Abstract

This work refines a recently formalized methodology proposed by D.J. Ewins consisting of ten steps for model validation of nonlinear structures. This work details, through a series of experimental studies, that many standard test setup assumptions that are made when performing dynamic testing are invalid and need to be evaluated for each structure. The invalidation of the standard assumptions is due to the presence of nonlinearities, both known and unrecognized in the system. Complicating measurements, many nonlinearities are currently characterized as constant properties instead of variables that exhibit dependency on system hysteresis and actuation amplitude. This study reviews current methods for characterizing nonlinearities and outlines gaps in the approaches. A brief update to the CONCERTO method, based on the accelerance of a system, is derived for characterizing a system’s nonlinearities. Finally, this study ends with an updated methodology for model validation and the ramifications for modeling assemblies with nonlinearities are discussed.

Introduction

There exist many types of nonlinearities in assembled structures, including geometric, material, and interfacial. Each of these nonlinearities present a unique set of challenges for characterization and model development. For instance, a fundamental question in the field of joint mechanics is: “How is a jointed interface best characterized and modeled?” Even if a metric to describe the behavior of a jointed system can be agreed upon, such as the change in stiffness and damping as a function of excitation amplitude, there is no consensus for how to measure this relationship accurately. Multiple signal processing techniques have been developed [19], but the variance between the estimated properties can exceed an order of magnitude [10,11]. This is further complicated by several issues; for instance, the experimental setup used for measuring the behavior of a jointed system can often contaminate the estimates of the joint properties with other sources of nonlinearity (such as a “clamped” boundary condition) or negligence (such as neglecting the effects of a 5-g accelerometer (Accel) on a 4-kg test structure) [1217]. For these, and other reasons, the measurement and characterization of joint properties has historically been a challenging effort as the variability in the measured properties often spanned too large of a range of values to be of use in modeling [10,18].

This challenge is further exacerbated by the lack of consensus on the appropriate modeling technique for jointed interfaces. High fidelity numerical analysis has demonstrated that point-wise (i.e., node-to-node) Coulomb friction models are inappropriate [10,18]. Alternative approaches that have gained traction in recent years include modal damping models [19] and discrete hysteretic models [2026] (which are often implemented as a single element per interface). Modal models presume that the nonlinearities affect only modal damping ratios, have a small effect on frequency and that the influence on the system’s mode shapes is negligible. This presumption of only nonlinear damping and a small effect on frequency is improper for joint mechanic applications where the stick-slip phenomenon reduces the stiffness of the joint [11]. Discrete hysteretic models, including Iwan elements [20,21], Jenkins elements [22], Buoc-Wen models [27], and three-dimensional friction models [2325], have shown promise at being able to reproduce the measured amplitude-dependent stiffness and damping parameters of a jointed structure, but state-of-the-art approaches are only able to model one or two modes accurately [26]. The small number of modes is due, in part, to the parameters necessary to populate discrete hysteretic models being, mostly, non-physical. Thus, the question remains “How is a jointed interface best characterized and modeled?”

Recent efforts to characterize and model an assembled structure have been driven by significant progress in the nonlinear system identification community. These approaches, however, treat the nonlinearities collectively as a black box that can be characterized with a low order model [28,29]. The ramification of this approach is that while the constitutive relationship of the structure can be well characterized for the regime of study, the spatial information of how the nonlinearity is engaged and interacts with the surrounding structure is lost. As a result, even the state-of-the-art methods for nonlinear model updating [26,3032] are unable to capture more than a few modes of the structural response accurately.

This is of particular importance when developing a nonlinear model of a system that cannot be tested using loads representative of the actual application environment. As an example, aerospace vehicles typically have large lift forces distributed across a wing (resulting in large deflections, significant internal tensions, localized buckling, etc.), unsteady aerodynamic loads, and aero-elastic coupling. During ground vibration testing, however, point loads are often applied via shakers. Thus, to have confidence in the performance of the vehicle in application environments, the nonlinearities need to be well understood.

A Methodology for Developing/Calibrating Nonlinear Models.

Historically, developing nonlinear models could be characterized by a four step procedure: formulation of the linear model, experiments to identify the nonlinearity, model updating to fit results, and some times validation with one more experiment. Other methods include: over-designing the structure to be linear, designing to a specific functional task and ignoring dynamic responses, tuning a linear model to a low level modal test, or tuning to a polynomial that resembles the nonlinearity [30,31]. To normalize the process for model validation, Ewins proposed a methodology which consists of the following ten steps [30,31]:

  1. Approaching Testing (known nonlinearities)

  2. Establishing the Underlying Linear Model

  3. Specifying Appropriate Representative Test

  4. Detecting the Nonlinearity

  5. Characterizing the Nonlinearity

  6. Locating the Nonlinearity

  7. Quantifying the Nonlinearity

  8. Upgrading the Linear Model with Nonlinear Elements

  9. Updating the Nonlinear Model

  10. Validating the Nonlinear Model

These ten steps focus on three phases: the development and characterization of an underlying linear model (Steps 1–3), the classification and characterization of the nonlinearity of interest (Steps 4–7), and updating the linear model with nonlinear elements (Steps 8–10). To develop nonlinear models that are able to predict the response of a structure under load cases (or boundary conditions) not included in an experimental study, several modifications to this methodology are proposed. First, two crucial steps must be added: “Deconvoluting Nonlinearities from the Test and System” (before Step 1) and “Resolving the Long Term Evolution of the Nonlinearity” (after Step 6). Step 5 “Characterization of Nonlinearity” has benefited tremendously from the recent progress in nonlinear system identification [7,33], but is still a current research challenge that necessitates significant investment to understand how to capture the physics of the nonlinearity for a multi-modal system in which modes are strongly coupled or closely spaced and how, in turn, an improved understanding of the physics can be used to efficiently update existing models of the nonlinearity itself [34]. Additionally, Step 6 “Locating the Nonlinearity” should be updated to include how the location (area in joint mechanics) evolves during dynamic loading, and it is proposed that Step 7 “Quantifying the Nonlinearity” be conceptually moved to the third phase. With these steps in mind, a new methodology is proposed and shown in Fig. 1.

Fig. 1
Updated methodology for model validation
Fig. 1
Updated methodology for model validation
Close modal

The order of “Locating the Nonlinearity” and “Characterizing the Nonlinearity” is interchangeable. As some nonlinearities need to be located before they can be fully characterized, while others can be characterized without, specifically, resolving the location/area (such as geometric nonlinearity). The phases of Ewins [30,31] methodology are shown in Fig. 1 as colored boxes. The individual steps can be performed separate from each other, while the steps in each phase typically occur at the same time as the other steps.

With the new methodology, the first step is to “Deconvolute the Nonlinearity from the Test and System,” which is crucial due to the uncertainties associated with any structure or experiment. Here, deconvolute means to separate the nonlinear characteristics of the system from the nonlinearities of the boundaries or next level assembly. The uncertainty that can hinder deconvoluting the nonlinearity, but be reduced, is epistemic. Epistemic uncertainty is attributed to the lack of adequacy and completeness of the parameters predicted or measured [31]. There are many assumptions that experimentalist may make that can introduce nonlinearities into the system. Many potential assumptions are rooted in the application of techniques developed for linear systems to nonlinear systems. These assumptions can include boundary conditions, loading/excitation conditions, and instrumentation configurations. As an example of these assumptions, the use of bungee cables (inherently nonlinear [35,36]) are used to simulate a free-free boundary condition. If the nonlinearity of interest is not isolated from other (potentially contaminating) sources of nonlinearities in the system, then the properties of the nonlinearity of interest will be misidentified leading to unintended modeling error. Thus, isolating the nonlinearity of interest is an endeavor that could pertain to the structure (which could have multiple nonlinearities) of study.

After deconvoluting the nonlinearities accounting for invalid assumptions and preliminary testing of the system and developing/validating the underlying linear model and determining the presence of nonlinearities, the next steps are to characterize, locate, and quantify the nonlinearities. “Locating the Nonlinearity” is trivial for some nonlinearities, such as geometric, material, or contact/ impact based interactions, and work is ongoing to locate nonlinearities by analyzing the responses of various locations to find the greatest influence [37,38]. For other nonlinearities, such as interfacial nonlinearities, the process of locating them is deceptively challenging. For distributed interfaces, parts of the interface behave linearly (and should be modeled as such) and parts of the interface, which corresponds to where wear or fretting failure is observed, exhibit strong damping and weak stiffness nonlinearities. Some models previously discussed discretize the nonlinearity into just a few elements [26], while others treat the whole system as a black box [28]. These models do not account for changes in area under dynamic loading, such as a joint. The tribomechadynamic nature of jointed interfaces is such that the local frictional and wear properties are directly related to the contact pressures and thus the system dynamics. Due to how joints are assembled and distribute the pre-load, the nonlinearity is not evenly distributed across the whole joint. Since the nonlinearity is not evenly distributed, the spatial resolution of the nonlinearity needs to be determined if the model is to be used to predict the response outside of the data used to update/ calibrate it.2 Thus, methods that include spatially distributed friction elements [2225] struggle to model the system level hysteresis, due to the need to identify hundreds, if not thousands, of nonlinear parameters to capture the non-uniform spatial distribution of frictional properties.

Current approaches focus on characterizing a nonlinearity at one point in time and assume that this characterization is valid for the lifetime [11]. Detailed models of interfaces (such as for the wear of a brake pad [39]) can capture the evolution of damage and frictional properties over time; however, the time scales necessary to see this evolution in a structural model are prohibitively difficult to simulate due to the high cost of transient simulations. Thus, the approaches adopted by the structural dynamics community all assume (with one or two exceptions [40]) that the constitutive response of the nonlinearity does not change over time. Real structures, though, exhibit wear and damage. Frameworks, such as tribomechadynamical models [41], tie the evolution of the structural (macro-scale) properties to micro-scale damage models. Therefore, an additional step necessary to capture the long term behavior of a structure is “Resolving the Long Term Evolution of the Nonlinearity.”

The last proposed modification to Ewins’ methodology [30,31] is shifting “Quantifying the Nonlinearity” to phase three. This step is where the variables necessary to describe the nonlinearity, previously identified, are quantified for the model form that will be used in the updated nonlinear model. Finding the parameters is an iterative process where this step is a first approximation that comes directly from experimental data, and subsequent steps use optimization routines to update the model to match the test data.

To qualify the proposed missing steps a benchmark nonlinear structure, commonly termed the Brake-Reuß Beam (BRB) [18], containing a three bolt lap joint is used. The beam, shown in Fig. 2, is designed to have a simple geometry that contains nonlinear effects from a lap-joint. The lap-joint is an ideal system due to the effect of the nonlinearity on the transfer function (TF) [18], which is not observable in all systems [4,42] (i.e., in the BRB, the nonlinearity is clearly observable even at low excitation amplitudes; this feature, however, is not common across other proposed benchmark systems). In all experiments, the BRB is assembled according to the procedure of Refs. [18,43], in which the outer bolts are first tightened to 70% of the desired torque (20 Nm unless otherwise stated), followed by the center bolt to 70% of the same desired torque, then all bolts are tightened to the desired torque in the same order.

Fig. 2
The simplistic geometry of the Brake-Reuß Beam, (a) side view and (b) top view with shaker and Accel attachment points
Fig. 2
The simplistic geometry of the Brake-Reuß Beam, (a) side view and (b) top view with shaker and Accel attachment points
Close modal

To “Deconvolute the Nonlinearities from the Test and System,” the epistemic uncertainties from the boundary conditions, equipment setup, and input methods need to be understood and quantified. “Deconvoluting Nonlinearities From the Test and System” section analyzes a monolithic beam, of the same dimensions as the BRB, to gain an understanding of the nonlinearities added by the test. The need for updating locating the nonlinearity is reviewed in the “Locating the Nonlinearity” section. “Characterizing the Nonlinearity” section discusses and outlines current work on “Characterizing the Nonlinearity.” “Resolving the Long Term Evolution of the Nonlinearity” is evaluated in “Resolving the Long Term Evolution of the Nonlinearity” section. Finally, “Conclusion” section concludes with the major outcomes from this work.

Deconvoluting Nonlinearities From the Test and System

A nonlinearity often manifests itself when a small change in the constitutive model generates a large change in the response [4446]. Because of this, experimental measurements often convolute the effects of the nonlinearity of interest with the experimental setup [10,12]. The very act of measuring the response of a system can fundamentally change the system (like the observer effect in quantum physics [47]). By creating a support structure or attaching a gauge or other instruments to a structure, the system is no longer the original structure, but rather a combination of the structure and the experimental setup. Boundary conditions have been shown to influence damping of lightly damped structures [48]. A shaker, for instance, can change the impedance of a system dramatically due to the method by which it is attached to the system. Further, a fixed boundary condition is difficult to achieve; in most cases, torsional stiffness, friction, and even gaps are present [12,17]. By contrast, a free boundary condition is often emulated via bungee cords or foam pads, which act as weak springs and dampers. These free boundary conditions are influenced by the structure as bungees and foam exhibit a stiffening nonlinearity (e.g., as they are stretched/ compressed they become stiffer than a linear model would suggest).

Another issue that arises from the structure itself is sensor loading. Attaching a 5-gram Accel to a solid cube that weighs 50 kg will have less of an effect than attaching the same Accel to a highly flexible, light weight body such as an aero-shell. For nonlinear systems, the effects of added mass are even more pronounced, especially at higher forcing amplitudes. Tutorials on dynamic testing [49,50] recommend methods to setup an experiment, but tend to ignore or neglect any nonlinearity that arises from the setup. This practice is acceptable for linear systems; however, the nonlinearities associated with the test setup can be convoluted with those of the system. To reduce this convolution, the test setup needs to be evaluated.

Sources of setup related nonlinearities include misalignment, pre-loads, cable rattling, and poor transducer mounting [51]. These setup related nonlinearities are more likely to affect measurements at high levels of excitation, and their effects need to be accounted for. The damping added due to the test setup needs to be measured and studied for any nonlinearities that may be added into the system [19,50]. This is of particular importance for joint-based nonlinearities, which are typically damping related phenomenon [18].

Several excitation methods have been evaluated to isolate their nonlinear effects. The initial method evaluated is impact hammer excitation, which is of interest since it provides the possibility to excite a large range of modes with a single impact. The downside to the method is that it requires advanced data analysis techniques, such as the Hilbert Transform, Short-time Fourier Transform (STFT), Peak-Finding and Fitting (PFF), to separate different modal contributions and to study the amplitude-dependent response of each mode [1,6,9]. The excitation of multiple modes may lead to complex behavior at the location of nonlinearity, which may lead to an unexpected response. Another downside to this method is the difficulty in achieving a repeatable excitation force.

The other method is shaker excitation, which can excite the system at a specified frequency or by sweeping through a range of frequencies. One advantage of the method is that the amplitude of force or excitation level can be controlled. The latter is considered the better option to investigate nonlinear behavior around resonance, as constant response amplitude leads to constant activation of the nonlinearity. This constant activation leads to a linearized response, which allows for the use of standard linear modal analysis tools. Unfortunately, with strong nonlinear behavior, the high amplitude levels required for activating the nonlinearity make this approach tedious and time consuming. Force control allows for easier control of the system, though it changes the response amplitude of the nonlinearity during sine sweep excitation. This change in amplitude can lead to a stronger activation of the nonlinearity, which potentially leads to greater variability due to the uncontrolled events in the nonlinearity [11]. As a result, impact hammer testing is extensively studied due to reduction in the variability of the shaker method, though it introduces other potential sources of uncertainty. These potential sources of nonlinearities from excitation methods are evaluated in what follows.

To study the effects of the test setup, a monolithic beam was fabricated out of stainless steel 304. This structure is intended to be a linear system; therefore, any evidence of nonlinearity observed on its response is most likely due to the experimental setup, shown in Fig. 3. To gain an understanding of the vibration of the monolithic beam, modal analysis using the roving hammer technique [52] was performed. These frequencies and damping ratios are taken as the baseline for comparison. The natural frequencies of the first five bending modes are 246.3, 674.5, 1308.5, 2130.7, and 3126.8 Hz, and the first two torsional modes are 1965.0 and 3931.8 Hz. For each of the first seven modes, the modal damping ratios are between 0.03 and 0.05% (generally increasing with frequency).

Fig. 3
Basic test setup for testing the effects of boundary conditions, excitation techniques, and sensor setup. Shown is the jointed beam evaluated in the “Locating the Nonlinearity” section.
Fig. 3
Basic test setup for testing the effects of boundary conditions, excitation techniques, and sensor setup. Shown is the jointed beam evaluated in the “Locating the Nonlinearity” section.
Close modal

The experimental setup components studied are grouped into three categories: (1) impact hammer excitation, (2) “free-free” boundary condition approximation (Fig. 4), (3) instrumentation, and (4) excitation attachment techniques. Most of the experimental results can be found in the work by Smith et. al. [53]. The results of the study are summarized in Table 1; all values given are for the fourth mode (first torsional), any other significant changes will be listed in the Notes column. The cases with the largest change are shown in Fig. 5.

Fig. 4
Alternative boundary conditions chosen for this study—(a) Bungee-fishing line hybrid, (b) fishing line, and (c) foam supports
Fig. 4
Alternative boundary conditions chosen for this study—(a) Bungee-fishing line hybrid, (b) fishing line, and (c) foam supports
Close modal
Fig. 5
The TFs of the most significant tests in Table 1 for (a) the entire frequency range, (b) the first and second bending modes, and (c) the first torsional and fourth bending modes
Fig. 5
The TFs of the most significant tests in Table 1 for (a) the entire frequency range, (b) the first and second bending modes, and (c) the first torsional and fourth bending modes
Close modal
Table 1

Severity of the nonlinear influence on the first torsional mode and any significant changes to other modes (recorded in the Notes column) from different experimental setups on the natural frequency and modal damping of the beam

CategoryApproximate effect on frequencyApproximate effect on damping ratioNotes
HammerMetal tipREFREF
White plastic tip0.02 Hz0.00
Metal tip with added mass0.50Hz0.00
BungeesFull length (0.318 m)REFREF
Half length0.50 Hz0.01
Position: OutsideREFREF
Position: Inside3.00 Hz0.0003
Boundary materialBungeeREFREF
Fishing line2.00 Hz0.0002Constrains vertical motion
Bungee-fishing hybrid2.00 Hz0.0003
Foam2.50 Hz0.0003Affects first bending mode (1.50 Hz, 0.15%)
Sensor size1 gREFREF
10.5 g40 Hz0.06000
Accelerometer tests2 Accels glued at L1REFREF
2 Accels attached at L11.00 Hz0.0004
2 Additional Accels at L216.5 Hz0.0019Affects second torsion mode (1.00 Hz, 0.00%)
2 Additional Accels at L30.00 Hz0.00Affects second torsion mode (17.0 Hz, 1.31%)
2 Additional Accels at L2 and L316.0 Hz0.0021Affects second torsion mode (15.0 Hz, 1.15%)
Cable orientation: above, across, unsupported1.00 Hz0.0001
Shaker attachmentNone (Hammer)REFREFImpact hammer test
10–32 UNF (o 4.8 mm L = 23 cm)0.40 Hz0.0003No torsional mode excited
M2 (o 2 mm L = 14 cm)8.00 Hz0.0002Stinger bending occurs
Wire (o 1 mm L = 7.62 cm)19.0 Hz0.0003Stinger bending occurs
CategoryApproximate effect on frequencyApproximate effect on damping ratioNotes
HammerMetal tipREFREF
White plastic tip0.02 Hz0.00
Metal tip with added mass0.50Hz0.00
BungeesFull length (0.318 m)REFREF
Half length0.50 Hz0.01
Position: OutsideREFREF
Position: Inside3.00 Hz0.0003
Boundary materialBungeeREFREF
Fishing line2.00 Hz0.0002Constrains vertical motion
Bungee-fishing hybrid2.00 Hz0.0003
Foam2.50 Hz0.0003Affects first bending mode (1.50 Hz, 0.15%)
Sensor size1 gREFREF
10.5 g40 Hz0.06000
Accelerometer tests2 Accels glued at L1REFREF
2 Accels attached at L11.00 Hz0.0004
2 Additional Accels at L216.5 Hz0.0019Affects second torsion mode (1.00 Hz, 0.00%)
2 Additional Accels at L30.00 Hz0.00Affects second torsion mode (17.0 Hz, 1.31%)
2 Additional Accels at L2 and L316.0 Hz0.0021Affects second torsion mode (15.0 Hz, 1.15%)
Cable orientation: above, across, unsupported1.00 Hz0.0001
Shaker attachmentNone (Hammer)REFREFImpact hammer test
10–32 UNF (o 4.8 mm L = 23 cm)0.40 Hz0.0003No torsional mode excited
M2 (o 2 mm L = 14 cm)8.00 Hz0.0002Stinger bending occurs
Wire (o 1 mm L = 7.62 cm)19.0 Hz0.0003Stinger bending occurs

The only test setup variations that resulted in a moderate or higher effect on the dynamic measurements were foam supports, added mass from additional or larger Accels, and thinner stingers. Previous results have also shown that attempting to create a fixed boundary condition resulted in additional nonlinearities or creating a different boundary condition all together [12,17,54]. The additional sensors significantly influenced the beam via increased mass and moment of inertia. Therefore, using the results from an experiment with multiple sensors for model validation, all sensors should be included in computational models to match the response of the system more accurately. The wire and M2 stingers excite the torsional mode, which should not have been excited because the attachment location is along the center-line of the beam. The excitation of the torsional modes indicates that these stingers are bending, which causes energy to be inputted into unintended modes; this bending could be from the force levels used or the shaker and beam not being perfectly aligned. Historically, the use of the 10-32 stinger is contradictory to experience by researchers and the recommendation of using a thin stinger by Ewins [55]. A thin stinger is desired to decouple the shaker from the structural motion not in the direction of excitation, thus removing the bending moment applied at the force transducer [56].

Locating the Nonlinearity

From Ewins’ methodology [30,31], “Locating the Nonlinearity” is updated to specifically add how the nonlinearity changes during excitation. Spatially resolving an interfacial nonlinearity is a difficult endeavor, as the extent to which the contact area changes during excitation is unknown [57]. From the numerical perspective, there is a significant variability in how an interface is modeled: from a single hysteretic element, to a small number (approximately five) contact patches, to upwards of thousands of nodes in contact all tied together with nonlinear elements between the two surfaces [26]. A reasonable question is “how many elements are needed to spatially resolve the nonlinearity?” Too few may not capture aspects of the nonlinearities, while too many require more parameters than are measurable. To help determine the number of elements, contact pressure films can provide some insight into what occurs inside of a joint. While adding the pressure film fundamentally changes the response of a joint [57], the film can be used to improve the spatial resolution of a nonlinearity during dynamic testing. In Ref. [57], an electronic pressure film was used to measure in real time the contact area changes in the BRB’s joint during shaker loading, shown in Fig. 6. Figures 7 and 8 show instances in phase of the contact area changing during a shaker test, and the pressure change along a line, respectively. These figures show that the contact area evolves during excitation. This evolution indicates that the contact patch models do not hold with a small number of patches because the patches force the pressure on all nodes to be equal.

Fig. 6
Change in contact area during shaker testing around the first bending mode (adapted from Ref. [57])
Fig. 6
Change in contact area during shaker testing around the first bending mode (adapted from Ref. [57])
Close modal
Fig. 7
Snap shots of pressure measured using electronic pressure film for four phases of vibration ranging from approximately 0 deg to 270 deg (adapted from Ref. [57])
Fig. 7
Snap shots of pressure measured using electronic pressure film for four phases of vibration ranging from approximately 0 deg to 270 deg (adapted from Ref. [57])
Close modal
Fig. 8
Change in pressure versus time and normalized position, along the red line in the sub-figure that is 5/16” from the edge of the pressure film
Fig. 8
Change in pressure versus time and normalized position, along the red line in the sub-figure that is 5/16” from the edge of the pressure film
Close modal

In Ref. [58], this research was expanded to show how adding electronic pressure film to the BRB joint changes the response of the system. While the results show a decrease in stiffness and an increase in damping, the key finding was the ability to capture the changing contact area in the joint during dynamic testing. Figure 9 [58] shows the changing contact area in four regions (Regions 1 and 4 are the outer most regions and Regions 2 and 3 the inner). The changing contact area in Regions 1 and 4 indicates the presence of chatter, which introduces additional damping that is not accounted for in many models [2225] (which only focus on the damping increase and stiffness reduction introduced by sliding of the joint interface). The results of the two studies mentioned indicate that existing contact patch models do not fully resolve the change in the contact area [57]. They also show that the assumption of no modal coupling is invalid [58]. These findings show spatial resolution of the nonlinearity is important for proper modeling. As an example the measurements indicated that the nonlinearity is most severe in Regions 1 and 4. Supporting this, Ref. [59] found that fretting damage occurs in these regions of the BRB during low cycle shaker testing.

Fig. 9
Contact regions and the contact area change during shaker excitation in (a) region 1, (b) region 2, (c) region 3, and (d) region 4 [58]
Fig. 9
Contact regions and the contact area change during shaker excitation in (a) region 1, (b) region 2, (c) region 3, and (d) region 4 [58]
Close modal

Characterizing the Nonlinearity

The next step that needs revisited is “Characterizing the Nonlinearity.” According to Kerschen et al., several questions need to be answered to determine the type of nonlinearity [7]:

  1. Does the nonlinearity come from stiffness, damping or both?

  2. Is the system hardening or softening?

  3. Is it a strong or weak nonlinearity?

  4. Is the restoring force symmetric?

  5. Is the restoring force smooth?

Some questions can be answered looking at the distortion of a TF and the other frequency domain plots. However, the distortion may not be the conclusive evidence of a particular nonlinearity due to multiple types of nonlinearity that can act the same in certain excitation ranges [7,46]. These nonlinearities have been characterized with impact hammer [60] and shaker excitation [54] data.

Characterizing Nonlinearities Using Impact Hammer Data.

Impact hammer data is useful for characterizing a nonlinearity because all modes of interest are excited. Exciting all modes is also a hindrance because the linear mode superposition assumption no longer holds, and many of the modes are now coupled. To separate the modes, conventional methods use band-pass or modal filters. There are issues when using filtering techniques: not enough spatial information (too few sensors) and closely spaced modes (1560 and 1570 Hz). Figure 10 shows this difficulty when using a band-pass filter for closely spaced modes. When applied to closely spaced modes, band-pass filtering breaks down and the Hilbert Transform returns meaningless data as it is fitting two peaks. There are methods that may be used, such as, modal filtering that can be used when enough spatial information to filter to the desired mode is available, exciting at a specific location to only excite desired mode, or switching excitation technique (an option outlined later).

Fig. 10
Hilbert Transform applied to a BRB with the center bolt removed (B2B) and then increasing the torque to maintain same contact force (B2F) at the first torsional mode. Shown are the band-pass filtered data of the torsional mode overlaid on the original TF (a) 0–2 kHz, (b) 1520–1600 Hz, (c) the natural frequency versus response amplitude, (d) the damping ratio versus response amplitude, and (e) the envelope of the response versus time.
Fig. 10
Hilbert Transform applied to a BRB with the center bolt removed (B2B) and then increasing the torque to maintain same contact force (B2F) at the first torsional mode. Shown are the band-pass filtered data of the torsional mode overlaid on the original TF (a) 0–2 kHz, (b) 1520–1600 Hz, (c) the natural frequency versus response amplitude, (d) the damping ratio versus response amplitude, and (e) the envelope of the response versus time.
Close modal

Once the data are filtered to the response of a single mode, methods such as the Hilbert transform [7], STFT [1], zeroed early fast fourier transform [61], and wavelet-based methods [7,8] can be applied. A recent study investigates a way to resolve the loss of amplitude due to sample frequency called the peak finding and fitting method (PFF). The method assumes that the response near a peak can be fit to a second-order polynomial with a great deal of accuracy [9]; this assumption becomes more accurate as the sample frequency increases. Once the peak is found the frequency and damping are extracted using the amplitude and space between the peaks.

Characterizing Nonlinearities Using Shaker Data.

One method to avoid using filter is to excite the system at the natural frequency of interest using a shaker. This data can then be processed using the aforementioned methods. This process, however, is more time consuming than hammer testing. Another way to avoid filters is to perform controlled stepped-sine testing with a shaker [54] or phase-lock loop control [62]. The advantage of using controlled stepped-sine testing is that the rate of stepping through frequencies can be controlled to construct the backbone curve of the system; however, modes too closely spaced will be hard to characterize and a different method will need to be utilized. Linear extraction methods could be used to extract the damping and natural frequency, but these methods can introduce error because the peaks in the TF are not symmetric, resulting in two possible frequencies and damping values. A solution for extracting properties from asymmetric peaks is the CONCERTO method [63], which linearizes the response at fixed amplitudes in order to estimate the amplitude dependent properties of the system. The method linearizes the system by assuming that at each amplitude level the system can be defined as
mx¨+c(X)x˙+k(X)x=Fosin(ωt)
(1)
The resulting receptance frequency response function (FRF) is
Hk(ω)=Ar+jBrωr2ωk2+jηrωrωk=Rk+jIk
(2)
where the subscript r indicates resonance, subscript k a point on the FRF, ω is the frequency, η is the loss factor which is twice the damping ratio, j=1, and R and I are the real and imaginary parts of the FRF, respectively. The receptance, though, requires the outputs to be displacement; however, most dynamic tests use Accels to acquire the response of the system. Acceleration could be integrated to get displacement, introducing numerical error. To avoid integration error, the accelerance FRF can be used
Hk(ω)=ωk2(Ar+jBr)ωr2ωk2+jηrωrωk=Rk+jIk
(3)
which is based on the acceleration response values. The FRF value, Ar + jBr, is assumed to be equal at the same response level independent of the drive frequency ωk. The two FRF values are then set equal and separated into real and imaginary parts giving the frequency and loss factor at resonance.
ωr2=ω12ω22(R^12(R2R1)I^12(I2I1))R^12(ω12R2ω22R1)+I^12(ω12I2ω22I1)η=R^12(Ω1I2Ω2I1)I^12(Ω1R2Ω2R1)μ(R^122+I^122)R^12=ω2R1ω1R2I^12=ω2I1ω1I2Ω1=ω12(ωr2ω22)Ω2=ω22(ωr2ω12)μ=ω1ω2ωr
(4)

Discussion.

The PFF and accelerance CONCERTO methods are used to analyze the first elastic bending mode of the BRB. The FRFs and associated backbone curves are shown in Fig. 11. The comparison shows that the methods return continuous trends, indicating that either method can be used. The drawback of all of these methods is that they extract parameters for one mode at a time. When there is modal coupling, such as in lap joints, these methods struggle to extract all parameters to describe what is occurring for a single mode. Despite these efforts, there still exists evidence of modal coupling in the extracted parameters (see for instance Refs. [26,64]). Nonlinear normal mode (NNM) is a method that has some capabilities to capture modal coupling. NNMs can be used to fit nonlinear parameters from shaker or hammer data. The method is based on the synchronous oscillations of a structure, which, by definition do not usually include dissipative features as they lead to decaying periodic responses [65,66]. If the data does not contain information for the higher order modes, then modal coupling may be part of the truncation error inherent in the method. These drawbacks, and only characterizing one mode at a time, show the need to continue developing methods to characterize the nonlinearity. The need for continuing work on this step in model validation is paramount as real structures often contain many closely spaced and repeated modes (when symmetries exist) [11].

Fig. 11
The comparison of hammer and shaker characterization method highlighted in this section*. Shown are (a) the analyzed FRFs, (b) the natural frequency versus response amplitude, and (c) the damping ratio versus response amplitude
Fig. 11
The comparison of hammer and shaker characterization method highlighted in this section*. Shown are (a) the analyzed FRFs, (b) the natural frequency versus response amplitude, and (c) the damping ratio versus response amplitude
Close modal

Resolving the Long-Term Evolution of the Nonlinearity

For structures where damage or wear is of concern, one additional step is needed to predict the lifetime or to schedule the optimal maintenance of a structure. This missing step is “Resolving the Long Term Evolution of the Nonlinearity;” which is needed to understand how a system will react at the onset of damage. Though the damage is typically small, it can generate a large change in the response of the system [4446]. This damage for jointed interfaces is often manifested as wear caused by fretting [40], and it can grow over time. To demonstrate this, long-term testing data have been collected on three sets of BRBs [43,60]. These tests consist of hammer impacts ranging from 60 to 500 N with one system tested up to 2 kN [43,60] and stepped sine shaker data ranging from 0.1 to 20 N. Figure 12 shows how the frequency of the BRBs changed during the course of the experiments.

Fig. 12
Frequency shift of three BRBs under hammer impacts, shaker excitation, and a combination of tests
Fig. 12
Frequency shift of three BRBs under hammer impacts, shaker excitation, and a combination of tests
Close modal

As can be seen once damage occurs, the mechanism that causes it persists (is permanent) and often continues to grow over time. In the test that only the hammer was used, the damage mechanism is plastic deformation caused by an approximately 2 kN impact. In the test that has some hammer impacts and then is excited with a shaker, the damage mechanism is fretting wear as directly observed on the interface after testing. However, when only shaker excitation was used no appreciable change was observed. This lack of change could be due to the level of excitation is relatively low (∼2N maximum) or could be due the initial damage (at a small scale) driving substantial long-term degradation of the structure. These results show how important it is to resolve the long term evolution (or lack thereof) of the nonlinearity as different loading profiles can drive different damage mechanisms.

Conclusion

From the results of research into the dynamics of a jointed interface, the state-of-the-art methodology for model validation was missing two steps that are necessary to improve the model accuracy. This is especially important for developing models of nonlinear systems when it is difficult to adequately span the real environment due to testing limitations (e.g., the difference between flight conditions and ground vibration testing). Fundamentally, “Deconvoluting Nonlinearities from the Test and System” is an important step as the test setup can affect the response of the structure being investigated. It was shown that the boundary condition and number and placement of sensors have the largest influence on the structure. Tutorials on dynamic testing tend to ignore or neglect the nonlinear effects that test setup could introduce to the data. Test setups currently being used in dynamic testing were analyzed with the result that many of the setups do have an influence on the response of structure. Next, “Resolving the Long Term Evolution of the Nonlinearity” is important as every change in the constitutive model results in a change in the response and the nonlinearity (especially a joint) can change over the duration of dynamic loading. The order of “Locating the Nonlinearity” and “Characterizing the Nonlinearity” have been switched, because some nonlinearities need to be located and spatially resolved over time before they can be characterized. From recent research, “Locating the Nonlinearity” needs to be updated to model a nonlinearity accurately, as its area of influence may evolve during dynamic inputs. The evolution of the contact area warrants further investigation as experimental methods mature. “Characterizing the Nonlinearity” needs to be revisited as a consequence of the improved understanding of the interface dynamics. Current methods are single mode based, while a nonlinearity may have significant modal coupling that requires higher order modes to be analyzed at the same time. Finally, an updated methodology incorporates the missing steps (as summarized in Fig. 1). Specific recommendations for nonlinear testing include:

  • Include all sensors in the computational models. Even sensors weighing less then 0.05% of the systems mass can cause a 1% change in the natural frequencies due to modifying the moment of inertia.

  • Most support structures introduce unintentional stiffness, damping, and nonlinearity. To best avoid this, use a bungee-fishing line configuration to reproduce “free” boundary conditions.

  • When testing with a shaker, use a stinger that will decouple the system from the structure. For large amplitude testing, care must be used when selecting a stinger: the stinger used needs be able to handle the desired loads while not buckling.

  • The nonlinearity needs to be spatially located, as the spatial distribution of nonlinear elements in models may not be uniform and may change during excitation.

  • The long-term evolution of a nonlinearity needs to be resolved by tracking the change in response over time.

Footnote

2

Often, models with nonphysical parameters gloss over details such as the spatial distribution of a nonlinearity. The resulting models are typically calibrated to experiments and have limited predictability.

Acknowledgment

The authors would like to thank Dr. Matthew Allen (University of Wisconsin-Madison), William Flynn (Siemens), Juan Carlos Bilbao-Ludena (Technical University of Berlin), Simone Catalfamo and Florian Morlock (University of Stuttgart), Pascal Reuß (Daimler Automotive), and Benjamin Pacini and Randall Mayes (Sandia National Laboratories) for their input during the tests. This work was conducted in part at the Nonlinear Mechanics and Dynamics (NOMAD) research institute hosted by Sandia National Laboratories during the summers of 2014 and 2015. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

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