Stepped-sine Testing Research Articles

Modal analysis of non-conservative systems with friction-induced strong nonlinear damping by using response-controlled testing

In recent years, the response-controlled stepped-sine testing (RCT) method has made important contributions to the evolution of the single nonlinear mode (SNM) method from a postulate to a well-established theory with the accurate modal identification of various nonlinear mechanical systems ranging from simple benchmark beams to real missiles. Since the RCT method resides on the SNM theory which is based on Rosenberg’s periodic motion definition of nonlinear modes, it has been regarded as limited to weakly damped mechanical systems that mainly exhibit strong conservative nonlinearities. An important contribution of this current paper is extending the RCT method to non-conservative mechanical systems with complex nonlinear modes. The extension is first numerically validated on a strongly friction-damped cantilever beam and then experimentally validated on the actuation mechanism of a real control fin that exhibits strong softening-stiffening nonlinearity (up to 50% shift of natural frequency) due to friction/backlash and very high and nonlinear damping (up to 30% modal damping ratio) due to friction. Another important contribution of this paper is to demonstrate the applicability of the superposition principle in nonlinear dissipative systems with closely spaced modes at an acceptable accuracy if the nonlinear modes are similar in shape to their corresponding linear modes and the modal coupling terms in the equation of motion can be ignored. The assumption of similar nonlinear modes is reasonable for most practical applications that exhibit dissipative nonlinearities due to mechanical joints. Although the assumption of ignorable coupling terms does not reside on a solid theoretical framework, numerical studies on a friction-damped lumped system with closely spaced modes give promising results, pointing to the significant potential of the modal superposition in nonlinear systems.

Reliability of Response-Controlled Stepped Sine Testing for Experimental Detection of Nonlinear Structure

Nonlinear structural dynamic analysis is required for mechanical structures experiencing nonlinearity through large force-vibration response ranges. Nonlinearities can be caused by large vibration displacements, material properties, or joints. Experimental modal analysis for nonlinear detection is achieved using conventional force-controlled stepped sine testing. However, this approach often encounters premature jumps in frequency response curves before reaching actual resonance peaks. In recent years, response-controlled stepped sine testing (RCT) has been introduced to precisely quantify resonant peaks. This approach, however, has only been limitedly utilised to detect and analyse nonlinearity in jointed structures and structures experiencing large displacement. In this paper, the reliability of the RCT approach is assessed for detecting nonlinearity from different sources. The experimental setup involves placing two magnets on opposite sides of a plate's free end to induce localised nonlinearity through magnet attraction. A low force magnitude of random excitation is employed to identify the frequency range of the first vibration mode using an electromagnetic shaker. Subsequently, RCT is performed within this range to measure the nonlinear forced response. Frequency response functions are measured at ten different controlled displacement amplitudes at the driving point. The analysis observed a symmetry curve of response in the measured FRFs. The results indicate that nonlinear hardening is detected at structures with localised magnet attraction. In conclusion, the reliability of applying the RCT approach for detecting nonlinearity from magnet attraction is achieved due to the absence of a jump issue in FRFs.

Stabilizing a strongly nonlinear structure through shaker dynamics in fixed frequency voltage control tests

Bifurcations are commonly encountered during force controlled swept and stepped sine testing of nonlinear structures, which generally leads to the so-called jump-down or jump-up phenomena between stable solutions. There are various experimental closed-loop control algorithms, such as control-based continuation and phase-locked loop, to stabilize dynamical systems through these bifurcations, but they generally rely on specialized control algorithms that are not readily available with many commercial data acquisition software packages. A recent method was developed to experimentally apply sequential continuation using the shaker voltage that can be readily deployed using commercially available software. By utilizing the stabilizing effects of electrodynamic shakers and the force dropout phenomena in fixed frequency voltage control sine tests, this approach has been demonstrated to stabilize the unstable branch of a nonlinear system with three branches, allowing for three multivalued solutions to be identified within a specific frequency bandwidth near resonance. Recent testing on a strongly nonlinear system with vibro-impact nonlinearity has revealed jumping behavior when performing sequential continuation along the voltage parameter, like the jump phenomena seen during more traditional force controlled swept and stepped sine testing. This paper investigates the stabilizing effects of an electrodynamic shaker on strongly nonlinear structures in fixed frequency voltage control tests using both numerical and experimental methods. The harmonic balance method is applied to the coupled shaker-structure system with an electromechanical model to simulate the fixed voltage control tests and predict the stabilization for different parameters of the model. The simulated results are leveraged to inform the design of a set of experiments to demonstrate the stabilization characteristics on a fixture-pylon assembly with a vibro-impact nonlinearity. Through numerical simulation and experimental testing on two different strongly nonlinear systems, the various parameters that influence the stability of the coupled shaker-structure are revealed to better understand the performance of fixed frequency voltage control tests.

Experimental Modal Analysis of Nonlinear Amplified Piezoelectric Actuators by Using Response-Controlled Stepped-Sine Testing

Experimental Modal Analysis of Nonlinear Amplified Piezoelectric Actuators by Using Response-Controlled Stepped-Sine Testing

Experimental Quantification and Validation of Modal Properties of Geometrically Nonlinear Structures by Using Response-Controlled Stepped-Sine Testing

Experimental Quantification and Validation of Modal Properties of Geometrically Nonlinear Structures by Using Response-Controlled Stepped-Sine Testing

Measurement and identification of the nonlinear dynamics of a jointed structure using full-field data, Part I: Measurement of nonlinear dynamics

Jointed structures are ubiquitous constituents of engineering systems; however, their dynamic properties (e.g., natural frequencies and damping ratios) are challenging to identify correctly due to the complex, nonlinear nature of interfaces. This research seeks to extend the efficacy of traditional experimental methods for linear system identification (such as impact testing, shaker ringdown testing, random excitation, and force or amplitude-control stepped sine testing) on nonlinear jointed systems, e.g., the half Brake–Reußbeam, by augmenting them with full-field data collected by high-speed videography. The full-field response is acquired using high-speed cameras combined with Digital Image Correlation (DIC), which enables studying the spatial–temporal dynamic characteristics of the system. As this is a video-based experiment, additional constraints are attached to the beam at the node points to remove the rigid body motion, which ensures that the beam is in the view of the camera during the entire test. The use of a video-based method introduces new sources of experimental error, such as noise from the high-speed camera’s fan and electrical noise, and so the measurement accuracy of DIC is validated using accelerometer data. After validating the DIC data, the measurements are recorded for several types of excitation, including hammer testing, shaker ringdown testing, fixed sine testing, and stepped sine testing. Using the DIC data to augment standard nonlinear system identification techniques, modal coupling and the mode shapes’ evolution are investigated. The suitability of videography methods for nonlinear system identification of nonlinear beams is explored for the first time in this paper, and recommendations for techniques to facilitate this process are made. This article focuses on developing an accurate data collection methodology as well as recommendations for nonlinear testing with DIC, which paves the way for video-based investigation of nonlinear system identification. In Part-II (Jin et al., 2021) of this work, the same data set is used for a rigorous assessment of nonlinear system identification with full-field DIC data.

Measurement and identification of the nonlinear dynamics of a jointed structure using full-field data; Part II - Nonlinear system identification

The dynamic responses of assembled structures are greatly affected by the mechanical joints, which are often the cause of nonlinear behavior. To better understand and, in the future, tailor the nonlinearities, accurate methods are needed to characterize the dynamic properties of jointed structures. In this paper, the nonlinear characteristics of a jointed beam is studied with the help of multiple identification methods, including the Hilbert Transform method, Peak Finding and Fitting method, Dynamic Mode Decomposition method, State-Space Spectral Submanifold, and Wavelet-Bounded Empirical Mode Decomposition method. The nonlinearities are identified by the responses that are measured via accelerometers in a series of experiments that consist of hammer testing, shaker ringdown testing, and response/force-control stepped sine testing. In addition to accelerometers, two high-speed cameras are used to capture the motion of the whole structure during the shaker ringdown testing. Digital Image Correlation (DIC) is then adopted to obtain the displacement responses and used to determine the mode shapes of the jointed beam. The accuracy of the DIC data is validated by the comparison between the identification results of acceleration and displacement signals. As enabled by full-field data, the energy-dependent characteristics of the structure are also presented. The setup of the different experiments is described in detail in Part I (Chen et al., 2021) of this research. The focus of this paper is to compare nonlinear system identification methods applied to different measurement techniques and to exploit the use of high spatial resolution data.

Experimental Identification of Backbone Curves of Strongly Nonlinear Systems by Using Response-Controlled Stepped-Sine Testing (RCT)

In stepped-sine testing of strongly nonlinear structures with the classical force-control strategy, corrective force perturbations of a standard controller used to capture the reference signal in the proximity of turning points of frequency response curves may often lead to a premature jump before reaching the actual resonance peak. Accordingly, a classical force-control approach is not suitable to identify backbone curves of strongly nonlinear structures. This paper shows that currently available commercial modal test equipment can accurately identify backbone curves of strongly nonlinear structures by using Response-Controlled stepped-sine Testing (RCT) and the Harmonic Force Surface (HFS) concept, both recently proposed by the authors. These methods can be applied to systems where there are many nonlinearities at several different (and even unknown) locations. However, these techniques are not applicable to systems where internal resonances occur. In RCT, the displacement amplitude of the driving point, rather than the amplitude of the applied force, is kept constant during the stepped-sine testing. Spectra of the harmonic excitation force measured at several different displacement amplitude levels are used to build up a smooth HFS. Isocurves of constant amplitude forcing on the HFS lead to constant-force frequency response curves with accurately measured turning points and unstable branches (if there are any), which makes it possible to identify backbone curves of strongly nonlinear structures experimentally. The validation of the proposed approach is demonstrated with numerical and experimental case studies. A five degree-of-freedom (DOF) lumped system with five cubic stiffness elements, which create strong conservative nonlinearity, is used in the numerical example. Experimental case studies consist of a cantilever beam and a control fin actuation mechanism of a real missile structure. The cantilever beam is supported at its free-end by two metal strips constrained at both ends to create strong stiffening nonlinearity. The control fin actuation mechanism exhibits very complex and strong nonlinearity due to backlash and friction.

Nonlinear vibrations and damping of fractional viscoelastic rectangular plates

Damping is largely increasing with the vibration amplitude during nonlinear vibrations of rectangular plates. At the same time, soft materials present an increase in their stiffness with the vibration frequency. These two phenomena appear together and are both explained in the framework of the viscoelasticity. While the literature on nonlinear vibrations of plates is very large, these aspects are rarely touched. The present study applies the fractional linear solid model to describe the viscoelastic material behavior. This allows to capture at the same time (i) the increase in the storage modulus with the vibration frequency and (ii) the frequency-dependent nonlinear damping in nonlinear vibrations of rectangular plates. The solution to the nonlinear vibration problems is obtained through Lagrange equations by deriving the potential energy of the plate and the dissipated energy, both geometrically nonlinear and frequency dependent. The model is then applied to a silicone rubber rectangular plate tested experimentally. The plate was glued to a metal frame and harmonically excited by stepped sine testing at different force levels, and the vibration response was measured by a laser Doppler vibrometer. The comparison of numerical and experimental results was very satisfactorily carried out for: (i) nonlinear vibration responses in the frequency and time domain at different excitation levels, (ii) dissipated energy versus excitation frequency and excitation force, (iii) storage energy and (iv) loss factor, which is particularly interesting to evaluate the plate dissipation versus frequency at different excitation levels. Finally, the linear and nonlinear damping terms are compared.

Experimental Characterization of Nonlinear Damping in the F-16 Wing Structure

Earlier work did not find any static stiffness nonlinearities in the F-16 wing structure, but suggested that dynamic nonlinearities may be present. Computational studies have demonstrated that small amounts of nonlinear structural damping are sufficient to change a diverging flutter oscillation into a limited amplitude oscillation. Experimental damping measurements of a fixture-mounted F-16 wing were made to determine whether damping nonlinearities are significant and to quantify the nature of any structural damping present in the built-up wing structure. Three fundamental component configurations of the wing were examined: wing box only, wing (i.e., wing box with leading- and trailing-edge control surfaces), and wing with wingtip launcher. A ground vibration test approach was used with a stepped-sine excitation technique over a range of excitation forces. Damping and frequency were extracted from measured frequency response functions of the stepped-sine test data using a modified single degree-of-freedom modal analysis technique. Nonlinear damping with respect to bending mode displacement was observed for all test configurations. The overall magnitude of the damping increased when control surfaces were installed on the wing box. The presence of a wingtip launcher decreased the damping at higher displacements.

Experimental modal analysis of nonlinear systems by using response-controlled stepped-sine testing

Although the identification and analysis of structures with a localized nonlinearity, either weak or strong, is within reach, identification of multiple nonlinearities coexisting at different locations is still a challenge, especially if these nonlinearities are strong. In such cases, identifying each nonlinearity separately requires a tedious work or may not be possible at all in some cases. In this paper, an approach for experimental modal analysis of nonlinear systems by using Response-Controlled stepped-sine Testing (RCT) is proposed. The proposed approach is applicable to systems with several nonlinearities at various different locations, provided that modes are well separated and no internal resonances occur. Step-sine testing carried out by keeping the displacement amplitude of the driving point constant yields quasi-linear frequency response functions directly, from which the modal parameters can be identified as functions of modal amplitude of the mode of concern, by employing standard linear modal analysis tools. These identified modal parameters can then be used in calculating near-resonant frequency response curves, including the unstable branch if there is any, for various untested harmonic forcing cases. The proposed RCT approach makes it also possible to extract nonlinear normal modes experimentally without using sophisticated control algorithms, directly from the identified modal constants, and also to obtain near-resonant frequency response curves experimentally for untested constant-amplitude harmonic forcing cases by extracting isocurves of constant-amplitude forcing from the measured Harmonic Force Surface (HFS), a new concept proposed in this paper. The key feature of the HFS is its ability to extract unstable branches together with turning points of constant-force frequency response curves directly from experiment, accurately. The method is validated with numerical and experimental case studies. The numerical example consists of a 5 DOF lumped system with strong several conservative nonlinear elements. Experimental case studies consist of a cantilever beam supported at its free-end by two metal strips which create strong stiffening nonlinearity, and a real missile structure which exhibit moderate damping nonlinearity mostly due to several bolted joints on the structure.

A frequency domain nonparametric identification method for nonlinear structures: Describing surface method

In this paper a new method called ‘Describing Surface Method’ (DSM) is developed for nonparametric identification of a localized nonlinearity in structural dynamics. The method makes use of the Nonlinearity Matrix concept developed in the past by using classical describing function theory, which assumes that nonlinearity depends mainly on the response amplitude and frequency dependence is negligible for almost all of the standard nonlinear elements. However, this may not always be the case for complex nonlinearities. With the method proposed in this study, nonlinearities which are functions of both frequency and displacement amplitude can be identified by using response-controlled stepped-sine testing. Furthermore, the nonlinearity does not need to be mathematically expressible in terms of response amplitude and frequency, which allows us to identify more complex nonlinearities nonparametrically. The method is applicable to real engineering structures with local nonlinearity affecting the boundary conditions, where modes are not closely spaced, and sub- and super-harmonics are assumed to be negligible compared to the fundamental harmonic. Multiple nonlinearities may coexist at the same location and a priori knowledge of nonlinearity type is not necessary. The method yields the describing surface of nonlinearity, real and imaginary parts of which correspond to the equivalent nonlinear stiffness and nonlinear damping at that location in the structure. Harmonic response of a nonlinear system to any force, including any existing unstable branch, can be calculated iteratively by using the describing surface representing the nonlinearity. Unstable branches captured by using Newton’s Method with arc-length continuation algorithm can be validated experimentally by using Harmonic Force Surface (HFS) concept. The validation of DSM is demonstrated with three experimental case studies: a cantilever beam with cubic stiffness at its tip point, a dummy mass on elastomeric vibration isolators, and a control fin actuation mechanism of a real missile structure which exhibits very complex (due to backlash and friction) and strong nonlinearity causing jump phenomenon in the frequency response. In each case, nonlinear FRFs calculated by using identified describing surface agree with measured FRFs at various force levels much better than FRFs calculated by using classical describing function method. It is observed in the experimental case studies that frequency dependence of nonlinearity occurs mostly in the imaginary part of the describing surface which represents nonlinear damping.

Novel Criterion for Evaluation of Stationarity in Nonlinear Structural Dynamics

AbstractThe evaluation of stationarity, such as the widely used steady-state response assessment in stepped-sine testing, is a critical issue in structural dynamics. Stepped-sine excitation is a cr...

An Experimental Investigation on Uncertainty in Measuring Vibration Deflection Shapes With Digital Image Correlation

3-D digital image correlation (DIC) is a widespread full-field displacement measurement technique based on stereo vision. It was formulated to deal with static problems in experimental mechanics in the early seventies, and since then, the applicability of the technique increased due to increasing frame rates and dropping prices of machine vision cameras. As a consequence, DIC became a feasible solution for general-purpose vibration testing. This paper, however, does not offer a comprehensive analysis of DIC accuracy when applied to vibration testing, and therefore, this paper offers a “Type A” evaluation of uncertainty when measuring vibration deflection shapes with DIC. Uncertainty is evaluated for different experimental conditions on a stepped sine test. Data show that the average value of uncertainty normally lies below 0.02 mm, but in resonant conditions, it can increase up to 0.05 mm (considering a field of view of about half a meter). This demonstrates a strong correlation between the deflection amplitude and the random uncertainty due to motion blur.

Identification of weak nonlinearities in MDOF systems based on reconstructed constant response tests

A novel strategy to characterize and identify structural nonlinearities in MDOF systems based on reconstructing constant response tests from constant excitation tests is developed in this paper. Constant displacement frequency response functions (FRFs) can be measured by a stepped sine test where the displacement is controlled at every frequency of interest. In these FRFs, the nonlinear restoring force is effectively linearized and natural frequencies can be estimated by linear modal analysis. Using a series of constant displacement tests, the relationship of equivalent stiffness versus displacement can be established by curve fitting, and hence, the nonlinear stiffness characterized. This paper proposes a method to reconstruct the constant displacement FRFs from stepped sine tests with constant excitation; this avoids the requirement to control either the response or force amplitude leads to a faster and more stable testing program. Similarly, damping nonlinearities in structures can be characterized and identified by constant velocity tests reconstructed in a similar way. This approach of FRF reconstruction is mathematically simple and suitable for structures with weak nonlinearities. The method is demonstrated on a framed structure with unknown weak nonlinearities, and the nonlinear stiffness and damping parameters of the structure are identified and validated. The results demonstrate the feasibility and effectiveness of the approach and also show the potential for practical applications in engineering.

Detection and Location of Nonlinearities using Reciprocity Breakdown

Damage in a structure often results in local stiffness nonlinearities and detecting these nonlinearities can be used to monitor the health of the structure. It is well-known that nonlinearities in structures lead to a breakdown in reciprocity, where the frequency response function between two points on the structure depends upon the forcing location. This paper proposes a measure to quantify the level of non-reciprocity in a structure and investigates the effect of the location and form of nonlinearity on this non-reciprocity measure. A simulated discrete mass-spring system was used to determine the effect of the excitation and response locations on the ability to detect the nonlinearity. Stepped-sine testing is commonly used to characterise a nonlinear system since harmonic excitation emphasises nonlinear phenomena and can, for example, allow the system to exhibit multiple solutions. Thus, a simulation of a stepped sine test was used as the benchmark to highlight reciprocity breakdown in the most favourable case. However, impact excitation is much easier and faster to implement in practice and consequently the effect of the type of excitation on the detection and location of nonlinearities was considered. Finally, the prospects for using a measure of reciprocity in a structural health monitoring system are discussed.

Integration of system identification and finite element modelling of nonlinear vibrating structures

Abstract The Finite Element Method (FEM), Experimental modal analysis (EMA) and other linear analysis techniques have been established as reliable tools for the dynamic analysis of engineering structures. They are often used to provide solutions to small and large structures and other variety of cases in structural dynamics, even those exhibiting a certain degree of nonlinearity. Unfortunately, when the nonlinear effects are substantial or the accuracy of the predicted response is of vital importance, a linear finite element model will generally prove to be unsatisfactory. As a result, the validated linear FE model requires further enhancement so that it can represent and predict the nonlinear behaviour exhibited by the structure. In this paper, a pragmatic approach to integrating test-based system identification and FE modelling of a nonlinear structure is presented. This integration is based on three different phases: the first phase involves the derivation of an Underlying Linear Model (ULM) of the structure, the second phase includes experiment-based nonlinear identification using measured time series and the third phase covers augmenting the linear FE model and experimental validation of the nonlinear FE model. The proposed case study is demonstrated on a twin cantilever beam assembly coupled with a flexible arch shaped beam. In this case, polynomial-type nonlinearities are identified and validated with force-controlled stepped-sine test data at several excitation levels.

Damping for large-amplitude vibrations of plates and curved panels, Part 1: Modeling and experiments

Theoretical and experimental non-linear vibrations of thin rectangular plates and curved panels subjected to out-of-plane harmonic excitation are investigated. Experiments have been performed on isotropic and laminated sandwich plates and panels with supported and free boundary conditions. A sophisticated measuring technique has been developed to characterize the non-linear behavior experimentally by using a Laser Doppler Vibrometer and a stepped-sine testing procedure. The theoretical approach is based on Donnell's non-linear shell theory (since the tested plates are very thin) but retaining in-plane inertia, taking into account the effect of geometric imperfections. A unified energy approach has been utilized to obtain the discretized non-linear equations of motion by using the linear natural modes of vibration. Moreover, a pseudo arc-length continuation and collocation scheme has been used to obtain the periodic solutions and perform bifurcation analysis. Comparisons between numerical simulations and the experiments show good qualitative and quantitative agreement. It is found that, in order to simulate large-amplitude vibrations, a damping value much larger than the linear modal damping should be considered. This indicates a very large and non-linear increase of damping with the increase of the excitation and vibration amplitude for plates and curved panels with different shape, boundary conditions and materials.

Kissing bond detection in structural adhesive joints using nonlinear dynamic characteristics

In this paper, the effect of kissing bond on nonlinear dynamic behavior of structures with flexible adhesive joint is investigated. Bilinear characteristic due to opening and closing in kissing bond region results in nonlinear dynamic behavior of the structure such as harmonic distortion in response to harmonic excitation. So, the higher-order harmonics can be considered as Nonlinear Damage Indicator Functions (NDIF) for the purpose of damage identification. A two-dimensional (2D) finite element model of a beam connected to a rigid support via a flexible adhesive layer is used to investigate the efficiency of the proposed NDIFs in kissing bond detection. Kissing bond is introduced to the model via contact elements. NDIFs are extracted for the finite element model using single tone stepped-sine test simulation. Parameters such as amplitude of excitation, size and location of kissing bond region as well as friction between kissing surfaces, are studied. The results proved that the NDIFs are sensitive to the size and location of kissing bond. Consequently, in an experimental damage identification procedure, NDIFs can be used as an indicator of kissing bond type damages in adhesive joints.

Nonlinear vibrations of laminated and sandwich rectangular plates with free edges. Part 2: Experiments & comparisons

Large-amplitude (geometrically nonlinear) forced vibrations of completely free sandwich rectangular plates are investigated experimentally. Harmonic excitation is applied by using an electro-dynamic exciter and the plate vibration is measured by using laser Doppler vibrometers. A scanning laser Doppler vibrometer is used for experimental modal analysis since it provides non-contact vibration measurements with very high spatial resolution. The large-amplitude vibration experiments are carried out by using a single point laser Doppler vibrometer and a stepped-sine testing procedure. The nonlinear frequency response curves are obtained by increasing and decreasing the excitation frequency in very small steps at specific force amplitudes controlled in a closed-loop. The experimental results are compared to numerical simulations obtained by reduced-order models and show very good agreement. The nonlinear damping is experimentally obtained as a function of the vibration amplitude.