Modal analysis of an isolated nonlinear response mode using the Nyquist circle properties: Numerical case

Modal analysis of an isolated nonlinear response mode

using the Nyquist circle properties: numerical case.

D. Di Maio1

1 _{University of Twente, Department of Applied Mechanics}

De Horst 2, 7522 LW, Enschede, the Netherlands e-mail: d.dimaio@utwente.nl

Abstract

Modal testing practitioners are well accustomed to FRFs which, nowadays, can be measured and analysed by a large variety of methods and tools. However, most of the tools for linear modal analysis are practically unusable when the frequency response is amplitude dependent. Hence, the methods based on the properties of the Nyquist circle become unavailable because of the loss of its circularity and completeness. Nevertheless, FRFs of well-isolated modes can still be processed by calculating the modal parameters for every pair of receptance points taken at equal amplitude either side the maximum response peak. The major limitation of this method is the absence of one branch of the response function due to unstable dynamics. The objective of this work is to show that incomplete FRF functions can be still processed to obtain the modal parameters. The method is benchmarked against a single degree of freedom both for linear and nonlinear response conditions.

1

Introduction

Modal testing is today a well-established practice for measuring and analysing frequency response functions. The modal parameters can be both used to generate a simple mathematical model (few degrees of freedom) or to validate a model generated by Finite Element (a large number of degrees of freedom). The groundwork was carried out by several scholars, who delivered several methods now packaged in high levels of technology readiness software suites [1], [2], [3]. However, experimental measurements showed that the linear modal analysis tools were unusable when the frequency responses presented a skewness typical of the nonlinear systems. Experimental methods, for analysing nonlinear vibrations, have to rely on measurements carried out by controlling the vibration amplitude of the response, which means “linearize” the FRF for a given constant response amplitude. By doing so, single DoF analysis methods can be used for processing FRFs and evaluate both natural frequency and damping nonlinear curves. This approach can be accurate, but its implementation is highly time consuming and therefore limited to some selected nonlinear responses[4]. The analysis was improved by using a method still based on single DoF approach, but which enabled fast nonlinear modal analysis. In fact, it analyses two pairs of frequency points at equal vibration amplitude either side the response peak to compute the four modal parameters. By sweeping pairs of frequency points from low up to high vibration amplitude, both the frequency and damping nonlinear curves were estimated [5]. Despite the intelligent approach to the data analysis, the method failed when one branch of the FRF was not present due to the unstable dynamic behaviour of the system, or commonly known as response jump. Furthermore, the analysis revealed poor accuracy in the identification of the damping nonlinear curve. Some minor issues were also observed on the constructed backbone that presented discontinuities when two modes were too close to each other. Many scholars investigated and successfully identified several methods based on direct analysis of time responses or frequency spectra, but no references in this work mainly based on steady-state measurements.

The objective of this research is to develop an identification method which can process FRFs measured under steady-state conditions when the vibration response becomes nonlinear. This method will start from the shortfalls of the method presented by Carrella [5] and aim to overcome those in the attempt to offer a

valuable tool to modal analysis practitioners. The paper is divided into two parts; the first one explains the basic idea developed to overcome analysis issues due to the frequency jump. Moreover, it proves that the method can intelligently calculate the system parameters of a linear single DoF system. The second part will present examples of a single DoF system included with stiffness and damping nonlinearity.

2

Theoretical background

This section will deal with the theoretical background which leads to the development of a novel method for performing the identification of nonlinear modal parameters. The case regards all those frequency response functions which present a response jump from high to low vibration amplitude, or the other way around. As already explained in section 1, the complex FRF analysis method, presented by Lin and Carrella [5], is unable to resolve the modal parameters when one branch of the frequency response curve is absent. However, the method paved the way to a simple approach to unbundling features from nonlinear system dynamics without going through expensive experiments or complex signal processing. This paper will show that it is possible to overcome the existing limitations and to derive the modal properties form an incomplete Nyquist dataset. The novel analysis method will be attempted by using a single degree of freedom system (SDOF).

The mobility FRF of an SDOF is described by equation (1), which by post-multiplying the numerator and denominator by the conjugate of the denominator will decouple the equation is the Real and Imaginary parts, in equations (2) and (3).

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2 2 2 2 2 2 2 2 2

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2 2 2 2 2 IM Y k m k m c

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The choice of using the mobility FRF for developing this new methodology is due to the circularity of the Nyquist circle whereby the viscous damping is the inverse of the diameter of the circle. Furthermore, the simulation of nonlinear FRFs is performed by ODE methods, where the use of viscous damping makes the solution of the equation of motion more practical than using other types of damping models. One should immediately notice from equations (2) and (3) three unknown parameters m, c and k are present But, the parameter c can be quickly determined by measuring the circle diameter in a single DoF system. Therefore, two equations are now necessary to resolve both m and k. The solution to this problem is sought from the fundamental principle of the geometry of the circle, which can be defined by a minimum of three points. To understand how to exploit this principle, one should look at Figure 1 which shows two Nyquist circles for linear (black trace) and nonlinear (red trace) systems, respectively. By first focussing on the linear system, one can notice that every three frequency points taken from the Nyquist dataset will always resolve the same circle, in fact, the circle parameters are unique. The explanation is sensible since the system is linear and, therefore, all the frequency points are equally spaced from the centre of such a circle. By switching the attention to the nonlinear system, described by the Nyquist dataset in red, one should notice that every three frequency points will define an amplitude dependent circle. It means that the circle parameters will depend

on the magnitudes of the three selected frequency points. How to exploit this feature will depend on how these three frequency points are selected.

Figure 1 Nyquist circle for linear (in black) and nonlinear (in red) system

Mechanical structures are known to exhibit a degree of nonlinearity depending on the material, geometrical and boundary condition properties and the level of vibration amplitude to which such systems are exposed to. However, in all cases, mechanical systems will always show an underlining linear behaviour for low levels of vibration amplitude. This behaviour is somewhat essential for understanding how to perform the selection of the frequency points from a Nyquist dataset. It means that to observe the evolution of the circle parameters depending on the vibration amplitude; one should select three frequency points two of which are references and let the last one sweeping the frequency range measured for the FRF. In practice, two frequency points are selected as references at the lowest vibration amplitude, at which the mechanical system is assumed to present an underlying linear behaviour. The third frequency point can be called the sweeper and used to sweep all, but the two references, data points forming the FRF. It is now possible to determine the circle’s parameters at the vibration amplitude of the sweeper. It is now possible to write a system of equations based on the equation (2) and (3). The system of equations (4) is defined by fixing two references at



_s (start frequency) and

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_e(end frequency) typically at very low vibration amplitude and



_iwhich is the sweeping frequency. One can notice that the system parameter c, the inverse of the diameter of the circle, depends on the point selected (the sweeper).

6.0E-3 -5.0E-3 -4.0E-3 -3.0E-3 -2.0E-3 -1.0E-3 0.0E+0 1.0E-3 2.0E-3 3.0E-3 4.0E-3 5.0E-3 Real 1E-2

0E+0 2.5E-3 5E-3 7.5E-3

LIN NLIN Nyquist

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(4)

It is trivial to note that only two of the equations in (4) are necessary for resolving the system parameters m and k. Another simple approach which will not require to measure the circle diameter is based on the use of the imaginary part of the inverse of the mobility as expressed in equations (5). By taking two equations of the three available in the system of equations (5), both m and k can be resolved for every set of three frequency points.

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2.1

Numerical simulations and system identification

This section is focussed on the numerical simulation, and the definition of the nonlinear systems developed for studying the newly developed identification method. A single degree of freedom system is initially attempted to prove that the approach can identify the system parameters correctly. However, before the system is integrated with nonlinear terms, a mobility FRF is calculated from equation 1 and solved by using two of the equations of the system in (4) and (5).

2.1.1 Evaluation of system parameters from linear vibrations

Figure 2 shows the Nyquist circle of equation (1) calculated with the following system parameters, m = 6

kg, c = 100 Ns/m and k = 1e8 _{Nm in a frequency range between 600 (0.1) 700 Hz. The natural frequency of}

Figure 2 Nyquist circle (left) and magnitude (right) of mobility for a linear system

Using two of the equations in the system (4) it is possible to calculate the modal parameters; these are shown in Figure 3. The plots of Figure 3 a, b, c and d only report the partial solutions calculated for the frequency range between 600 and 700 Hz using the real part of the system of equations (4). The damping value c is directly calculated from the measurement of the circle diameter. The other two parameters m and k are calculated by using two equations which, when resolved, will give four sets of solutions. These are visible from the colours (red, black, blue and green) used to describe the curves in Figure 3. Two of the four set of solutions are meaningful, and one part of these two (see the black arrows in Figure 3) describes the correct system parameters m and k. Hence, the method to fix two references and sweep the entire Nyquist mobility dataset enables the recovery of the system parameters, which match the initial ones used for simulating the mobility FRF. The result should have been evident at the outset since whichever selection of three frequency points would fit the same circle. Nevertheless, the nonlinear system will present a quasi-linear response for low vibration amplitude which will change to nonlinear as the amplitude is increased. Therefore, given two references the sweeper will determine a new circle the parameters of which can be calculated. Finally, by tracking the behaviour of the system parameters against the vibration amplitude one can determine the full dynamics of a nonlinear single degree of freedom system.

(a) Solutions calculated for the natural frequency (b) Solutions calculated for the mass m

5.0E-3 -5.0E-3 -4.0E-3 -3.0E-3 -2.0E-3 -1.0E-3 0.0E+0 1.0E-3 2.0E-3 3.0E-3 4.0E-3 Real Y(w) 1.1E-2 0E+0 2E-3 4E-3 6E-3 8E-3 1E-2

dataset Nyquist 1.0E-2 1.0E-4 1.0E-3 Frequency [Hz] 700.0 600.0 610.0 620.0 630.0 640.0 650.0 660.0 670.0 680.0 690.0 dataset Magnitude 700.00 600.00 610.00 620.00 630.00 640.00 650.00 660.00 670.00 680.00 690.00 Displacement [m] 2.50E-6 0.00E+0 5.00E-7 1.00E-6 1.50E-6 2.00E-6

Plot 4 4 plot Frequency 10.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Displacement [m] 2.50E-6 0.00E+0 5.00E-7 1.00E-6 1.50E-6 2.00E-6

Plot 3 3 plot m-X 101.0 99.0 99.2 99.4 99.6 99.8 100.0 100.2 100.4 100.6 100.8 Displacement [m] 2.50E-6 0.00E+0 5.00E-7 1.00E-6 1.50E-6 2.00E-6

NLIN 1 plot C 1.5E+8 -1.5E+8 -1.2E+8 -1.0E+8 -8.0E+7 -6.0E+7 -4.0E+7 -2.0E+7 0.0E+0 2.0E+7 4.0E+7 6.0E+7 8.0E+7 1.0E+8 1.2E+8 Displacement [m] 2.50E-6 0.00E+0 5.00E-7 1.00E-6 1.50E-6 2.00E-6

Plot 3 3 plot k-X

(c) Solutions calculated for the damping c (d) Solutions calculated for the stiffness k Figure 3 Calculation of the system parameters and natural frequency

Instead, by using the equation in (5) one can notice that the parameter c is not necessary for resolving the equations and the parameters m and k are easy to find. Figure 4 shows the natural frequency of the single DoF system, which is a straight line because every three sets of frequency points will always resolve the same circle parameters.

Figure 4 Calculation of the natural frequency by using the system of equation (5)

2.1.2 Evaluation of system parameters from nonlinear vibrations

This section deals with the identification of the system parameters which are amplitude dependent. As already explained a set of three frequency points would define a circle from which the system parameters can be calculated. Repeating this calculation for as many circles as defined by the set of Nyquist data, it is possible to track the evolution of the system parameters over the vibration amplitude. Two nonlinear systems were created to carry out the validation of the proposed identification method, such as one with a cubic stiffness nonlinearity and one with both cubic stiffness and quadratic damping nonlinearity.

Equation (6) represents a nonlinear single DoF with cubic nonlinearity. The parameter m, c, and k are the same already used for the linear system. Instead, the knl = 1e18N/m3 is the coefficient of the stiffness

nonlinearity. Equation (7) represents a single DoF system with a quadratic damping and cubic stiffness nonlinearity. The nonlinear stiffness coefficient is the same used for equation (6) instead the damping nonlinear coefficient is cnl = 10000N(s/m)2.

 

3 sin nl mxcxkxk x  f  t (6)

 

3 sin nl nl mxcxc x x kxk x  f  t (7) Equations (6) and (7) were calculated by integration method and hundred cycles of steady-state vibration were used for the analysis. The frequency range simulated was between 640Hz and 660Hz with a frequency spacing of 0.1Hz. The sample rate and the length of the time vector were 30kHz and 2 seconds, respectively. The FRF was calculated by the ratio between the response and the excitation force of the first fundamental component. The higher order harmonics from the response were neglected because their very tiny magnitudes. 650.00 649.00 649.10 649.20 649.30 649.40 649.50 649.60 649.70 649.80 649.90 Displacement [m] 2.50E-6 0.00E+0 5.00E-7 1.00E-6 1.50E-6 2.00E-6

LIN 0

2.1.3 Stiffness nonlinearity case study

Figure 5 shows the four FRFs, both Magnitude and Nyquist format, of a single DoF system with stiffness nonlinearity. These FRFs were simulated with excitation forces of 0.1N, 0.4N, 0.7N and 1N, respectively. The system presents hardening behaviour with typical jump caused by its unstable dynamic behaviour, and with the Nyquist circles presenting no loss of circularity due to the sole inclusion of stiffness nonlinearity. By closer inspection, one can notice that the frequency points rotate clockwise concerning a reference, such as a frequency point measured at the lowest forcing. This supports the ideas that by selecting two references and sweeping the third point, one can observe the nonlinear properties of the system as the vibration amplitude is increased. The primary objective is to trace the natural frequency of the system against the vibration amplitude. The analysis can be done by using two of the equations presented in the system of equation (6), which require the calculation of the damping, c. This parameter can be calculated merely because of a mobility FRF of a single DoF system would correspond to the inverse of the diameter of the circle. This diameter is calculated by fixing two frequency points at the lowest vibration amplitude and sweeping the third point over the number of points measured for the Nyquist.

Figure 6 shows that the diameter is approximately constant for the four FRFs simulated (alternated black-red) and it matches the original value set for the simulation (c = 100Ns/m). One would notice a numerical error of approx. 5% for the lowest vibration amplitudes.

(a) FRF magnitude (b) FRF Nyquist

Figure 5 FRF of a single DoF with stiffness nonlinearity

Figure 6 Damping, c, of the single DoF system

1.0E-2 0.0E+0 2.0E-3 4.0E-3 6.0E-3 8.0E-3 Frequency [Hz] 660.0 640.0 642.0 644.0 646.0 648.0 650.0 652.0 654.0 656.0 658.0 dataset Magnitude 6.0E-3 -6.0E-3 -5.0E-3 -4.0E-3 -3.0E-3 -2.0E-3 -1.0E-3 0.0E+0 1.0E-3 2.0E-3 3.0E-3 4.0E-3 5.0E-3 Real 1.5E-2 0E+0 2.5E-3 5E-3 7.5E-3 1E-2 1.25E-2

ALL dataset Nyquist selected 110.00 90.00 92.00 94.00 96.00 98.00 100.00 102.00 104.00 106.00 108.00 Displacement [m] 2.00E-6 0.00E+0 5.00E-7 1.00E-6 1.50E-6

Plot 3 3

It is possible to use two of the system of equation (4) to calculate both m and k and therefore possible to calculate a natural frequency for the equivalent linear system. By repeating this calculation for ever set of three frequency points, as explained earlier, it is possible to track the natural frequency against the vibration amplitude. Figure 7a shows the four traces of the natural frequency of the system. The red branch refers to the natural frequencies calculated by sweeping the FRF branch before the jump, from low to high frequency. The black branch refers to the frequency points after the jump, from high to low amplitude. Figure 7b shows a similar behaviour of the natural frequency, but the curves were calculated using equation (5). The difference between the two plots is about the way the curves are constructed, which requires to isolate the meaningful solution from the results of equation (4) and, therefore, the curve is built by composition of the upper and lower branch. The use of equation (5) leads to the complete calculation of the natural frequency curve. One can observe in Figure 7b the red curve. Interestingly, this curve also shows that by sweeping the data points the two branches before and after the jump do not overlap but create a kind of hysteresis. This behaviour will be more evident during the analysis of the single DoF system including both stiffness and damping nonlinearity.

(a) Analysis carried out by using equation (6) (b) Analysis carried out by using equation (7)

Figure 7 Natural frequency traces

The data of the natural frequency from the graph in Figure 7b was curve-fitted by a polynomial function to identify if the natural frequency of the underlying linear system was correctly identified and if the nonlinearity was also adequately identified. Figure 8 shows the four natural frequency curves fitted by a polynomial function, depicted in red colour. The nonlinear polynomial function identifying the nonlinearity is:

 

18 3 12 2 5

2.97 5.75 2.8 649.75

nl x e x e x e x

      (8)

The polynomial function of equation (8) captures the underlying linear natural frequency of the system correctly, 649.75Hz. It also identifies the cubic nonlinearity as initially used for the simulating the FRFs of the single DoF system but with an overestimated coefficient, -2.97e18_x3 _{instead of 1e}18_x3_.

658.00 649.00 649.50 650.00 650.50 651.00 651.50 652.00 652.50 653.00 653.50 654.00 654.50 655.00 655.50 656.00 656.50 657.00 657.50 Displacement [m] 2.00E-6 0.00E+0 5.00E-7 1.00E-6 1.50E-6

Plot 7 7 plot Frequency 658.00 649.00 649.50 650.00 650.50 651.00 651.50 652.00 652.50 653.00 653.50 654.00 654.50 655.00 655.50 656.00 656.50 657.00 657.50 Displacement [m] 1.75E-6 0.00E+02.50E-7 5.00E-7 7.50E-7 1.00E-6 1.25E-6 1.50E-6

LIN 0

Figure 8 Natural frequency traces fitted by polynomial function in red colour

The functions of the stiffness and the mass, which are used to generate the natural frequency of the system, present some interesting features. For instance, the mass function would be assumed constant for all the vibration amplitudes, but this is not the case. Furthermore, being a hardening nonlinearity, one would suppose that the stiffness function would also present higher stiffness for higher vibration amplitudes. Again, this is not the case either and Figure 9a&b show what was just explained. Both behaviours are presented for an FRF measured with an input force of 0.1N. Both stiffness and mass functions tend to converge to the

underlying linear system properties, such as m = 6kg and k = 1e8 _{N/m. In principle, the method identifies}

the nature of the nonlinearity correctly, but it is yet to understand why the mass is not a constant function and the stiffness presents an opposite behaviour than expected for a hardening nonlinearity.

(a) Stiffness function (b) Mass function

Figure 9 Stiffness and mass functions obtained from equation (4)

2.1.4 Stiffness and damping nonlinearity case study

The second case presented in this paper is the inclusion of both stiffness and damping nonlinearity in the single DoF system. Figure 10 shows both the FRF magnitudes in (a) and the Nyquist circles in (b). The inclusion of damping nonlinearity determines the change of the circle diameter for every excitation force, but it does not rotate the frequency points on the Argand diagram. The combination of stiffness and damping

660.00 649.00 650.00 651.00 652.00 653.00 654.00 655.00 656.00 657.00 658.00 659.00 Displacement [m] 2.00E-6 0.00E+0 5.00E-7 1.00E-6 1.50E-6

1.000E+8 9.760E+7 9.780E+7 9.800E+7 9.820E+7 9.840E+7 9.860E+7 9.880E+7 9.900E+7 9.920E+7 9.940E+7 9.960E+7 9.980E+7 Displacement [m] 2.50E-7 5.00E-8 1.00E-7 1.50E-7 2.00E-7

6.000E+0 5.860E+0 5.870E+0 5.880E+0 5.890E+0 5.900E+0 5.910E+0 5.920E+0 5.930E+0 5.940E+0 5.950E+0 5.960E+0 5.970E+0 5.980E+0 5.990E+0 Displacement [m] 2.50E-7 5.00E-8 1.00E-7 1.50E-7 2.00E-7

will rotate the circles and change its diameters. The procedure for analysing the frequency points is precisely performed as described in the previous case, and both equations (4) and (5) can be used for this.

(a) FRF magnitudes (b) Nyquist circles

Figure 10 FRF of a single DoF with stiffness and damping nonlinearity nonlinearity

The results for this stiffness and damping nonlinear system are presented in Figure 11 a&b. The observation should be drawn on the “hysteresis” feature of the curves as the sweeper inspects the frequency points of the FRF. One possible explanation is that the energy of the nonlinear response is not concentrated in its first fundamental frequency but spread across multiple harmonics. By neglecting the high order harmonics when generating the FRF one could claim that an energy loss is artificially created. Figure 11a is created by using the real part of equation (4). Instead, the Figure 11b shows the results obtained by using equation (5), which takes into account both real and imaginary part of the frequency points. The red curve in Figure 11b shows the analysis carried out on FRF simulated with excitation force 0.4N. This behaviour is understandable since the imaginary part which carries information about the damping is not included in the analysis. One could argue that for frequency identification the system of equation (4) could be sufficient.

(a) Analysis carried out by using equation 4 (b) Analysis carried out by using equation 5

Figure 11 Natural frequency traces

Finally, the viscous damping is calculated by measuring the circle diameter for every three frequency points selected. Such a calculation is possible because of the single DoF system analysis whereby the parameter c is: 1.0E-2 0.0E+0 2.0E-3 4.0E-3 6.0E-3 8.0E-3 Frequency [Hz] 660.0 640.0 642.0 644.0 646.0 648.0 650.0 652.0 654.0 656.0 658.0 dataset Magnitude 6.0E-3 -6.0E-3 -5.0E-3 -4.0E-3 -3.0E-3 -2.0E-3 -1.0E-3 0.0E+0 1.0E-3 2.0E-3 3.0E-3 4.0E-3 5.0E-3 Real 1.5E-2 0E+0 2.5E-3 5E-3 7.5E-3 1E-2 1.25E-2

ALL dataset Nyquist selected 656.00 649.00 649.50 650.00 650.50 651.00 651.50 652.00 652.50 653.00 653.50 654.00 654.50 655.00 655.50 Displacement [m] 2.00E-6 0.00E+0 5.00E-7 1.00E-6 1.50E-6

Plot 7 7 plot Frequency 656.00 649.50 650.00 650.50 651.00 651.50 652.00 652.50 653.00 653.50 654.00 654.50 655.00 655.50 Displacement [m] 1.75E-6 0.00E+02.50E-7 5.00E-7 7.50E-7 1.00E-6 1.25E-6 1.50E-6

Plot 3 3

1

c _D (9)

where D is the circle diameter. Figure 12 shows the damping behaviour against the vibration displacement. This paper will not go through the simulation of a nonlinear FRF based on the nonlinear parameters so far identified. Instead, it will select three values from the mass, damping and stiffness nonlinear functions and use them to calculate the regenerated FRF. The analysis is carried out on the assumption of linearity, which means that every three frequency points define a linear single DoF system for the given mi, ci, and ki.

Therefore, by using equation (1) to regenerate the FRF, the match with the nonlinear FRF will be at the extremes and one frequency point, as it is showed in Figure 13 for nonlinear FRF simulated with 0.4N excitation force. One can observe that such a match is qualitatively poor, but here the aim is to evaluate the peak response at any point of the natural frequency and nonlinear damping function.

Figure 12 Viscous damping trace

Figure 13 Comparison between simulated nonlinear FRF and a regenerated linear one

3

Conclusions and future work

This paper shows for the first time how to overcome the issue of the vibration response jump in the analysis of FRF Nyquist data. The method proposed is relatively simple, it is based on the idea that both the natural frequency and damping can be observed when two references and one sweeper are selected from the data set. Hence, three frequency points will always be available to define Nyquist circle of a linear system, from

155.00 95.00 100.00 105.00 110.00 115.00 120.00 125.00 130.00 135.00 140.00 145.00 150.00 Displacement [m] 2.00E-6 0.00E+0 5.00E-7 1.00E-6 1.50E-6

Plot 3 3 plot Frequency 0.01 1E-6 1E-5 0.0001 0.001 Frequency [Hz] 700 600 610 620 630 640 650 660 670 680 690 FRF- Magnitude measured and

which the modal parameters can be extracted. This new methodology is applied to two numerical cases of a single DoF system which was included with stiffness and damping nonlinearity. The results from the analysis revealed that the nonlinear functions describing the natural frequency and the damping could be retrieved from the simulated data. The method based on the proposed equations work adequately, but these are not sufficiently accurate. A forthcoming publication will present a new algorithm which will resolve both the frequency and nonlinear damping functions as well as the modal constant.

References

[1] D. J. Ewins, Modal Testing: theory, practice and application, no. 2000. 1970.

[2] P. Avitabile, “Insight into Modal Parameter Estimation,” Modal Test. A Pract. Guid., pp. 327–364.

[3] “ICATS.” ICON Suite, 58 Prince’s Gate Exhibition Road, London SW7 2PG.

[4] D. Göge, M. Sinapius, U. Füllekrug, and M. Link, “Detection and description of non-linear

phenomena in experimental modal analysis via linearity plots,” Int. J. Non. Linear. Mech., vol. 40, no. 1, pp. 27–48, Jan. 2005.

[5] A. Carrella and D. J. Ewins, “Identifying and quantifying structural nonlinearities in engineering applications from measured frequency response functions,” Mech. Syst. Signal Process., vol. 25, pp. 1011–1027, 2011.