A new approach based on Gauss-Newton iteration is proposed to estimate modal damping. Noise resistance of the proposed method and half-power bandwidth method are analyzed and compared by plenty of simulations with different signal-to-noise ratios (SNR). The proposed method is more accurate and stable than half-power bandwidth method in all SNRs, especially when the noise level is high. If SNR ≤ 30 dB, the proposed method should be used for damping estimation instead of half-power bandwidth method. A damping estimation experiment is carried out with both methods, and the results indicate and verify that there is smaller variability for the proposed method

Jintao Gu

Meiping Sheng

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 11 
A method for modal loss factor estimation based on 
Gauss-Newton iteration 
Jintao Gu1, Meiping Sheng2 
1First Aircraft Institute, Xi’an, 710089, China 
2School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, 710072, China 
1Corresponding author 
E-mail: 1gujintao@yeah.net, 2smp@nwpu.edu.cn 
(Received 27 July 2016; accepted 29 July 2016) 
Abstract. A new approach based on Gauss-Newton iteration is proposed to estimate modal 
damping. Noise resistance of the proposed method and half-power bandwidth method are analyzed 
and compared by plenty of simulations with different signal-to-noise ratios (SNR). The proposed 
method is more accurate and stable than half-power bandwidth method in all SNRs, especially 
when the noise level is high. If ܴܵܰ ≤ 30 dB, the proposed method should be used for damping 
estimation instead of half-power bandwidth method. A damping estimation experiment is carried 
out with both methods, and the results indicate and verify that there is smaller variability for the 
proposed method. 
Keywords: Gauss-Newton iteration, damping estimation, half-power bandwidth. 
1. Introduction 
Since damping is a valuable parameter for resonant response of structures or systems, damping 
is of great significance for structural dynamics. It is difficult to obtain damping through a 
theoretical method so that the universal way for damping estimates is experimental investigation 
[1, 2]. Half-power bandwidth method [3, 4], which calculates resonant damping with frequency 
bandwidth for vibration energy decreases 3dB and the resonant frequency, is a widely used 
approach for damping estimation. Half-power bandwidth method is introduced as main method 
for damping estimates in the American test standard ASTM E756-05. Bertha [5] obtained damping 
of a system that does not possess real modes with half-power bandwidth method. Guo [6] 
improved the half-power bandwidth method and proposed a new method based on integral  
opinion. Badsar [7] determined the material damping ratio in shallow soil layers with the 
half-power bandwidth method. 
As the half-power bandwidth method uses only three data points, its accuracy will be affected 
by the signal-to-noise ratio (SNR). When some of the three data points are seriously affected by 
noise, a large error will appear and cannot be neglected. The aim of this work is to put forward a 
new method which has high accuracy and strong noise resistance by using the amplitude of 
frequency response. 
2. Half-power bandwidth method 
A widely used method, namely half-power bandwidth method, is introduced as follows. The 
damping loss factor can be obtained by the quotient of half-power bandwidth and the resonant 
frequency, as shown in Eq. (1): 
ߟ = Δ݂
଴݂
. (1)
The diagrammatic sketch of the method is shown in Fig. 1. 
The half-power bandwidth method is a classical and widely used method, but there is a fatal 
short coming that if the signal-to-noise ratio (SNR) is not high enough. It is difficult to recognize 
the half-power bandwidth, seen in Fig. 2. 
A METHOD FOR MODAL LOSS FACTOR ESTIMATION BASED ON GAUSS-NEWTON ITERATION.  
JINTAO GU, MEIPING SHENG 
12 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
It can be seen in Fig. 2 that the half-power bandwidth is hard to decide because there are more 
than two frequencies related to half-power response. 
 
Fig. 1. Half-power bandwidth method 
 
Fig. 2. Half-power bandwidth method with low SNR 
3. Damping estimation based on Gauss-Newton iteration 
The parameter of damping is often estimated with data near the resonant frequency. The widely 
used half-power bandwidth method for damping investigation only uses three data points of 
frequency response so that it will cause a large error when the experimental data is affected by 
noise. It is expected to obtain a more accurate damping if more data points of frequency function 
are used. Therefore, a new approach based on Gauss-Newton iteration is proposed in which a 
numbers of data points near resonant frequency are used. 
The amplitude of frequency function is: 
|ܪ| = ܨ ܭ⁄
ඨ൬1 − ߱ଶ߱଴ଶ൰
ଶ
+ ߟଶ
݀,
(2)
where ܨ is excitation force, ܭ the stiffness, ߱଴ natural circle frequency, and ߟ the loss factor.  
It is obvious that the maximum amplitude can be obtained: 
|ܪ|୫ୟ୶ =
ܨ
ܭߟ. (3)
The dimensionless amplitude is defined as: 
|ܪഥ| = |ܪ||ܪ|୫ୟ୶ =
ߟ
ඨ൬1 − ߱ଶ߱଴ଶ൰
ଶ
+ ߟଶ
.
(4)
Rewrite Eq. (4) as: 
ݕ = ݂ሺݔ, ߟሻ = |ܪ| = ߟඥሺ1 − ݔሻଶ + ߟଶ, (5)
where: 
0.95 1 1.050.5
0.6
0.7
0.8
0.9
1
Dimensionless frequency
Di
me
ns
ion
les
s a
mp
litu
de
A METHOD FOR MODAL LOSS FACTOR ESTIMATION BASED ON GAUSS-NEWTON ITERATION.  
JINTAO GU, MEIPING SHENG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 13 
ݔ = ߱ଶ ߱଴ଶ⁄ . (6)
ݕ is dimensionless parameters and represent placement. 
A number of data points ሺ ௜ܺ, ௜ܻሻ (݅ = 1, 2,…, ݊) can be obtained according to the experimental 
frequency response. The residual error is defined as: 
݁ݎݎሺߟሻ = ෍ሾ ௜ܻ − ݂ሺ ௜ܺ, ߟሻሿଶ
௡
௜ୀଵ
. (7)
The damping identification can be transferred into getting a proper value of ߟ to make the 
residual error the most minimum. Gauss-Newton iterative method is used to find the optimal 
parameter of damping.  
Expand Eq. (5) to Taylor series at point of initial value ߟ଴ as: 
ݕ = ݂ሺ ௜ܺ, ߟሻ =
ߟ଴
ඥሺ1 − ௜ܺሻଶ + ߟ଴ଶ
+ ሺ1 − ௜ܺሻ
ଶ
ሺሺ1 − ௜ܺሻଶ + ߟ଴ଶሻ
ଷ
ଶ
ሺߟ − ߟ଴ሻ, (8)
where ݂′ሺݔ௜, ߟ଴ሻ = ௗ௙ሺ௫೔,ఎሻௗఎ ห ఎୀఎబ. 
Rewrite Eq. (7) as: 
݁ݎݎ݋ݎሺߟሻ = ෍ሾ ௜ܻ − ݂ሺ ௜ܺ, ߟ଴ሻ − ݂′ሺ ௜ܺ, ߟ଴ሻ ⋅ ሺߟ − ߟ଴ሻሿଶ
௡
௜ୀଵ
= ෍ሾݕ෤௜ − ݃ሺ ௜ܺ − ߟ଴ሻߟሿଶ
௡
௜ୀଵ
, (9)
where ݕ෤௜ = ௜ܻ − ݂ሺ ௜ܺ, ߟ଴ሻ + ݃ሺ ௜ܺ, ߟ଴ሻ ⋅ ߟ଴,݃ሺ ௜ܺ, ߟ଴ሻ = ݂′ሺ ௜ܺ, ߟ଴ሻ. Eq. (9) is a typical least square 
optimization problem. The estimated ߟଵ  can be obtained for the first generation by solving 
Eq. (10): 
ܣ்ܣ ⋅ ߟ = ܣ்ܾ, (10)
where ܣ = ൫݃ሺݔଵ, ߟ଴ሻ, ݃ሺݔଶ, ߟ଴ሻ, ⋯ , ݃ሺݔ௡, ߟ଴ሻ൯், ܾ = ሺݕ෤ଵ, ݕ෤ଶ, … , ݕ෤௡ሻ. 
After ߟଵ is obtained, ߟ଴ will be replaced by ߟଵ and repeat the above step to get the estimated 
value ߟଶ for the second generation. The iteration will be finished if the difference between the 
neighbor generations of ߟ is small enough for satisfactory demand. The last generation of ߟ is the 
estimated damping. The experimental and fitting response are shown in Fig. 3 and damping can 
be obtained in the fitting process. 
 
Fig. 3. Experimental and fitting response 
0.95 1 1.050.5
0.6
0.7
0.8
0.9
1
Dimensionless frequency
Di
me
ns
ion
les
s a
mp
litu
de
 
 
experimental response
fitting response
A METHOD FOR MODAL LOSS FACTOR ESTIMATION BASED ON GAUSS-NEWTON ITERATION.  
JINTAO GU, MEIPING SHENG 
14 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
4. Resistance to noise 
This simulation is conducted to illustrate the noise sensitivity of the proposed methods. A 
single-degree-of-freedom system is adopted to verify the validity of Gauss-Newton iteration. A 
comparison is made between the proposed method and half-power bandwidth method. Gaussian 
random noise is applied to the simulated frequency response, and the signal-to-noise ratio (SNR) 
is defined as: 
SNR = 20logଵ଴൫|ܪ|௦௜௚௡௔௟ |ܪ|௡௢௜௦௘⁄ ൯, (11)
where |ܪ|௦௜௚௡௔௟ is amplitude of effective signal and |ܪ|௡௢௜௦௘ is amplitude of noise. 
The simulated signal parameters are ߟ = 0.001, ଴݂ = 500 Hz, ߱଴ = 2ߨ ଴݂, and Δ݂ = 0.1 Hz. It 
is simulated on 1000 samples with the SNR in the range from 10 dB to 50 dB. Errors of estimated 
damping with the two methods are shown in Fig. 4.  
In Fig. 5, each method is tested for 41 noise levels, which is from 10 dB to 50 dB with 1 dB 
interval, with 1000 samples in each noise level. The average error is calculated by Eq. (12): 
ܽݒ݁ݎܽ݃݁  ݁ݎݎ݋ݎ = 1ܰ ෍
|ߟ௜ − ߟ௘௫௔௖௧|
ߟ௘௫௔௖௧
ே
௜ୀଵ
, (12)
where ߟ௜ is the identified damping, ߟ௘௫௔௖௧ is the exact loss factor set in the simulation, ܰis the 
number of samples. 
 
Fig. 4. Error of the estimated damping versus SNR: a) proposed method, b) half-power bandwidth method 
 
Fig. 5. Average error for the two methods 
It can be seen in Fig. 4 that identified damping by the two methods are more and more accurate 
10 20 30 40 50
-80
-60
-40
-20
0
20
40
60
err
or%
SNR/dB
10 20 30 40 50-80
-60
-40
-20
0
20
40
60
err
or%
SNR/dB
(a) (b)
10 20 30 40 500
10
20
30
40
Er
ror
%
SNR/dB
 Half-power bandwidth
 Proposed method
A METHOD FOR MODAL LOSS FACTOR ESTIMATION BASED ON GAUSS-NEWTON ITERATION.  
JINTAO GU, MEIPING SHENG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 15 
with the increased SNR. The accuracy of proposed method is more than half-power bandwidth 
method at all SNRs. It may also be noticed that there is considerable variability of half-power 
bandwidth method especially in the lower SNR. 
As seen in Fig. 5, both methods are of satisfied accuracy if ܴܵܰ > 30 dB. But if ܴܵܰ ≤ 30 dB, 
error of half-power bandwidth is much larger than proposed method. It is clear that the proposed 
method should be used to estimate damping instead of half-power bandwidth if ܴܵܰ ≤ 30 dB. 
5. Conclusions 
A novel method based on Gauss-Newton iteration is proposed to estimate damping in this 
paper. There is smaller variability for the proposed method than half-power bandwidth method. 
The proposed method is more accurate than half-power bandwidth method, especially in the lower 
SNR circumstances. The proposed method should be used instead of half-power bandwidth 
method for damping estimation if ܴܵܰ ≤ 30 dB. 
References 
[1] Guasch Oriol A direct transmissibility formulation for experimental statistical energy analysis with 
no input power measurements. Journal of Sound and Vibration, Vol. 330, 2011, p. 6223-6236. 
[2] Rak Michal, Ichchou Mohamed, Holnicki Szulc Jan Identification of structural loss factor from 
spatially distributed measurements on beams with visoelastic layer. Journal of Sound and Vibration, 
Vol. 310, 2008, p. 801-811. 
[3] Papagiannopoulos George A., Hatzigeorgiou George D. On the use of the half-power bandwidth 
method to estimate damping in building structures. Soil Dynamics and Earthquake Engineering, 
Vol. 31, Issue 7, 2011, p. 1075-1079. 
[4] Wang P. W., Zhong W. W., Hsu J. F. Investigation of multi-layer sandwich beams through single 
degree-of-freedom transformation. Applied Acoustics, Vol. 74, Issue 4, 2013, p. 521-525. 
[5] Olmos Bertha A., Roesset Jose M. Evaluation of the half-power bandwidth method to estimate 
damping in systems without real modes. Earthquake Engineering and Structural Dynamics. Vol. 39, 
14, p. 1671-1686. 
[6] Guo Zhiwei, Sheng Meiping, Ma Jiangang, Zhang Wulin Damping identification in frequency 
domain using integral method. Journal of Sound and Vibration, Vol. 338, 2015, p. 237-249. 
[7] Badsar S. A., Schevenels M., Haegeman W., Degrande G. Determination of the material damping 
ratio in the soil from SASW tests using the half-power bandwidth method. Geophysical Journal 
International, Vol. 182, Issue 3, 2010, p. 1493-1508. 


A method for modal loss factor estimation based on Gauss-Newton iteration

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A method for modal loss factor estimation based on Gauss-Newton iteration

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