Validation of a model calibration method through vibrational testing …michael.friswell.com/PDF_Files/C407.pdf · 2016. 10. 4. · Validation of a model calibration method through

Validation of a model calibration method through vibrational testing of a mechanical system with local clearance

Y.Chen1, A. Linderholt1, T. Abrahamsson2, Y. Xia3, M.I. Friswell4 1 Linnaeus University, Department of Mechanical Engineering 35195, Växjö, Sweden e-mail: [email protected] 2 Chalmers University of Technology, Department of Applied Mechanics 41296 Göteborg, Sweden 3 University of the West of England, Department of Engineering Design and Mathematics BS16 1QY, Bristol, UK 4Swansea University, College of Engineering Swansea SA1 8EN, UK

Abstract Nonlinear finite element models are often validated using experimental data. A previously proposed calibration method, which concerns pre-test planning, multi-sinusoidal excitation and an effective optimization routine, is improved with an extended version of the pre-test planning. The improved method is validated using a test structure with a clearance type nonlinearity. From the pretest planning, an optimal configuration for the data acquisition is determined. The multi-harmonic nonlinear frequency response functions (FRFs) of the structure under test are then generated by a multi-sinusoidal excitation. Model calibration is conducted by minimizing the difference between the experimental multi-harmonic nonlinear FRFs and their analytical counterparts. The uncertainties of the estimated parameters are assessed by a k-fold cross validation, which confirm that the uncertainties of the estimated parameters are small when the optimal configuration is applied.

1 Introduction

All mechanical systems display a certain degree of nonlinearity due to nonlinear material properties, large displacement, clearance, etc. In industry, linear finite element (FE) models are extensively used to represent the global behavior of mechanical systems, in which nonlinear effects are often neglected to simplify the model. When test data show significant nonlinear behaviour, the prediction of a linear FE-model may become erroneous. In this case, it is essential to take the nonlinearity into account. The nonlinearities are often localized at boundaries and connections of mounting interfaces. To present the dynamic behavior of structures with local nonlinearities, a common approach is to extend linear FE-models by adding nonlinear elements. The nonlinear parameters often suffer from a large amount of uncertainty and they must be calibrated using experimental data. This procedure is called nonlinear FE-model calibration and has interested researchers in the field of structural dynamics in the past decades [1-3]. Pretest planning is a vital part of successful vibrational testing especially for a complex structure and a limited number of sensors and actuators. Finding the optimal sensor placement (OSP) is one of the most important pretest steps, which has been extensively conducted in modal analysis. Many sensor placement

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techniques based on the information obtained from the nominal FE model have been developed [4-6]. The effective independence method (EFI), based on the Fisher information matrix (FIM), is one of the most well-known methods to find the OSP. In [4], the FIM is defined as the product of the modal matrix and its transpose. While much research has focused on how to best set-up and perform modal property identification, there has been little discussion on pretest planning for nonlinear system identification. In [7], the OSP algorithm, based on principal component analysis, has been applied for the detection of nonlinear structural behaviour. The measured responses of a nonlinear system depend not only on the placement of sensors and actuators but also on the excitation levels and the excitation frequency range. Therefore, they should be carefully selected to obtain the most important dynamic information for nonlinear model calibration. In this paper, the previously proposed model calibration method [8] is improved by selecting an optimal configuration, involving the location of the sensors and actuators, the excitation levels and the frequency ranges, ranked by the FIM. The nonlinear model calibration method consists of three parts; the pretest planning, multi-harmonic excitation and an efficient optimization routine. The pretest planning gives a test design for measurements and forms a calibration metric for the optimization. The optimization is conducted by minimizing the difference between the calculated and measured multi-harmonic nonlinear FRFs using a selected starting point. To calculate the steady-state response, the time domain method may be inefficient for a lightly damped system; therefore the frequency domain multi-harmonic balance (MHB) method [9-11] is used here to calculate the numerical nonlinear FRFs. The implementation and experimental verification of the proposed model calibration method are demonstrated on a test structure with a clearance nonlinearity.

2 Model calibration method

The previously proposed model calibration method [8], which requires pretest planning, uses a multi-sinusoidal excitation and an effective optimization routine. The pretest planning part of the calibration method is extended in this paper, which is shown in section 2.1. To increase the accuracy of the prediction of the locally nonlinear model representing the nonlinear dynamics of the structure under test, the calibration is here divided into two phases; first a calibration of the underlying linear system (linear calibration) and then a calibration of the nonlinear part (nonlinear calibration). The linear part of the model is assumed to be captured after the linear calibration. Only the parameters representing nonlinear elements are then calibrated in the nonlinear calibration phase. The nonlinear calibration consists of the pretest planning, the measurements and model calibration/validation. In the pretest planning, the parameter identifiability is examined by use of the FIM and correlation indices. A test design for the measurements and a calibration metric for the optimization are defined. Multi-harmonic frequency response test data are then generated using a multi-sinusoidal excitation. Finally, a parameter starting point, selected among starting point candidates found by the Latin hypercube sampling method, is used for calibration and the optimal parameter setting found from the calibration is used as the starting point for a Ƙ-fold cross validation from which uncertainties of the parameters calibrated are obtained. The improved version of the previously proposed model calibration method is systematically shown in Fig. 1.

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Figure 1: An overview of the nonlinear model calibration process

2.1 Pretest planning

Generally, the aim of a pretest planning is to increase the likelihood of achieving valuable data from the tests planned. Such data maximize the identifiability of the uncertain model parameters. At this stage, the experimental data are not available and the baseline FE-model representing the test object has to be used in the pretest planning.

2.1.1 The extended pretest planning framework

For a nonlinear system, steady-state frequency response data depend on the excitation levels. It is computationally expensive to calculate or measure the nonlinear FRFs within a broad frequency range and frequency ranges where the nonlinear behaviour is not pronounced may be considered as non-informative with respect to the nonlinear parameters. Therefore, it is desirable to use a narrow and informative excitation frequency range for model calibration. The previously proposed pretest planning method mainly concerned the selection of excitation levels and frequency ranges [8]. Due to the limitations in number of sensors and actuators to be used, their configurations should also be considered in the pretest planning. In this paper, the four variables considered in the pretest planning are the actuator positions, the sensor locations, the excitation levels and the excitation frequency ranges. Examining all the possible combinations which can be used in the tests, removing one data set at a time, and recalculating the FIM based on the remaining data sets, is sub-optimal, and may result in measuring more data than necessary in the calibration. To use measured data more efficiently, the optimal configuration is found by three procedures:

2.1 Pretest planning 2.2 Measurements

2.3 Model calibration / validation

Latin hypercube sampling Calibration Ƙ-fold cross-validation

Calibrated locally nonlinear FE-model with parameter

uncertainties

Baseline FE model Structure under test

1. Calibration of the underlying linear part

2. Calibration of the nonlinear part

Parameter identifiability Correlation indices Fisher information matrix

Multi-sinusoidal excitation Test

design

Test data

Calibration metric

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1. Select candidates for actuators, sensors, frequency ranges and excitation levels. 2. Combine the candidates from all the variables to obtain the number of the possible configurations. 3. Calculate the resulting FIM for each candidate configuration. The configuration that results in the FIM which has the largest determinant is considered as the optimum. An overview of the pretest planning procedure is shown in Fig. 2.

Figure 2: An overview of the pretest planning procedure

2.1.2 Calculation of the FIM

Assume that the test data 𝒛𝒛𝑛𝑛 can be predicted without bias with a parameterized model in its calibrated setting 𝑷𝑷∗. In addition, assume a stationary, Gaussian residual 𝜺𝜺𝑛𝑛, with known and identical standard deviation 𝜎𝜎𝑛𝑛 for all entries. Let the predicted data from the model be 𝒛𝒛�𝑛𝑛, then the test data can be presented as

𝒛𝒛𝑛𝑛 = 𝒛𝒛�𝑛𝑛(𝑷𝑷∗) + 𝜺𝜺𝑛𝑛 (1)

The covariance matrix of the estimation error

𝜮𝜮 = 𝐸𝐸[(𝒛𝒛𝑛𝑛 − 𝒛𝒛�𝑛𝑛)(𝒛𝒛𝑛𝑛 − 𝒛𝒛�𝑛𝑛)T] (2)

should be small for the test data to be valuable. The inverse of the lower bound of the covariance matrix is known as the FIM, 𝓕𝓕, and can be expressed as

𝓕𝓕 = 𝜮𝜮−1 = �𝜎𝜎𝑛𝑛−2 �𝜕𝜕𝜕𝜕𝑷𝑷

𝒛𝒛�𝑛𝑛(𝑷𝑷)� �𝜕𝜕𝜕𝜕𝑷𝑷

𝒛𝒛�𝑛𝑛(𝑷𝑷)�T𝑁𝑁

𝑛𝑛=1

(3)

The derivation of Eq. (3) can be found in [12]. The FIM is useful in the study of data informativeness and parameter identifiability and it is here used to rank the usefulness of data sets.

2.2 FRFs of a multi-harmonically excited structure

The governing equation of an N degrees of freedom nonlinear mechanical system can be expressed as

𝑛𝑛A 𝑛𝑛S 𝑛𝑛F 𝑛𝑛E No. of candidates

𝑛𝑛A ∙ 𝑛𝑛S ∙ 𝑛𝑛F ∙ 𝑛𝑛E

No. of the possible configurations

Optimal configuration

Ranked by det(FIM)

Actuators Sensors Frequency ranges Excitation levels


𝑴𝑴�̈�𝒒 + 𝑽𝑽�̇�𝒒 +𝑲𝑲𝒒𝒒 + 𝒇𝒇NL(𝒒𝒒, �̇�𝒒) = 𝒇𝒇 (4)

where M, K and V are the N× N mass, stiffness and damping matrices, 𝒒𝒒, �̇�𝒒, �̈�𝒒 are the nodal displacement, velocity and acceleration vectors, 𝒇𝒇NL denotes the nonlinear force vector and 𝒇𝒇 is the external load vector. The excitation force, at degree of freedom (DOF) j, is designed to have harmonics of the fundamental frequency 𝛺𝛺, and can be expressed as

𝑓𝑓𝑗𝑗(𝑡𝑡) = Re��𝐹𝐹𝜐𝜐𝑒𝑒𝜐𝜐𝜐𝜐Ω𝑡𝑡𝜐𝜐�

𝜐𝜐=1

+ �𝐹𝐹1 𝜐𝜐⁄ 𝑒𝑒𝜐𝜐Ω𝑡𝑡 𝜐𝜐⁄𝜐𝜐

𝜐𝜐=2

�

(5)

where Re denotes the real part, 𝐹𝐹𝜐𝜐 and 𝐹𝐹1 𝜐𝜐⁄ are complex numbers, and (υ� − 1) and �𝜐𝜐 − 1� are the number of super-harmonic and sub-harmonic components included in the excitation force. The periodic steady-state time response at DOF 𝑖𝑖, if such exists, can be approximated as the series

𝑞𝑞𝜐𝜐(𝑡𝑡) = Re�� 𝑄𝑄𝜐𝜐𝑒𝑒𝜐𝜐𝜐𝜐Ω𝑡𝑡𝑁𝑁�

𝜐𝜐=1

+ �𝑄𝑄1 𝜐𝜐⁄ 𝑒𝑒𝜐𝜐Ω𝑡𝑡 𝜐𝜐⁄𝑁𝑁

𝜐𝜐=2

�

(6)

where 𝑄𝑄𝜐𝜐 and 𝑄𝑄1 𝜐𝜐⁄ are the harmonic coefficients of the displacement, and (𝑁𝑁� − 1) and �𝑁𝑁 − 1� are the number of super-harmonic and sub-harmonic components in the displacement. For a harmonic excitation, the nonlinear FRF between the response at DOF 𝑖𝑖 and the excitation at DOF 𝑗𝑗 is defined as

𝐻𝐻𝜐𝜐𝑗𝑗𝜐𝜐 (𝛺𝛺) =𝑄𝑄𝜐𝜐𝐹𝐹𝜐𝜐

(7)

𝐻𝐻𝜐𝜐𝑗𝑗1 𝜐𝜐⁄ (𝛺𝛺) =

𝑄𝑄1 𝜐𝜐⁄𝐹𝐹1 𝜐𝜐⁄

(8)

in which 𝐻𝐻𝜐𝜐𝑗𝑗𝜐𝜐 (𝛺𝛺) denotes the 𝜐𝜐:th-order nonlinear FRF with the fundamental frequency 𝛺𝛺 and 𝐻𝐻𝜐𝜐𝑗𝑗1 𝜐𝜐⁄

represents the 1 𝜐𝜐:⁄ th order harmonic nonlinear FRF. These harmonic FRFs are calculated by the receptance based MHB method described in [10].

2.3 Model calibration and validation

The nonlinear dynamics of a locally nonlinear structure can be captured more efficiently if the underlying linear system of the structure is calibrated first, since the number of unknown parameters is then reduced in the nonlinear calibration. Thus, the model calibration of nonlinear FE-models is done in two phases; the calibration of the linear part and the calibration of the nonlinear part. For the calibration of the linear part, a random excitation with a low excitation level is used to excite the linear dynamic of the structure under test, and the linear FRFs are measured. The linear calibration is done by minimizing the deviation between the calculated and measured linear FRFs. When calibrating the nonlinear part, many starting points are examined to avoid a local minimum. However, the number of starting points that can be evaluated is limited due to the computational expense. Therefore, the Latin hypercube sampling method is used to select a limited number of starting seed candidates that span the parametric space. Then the candidate for which the calibration metric has the smallest value is used as a starting point for the optimization of the nonlinear calibration. Linear and nonlinear calibrations are both done through optimizations using the gradient-based Levenberg-Marquardt optimization method [13].


2.3.1 Nonlinear calibration metric

The nonlinear model calibration minimizes the deviation between the analytical multi-harmonic nonlinear FRFs and the experimental counterparts. The objective function is defined as

𝛿𝛿 = 𝜺𝜺𝐻𝐻𝑾𝑾𝜺𝜺 (9)

where W is a positive definite weighting matrix (an identity matrix is used for W in this paper) and 𝜺𝜺 is

𝜺𝜺 = 𝒗𝒗HA − 𝒗𝒗Hx (10)

where superscripts A denotes analytical and X denotes experimental and 𝒗𝒗𝐻𝐻 is

𝒗𝒗𝐻𝐻 = �Re �𝑣𝑣𝑒𝑒𝑣𝑣𝑡𝑡�𝐻𝐻1 𝜐𝜐⁄ ;⋯ ;𝐻𝐻1 2⁄ ;𝐻𝐻1;𝐻𝐻2;⋯ ;𝐻𝐻𝜐𝜐�� ; Im �𝑣𝑣𝑒𝑒𝑣𝑣𝑡𝑡�𝐻𝐻1 𝜐𝜐⁄ ;⋯ ;𝐻𝐻1 2⁄ ;𝐻𝐻1;𝐻𝐻2;⋯ ;𝐻𝐻𝜐𝜐�� (11)

In Eq.(11), Im denotes the imaginary part, vect denotes vectoring and 𝐻𝐻𝜐𝜐� includes all the �̅�𝜐:th order nonlinear FRFs resulting from different measurement points.

2.4 Ƙ-fold cross-validation

A Ƙ-fold cross-validation is used to obtain an estimate of the parameter statistics after the nonlinear parameters are calibrated. In the Ƙ-fold cross-validation, the available data set is partitioned into Ƙ equally sized subsamples. Here, one subsample is used for validation while the remaining Ƙ-1 subsamples are used for calibration, which results in Ƙ calibration runs. From these, Ƙ sets of calibrated parameter settings that minimize the calibration deviation metric for the Ƙ different partitions of the available data are obtained. These can then be used for statistical evaluation of the mean and covariance of the parameter estimates. The optimum parameter setting found by the nonlinear calibration is used as the starting point for the Ƙ-fold cross-validation and the optimizations are done by using the undamped Gauss-Newton method [13].

3 Case study

The improved nonlinear model calibration method is used to experimentally calibrate a nonlinear FE-model representing a structure with a clearance nonlinearity.

3.1 The structure under test and experimental setup

The test rig used to study the method consists of a cantilever beam with a rubber bump stop at its free end, see Fig. 3. The cantilever beam is supported by a U-shaped steel frame, mainly made of square hollow tubes with an outer side length of 80 mm and a wall thickness of 6 mm. The cantilever beam is 633 mm long with a cross section of 12 x 30 mm2. The test structure is supported by four soft rubber mounts. The test structure is excited using a modal shaker of type 2025E at the measurement point number 3, and 13 PCB accelerometers of type 352A56 are used to measure the responses. The measurement points are marked with red numbers in Fig. 3 and their positions are presented in Table 1. The nonlinear FE-model of the test rig is shown in Fig. 4.


3.2 Modelling and calibration of the linear part of the test structure

The linear part of the test structure is calibrated prior to the full system calibration by minimizing the deviation between the analytical and measured FRFs with damping equalization, see [14]. The test structure is excited, using the shaker with a random signal, at point number 3, and all 13 accelerometers are used to measure the responses. The root mean square (RMS) value of the random excitation is selected to ensure that the gap nonlinearity is not engaged.

Figure 3: (a) The structure under test (b) Closer perspective on the gap mechanism

Figure 4: The FE-model of the structure under test

The flexibility of the frame is included in the model. The frame is modelled using 42 simple beam elements (CBAR), within MSC Nastran, while the cantilever beam is modelled by 13 CBAR elements. The cantilever beam is rigidly connected to the frame by a Nastran RBE2-elelment. The supports (rubber mounts) are modelled using two linear springs with the same stiffness coefficient 𝑘𝑘𝑠𝑠. The mass of the frame is 20.78 kg and the mass of the cantilever beam is 1.98 kg. To obtain the same mass of the model, a density of 7795 kg m3⁄ is used for the frame and a density of 7901 kg m3⁄ is used for the cantilever beam. There are some uncertainties in the mass and stiffness properties of the connection corners of the frame (each corner is represented using three beam elements, which are plotted using red lines in Fig. 4)

𝒇𝒇(𝑡𝑡)

𝒒𝒒6 𝒒𝒒4 𝒒𝒒7

𝑘𝑘𝑠𝑠 𝑘𝑘𝑠𝑠

𝒒𝒒5 𝒒𝒒3 𝒒𝒒2 𝒒𝒒1

𝒇𝒇NL

�𝒇𝒇NL = 𝑘𝑘𝑐𝑐(|𝒒𝒒7| − 𝑑𝑑)1.5 − 𝑣𝑣�̇�𝒒7|�̇�𝒒7|, for 𝒒𝒒7 < −𝑑𝑑

𝒇𝒇NL = 0, otherwise

𝑑𝑑 𝑘𝑘𝑐𝑐 𝑣𝑣

x

y 1 2 3 4 5 6 7

8

9 10 11 12

13

(a) (b)


Young’s moduli of the frame, the connection corners and the cantilever beam, the density of the connection corner and the stiffness coefficient 𝑘𝑘𝑠𝑠 are used as the five parameters in the calibration. The nominal and the calibrated parameter values are shown in Table 2. Nine out of 13 measured FRFs are used in the calibration and the FRFs from the rest of the measurement points are used for validation. An example of FRFs from the test data, the calibrated and nominal FE-model of the linear part of the structure is shown in Fig. 5a and the corresponding validation result is shown in Fig. 5b.

Sensor No. 1 2 3 4 5 6 7 Coordinates (100, 0) (200, 0) (300, 0) (400,0) (500, 0) (600,0) (633, 0)

Sensor No. 8 9 10 11 12 13 Coordinates (0,-47) (0,-179) (35,-219) (342,-219) (670,-179) (670,-47)

Table 1: Sensor distances given as (∆𝑥𝑥,∆𝑦𝑦) in relation to the origin shown in Fig. 3a

3.3 Modelling and calibration of the nonlinear dynamics of the structure under test

From the calibration results above, the characteristics of the linear part of the test structure are assumed to have been captured. The next task is to model the nonlinearity due to the contact. The contact interface between the tip of the beam and the bump stop is represented using a Hertzian contact model with contact stiffness 𝑘𝑘𝑐𝑐 and a quadratic damping presented in Fig 4. The contact stiffness 𝑘𝑘𝑐𝑐 (parameter 1), gap d (parameter 2) and quadratic damping coefficient 𝑣𝑣 (parameter 3) are considered as the three unknown parameters to be calibrated. The nominal values of the parameters are shown in Table 5. An example of the nonlinear multi-harmonic FRFs between response at measurement point 7 and a multi-harmonic external force, which has the fundamental amplitude 8N, applied at measurement point 3, is shown in Fig 6.

3.3.1 Pretest planning

Before measuring the nonlinear dynamics of the test structure, the nominal nonlinear model is used to select an optimal configuration to obtain the most valuable test data to be used for the nonlinear calibration. In this case, the location of three sensors, the location of one actuator, three different excitation levels and one frequency range are selected from the candidate sets shown in Table 3. The first resonance of the underlying linear system appears in the first frequency range, [21-40] Hz, while the second resonance is within the last set [141-160] Hz. When an actuator operates with three different excitation levels in a frequency range, the nonlinear multi-harmonic FRFs, measured using three sensors, are considered as one data set. To select 3 sensor locations from 7 candidates gives 35 combinations. Similarly, 3 combinations are provided by the candidate actuator positions, 20 combinations of the excitation levels and 7 combinations of frequency range. Combining these 4 terms (selections of the sensor locations, actuator location, excitation levels and frequency ranges) gives 14700 data sets in total. The FIM is calculated for each data set. The first five data sets with the largest determinant of the resulting FIM, which gives the smallest covariance matrix of the estimated parameters, are shown in Table 4. It is noted that No. 2 and No. 1 configurations give nearly the same determinant of the FIM. Additionally, the shaker is connected to an impedance head which can measure force and acceleration at the same time. Therefore, configuration No. 2 is selected for the nonlinear model calibration. After the measurement configuration is selected in the pretest planning process, the inverse of the FIM and the MAC of the corresponding Jacobian matrix are calculated. The inverse of the FIM is


FIM−1 = 1 ∙ 10−10 ∙ �0.35 −0.16 −0.19−0.16 0.13 0.02−0.19 0.02 0.78

� σ2

The inverse of the FIM, the parameter covariance matrix, shows that the unknown parameter values can be calibrated with small uncertainties under the assumption that σ2 is small. From the measured data the standard deviation, σ, of the measurement noise is found to be smaller than 2% for the studied test design. Figure 7a shows that 𝑃𝑃1 is correlated with 𝑃𝑃2 whereas 𝑃𝑃1 has a very small correlation with 𝑃𝑃3. The correlation value between 𝑃𝑃1 and 𝑃𝑃2 is equal to 0.56 To improve the identifiability of 𝑃𝑃1 and 𝑃𝑃2, the steps of excitation frequency are decreased to 0.1 Hz around the first resonance of the underlying linear system; i.e. the range 24-28 Hz. Then, the correlation value between 𝑃𝑃1 and 𝑃𝑃2 is reduced to 0.45.

E of the frame

(GPa)

E of the connection

corners (GPa)

E of the beam

(GPa)

Density of the connection

corners (kg/m3)

Stiffness of the rubber mounts

(N/m)

Nominal 210 150 210 4000 10000 Calibrated 205 128 201 5175 57989

Table 2: The nominal and calibrated parameter values used for the linear parts of the test structure

Figure 5: An example of the FRFs from the nominal, calibrated model representing the underlying linear system compared with the corresponding experimental FRFs. (a) calibration result (b) validation result

3.3.2 Measurement of nonlinear dynamics of the structure under test

Using the result from the pretest planning, the multi-harmonic nonlinear FRFs excited using multi- sinusoidal excitations were obtained using a National Instruments data acquisition system with a Matlab interface. The experimental setup for obtaining the multi-harmonic nonlinear FRFs is shown in Fig. 8. The gap between the free tip of the cantilever beam in rest and the bump stop was kept to 0.60 mm during the measurements. The known value of the gap, which is a parameter used in the calibration, is used to validate the success of the calibration. Point 3 was used for the excitation and the responses at points 3, 6 and 7 were measured. The multi- sinusoidal excitation was designed to contain 2 super-harmonics (the second and third harmonics) together

(a) (b)


with the fundamental harmonic, in which the fundamental harmonic stepped from 21 to 40 Hz, with steps of 1 Hz. The excitation frequency steps are decreased to 0.1 Hz for the frequency range 24-28 Hz which is around the system’s first resonance. The magnitudes of the two side harmonics are small compared to the fundamental but large compared to the expected noise level [8]; the magnitude of the side harmonics’ excitation is chosen as 2% of the magnitude of the fundamental. The measurement was repeated three times for the fundamental excitation levels 2, 4 and 6N. Sensor locations Actuator positions Excitation levels [N] Frequency ranges [Hz]

[1, 2, 3, 4, 5, 6, 7] [1, 2, 3] [1, 2, 3, 4, 5, 6] [21-40], [41-60], [61-80], [81-100], [101-120],[121-140],[141-160] Table 3: The candidate sets for the sensor locations, actuator positions, excitation levels and frequency ranges.

Figure 6: An example of the multi-harmonic nonlinear FRFs compared with the underlying linear FRFs.

The nonlinear FRFs are plotted by red dots while the linear FRFs are plotted as blue solid lines. (a) fundamental nonlinear FRF (b) second order nonlinear FRF (c) third order nonlinear FRF.

Actuator position Excitation levels

(N) Frequency range

(Hz) Sensor position Determinant of the FIM

1 3 [2, 4, 6] [21-40] [4, 6, 7] 7.57e+31 2 3 [2, 4, 6] [21-40] [3, 6, 7] 7.36e+31 3 3 [2, 4, 6] [21-40] [5, 6, 7] 6.72e+31

4 3 [2, 4, 6] [21-40] [3, 4, 7] 5.69e+31 5 3 [2, 4, 6] [21-40] [4, 5, 7] 5.12e+31

Table 4: The first five selected configurations from the nominal model


(a) (b)

Figure 7: The correlation indices of the parameters included in the nonlinear calibration. (a) correlation indices with step size 1 Hz. (b) The correlation indices with a refined step size

It is difficult and time consuming to control the amplitude of the harmonics of the excitation force to be perfectly constant for each frequency step. Therefore, the amplitude is controlled with a few iterations by an off-line feedback force controller which dramatically shortens the measurement time and the amplitudes of the applied force are kept almost constant for each excitation level. To keep the accuracy of the measurement data, the applied force amplitudes were measured and used to calculate the analytical multi-harmonic nonlinear FRFs.

Figure 8: The experimental setup for nonlinear testing

3.4 Nonlinear calibration

For the evaluation of optimization starting points, it is assumed that the contact stiffness (𝑃𝑃1) has a uniform probability distribution between 105 and 107, the gap (𝑃𝑃2) follows a beta distribution with a mean values equal to the corresponding value of the nominal model and the standard deviation is set to 50% of the mean value. Further, the quadratic damping coefficient (𝑃𝑃3) has a uniform distribution between 1 and 102. The Latin hypercube sampling method is used to select one hundred parameter realizations which span the parametric space. The multi-harmonic nonlinear FRFs of these realizations are calculated and compared to the experimental nonlinear FRFs. Among the 100 candidate starting points, the point giving the smallest value of the objective function is used as the starting point for calibration. The calibrated parameter values

Matlab

Off-line Force Controller

Signal Generator

Amplifier Shaker National Instruments Data Acquisition System

Force Transducer and Accelerometers

Multi-sinusoidal input


and the measured gap value are shown in Table 5. An example of comparisons of the calculated multi-harmonic nonlinear FRFs from the calibrated model and the measured ones is shown in Fig. 9. Evidently, the calculated multi-harmonic nonlinear FRFs show good agreement with the measured FRFs and the calibrated value of the gap parameter deviate from the known value by less than 1.7%, indicating that the calibrated nonlinear model can be used to represent the nonlinear dynamics of the test structure. To validate the calibrated nonlinear model, the multi-harmonic nonlinear FRFs are calculated for the excitation level 8N, which was not included in the calibration, and compared with the measured counterparts. The validation results are shown in Fig. 10.

3.5 Cross validation results

The data set is equally divided into 9 subsamples by the excitation levels and the sensors; one subsample is used for validation, and the remaining 8 subsamples are used to calibrate the model which results in 9 calibration runs. Each subsample consists of the multi-harmonic nonlinear FRFs measured from one measurement point excited by one excitation level. The cross validation results can be seen in Fig. 11. From the calibration results shown in Fig. 9 and the validation results shown in Fig. 10, the improved calibration framework is able to accurately identify unknown parameter values from the experimental data. The cross validation results show that the calibrated parameters have small uncertainties.

4 Conclusions

The previously proposed nonlinear model calibration method, which consists of a pretest planning, a multi- sinusoidal excitation and an effective optimization routine, is improved and validated in this paper. The improvement is done through calibrating linear and nonlinear parts separately, and extending the pretest planning process. The improved method is validated through model calibration and validation using experimental data from a structure with clearance. The calibration of the linear part used the linear FRFs from a test with a random excitation at a small force level. The optimal configuration selected by the extended pretest planning method, was used to determine a test design for measuring the nonlinear dynamics of the structure under test. Then the parameters that represent the gap mechanism were calibrated using the multi-harmonic nonlinear FRFs found from tests where the structure was excited by a multi- sinusoidal force. The calibration and validation results show that the improved method is able to calibrate a nonlinear model, which represents a structure with a gap nonlinearity, with a good prediction accuracy and small uncertainties.

𝑃𝑃1 (contact stiffness) [N/m1.5]

𝑃𝑃2 (gap) [mm]

𝑃𝑃3 (quadratic damping coefficient)

[N/(m/s2)2] Nominal 1.00 × 106 1.00 10.00

Calibrated 4.51 × 105 0.61 25.21 Measured - 0.60 -

Table 5: The nominal, calibrated and measured parameters of the nonlinear model


(a)

(b)

(c)

Figure 9: Comparison of calculated and measured multi-harmonic nonlinear FRFs for the excitation levels that are included in the calibration process. (a) excitation level 2N (b) excitation level 4N (c) excitation level 6N


Figure 10: Comparison of calculated and measured multi-harmonic nonlinear FRFs for the excitation level 8N.

Figure 11: The calibration results from the 9-fold cross-validation. The nine estimated parameter

realizations are normalized to the calibrated parameter values. The blue bars represent the normalized estimated parameter values for each calibration and the red line represents the normalized values of the

parameters calibrated.


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