China Earthquake Administration and a data acquisition instrument INV3060T produced by China Orient Institute of Noise & Vibration.
First, we beat on the load with a rubber hammer, and a timedomain attenuation curve in the vertical direction can be obtained. The measured curve for 5 s is shown in Figure 9, and the values of consecutive peaks are shown in Table 3. The average attenuation rate can be calculated:(7)
Then, the damping ratio of the isolation system in the vertical direction can be estimated using equation (8) [11].(8)
The above timedomain attenuation method is a very accurate approach. Therefore, the estimation result ζ_{0} can be regarded as the standard value of the damping ratio.
Finally, the measured transmissibility curve in the vertical direction is shown in Figure 10. The resonance peak T_{r} is 13.58 dB, and then we can estimate the damping ratio ζ_{1} and ζ_{2} by the halfpower bandwidth method [5,6] and the new method using equation (6), respectively. These three estimation results of the damping ratio are shown in Table 4. It is obvious that the error of ζ_{2} is significantly less than the error of ζ_{1}.
Fig. 8
Experiment platform supported by six air spring isolators. 
Fig. 9
Measured timedomain attenuation curve for 5 s. 
Values of 5 consecutive peaks.
Fig. 10
Measured transmissibility curve in the vertical direction. 
Three estimated values of the damping ratio.
4 Conclusions
This paper has presented an efficient approach called the “ζT_{r}” method to estimate the damping ratio of a vibration isolation system whose damping model is linear. According to an analysis of the relationship between the resonance peak of the transmissibility curve and the damping ratio, a mathematical expression for the onetoone correspondence between the resonance peak of the transmissibility curve and the damping ratio can be obtained. Then, the damping ratio can be estimated on the basis of the recognition of the resonance peak from the measured transmissibility curve. In contrast to other methods, the “ζT_{r}” method has little approximations, simplifications or limitations and requires fewer steps. Consequently, the estimation results are more accurate, and the “ζT_{r}” method is more efficient and easier to use in engineering applications.
Acknowledgments
This work was supported by Major Scientific and Technological Project of China National Machinery Industry Corporation Ltd (SINOMASTZDZX201705), National Science and Technology Major Project (2013ZX02104003) and the Natural Science Foundation of Hubei Province (2018CFC889).
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Cite this article as: Q. Yu, D. Xu, Y. Zhu, G. Guan, An efficient method for estimating the damping ratio of a vibration isolation system, Mechanics & Industry 21, 103 (2020)
All Tables
All Figures
Fig. 1
Procedure of the halfpower bandwidth method. 

In the text 
Fig. 2
Procedure of the timedomain attenuation method. 

In the text 
Fig. 3
Massspringdamper system. 

In the text 
Fig. 4
Variation in T(g) with g. 

In the text 
Fig. 5
Variation in ζ with T_{r}. 

In the text 
Fig. 6
Procedure of the “ζT_{r}” method. 

In the text 
Fig. 7
Variation in T with ω for different damping ratios ζ. 

In the text 
Fig. 8
Experiment platform supported by six air spring isolators. 

In the text 
Fig. 9
Measured timedomain attenuation curve for 5 s. 

In the text 
Fig. 10
Measured transmissibility curve in the vertical direction. 

In the text 
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