Nonlinear Frequency Response Functions Research Articles

Nonlinear frequency response function: experimental study on gearbox fault detection under step load

For rotating machinery, excitation signals are difficult to measure, making it impossible to detect fatigue damage using nonlinear frequency response function. The purpose of this study is to propose a fatigue damage detection method for gearboxes under stepped load conditions based on normalized nonlinear output frequency response functions (NOFRFs). That is, using the vibration signals from the gearbox during the step loading to calculate the cross power spectrum, and therefrom estimating the normalized NOFRFs. This method is used to detect gearboxes containing prefabricated damaged gears or bearings. The research result shows that gear and bearing damage can lead to an increase in the index value of normalized NOFRFs. The different type of damage lead to the significant diversification of higher-order normalized NOFRFs, which further demonstrates that the method proposed in this article is not only effective in gearbox damage detection, but also has certain potential in fault classification. Then, experimental research for accumulated fatigue damage of gear was conducted. The results show that the normalized NOFRFs index is a better indicator of fatigue damage than the kurtosis index.

Exploratory Structural Modification and Nonlinear Based Analysis of Tuned Mass Dampers

This article presents a structural modification-based analysis of tuned mass dampers that motivates to investigate the performance of a cubic stiffness absorber. The rationale behind this investigation is that the analysis of vibration absorber using structural modifications could pave the way for the use of nonlinear mechanical elements in vibration absorption applications. To achieve this, the article reviews the fundamentals of structural modification-based analysis of absorbers. It then examines the effects of using the Duffing oscillator as an absorber. The Duffing oscillator is integrated into the primary system using the describing function and the amplitude-dependent nonlinear frequency response functions are computed. The results of this analysis are compared with those of a linear absorber in order to clarify the differences and point out possible directions for further development.

Experimental and Numerical Investigations on the Dynamic Response of Blades with Dual Friction Dampers

Friction dampers are widely employed to reduce blade resonance vibration amplitude in turbomachinery. In this paper, a study was performed on the forced response of two blades with dual friction dampers. Numerical simulation and experimental testing were conducted. Firstly, the dynamics of the blade and dual friction damper system assembly are modeled. A nonlinear code based on the multi-harmonic balance method was developed to calculate the resonance response. In this analysis, both the blade and the damper are modeled with the finite element and the matrices reduced with the component mode synthesis method, while the contact forces are modeled with a one-dimensional variable normal load array element. Secondly, a test rig made of two blades and dual friction dampers, the material of which was steel, was established to measure the nonlinear frequency response function curves of the blade system. The results indicate that when a dual friction damper is applied, superior vibration reduction characteristics are demonstrated, with the system exhibiting an average 21% reduction in the response amplitude levels and an increase of 3% in the frequency shifting range compared to a single damper. Dampers positioned at relatively higher locations contribute significantly to the vibration reduction process. In the end, the numerical predictions match very well with the experimental ones.

Theoretical dynamic reduction for steady-state nonlinear vibration responses of complex jointed structures having hysteretic contact behavior

A novel nonlinear dynamic reduction method was developed to determine the steady-state vibration responses of complex jointed structures having hysteretic contact behavior. By using the harmonic balance method to reformulate nonlinear dynamic equilibrium equations into a set of nonlinear algebraic ones, a dynamic reduction technique based on nonlinearity transformation was theoretically developed to iterate nonlinear vibration solutions in the local coordinate that was related only to the degree-of-freedoms of nonlinear joints. Only odd-order harmonic components were truncated to approximate the hysteretic nonlinear contact forces of joint interfaces, which was conducive to further dimension reduction of the nonlinear algebraic equations and iteration matrix. Then, a nonlinear dynamic reduction solver was developed to associate the steady-state nonlinear vibration responses of overall structures with the dynamic characteristics of underlying linear substructures, nonlinear joint models and external excitations. Combined with finite element analysis, the steady-state nonlinear vibration responses of a complex assembled structure with four reduced-order nonlinear joint models were numerically simulated to validate the proposed nonlinear dynamic reduction method by comparing with the literature. The comparative results of nonlinear frequency response functions exhibited good agreement, and the proposed method indicated higher computational efficiency. The experimental investigations of a rubber isolator system were also performed to validate the proposed method, which demonstrated good performance.

Data-driven modeling and analysis of nonlinear isolated mechanical system

This study aims to identify nonlinearities in the stiffness and damping characteristics of a passive pneumatic isolator connected to an ultra-precision manufacturing prototype in the uncoupled transverse and horizontal directions. Experimental data are collected using sine sweep tests and is used to plot nonlinear frequency response functions. The resulting analysis reveals a quadratic polynomial relationship between the equivalent stiffness and the system’s dynamic response, indicating a combination of quadratic and cubic nonlinearity. Similarly, the nonlinear damping characteristic is determined using the half-power bandwidth method, which exhibits a quadratic polynomial relationship between the equivalent damping and dynamic response. The linear stability analysis of the system for the identified nonlinear stiffness and damping characteristics reveals that the system remains linearly stable for the given range of frequency and excitation amplitudes. Furthermore, bifurcation analysis reveals the existence of complex solutions, such as period-2 and period-4 solutions, for higher excitation amplitudes.

Non-intrusive frequency response analysis of nonlinear systems with interval uncertainty: A comparative study

This paper investigates the non-intrusive frequency response function (FRF) computation of nonlinear vibration systems subject to interval uncertainty. The arc-length ratio (ALR) method is generalized into non-probabilistic nonlinear problems and the interpolation technique is introduced to adapt the ALR for classic predictor-corrector algorithms. The ALR method is comparatively studied with the polar angle interpolation (PAI), which is another state-of-the-art technique for uncertainty propagation in nonlinear FRFs. Applications of the two techniques to two nonlinear systems, i.e., a dual-rotor with rotor/stator contact and a two degrees-of-freedom system with cubic nonlinearity, are presented. The in-depth case studies reveal the essence of the two methods and their strengths and limitations. It is found that the ALR is a powerful tool for various problems while the PAI may fail in nonlinear systems whose FRFs have extreme curvatures or complex shape structures. However, the PAI can transform the uncertain FRF bands into a traditional frequency sense. As the uncertainty propagation methods constructed in the present study are non-intrusive, they can be conveniently generalized into other nonlinear problems. Thus, the findings and conclusions in this study will guide future research on the dynamic responses of general nonlinear mechanical systems.

Online Rotor Systems Condition Monitoring Using Nonlinear Output Frequency Response Functions Under Harmonic Excitations

Operation conditions such as rub-impact and misalignment usually accumulate and can eventually induce severe faults in rotor systems. These abnormal conditions often make rotor systems behave nonlinearly. Consequently, nonlinear signal and systems analyses have been applied to address rotor system condition monitoring and fault diagnosis problems. Nonlinear frequency response functions (NOFRFs) are a concept proposed to extend the well-known concept of frequency response function to the nonlinear case, and have recently been exploited to solve problems with signal analysis based rotor system condition monitoring and fault diagnosis. However, existing NOFRFs based analyses require conducting offline experiments on rotor systems. This cannot satisfy the industrial need of continuous monitoring of rotor system operational conditions, which is important for online fault diagnosis. In order to resolve this problem, the present article proposes a novel approach that performs the rotor system condition monitoring using the NOFRFs determined directly from the rotor system online operational data which are inherently harmonics. This approach involves the identification of a data-driven NARX (nonlinear auto regressive with eXegenous input) model of rotor systems from harmonic excitations and corresponding responses, evaluation of rotor system NOFRFs using the identified NARX model, and the use of evaluated NOFRFs for rotor system operational condition monitoring and fault diagnosis. The approach, for the first time, enables the NOFRFs to be applied to perform online monitoring of rotor systems. Experimental studies have been conducted to verify the effectiveness and potential applications of the proposed approach as well as its advantages over commonly used signal analysis based techniques.

Frequency response function method for dynamic gas flow modeling and its application in pipeline system leakage diagnosis

Timely diagnosing leakages is significant for the safety and reliability of gas pipeline systems, where an efficient and accurate modeling and solution method is critical to identify the leakage rate and location. Here, a frequency response function-based modeling method for dynamic gas flow in pipelines is proposed. Based on the frequency-domain flow resistance of pipelines and Fourier Transform, original dynamic flow model is transformed into the frequency response function model consisting of linear lumped transmission constraints, nonlinear frequency response function, and nonlinear pipeline resistance characteristics, which are efficiently solved according to their different mathematical properties. Dynamic simulation cases show the proposed modeling and solution method greatly improves the computational efficiency keeping the accuracy. Besides, taking the difference between measurements and simulation results under leakage condition as objective, an iterative bilayer optimization algorithm is proposed based on the mathematical categorization of system constraints to efficiently diagnose leakages. Four leakage diagnosis cases in single pipeline and pipeline network with measurement noise and transient boundary conditions are studied to validate the leakage diagnosis method. Absolute location errors in the results are all within 1%, demonstrating the accuracy and robustness of the proposed method for single leakages and multiple leakages occurring successively or simultaneously.

Run-Time Cutting Force Estimation Based on Learned Nonlinear Frequency Response Function

Abstract The frequency response function (FRF) provides an input–output model that describes the system dynamics. Learning the FRF of a mechanical system can facilitate system identification, adaptive control, and condition-based health monitoring. Traditionally, FRFs can be measured by off-line experimental testing, such as impulse response measurements via impact hammer testing. In this paper, we investigate learning FRFs from operational data with a nonlinear regression approach. A regression model with a learned nonlinear basis is proposed for FRF learning for run-time systems under dynamic steady state. Compared with a classic FRF, the data-driven model accounts for both transient and steady-state responses. With a nonlinear function basis, the FRF model naturally handles nonlinear frequency response analysis. The proposed method is tested and validated for dynamic cutting force estimation of machining spindles under various operating conditions. As shown in the results, instead of being a constant linear ratio, the learned FRF can represent different mapping relationships under different spindle speeds and force levels, which accounts for the nonlinear behavior of the systems. It is shown that the proposed method can predict dynamic cutting forces with high accuracy using measured vibration signals. We also demonstrate that the learned data-driven FRF can be easily applied with the few-shot learning scheme to machine tool spindles with different frequency responses when limited training samples are available.

Highly Sensitive Nonlinear Identification to Track Early Fatigue Signs in Flexible Structures

Abstract This study describes a physics-based and data-driven nonlinear system identification (NSI) approach for detecting early fatigue damage due to vibratory loads. The approach also allows for tracking the evolution of damage in real-time. Nonlinear parameters such as geometric stiffness, cubic damping, and phase angle shift can be estimated as a function of fatigue cycles, which are demonstrated experimentally using flexible aluminum 7075-T6 structures exposed to vibration. NSI is utilized to create and update nonlinear frequency response functions, backbone curves and phase traces to visualize and estimate the structural health. Findings show that the dynamic phase is more sensitive to the evolution of early fatigue damage than nonlinear parameters such as the geometric stiffness and cubic damping parameters. A modified Carrella–Ewins method is introduced to calculate the backbone from nonlinear signal response, which is in good agreement with the numerical and harmonic balance results. The phase tracing method is presented, which appears to detect damage after approximately 40% of fatigue life, while the geometric stiffness and cubic damping parameters are capable of detecting fatigue damage after approximately 50% of the life-cycle.

Extension of the Harmonic Balance Method for dynamic systems with Iwan joints

In many applications it would be advantageous to be able to compute the nonlinear Frequency Response Functions (FRF) of structures containing bolted interfaces, but to do this with time integration is expensive so more efficient methods are desired. Harmonic Balance methods are typically more efficient, yet they are difficult to apply to systems with joints because one must track the state of many points on a surface and their highly nonlinear evolution from stick to slip in each cycle of vibration. This paper proposes an adaptation of the alternating frequency time harmonic balance method that is able to compute the steady-state response and consequently the nonlinear FRFs of systems with Iwan elements in an accurate and efficient manner. Rather than including the state of all of the sliders in the state vector, the points at which the displacement across the joint reverses are identified and used as a surrogate, greatly reducing the number of state variables and making the state variables more continuous. Furthermore, the nonlinear force and the Jacobian only need to be computed for the degrees of the freedom related to the joints, improving the efficiency of the approach. While the damping in the hysteretic system is not velocity dependent, an effective damping is introduced and added to the Jacobian to improve convergence of the Newton–Raphson algorithm for frequencies near the resonance. The proposed method, dubbed as “Hysteresis Identification via Reversal Points or (HIRP)”, is tested on various models with multiple DOF and Iwan elements, including a relatively high order model of two beams with two bolted joints. The results obtained show the capability of the proposed method and its efficiency and accuracy for realistic, industrially relevant models.

Exploration of the Nonlinear Effect of Pendulum Tuned Mass Dampers on Vibration Control

Understanding the nonlinearity of pendulum tuned mass dampers (PTMDs) under large-amplitude vibration is important to enhance control performance. Here, the nonlinear effects of PTMDs on vibration control of a single-degree-of-freedom (SDOF) system under harmonic excitation are explored. The nonlinear governing equations of the linear SDOF structure coupled with the PTMD are formulated. The results obtained via the method of harmonic balance under various excitation amplitudes are compared with those obtained employing the linearized system. It is shown that the effects of PTMD nonlinearity can be ignored when the PTMD rotation angle is below about 9°. Upon increasing the excitation amplitude, remarkable differences are observed between the nonlinear and linear frequency response functions (FRFs). Large mass ratios of PTMD are found to lead to reduced PTMD motion and attenuated nonlinear effects. A parametric study is performed to find the optimal parameters of the nonlinear PTMD under various excitation amplitudes. The results indicate that the optimal tuning frequency ratio increases significantly with excitation amplitudes, whereas the optimal damping is less sensitive to excitation amplitudes. Finally, a 61-m-high steel chimney structure subjected to harmonic and random wind loadings is investigated to verify the proposed PTMD design method.

Time-domain numerical continuation of periodic orbits for harmonically forced hysteretic nonlinear systems with Iwan joints

Coulomb sliders are used widely in order to model frictional interfaces. In many applications, the nonlinear Frequency Response Functions (FRFs) of these systems is desired, but it is extremely computationally expensive to use time-domain integration methods to compute the steady-state response. The nonlinear force and stiffness of these devices depends on the position of sliders, and the implicit nature of the slider positions makes these systems hysteretic and therefore non-trivial to address using conventional continuation methods. This paper proposes a novel numerical method, dubbed ”Hysteresis Identification via Reversal Points or (HIRP)” that can compute the steady-state harmonically forced response of frictional nonlinear systems. For this purpose, a quasi-static algorithm is introduced which can evaluate the state of sliders at any time based on the reversal points over a period. This reduces the number of state variables needed in the continuation routine by at least an order of magnitude. The method has been tested by computing the harmonic response of a hysteretic system with multiple Coulomb sliders. The method is evaluated on multi-degree of freedom systems with Iwan joints, demonstrating that the method can be a valuable tool for predicting the nonlinear, steady-state response of hysteretic systems.

Measured and Simulated Forced Response of a Rotating Turbine Disk With Asymmetric and Cylindrical Underplatform Dampers

Abstract The dynamics of turbine blades with underplatform dampers (UPDs) is often experimentally explored by using small test rigs like two-blade models for cost and complexity reasons. In this paper, the dynamics of a large-scale academic turbine disk is measured on a special rotation test rig. Such measurements have rarely been published so far. The test rig supports speeds up to 3600 rpm and turbine disks up to a diameter of 1.2 m. The turbine disk is tested linearly as well as with asymmetric and cylindrical UPDs. The excitation forces and the excitation order are varied. The results prove the damper effectiveness by lowering resonance amplitudes. Additionally, the mistuning influence on the result depiction is discussed. The measurements are compared to simulations of the nonlinear frequency response functions (FRFs), showing good agreement.

Parameter Variation on Nonlinear Energy Sink attached to Multiple Degree of Freedom System

AbstractThe importance of lightweight structures has increased in the last years motivated by the goals of improving efficiency of vehicles and conserving resources. In most cases the loss of weight by using fewer or lighter materials is accompanied by a reduction of the structures stiffness having a strong effect on occuring vibrations. Even if the damping ratio of the structure remains unchanged, by reducing weight and stiffness the vibration amplitudes might increase if the vibration excitation stays the same. Damping elements might be introduced to the structure mitigating critical resonance gains of the structure to reduce these vibrations. In this work the dynamics of a linear multiple degrees of freedom structure with a nonlinear damping element are investigated. The damping element is chosen to have a nonlinear characteristic to allow for a situation adaptive design. Numerical parameter studies are carried out to optimize the resulting damping of the entire structure. The nonlinear equation of motion is solved by applying the Harmonic Balance Method. The evaluation of the performance of the nonlinear damping element is done by consideration of the nonlinear frequency response functions of the structure.

Geometrically non-linear forced vibrations of fully clamped functionally graded beams with multi-cracksresting on intermediate simple supports

The objective of this work is to investigate the non-linear forced dynamic response at large vibration amplitudes of fully clamped functionally graded beams containing a multiple open edge cracks and resting on intermediate simple supports. The theoretical model is based on the Euler-Bernoulli beam theory and the crack rotational spring model. The functionally graded beam properties are supposed to vary continuously through the beam thickness. A homogenization procedure based on the neutral surface approach taking into account the presence of the crack is developed to reduce the problem examined to that of an equivalent isotropic homogeneous multi-cracked beam. In the non-linear analysis, harmonic motion is assumed and the discretized expressions for the beam total strain and kinetic energies are derived by applying Hamilton’s Principle, so as to reduce the problem to a non-linear algebraic system solved using an approximate explicit method previously applied to various non-linear structural vibration problems. Considering the forced vibration case, the non-linear frequency response functions are obtained numerically in the neighbourhood of the predominant non-linear mode shape using a single mode approach. The effects of crack, the beam material gradient and the applied harmonic force are presented and discussed.

Learning-based frequency response function estimation for nonlinear systems

ABSTRACTIn this paper, we perform the nonlinear frequency response function (FRF) estimation for a class of nonlinear systems. Two non-parametric estimation techniques are considered: radial basis function neural network (RBF-NN)-based estimation and support vector machine (SVM)-based estimation. Based on the system's available observations, the proposed estimation models are used to predict its frequency response. Simulation results are provided to demonstrate the model implementation. Finally, a comparative study is carried out to evaluate the effectiveness of the RBF-NN and SVM schemes, which has demonstrated that the SVM outperformed RBF-NN in the FRF estimation.

Geometrically nonlinear free and forced vibrations of Euler-Bernoulli multi-span beams

The objective of this paper is to establish the formulation of the problem of nonlinear transverse forced vibrations of uniform multi-span beams, with several intermediate simple supports and general end conditions, including use of translational and rotational springs at the ends. The beam bending vibration equation is first written at each span and then the continuity requirements at each simple support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. The formulation is based on the application of Hamilton’s principle and spectral analysis to the problem of nonlinear forced vibrations occurring at large displacement amplitudes, leading to the solution of a nonlinear algebraic system using numerical or analytical methods. The nonlinear algebraic system has been solved here in the case of a four span beam in the free regime using an approximate method developed previously (second formulation) leading to the amplitude dependent fundamental nonlinear mode of the multi-span beam and to the corresponding backbone curves. Considering the nonlinear regime, under a uniformly distributed excitation harmonic force, the calculation of the corresponding generalised forces has led to the conclusion that the nonlinear response involves predominately the fourth mode. Consequently, an analysis has been performed in the neighbourhood of this mode, based on the single mode approach, to obtain the multi-span beam nonlinear frequency response functions for various excitation levels.

Optimum Design of a Nonlinear Vibration Absorber Coupled to a Resonant Oscillator: A Case Study

This paper presents the optimal design of a passive autoparametric cantilever beam vibration absorber for a linear mass‐spring‐damper system subject to harmonic external force. The design of the autoparametric vibration absorber is obtained by using an approximation of the nonlinear frequency response function, computed via the multiple scales method. Based on the solution given by the perturbation method mentioned above, a static optimization problem is formulated in order to determine the optimum parameters (mass and length) of the nonlinear absorber which minimizes the steady state amplitude of the primary mass under resonant conditions; then, a PZT actuator is cemented to the base of the beam, so the nonlinear absorber is made active, thus enabling the possibility of controlling the effective stiffness associated with the passive absorber and, as a consequence, the implementation of an active vibration control scheme able to preserve, as possible, the autoparametric interaction as well as to compensate varying excitation frequencies and parametric uncertainty. Finally, some simulations and experimental results are included to validate and illustrate the dynamic performance of the overall system.

Identification of mechanical properties of adhesive joints using ANN

PurposeThis paper aims to present a new approach to the fast determination of the effective, dynamic, mechanical properties of an adhesive for linear and nonlinear regions of the adhesive response, for both healthy and damaged states of the bond.Design/methodology/approachThe proposed approach is based on the measurement of the linear and nonlinear frequency response function (FRF) of adhesive-bonded structure and using artificial neural network identification technique. For this purpose, linear and nonlinear FRFs are measured for several single-lap joint specimens that are fabricated in healthy and damaged configurations of the bond. The measured FRFs of healthy and damaged specimens are then used to identify the natural frequencies of the specimens. The experimental natural frequencies, in turn, would be used to train artificial neural network (ANN) which would be able to predict the effective Young’s and shear moduli and damping of adhesive in healthy and damaged specimens, for any given excitation level and frequency, within the training domain.FindingsSimultaneous identification of the effective mechanical properties of adhesive for linear and nonlinear response regions, as well as healthy and damages states of the adhesive bond.Practical implicationsThe introduced method is effective to model the assembled structures with the viscoelastic adhesive joints, for linear and nonlinear regions.Originality/valueA fast methodology, using ANN, for identification the effective mechanical properties of adhesives, compared to other methods for both linear and nonlinear regions.