power law damping parameter identification

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Journal of Sound and Vibration

Journal of Sound and Vibration 330 (2011) 5878–5893

0022-46

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/jsvi

Power law damping parameter identification

Nikola Jaksic

Turboinstitut, Rovsnikova 7, SI-1210 Ljubljana, Slovenia

a r t i c l e i n f o

Article history:

Received 14 March 2011

Received in revised form

19 July 2011

Accepted 20 July 2011

Handling Editor: M.P. CartmellThe identification procedure was found to be robust on the guessed value of

Available online 9 August 2011

0X/$ - see front matter & 2011 Elsevier Ltd.

016/j.jsv.2011.07.029

ail addresses: [email protected], nik

a b s t r a c t

The parameter estimation of a nonlinear power damping system is studied. The

parameter identification method used here assumes a priori the equation of motion

describing the system dynamics. The method, which is based on the measured data

(acceleration), was applied to the free and forced vibrations.

parameters at the numerical experimentation. The parameter values were estimated

with a good accuracy for both modes of system operation (free and forced) if only the

measured time history was sampled at a high enough rate for the noise level contained

within. It was shown that the steady state of the harmonically excited system is not the

best region for the parameter identification with this method.

During the experimentation the method was applied to the free vibrations in

different media (air and water). The results obtained by the parameter identification

method were compared to the ones obtained by separate tests and good agreement was

found. The identification procedure was found to work fine for all models under

consideration and the models’ responses correspond well to the measured acceleration

time histories.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Energy dissipation is usually modelled as viscous damping, where amount of dissipated energy is proportional to thevelocity, or as for frictional damping, where the energy dissipation is constant [1–3]. There have been other energydissipation models developed to simulate different mechanisms of the energy dissipation such as fractional derivativeapproach [4–6] or the hysteretic damping [7] as an integral-differential equation in the time domain which is differentfrom the conventional complex stiffness model [8,9].

The intention of this paper is to demonstrate that the parameter identification method [10,11] is applicable to thepower law damping model. The method is tested by the numerical experiments for free and forced vibrations and by theexperimental work.

1.1. Statement of the problem

The power dissipation model [12–14] covers a group of energy dissipation models that are proportional to velocity. Thesingle degree of freedom (SDOF) system is considered here in a configuration where the stiffness and the dissipation of theenergy can be separated. The equation of motion for such a system can be formulated as

m €xþGðxÞþHð _xÞ ¼ FðtÞ, (1)

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dx.doi.org/10.1016/j.jsv.2011.07.029

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dx.doi.org/10.1155/2010/695025

N. Jaksic / Journal of Sound and Vibration 330 (2011) 5878–5893 5879

where m stands for the system inertia, G(x) for the stiffness function of the system where the linear and nonlinear stiffnessare taken into account, Hð _xÞ for the energy dissipation mechanism and a power law for damping is taken into account, andF(t) for the excitation force. Hence, the mathematical model can be rewritten as

m €xþklxþkcx3þc sgnð _xÞ9 _x9r¼ FðtÞ, (2)

where kl stands for the coefficient of the linear stiffness, kc for the coefficient of the cubic stiffness, c for the dampingcoefficient and r for the exponent defining the nature of the damping mechanism. The system governed by Eq. (2) isgeneralisation of the Duffing system [15].

2. The parameter identification method

The approach to the parameter identification used throughout this paper has been presented in [10,11] and used in[16]. The approach is based on parameter identification by utilizing the equation of motion as a model of the system underconsideration, and measured acceleration response of the system.

The main advantages of the method are:

1.
simplicity based on constructing an objective function from the equation of motion, 2. applicability to short time histories and 3. capability of computing initial conditions in addition to the model parameters.
2.1. Free vibrations

Free vibrations of the SDOF system are governed by the equation of motion derived from Eq. (2) as

€xþaxþdx3þp sgnð _xÞ9 _x9r

¼ 0, (3)

where a¼ kl=m¼o2 stands for the normalized coefficient of the linear stiffness, d¼ kc=m stands for the normalizedcoefficient of the cubic stiffness and o for the natural frequency, p¼c/m for the normalized damping coefficient and r forthe exponent defining the nature of the damping mechanism. Normalization is achieved by means of dividing the equationof motion by the system inertia. The equation of motion is the basis for designing the objective function for optimalparameter estimation procedure. The Gaussian nature of the random spread around the true values of the measured timehistory is assumed and thus the least-mean-squared-error method is chosen [17] for creating the objective function

C¼Xn

i ¼ 1

½ €xiþaðxiþx0þ _x0tiÞþdðxiþx0þ _x0tiÞ3þp signð _xiþ _x0 Þ9 _xiþ _x0 9

r�2, (4)

where n denotes the number of points of measured time history, and ti, xi, _xi, €xi denote time, displacement, velocity andacceleration at the ith sampling point, respectively. a, d, p and r denote four parameters to be identified. And, since the timehistories of velocity and displacement are numerically reconstructed (integrated) from the acceleration time history, twonew unknowns appear: x0 and _x0, which denote initial displacement and initial velocity, respectively. The parametervalues are identified by minimization of the objective function C, Eq. (4), which may not be the easiest one to handle [18].The time-window approach [10] was not used in the numerical experimentation. On the contrary, it has been used whenanalyzing free responses of the system in the experimental work.

The time-window approach [10] means practically that only a part of the time history enters the identificationprocedure described above. The time window defines its length and position on the entire time history. This is similar tothe windowed spectral analysis approach with a rectangular window and overlapping. When time window of the certainlength (TWL) slides along the measured signal, different parameter values are estimated for each time window. Supposingthat the parameters under considerations are invariant in time then the results of the time-window approach for each ofthem are considered to be members of a population of an unknown probability distribution. The average population valuecould be computed as the parameter value. However, when noise levels are high, outliers will occur, predominantly due tofailed convergence of the minimization procedure or due to the fact that the minimization procedure may converge to awrong minimum [17]. In this case, just averaging the time-window results will produce parameter estimates heavilyinfluenced by the outliers. In order to avoid that the median is used instead of the mean of the whole data set.

2.2. Forced vibrations

The forced vibration of the SDOF system excited by the harmonic force is governed by Eq. (2). The normalized equationof motion can be written as

€xþaxþdx3þp sgnð _xÞ9 _x9r

¼ f sinðof tÞ, (5)

N. Jaksic / Journal of Sound and Vibration 330 (2011) 5878–58935880

where the additional parameters to Eq. (3) are defined as: f¼F/m for the normalized amplitude of the excitation/drivingforce and of for the excitation/driving frequency. The normalization and the objective function derivation are done in thesame way as for the free vibrations. Hence, the objective function can be written as

C¼Xn

i ¼ 1

½ €xiþaðxiþx0þ _x0tiÞþdðxiþx0þ _x0tiÞ3þp sgnð _xiþ _x0 Þ9 _xiþ _x0 9

r�f sinðof tÞ�2: (6)

The parameter values are identified by minimization of the objective function C, Eq. (6). The time-window approach wasused here.

2.3. Summary of the method

The method can be summarized as follows:

�
The time window approach is used to select a certain part of or a whole of the entire time history. � For each time window:
J The acceleration time history belonging to the window is integrated twice, the time histories of velocity anddisplacement are therefore generated with the zero initial conditions.

J The objective function is created by using the differential equation of motion.J All three time histories are fed into the objective function and the minimization procedure is carried out yielding

estimates of the parameters.

�
In the case of multiple time windows the median is used for parameter value estimation to eliminate poorly estimatedparameters.
3. Numerical simulations

Numerical simulation of an experiment gives us a controllable environment for testing the method with the same set-up of input data as for the real experiment.

In order to mimic an experiment, the acceleration response of the system the equation of motion is numericallyintegrated (solved) based on the chosen parameter set. That procedure yields time histories of displacement and velocity.The acceleration time history is computed from the equation of motion in which the integrated time histories are fed.Gaussian noise is then added to the acceleration time history. Only the noisy acceleration history and the equation ofmotion (the model) enter the identification procedure, see Section 2.3. This acceleration time history will be addressed asmeasured acceleration throughout this section.

3.1. Free vibrations

The free vibration of the SDOF system is governed by the equation of motion (3), where the nonlinear stiffness wasneglected, hence, d¼0.0. The basic set of parameter values is a¼1.0, p¼0.2 and r¼0.7 and the initial conditions are x0¼2.0and _x0 ¼ 3:0. The parameter values are chosen arbitrarily.

Several analysis were undertaken in order to established feasibility of the approach to the problem of parameteridentification of systems with the power damping law. First the basic signal properties were varied-level of noisecontained in the time history and the sampling rate of the time history. Then the convergence of the identificationprocedure towards correct minimum is investigated by using different guessed values of the parameters and initialconditions. The different natures of damping mechanism were investigated through different values of the p and r

parameters.

3.1.1. Variation of the signal properties

The acceleration response was numerically integrated and Gaussian noise was added afterwards. The amount of addednoise was measured with a signal to noise ratio (SNR) [17] of 1 (no noise added), 40, 20, 10. The integrated signal wasresampled to three different values of the samples per cycle (SPC): 20, 80 and 320. The signals for SPC¼20 and SPC¼80 andfor all noise levels are presented in Fig. 1(a) and (b), respectively.

The results of the parameter identification are presented in Table 1. The guessed values of the parameter estimateswere set to ag¼0.5, pg¼1.0 and rg¼1.0 and the guessed values of the initial conditions to x0g ¼ 0:0 and _x0g ¼ 0:0.

The results in Table 1 support the findings in [10,11] that more points are needed when noise levels are relatively highfor the same quality of results as achieved with lower noise levels. The parameter identification procedure estimatesrelatively accurate parameter values until SNRZ20 dB when sampling of a signal is relatively low; SPCr80. Highersampling frequencies of a measured signal enables us to estimate parameter values with higher accuracy in spite of thenoise in the signal.

The model responses are presented in Fig. 2 for parameters identified from the measured acceleration responses forSPC¼20 and SNRs of 40, 20 and 10 dB. The responses (1) and (2) in Fig. 2 are practically coincident. The response (3) is still

(b)(a)

Fig. 1. Basic properties of the measured acceleration response of the system under free vibrations: (a) SPC¼20 and (b) SPC¼80.

Table 1Parameter identification results on the free system response.

SNR¼40 dB SNR¼20 dB SNR¼10 dB

SPC Par. Val. Est. Err. (%) Est. Err. (%) Est. Err. (%)

20 a 1.0 0.9984 �0.2 1.0122 1.2 0.8095 �19.0

p 0.2 0.1988 �0.6 0.1502 �24.9 0.2069 3.5

r 0.7 0.7162 2.3 1.0830 54.7 �0.0357 �105.1

x0 2.0 2.0170 0.9 1.5483 �22.6 0.8089 �59.6_x0 3.0 2.9925 �0.3 3.0353 1.2 3.2683 8.9

80 a 1.0 0.9982 �0.2 1.0225 2.5 0.9722 �2.8

p 0.2 0.1970 �1.5 0.2032 1.6 0.2608 30.4

r 0.7 0.7260 3.7 0.7345 4.9 0.1789 �74.4

x0 2.0 2.0293 1.5 1.5060 �24.7 1.0362 �48.2_x0 3.0 2.9864 �0.5 3.2197 7.3 3.3825 12.8

320 a 1.0 0.9990 �0.1 1.0114 1.1 1.0100 1.0

p 0.2 0.1992 �0.4 0.2103 5.1 0.2439 21.9

r 0.7 0.7076 1.1 0.6292 �10.1 0.4004 �42.8

x0 2.0 2.0148 0.7 1.7565 �12.2 1.6087 �19.6_x0 3.0 2.9924 �0.3 3.0894 3.0 3.1572 5.2


reasonably close to the model true response (1). It is clear that identification at high noise content in the response yields illdefined parameter values and thus the response (4) is not an appropriate representation of (1). The energy dissipationmodel is wrongly identified by (4). However, the frequency and the amplitude of responses (1) and (4) remain inreasonably good agreement.

3.1.2. Convergence of the identification process

The convergence of the identification process was checked for different sets of the guessed values of the initialconditions. The guessed values of x0g were set in the interval [�10, 10] and the same interval was used for _x0g . Theprocedure converged to the correct minimum in all cases for two combinations of the (SPC, SNR): (20, 40) and (80, 20).

The convergence was checked also for variations of the parameters p and r. The guessed values pg of the parameter p

were varied in the interval [�1, 2] and the ones for the parameter r in the interval [0, 2]. Three sets of the guessed values ofthe initial conditions ðx0g , _x0gÞ were used: (0, 0), (1, 1) and (�1, �1). The process converged to a different minimum onlyfor the guessed initial conditions (0, 0) and for combinations of parameters (pg, rg) of: (0.1, 0.0), (0.3, 0.0) and (0.4, 0.0)for the combination (SPC, SNR) ¼ (20, 40) and for combinations of parameters (pg, rg) of: (0.1, 0.0), (0.4, 0.0), (1.4, 0.1) and(1.5, 0.1) for the combination (SPC, SNR) ¼ (80, 20).

The identification procedure proved to be robust on the initial guessed values of the parameters.

3.1.3. Variation of the parameter p values

The values taken for the parameter p were chosen to cover a wide range of possible scenarios—from positive tonegative damping effects. The results are presented in Table 2 for three pairs of signal parameters ðSPC,SNRÞ.

Fig. 2. Comparison of different model responses based on the true parameter values (1) and the identified ones at SPC¼20, (2) SNR¼40 dB,

(3) SNR¼20 dB, and (4) SNR¼10 dB.

Table 2Parameter identification results on the free system response for different values of the p parameter.

SPC 20 80 320

SNR 40 dB 20 dB 10 dB

Par. Val. Est. Err. (%) Est. Err. (%) Est. Err. (%)

a 1.0 0.9827 �1.7 1.1563 15.6 1.1526 15.3

p 1.1 1.1022 0.2 1.0809 �1.7 1.0042 �8.7

r 0.7 0.6948 �0.7 0.8634 23.3 0.8952 27.9

x0 2.0 2.0665 3.3 1.3291 �33.6 1.3826 �30.9_x0 3.0 2.9781 �0.7 3.1852 6.2 3.1495 5.0

a 1.0 1.0004 0.0 0.9926 �0.7 0.9859 �1.4

p 0.0 �0.0027 / 0.0064 / 0.0485 /

r 0.7 �1.3003 / 1.2579 / �0.2892 /

x0 2.0 2.0306 1.5 1.4084 �29.6 1.5379 �23.1_x0 3.0 2.9819 �0.6 3.2922 9.7 3.2074 6.9

a 1.0 1.0076 0.8 0.9226 �7.7 0.9218 �7.8

p �1.1 �1.1059 0.5 �0.7975 �27.5 �1.0558 �4.0

r 0.7 0.6958 �0.6 0.8070 15.3 0.7022 0.3

x0 2.0 2.1851 9.3 �1.1361 �156.8 �0.4072 �120.3_x0 3.0 2.8655 �4.5 5.1767 72.6 4.5123 50.4


The results in Table 2 show again that a denser signal raster is needed to estimate the parameter values with reasonablequality if the noise levels are high. As noted in [10] the parameter that influences the response the most can be estimatedbetter than the parameters to which the system is not particularly sensitive. This can be clearly seen in Table 2 for a zerovalue of parameter p and for negative values of the same parameter. In the first case the parameter r has no influence onthe shape of the system response, and hence, its value is impossible to estimate. However, since the estimated parameter p

value is not zero, the parameter r value is given some value in order to minimize the objective function. The values for theinitial condition parameters for the negative value of the parameter p are less well estimated than in the case of thepositive p values. Negative viscous damping is normally associated with self-exciting systems [19]. They are mostcommonly found in solid–fluid interactions with fluid induced vibrations [20,21], but certainly not exclusively there [22].There are two main reasons for using the self-excited model: either the excitation itself is too difficult to model or only theconsequences of the energy-input mechanism are important for the problem under consideration. Due to the energy inputmechanism of negative damping the amplitudes of the response are rising in time and thus make the values of the initialconditions less important for the overall shape of the response.

3.1.4. Variation of the parameter r values

The values taken for the parameter r were chosen to cover frictional damping r¼0, linear viscous damping r¼1and quadratic viscous damping r¼2. The results are presented in Table 3 for three pairs of signal parameters ðSPC,SNRÞ.

Table 3Parameter identification results on the free system response for different values of the r parameter.

SPC 20 80 320

SNR 40 dB 20 dB 10 dB

Par. Val. Est. Err. (%) Est. Err. (%) Est. Err. (%)

a 1.0 0.9990 �0.1 1.0066 0.7 0.9964 �0.4

p 0.2 0.2009 0.4 0.1940 �3.0 0.2073 3.6

r 0.0 0.0406 / 0.1367 / 0.0989 /

x0 2.0 1.9199 �4.0 1.4853 �25.7 1.5911 �20.4_x0 3.0 3.0583 1.9 3.2434 8.1 3.1787 6.0

a 1.0 0.9976 �0.2 1.0296 3.0 1.0179 1.8

p 0.2 0.2003 0.1 0.1874 �6.3 0.2485 24.3

r 1.0 1.0038 0.4 1.1597 16.0 0.6313 �36.9

x0 2.0 2.0267 1.3 1.5135 �24.3 1.6037 �19.8_x0 3.0 2.9869 �0.4 3.2091 7.0 3.1514 5.0

a 1.0 0.9958 �0.4 1.0476 4.8 1.0323 3.2

p 0.2 0.2020 1.0 0.1777 �11.1 0.1699 �15.0

r 2.0 1.9883 �0.6 2.2687 13.4 2.2162 10.8

x0 2.0 2.0298 1.5 1.4810 �26.0 1.5931 �20.3_x0 3.0 2.9912 �0.3 3.1882 6.2 3.1359 4.5

(b)(a)

Fig. 3. Basic properties of the measured acceleration response of the harmonically excited system: (a) SPC¼20 and (b) SPC¼80.


The values of the estimated parameters are in good agreement with true values for all three basic damping mechanisms.The quality of estimated values is subject to the SPC and SNR combinations as already noted.

3.2. Forced vibrations

The forced vibration of the SDOF system is governed by the equation of motion (5) with the parameter values ofa¼o2 ¼ 1:0, p¼0.2, r¼0.7 and f¼1.0, of ¼ 0:7 and also the initial conditions of x0¼2.0 and _x0 ¼ 3:0. The nonlinearstiffness was neglected in this case. Hence, d¼0.0. The parameter values are chosen arbitrarily.

Again, the basic signal properties were varied – level of noise contained in the time history and the sampling rate of thetime history – to establish the robustness of the method on noisy time histories of the excited systems. After that, theparameters were estimated for different values of the driving frequency of .

The measured acceleration response was numerically integrated and the Gaussian noise added afterwards in the samefashion as for the free vibrations. The same amount of noise level (SNR) was used as well as the same sampling rates (SPC).The time histories for SPC¼20 and SPC¼80 and for all noise levels are presented in Fig. 3(a) and (b), respectively.The results of the identification procedure are presented in Table 4.

When comparing Tables 1 and 4 it becomes clear that the time window approach greatly improves the quality of theresults. The findings of the free vibrations are confirmed here as well (Table 5).

Table 4Parameter identification results of the harmonically excited system.

SNR¼40 dB SNR¼20 dB SNR¼10 dB

SPC Par. Val. Est. Err. (%) Est. Err. (%) Est. Err. (%)

20 a 1.0 0.9998 �0.0 1.0040 0.4 1.0151 1.5

p 0.2 0.2009 0.4 0.1969 �1.5 0.1803 �9.9

r 0.7 0.6865 �1.9 0.7398 5.7 1.3876 98.2

80 a 1.0 1.0000 �0.0 0.9988 �0.1 0.9958 �0.4

p 0.2 0.2001 0.0 0.2047 2.3 0.2022 1.1

r 0.7 0.7000 �0.0 0.6177 �11.8 1.1888 69.8

320 a 1.0 1.0002 0.0 1.0018 0.2 0.9932 �0.7

p 0.2 0.1998 �0.1 0.2008 0.4 0.2111 5.6

r 0.7 0.7025 0.4 0.6591 �5.8 0.7477 6.8

Table 5Parameter identification results of the harmonically excited system with different driving frequencies for SPC¼80 and SNR¼20 dB.

a true value: 1.0 p true value: 0.2 r true value: 0.7

of =o Est. Err. (%) Est. Err. (%) Est. Err. (%)

0.1 0.9933 �0.7 0.2191 9.6 0.6503 �7.1

0.2 0.9905 �0.9 0.2874 43.7 0.7540 7.7

0.3 1.0072 0.7 0.2163 8.2 0.6925 �1.1

0.4 1.0014 0.1 0.1959 �2.0 0.6897 �1.5

0.5 0.9982 �0.2 0.2117 5.8 0.7678 9.7

0.6 0.9986 �0.1 0.1982 �0.9 0.7204 2.9

0.7 1.0001 0.0 0.2004 0.2 0.6549 �6.4

0.8 0.9992 �0.1 0.2019 0.9 0.6726 �3.9

0.9 1.0001 0.0 0.2045 2.3 0.6770 �3.3

1.0 0.9992 �0.1 0.1872 �6.4 0.7571 8.2

2.0 0.9869 �1.3 0.2043 2.2 0.7783 11.2

3.0 0.9780 �2.2 0.1719 �14.1 0.5345 �23.6

4.0 0.9627 �3.7 0.1993 �0.4 0.7401 5.7

5.0 0.9641 �3.6 0.2998 49.9 0.9114 30.2

6.0 0.8736 �12.6 0.1935 �3.2 0.7393 5.6

7.0 0.8776 �12.2 0.2020 1.0 0.7518 7.4

8.0 0.8497 �15.0 0.1844 �7.8 0.6194 �11.5

9.0 0.8509 �14.9 0.1925 �3.7 0.7244 3.5

10.0 0.8490 �15.1 0.2274 13.7 0.8737 24.8


3.2.1. Variation of the driving frequency

The driving frequency of was varied in the interval of =o¼ ½0:1,10:0� in order to cover resonant behaviour of thesystem as well as under- and above the resonant region of system’s operation.

The results obtained in the above-resonant region are estimated with lower accuracy than those in the under-resonantregion. The values of the parameters varies much more in the above-resonant region, Fig. 6, when compared to ones in theunder-resonant region, Fig. 4. However, the identification procedure fails in several instances also at the under-resonantregion. The analysis shows us that the steady state is not the best area of the time history to perform parameteridentification. The identification procedure is more sensitive to noise in the steady state region than at the transient regionof the time history. The ratio between steady state amplitude and instantaneous transient amplitude plays an importantrole when identifying the parameters. The smaller the ratio the better. Figs. 4–6 are presented with the same scales foreasy comparison.

4. Experiment

The experimental work was undertaken on a purpose-made experimental rig which resembles the features of a Duffingoscillator [15] by allowing large amplitude oscillations [10,11]. The identification method is relatively simple to apply, and,hence, the simple separate tests of the system were considered by estimating the system’s spring characteristic by statictesting and by estimating the amount of dissipated energy by the logarithmic decrement. The comparison of bothapproaches is presented. After that the identification method was applied to measured responses of the rig whit a rakeallowed to oscillate in water.

Fig. 4. System response €x and values of the parameters a, p and r for different time-window positions, of =o¼ 0:4 SNR¼20 dB, SPC¼80.

Fig. 5. System response €x and values of the parameters a, p and r for different time-window positions, of =o¼ 1:0, SNR¼20 dB, SPC¼80.


4.1. Experimental rig and separate tests

The experimental rig and the separate tests have been thoroughly described in [10,11]. This section shortly summarizesthe facts.

Fig. 6. System response €x and values of the parameters a, p and r for different time-window positions, of =o¼ 4:0 SNR¼20 dB, SPC¼80.

Fig. 7. Experimental rig.


The experimental rig is composed of two parallel but separated leaf springs clamped at one end and attached to aninertial mass at the other end. The complete inertial mass was estimated to be mc¼1.971 kg. A rake like structure isattached to the inertial mass. The rake acts as a viscous damper when submerged into water. The experimental rig isschematically shown in Fig. 7.

The spring characteristic was determined by the static test, Fig. 8. It was approximated with the linear Eq. (7), and thecubic Eq. (8), functions. Both are shown in Fig. 8. The value of the coefficient kl of the characteristic Eq. (7) is kl¼71.172 N/m.The values of the coefficients kc and c of the characteristic Eq. (8) are kc¼78.072 N/m and c¼�2470.504 N/m3:

FðxÞ ¼ klx, (7)

FðxÞ ¼ kcxþcx3: (8)

The static test is pointing towards the nonlinear nature of the system stiffness which applies the need for introducingthe nonlinear (cubic) features when modelling the system.

If coefficients of the static characteristics, Eqs. (7) and (8), are divided by the total inertia, they fit into the modelparameters; for the linear characteristic als ¼ kl=mc ¼ 36:11 and for the nonlinear characteristic acs ¼ kc=mc ¼ 36:61 anddcs ¼ c=mc ¼�1253:43.

The acceleration time history was measured by an accelerometer on the dynamic rig test. The system response in air isdepicted in Fig. 9.

Fig. 8. Statically determined spring characteristic.

Fig. 9. Measured acceleration response.


The equivalent viscous damping ratio d was estimated by using the logarithmic decrement of the linear modelapproach. The damping ratio was estimated to have a mean value of d¼ 0:000869 and is largely time independent, Fig. 10.The values of coefficients of the linear fit d¼ aþbt in Fig. 10 are a¼0.000852 and b¼ 3:35� 10�7.

The instantaneous frequency o of the acceleration time history, Fig. 9, was extracted by the simple method ofmeasuring time between the zeros of the acceleration time history. The instantaneous frequency o as a function of time ispresented in Fig. 11. The instantaneous frequency rises with time approximately quadratically which is in line with thedegressive spring characteristic established in the static test. The mean value of the instantaneous frequency iso¼ 6:047 rad=s. The parameters of the polynomial o¼ aþbtþct2 fitted to the instantaneous frequency data, Fig. 11,are a¼5.995, b¼0.00141 and c¼�5:612� 10�6.

The parameter values obtained by the dynamic test can be translated into model parameters. The parameter a

describing the stiffness of the model is ad ¼o2 ¼ 36:57 which is in a good agreement with the static test. The parameter p

describing the linear viscous damping is pd ¼ 2od¼ 0:0105 and the exponent is rd¼1 for the linear model.

4.2. Parameter identification in air

The water container of the rig, Fig. 7, was removed here as was the case for the separate tests. A typical system responseis presented in Fig. 9. The identification procedure was carried out on the data of the acceleration time history presented inFig. 9 for four different models. The results are presented in Table 6. They are consistent when comparing different models.The introduction of the nonlinear stiffness to the model influences the value estimation of the coefficient of the linear

Fig. 10. Instantaneous damping ratio d as a function of time.

Fig. 11. Instantaneous frequency o as a function of time.

Table 6Parameter identification results on the rig’s response.

Model Equation of motion a d p r

1 €xþaxþp _x ¼ 0 36.50 0.0113

2 €xþaxþp signð _xÞ9 _x9r¼ 0 36.51 0.0039 0.4192

3 €xþaxþdx3þp _x ¼ 0 37.09 �500.9 0.0111

4 €xþaxþdx3þp signð _xÞ9 _x9r

¼ 0 37.09 �505.6 0.0040 0.4164


stiffness as well. When comparing the linear and power damping model, Fig. 12, it is clear that the identification proceduredefines the nature of the model of the energy dissipation as one being between frictional and linear viscous damping.

The comparison of the responses of the models in Table 6 is presented in Fig. 13. All responses are depicted at the top ofthe figure for a whole measurement. The details of the first and last second are presented at the bottom of the figure. Allmodel responses are in excellent agreement with the measured one at the beginning. However, at the end the best fit canbe found for models with cubic spring characteristics, which enables them to accommodate the instantaneous frequencychanges. The amplitudes are well captured by all model responses over a whole region of the measured time history.

The repeatability of the experimentation is confirmed in Table 7. The parameters d and r experience the largest spreadaround the mean value. This is in line with the findings of [10], which stated that the parameters that are influencing thesystem’s response the most are the easiest to estimate. The spread of these parameters is low. On the contrary, theparameters to which the system is not very sensitive are more difficult to estimate and their spread is larger.

Fig. 12. Linear and power damping model comparison.

Fig. 13. Comparison of the responses of the models listed in Table 6: measured (thick � � �Þ, model 1 (thin � � �Þ, model 2 (– –), model 1 (– � –) and

model 1 (–).

Table 7Comparison of the estimated values of the parameters for five independent measurements.

Measurement a d p r

1 37.09 �505.6 0.0040 0.4164

2 37.13 �546.1 0.0038 0.3978

3 37.14 �503.8 0.0038 0.3979

4 36.96 �468.9 0.0044 0.2608

5 37.09 �502.0 0.0040 0.4141

Average 37.08 �505.3 0.0040 0.3774


The values for the instantaneous frequency are similar when comparing the time dependence of the estimated parameters,Fig. 14, for the linear model (model 1 in Table 6) with ones estimated by the separate tests, Fig. 11. The instantaneous dampingratios d obtained in the same way, Figs. 10 and 14, are much less similar than instantaneous frequencies.

Fig. 14. Time dependence of the estimated parameters by the linear model.

Table 8Comparison of the estimated values of the parameters for five independent measurements.

a d p r

Separate tests

Static 36.61 �1253.4

Dynamic 36.57 0.0105

Parameter identification

€xþaxþdx3þp _x ¼ 0 37.09 �500.9 0.0111

€xþaxþdx3þp signð _xÞ9 _x9r

¼ 0 37.08 �505.3 0.0040 0.3774


4.2.1. Comparison of the estimated parameter values for separate tests with values estimated by the identification procedure

The comparison is presented in Table 8. The differences in parameter a are relatively small due to high systemsensitivity on the variation of this parameter. On the contrary, the differences in values of the parameter d are relativelylarge due to the low sensitivity of the system to the variation of this parameter. The linear viscous damping coefficient ofthe dynamic separate test and that the parameter identification of the Duffing system with the viscous damping are in verygood agreement. The difference between the linear viscous damping model and the power one is presented in Fig. 12 andin the accompanying text.

4.3. Parameter identification in water

The water container of the rig remains in place here as presented in Fig. 7. A typical system response is presented inFig. 15. The subharmonics visible in the second half of the response are due to the returning water waves. The canistercontaining water did not allow for water spillover and thus incoming waves were reflected from the canister walls. This isthe reason for estimating the parameter values only at the initial section of the measured acceleration responses. Theresponses are shown in Fig. 16 and the estimated values of the parameters in Tables 9 and 10.

The results are in good agreement between two different measurements and also between models with regards to thealready mentioned differences between models.

The comparison of the responses of the models in Table 10 is presented in Fig. 17. All responses are depicted at the topof the figure for a whole measurement. The details of the first and last second are presented at the bottom of the figure. Allmodel responses are in good agreement with the measured one at the beginning. Again, at the end the best fit can be foundfor the models with the cubic spring characteristic, which enables them to accommodate the instantaneous frequencychanges. However, the fit in water is not as good as in air due to the returning waves. The amplitudes are well captured byall model responses over a whole region of the measured time history.

Fig. 15. The acceleration response of the experimental rig with the rakes in water.

Fig. 16. The acceleration responses used in the identification procedure.

Table 9Parameter identification results—measurement 1.


1 €xþaxþp _x ¼ 0 36.53 0.0363


3 €xþaxþdx3þp _x ¼ 0 36.98 �416.8 0.0368


¼ 0 36.97 �414.3 0.0102 0.5376


5. Conclusions

In this paper the dynamic parameters of nonlinear power law damping together with the system’s stiffness parameterswere estimated by a method developed in [10,11]. The method assumes a priori the equation of motion describing the

Table 10Parameter identification results—measurement 2.


1 €xþaxþp _x ¼ 0 36.65 0.0345


3 €xþaxþdx3þp _x ¼ 0 37.14 �572.9 0.0351


¼ 0 37.14 �570.4 0.0092 0.5288

Fig. 17. Comparison of the responses of the models listed in Table 10: measured (thick � � �Þ, model 1 (thin � � �Þ, model 2 (– –), model 1 (– � –) and model 1 (–).


system dynamics. This method, which is based on the measured data (acceleration), was applied for free and forcedvibrations.

The identification procedure was found to be robust to the guessed value of parameters in the numerical simulations.The parameter values were estimated with good accuracy for both modes of system operations (free and forced) if only themeasured time history was sampled high enough rate for the noise level contained within. Hence, the parameteridentification procedure estimates parameter values relatively accurately until SNRZ20 dB when sampling of a signal isrelatively low SPCr80. Higher sampling frequencies of a measured signal enable us to estimate parameter values withhigher accuracy in spite of the noise in the signal. It was shown that steady state of the harmonically excited system is notthe best region for the parameter identification with this method.

The identification procedure was tested on a real system as well. An experimental rig was built for this purpose. Theresults of different models were compared to the measured acceleration time history. Due to the system’s nonlinear springcharacteristic, the models that are Duffing type by nature proved themselves to deliver best results. Hence, theinstantaneous frequency was captured best when nonlinear stiffness is taken into account. In all cases the amplitudesof the measured and simulated responses were in good agreement.

The power energy dissipation law together with the parameter identification method proved that it is possible toidentify parameters with a high accuracy.

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power law damping parameter identification

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