Detection and characterization of local nonlinearity in a clamped‐clamped beam with state space realization

The state space representation of linear and nonlinear systems is widely used in the literature for system characterization, system identification, and model‐based control synthesis. In the case of systems with local nonlinearity, the realization of a state space model where the states are physically interpretable as a requirement has been a challenging task. This requirement becomes more emphasized, especially in the scope of industrial high‐precision systems. Consequently, the class of black‐box system identification approaches becomes less attractive. In the scope of this paper, we are interested in modeling systems with dominant local nonlinearity where employing linear models can only cover a limited range of system dynamics. More specifically, geometric nonlinearities which are present in the joints (bolted) of structural interfaces are analyzed. The main goal is to provide systematical modeling of such systems without using a sparse nonlinear representation. Such a low‐order nonlinear model can improve the simplicity of analyzing the nonlinear system and can be used for structural vibration and noise control. In order not to neglect the sophisticated linear modeling technique, the linear model is proposed to be extended by means of smooth nonlinear terms. The systematic approach contains the modeling of the linear counterpart followed by the localization and characterization steps in the well‐known three‐step paradigm. For the characterization step, this work relies on the acceleration surface method (ASM). The experimental setup under study, as a benchmark, is a set of two beams of different lengths and thicknesses connected by a screw that is excited by a mechanical shaker. The axes are oriented in the transverse direction of the beams, while the boundaries are realized as imperfect clamped‐clamped boundary conditions at two ends in the model. Consequently, by selecting the excitation amplitude, we can control the dominant dynamics at lower excitation amplitudes and invoke the local nonlinearity at higher amplitudes. For higher excitation levels using sine sweep signals, the phase space information of the shaker and accelerometer sensors is used to detect the local nonlinearities along the clamped‐clamped beam. The detected and characterized nonlinearities are incorporated into the linear system as a systematic approach for modeling such a structurally nonlinear system.

nonlinearities along the clamped-clamped beam.The detected and characterized nonlinearities are incorporated into the linear system as a systematic approach for modeling such a structurally nonlinear system.

INTRODUCTION
Structural vibration and noise control are important fields of engineering that focus on minimizing unwanted vibrations and noise in structures.In various industries, including aerospace, automotive, construction, and manufacturing, where excessive vibrations and noise can negatively impact performance, safety, and human comfort, structural vibration and noise control have become more essential.One of the key concepts related to structural vibration and noise control is the active control system.It utilizes sensors, actuators, and control algorithms to counteract vibrations or to reduce noise actively.These systems continuously monitor the vibrations or noise levels and generate appropriate signals to minimize their effects [1].Advantages of designing control strategies based on a thorough understanding of the system dynamics, of the flexibility to optimize control algorithms before implementation, and the potential for adaptive control strategies that can adjust to changing conditions lead to the development and implementation of mathematical models in the framework of the controller design became an important research topic.The so-called model-based control engineering related to more efficient and effective enhancement in minimizing vibrations and noise in various structures [2].
The requirements for the mathematical models in controller design have ascended, and now there are multitudinous number of modeling techniques [3].In the following paper, the focus will be developing a low-order model that captures the dominant linear and locally nonlinear dynamics which can be employed for reducing the modeling uncertainties in model-based control development.Hence, the nonlinear model is computationally cheap and in a closed-loop scheme can represent enhanced robustness against unwanted disturbances.Compared to high-order models, low-order models have fewer state variables and parameters.This results in simpler mathematical equations and reduces the computational burden associated with analyzing and simulating the model.It allows for faster computations, making it feasible to implement real-time control algorithms.However, the parameter identification of the mathematical model can be challenging and requires a large amount of data.Non-parametric system identification presents a formidable challenge due to its dependence on the meticulous design of experiments [4], ensuring the excitation of all nominal system dynamics.In the frequency domain, the applicable frequency range may extend, for instance, from 0 to 1000 Hz, as seen in active vibration control scenarios.Consequently, the acquisition of time-domain signals necessitates high sampling rates over extended durations to encompass both low and high-frequency dynamics effectively.Moreover, in the context of Multiple Input Multiple Output (MIMO) systems, the intricacies of experiment design become considerably more demanding due to the intricate interactions among individual inputs.This complexity often results in the accumulation of vast datasets within non-parametric nonlinear MIMO systems, posing a significant challenge.In contrast, the utilization of low-order nonlinear parametric models offers an elegant and efficient means of data representation.Such models not only provide a more graceful explanation of the data but also facilitate their application in tasks such as model-based control synthesis.
In this contribution, the low-order model of the clamped-clamped beam will be developed to capture the linear and nonlinear characteristics.To utilize the sophisticated linear modeling technique, the linear model is proposed to be extended through smooth nonlinear terms.The selection of the appropriate nonlinear model order depends on a trade-off between the model accuracy and the computational complexity of the local nonlinearity, that is, in the context of this paper this is realized by a screw connection.Applying the acceleration surface method (ASM) offers insight into this trade-off, allowing one to choose the proper model order for describing the nonlinear behavior.The determination of the nonlinear coefficients will be carried out by the conditioned reverse path (CRP) method.This approach flourishes by leveraging the principles of the reverse path (RP) methodology, thereby mitigating the necessity of exciting each individual response location.In the framework of the CRP method, the local nonlinearity is far away from the excitation point.A prerequisite for the utilization of the CRP method is that the excitation of the system exists at a significant distance from localized nonlinearity, a condition met by the experimental arrangement involving a beam clamped at both ends.Employing the CRP method enables the computation of nonlinear coefficients corresponding to the nonlinear function acquired through the ASM.Additionally, the CRP method facilitates the computation of the Frequency Response Function (FRF).

EXPERIMENTAL PROCEDURE
The development of a low-order model is illustrated on a clamped-clamped beam, presented in

LINEAR LOW-ORDER MODELING USING GENETIC ALGORITHM
The aim is the state space representation of the nonlinear system.In this section, the focus is first on the linear statespace presentation based on mass, stiffness, and damping matrix.For the initial value of the system parameters restricted in the space of the sensors, the substructure is generated using Abaqus Finite Element Analysis (FEA).For this purpose, preparations such as the geometric modeling of the clamped-clamped beam, assignment of the material parameters, and definition of the boundary condition are performed.Choosing the retained nodes, which are the location of the five sensors, and allowing their transversal orientation of the beam, the substructure procedures are performed to obtain reduced-order mass and stiffness matrix.Calculating the Rayleigh damping based on the mass and stiffness matrix, the FRF is calculated based on the state space model in conjugation with system parameters.
To generate the validation data, the experimental modal analysis is performed.In the framework of linear modeling, the excitation signal has to be chosen in such a way that the nonlinearity of the system remains unaffected.However, Experimental studies have shown that nonlinear motions are detected at small excitations.For this reason, the concept of best linear approximation (BLA) developed by Pintelon and Schoukens [5] for calculating the FRF is used.This method quantifies the level of nonlinear distortion in the FRF and analyzes the impact of the process and measurement noise on the BLA.One requirement of the excitation signal is the application of designed periodic excitation like the multisine excitation This equation calculates the time-domain representation of a signal () by summing up contributions from different frequency components  obtained from the Fourier domain at different time instances , using complex exponentials.The complex exponential term is the kernel of the summation and represents sinusoidal functions at discrete frequencies:  represents the imaginary unit √ −1, 2 is used to ensure that the frequency is specified in radians per second,   represents the maximum frequency component you are interested in within the signal,  is the total number of samples in the signal, the frequency index  is currently being summed over,  is the time variable at which you are evaluating the signal.The sum formula of the equation indicates that you are taking a summation over a finite range of values for the index variable .The range is from − to , where  is a positive integer.The fraction at the beginning of the equation ensures that the amplitude of the output signal is normalized when converting from the frequency domain to the time domain.
In Figure 2, The FRF is shown together with the noise variance and total variance.The latter are the dominating errors in this benchmark.Two nonlinear distortion curves are plotted.The first shows the actual level of the distortions as they are present at the output of the system.However, to estimate the model, the data that are averaged over the five realizations.Although averaging does not eliminate the presence of the nonlinear distortions, their impact on the variance of the estimated FRF of   is reduced and it is the latter variance that should be used during the identification since the averaged data are used in the identification step [6].However, in order to mitigate the disparity between the measured and calculated FRFs, the genetic algorithms was employed within the Matlab software environment.This approach facilitated the optimization of system parameters (mass , stiffness, and damping matrix denoted as M, K, and D respectively) to minimize the divergence between the FRFs.The experimental data in Figure 1, consisting of one input and five corresponding outputs, were first implemented into the software.This input-output dataset was then utilized to compute the five linear FRFs.These FRF served as the objective functions for minimization within the Genetic Algorithm.Subsequently, a series of arrays were defined encompassing potential values for the mass, stiffness, and damping matrices.Initial values were assigned to these matrices, acquired through the Finite Element Method.Utilizing these matrices, a state space model was formulated in the subsequent step.From this model, the five FRFs were derived in the final stage.Employing the defined objective functions, the deviation was assessed via the root mean square approach.Should the deviation be less than 30%, the calculated mass, stiffness, and damping matrix values were deemed satisfactory.If the deviation exceeded this threshold, a Genetic Algorithm optimization process was initiated.Within the Genetic Algorithm process, steps 1 through 5 were reiterated.Noteworthy parameters governing the Genetic Algorithm include a Crossover Fraction of 0.8, a Population Size of 800, a Stall Generation count of 900, and a maximum number of Generations set at 1000.The outcomes of the Genetic Algorithm optimization are visually depicted in Figure 3.It demonstrates that the FRF derived from the state-space model closely approximates the FRF established through experimental modal analysis (Figure 4).Given the intended application of this low-order model within an active real-time control system, the disparity in accelerations within the time domain was also examined, particularly within the time interval spanning from 75.5 to 80 s, as depicted in Figure 5. Collectively, the level of concurrence observed amounts to approximately 76%.
F I G U R E 3 FRF based on the optimized system parameters closely approximates the FRF based on the experimental modal analysis.FRF, Frequency Response Function.

F I G U R E 4
The genetic algorithm for optimizing the mass, stiffness and damping matrix.

F I G R E 5
The accelerations measured along the beam (blue line) are compared with those from the simulation model (red dashed line).The quantitative assessment of agreement between these two datasets yields an approximate concurrence of 76%.

ACCELERATION SURFACE METHOD
The nonlinear modeling consists of the characterization step of the local nonlinearity and the estimation step for estimating the nonlinear coefficients.In this section, the focus is on the characterization of local nonlinearity around the screw connection.Therefore, the ASM [7] is suggested enabling a qualitative evaluation of the nonlinearity through visualization of the acceleration over the surface spanned by the relative displacement and velocity.The foundation of the ASM is Newton's second law of motion.In the vicinity of the nonlinearity, Newton's second law of motion can be formulated as follows: where   describes the total amount of Degrees of freedoms (DOF) in the system,  , are the mass matrix coefficients, , ẋ, ẍ describe the displacement, velocity and acceleration vectors,  is the restoring force vector and  is the external force vector.Neglecting all the inertia and restoring force contributions that have no direct influence on the nonlinear component and focussing on the local nonlinearity between two locations, the equation will simplify to If no force is applied to DOF  and  and if the mass coefficient is neglected, a simple transformation results in The nonlinear order can be determined graphically by evaluating (4) near the local nonlinearity, plotting the accelerations to their corresponding relative displacements and velocities.A three-dimensional figure is obtained whose axes describe the acceleration, relative velocity, and relative displacement.To allow interpretation of this figure, a cross-section is generated along the velocity axis and corresponding accelerations and displacements are plotted that are relatively small to a threshold value (typically a few percent of the maximum velocity values).For the clamped beam, the accelerations and relative velocities are plotted on a graph within the threshold value.For this purpose, measured data are generated with an The ASM is applied for different excitation levels (3.5, 6, 14, and 17 N) to investigate the smooth nonlinear behavior of the screw connection.The point cloud is based on displacement and acceleration data that pertain to within 1% of the maximum velocity (0.0448, 0.0786, 0.1441 and 0.2367 m/s).ASM, acceleration surface method.
excitation signal of 3.5, 6, 14 and 17 N and in a frequency range of 22 to 26 Hz, which includes the first resonance frequency.
In the next step, the point cloud is approximated by a various order of polynomial function.The square error is calculated between the point cloud and each polynomial function of a different order.The best fit between the point cloud and the polynomial function is selected, and dependent on the chosen order of the polynomial function, the linear equation will be extended.In the experiment of the clamped-clamped beam, the best fit of the point clouds is approximated by the second and third polynomial order functions, presenting in Figure 6.

CRP METHOD
The CRP method [8] extends the application of the RP algorithm [9] by relaxing the condition of exciting every measurement point.Both methods are proper for calculating the FRF of the underlying linear system and for estimating the nonlinear coefficients.Concerning the clamped-clamped beam, the smooth nonlinearity of the screw connection can be described with the second and third polynomial order function according to the ASM, and the CRP method is applied to determine the nonlinear coefficients.For the sake of completeness, the nonlinear equation of motion is presented: The equation of motion ( 5) is transformed into the frequency domain with the help of the Fourier transform.The requirement is the existence of the Fourier transform for each term of the equation of motion: where  is the linear dynamic stiffness matrix,  is the Fourier transform of the generalized displacement vector,  Fourier transform of the generalized force vector.The nonlinear term of the equation of motion is expressed as the sum of  components, and each of them depends on a nonlinear function vector  through a coefficient matrix .In the framework of the RP method, every response location has to be excited, otherwise, the method is not applicable.The CRP method relaxes this condition and estimate the nonlinear coefficients if the excitation is far away from the local nonlinearity.The CRP method used the relationship among the power spectral densities of displacement, nonlinear vectors and forces, and the cross-spectral density matrix between the th and th nonlinear vector () to construct the hierarchy of uncorrelated response components in the frequency domain.So the conditioned power spectral density matrices  (−1∶) [10] between two nonlinear vectors  and  can be calculated by recursive algorithm starting with  = 1: where  < , , the subscript −1 ∶  − 1 means PSD of the part uncorrelated with the spectra of nonlinear vectors from the first through the ( − 1)th.Conditioned PSD matrices involving the excitation vector and response vector are obtained in a F I G U R E 7 Amplitude and phase of the conditioned and unconditioned FRFs.FRF, Frequency Response Function.
similar manner, substituting the subscripts i and/or j with the subscripts X and F in the equation.The dynamic compliance matrix of the system can be finally computed by means of  1 or  2 estimation procedures: 2 ∶   =  −1 (−1∶)  (−1∶) The coefficient matrices can be estimated by Finally, the underlying linear model (ULM) based on the classical  1 function is compared w.r.t. the recovered conditioned FRFs using the CRP method.The magnitude and the phase are shown in Figure 7 for the outputs  5 .As expected, the classical estimation of the FRFs based on  1 , is distorted under the effect of invoked nonlinearity.However, the conditioned spectral analysis based on the CRP recovers the ULM correctly.

CONCLUSION
In this contribution, a nonlinear low-order model in state-space representation is successfully developed, representing the dynamic of the clamped-clamped beam in the space of the sensor.The presented technique is a superposition of the linear and nonlinear model.A linear low-order model is generated using the substructure generation procedure of

F I G U R E 1
Schematic configuration of the measurement setup (left), picture of the clamped-clamped beam (right).

Figure 1 .
The experimental setup studied consists of two beams of different lengths and thicknesses connected by three screws.The boundary conditions for the respective ends of the beam are realized as imperfect clamped boundary conditions.For the low-order modeling, different force excitation () are generated for the electrodynamical shaker, exciting the clampedclamped beam.The resulting accelerations  1 (),  2 (),  3 (),  4 () and  5 () are recorded at five locations along the clamped-clamped beam.The acquired signals are the basis for the subsequent approaches in this contribution.

F I G U R E 2
The FRF computed from the I/O signal (black line) is compared with the FRF of the initial state space model (violet dashed line) based on the FEA substructure generation.FRF, Frequency Response Function; I/O, Input/Output.