1. Introduction
By virtue of various advanced techniques developed in the field of linear experimental modal analysis over the last 40 years, the modal survey of aerospace structures, specifically named as ground vibration testing, was established as an industry standard in the new millennium. On the other hand, the increasing competition in the industry to achieve higher performance in aircraft, missiles, and satellites inevitably increased the frequency of occurrence of non-negligible structural nonlinearities in aerospace applications (e.g., References [
1,
2,
3]), which led to a shift of emphasis towards the development of nonlinear system identification techniques in structural dynamics [
4,
5].
The modal testing of linear structures is accomplished either by using broad band random testing or by using classical force-controlled stepped-sine testing, where the force amplitude is kept constant. Unfortunately, direct implementation of these conventional modal testing techniques to structures that exhibit strong nonlinearity does not give satisfactory results. Random testing, which requires taking many averages to increase the coherence of measured frequency response functions (FRFs), is not suitable for nonlinear system identification because averages do their best to hide nonlinearity, as stated in Reference [
6]. On the other hand, constant-force sine testing often fails to capture turning points and unstable branches of nonlinear frequency response curves. The constant-force test algorithm available in commercial equipment can measure only the stable branches of the nonlinear frequency responses, which requires running multiple sine tests in opposite sweep directions. In the case of standard constant-force sine testing, the controller unavoidably jumps from one stable branch to another, as shown in
Figure 1a. Moreover, in some cases, a premature jump may occur, as shown in
Figure 1b. Early methods [
7,
8] attempted to identify amplitude-dependent nonlinear modal parameters by developing curve fitting procedures applicable to measured constant-force FRFs based on the single nonlinear mode theory [
9]. However, their applications remained limited to simple benchmark structures due to the computational burden in the case of complex structures and due to the missing frequency response data caused by the jump phenomenon.
Contrary to the early attempts based on phase separation testing mentioned above, most of the techniques proposed in succeeding years were inspired by the phased resonance testing approach. Two interesting approaches based on normal mode force appropriation and on the application of the restoring force surface in modal space were proposed in References [
10,
11]. Experimental application of the nonlinear resonant decay method proposed in Reference [
11] to identify the backbone curve of a single degree-of-freedom oscillator was demonstrated in Reference [
12]. Alternatively, the phase lag quadrature criterion was implemented in nonlinear structures in order to isolate a single nonlinear normal mode (NNM) in Reference [
13]. In this approach, the NNM appropriation is succeeded by time-frequency analysis of the free-decay response data to determine the frequency-energy dependence of the nonlinear mode of interest. However, an important drawback of this method is the manual tuning of the phase lag between response and excitation, which introduces difficulty in the experiment and requires longer experimentation time. The two recently proposed experimental continuation techniques [
14,
15] eliminated this drawback by automating the tuning of the phase lag throughout the complete backbone curve. The phase-locked-loop (PLL) control algorithm [
14] is capable of tracing out backbone curves and also captures the unstable branches of frequency response curves, if there are any. Similarly, the control-based continuation (CBC) approach [
15] relies on phase-quadrature condition to trace out the backbone curve. In CBC, unstable branches of constant-force frequency response curves are obtained by processing S-curves measured at several fixed frequencies by varying the response level. However, these two state-of-the-art techniques cannot utilize commercial modal testing equipment, and although the determination of NNMs and corresponding modal frequencies is straightforward, identification of nonlinear modal damping is still a considerable problem.
As an alternative to these experimental continuation techniques, a systematic approach called Response-Controlled stepped-sine Testing (RCT) was proposed quite recently by the authors of this paper, which constitutes the main step of a new experimental modal analysis method for nonlinear systems [
16]. The method can be applied to systems where there are many nonlinearities at several different (and even unknown) locations. However, this technique is not applicable to systems with internal resonances. In other words, the effects of sub- and super-harmonics are assumed to be negligible. In the RCT strategy, the displacement amplitude of the test point (equivalently, the modal amplitude) is kept constant during stepped-sine testing, which leads to quasi-linear FRFs even in the case of strongly nonlinear structures. Accordingly, conventional linear modal analysis techniques can be employed to identify nonlinear modal parameters as functions of modal amplitude. Unmeasured constant-force FRFs, which may even have unstable branches, can then be synthesized by using these modal parameters based on the single nonlinear mode theory [
9]. Alternatively, constant-force FRFs can be directly extracted from the measured Harmonic Force Surface (HFS), an innovative concept proposed in Reference [
16]. The key feature of the RCT approach is that it simply uses standard equipment, which makes it very attractive especially for industrial applications. Furthermore, the identification of nonlinear modal damping and of mass normalized NNMs is straightforward via applying standard linear modal analysis methods to measured quasi-linear FRFs. The proposed HFS concept was employed as a validation tool for the constant-force FRFs synthesized by using experimentally extracted nonlinear modal parameters. The current paper is a complementary work specifically dedicated to demonstrating the performance of HFS in experimentally identifying backbone curves of strongly nonlinear systems by emphasizing its ability to accurately identify turning points of frequency response curves with unstable branches. It is important to note that in the context of this paper,
strong nonlinearity terminology is used to refer to nonlinear systems with overhanging unstable branches in the frequency response curves which result in jump phenomenon in standard constant-force stepped-sine testing. This does not necessarily imply a significant effect of higher harmonics, as illustrated in the numerical and experimental case studies.
The paper is organized as follows. In
Section 2, the RCT approach and HFS concept are briefly summarized.
Section 3 is dedicated to the analytical illustration of the HFS method on a 5 DOF system with strong conservative nonlinearity. In
Section 4, HFS is successfully applied to identify the backbone curves of a benchmark beam with cubic stiffness and of a control fin actuation mechanism of a real missile structure, both of which exhibit strong nonlinearity causing jump phenomenon during classical constant-force stepped-sine testing. Finally, conclusions are given in
Section 5.
2. Experimental Modal Analysis with RCT and HFS
Experimental modal analysis using response-controlled stepped-sine testing was recently proposed by the authors of this paper in Reference [
16], where the theoretical background is explained in full detail. Here, only a brief summary of the experimental methodology will be given, with the emphasis being on important key features.
The flow chart of the proposed experimental methodology by using RCT is shown in
Figure 2. The right column of the flow chart consists of the experimental extraction of nonlinear modal parameters and of the synthesis of constant-force FRFs by using these parameters. The identification of nonlinear modal parameters is straightforward via applying standard linear modal analysis techniques to quasi-linear constant-response FRFs measured by RCT. Quasi-linearization of FRFs by keeping the displacement of the test point (equivalently, the modal amplitude) constant is based on the Nonlinearity Matrix concept [
17] and the single nonlinear mode theory [
9] as explained in Reference [
16]. The key formulation of the quasi-linearization concept is as follows:
where
is the near-resonant receptance at point
for a given excitation at point
and
is the excitation frequency.
,
and
are the modal constant, natural frequency, and modal damping ratio of the
th mode, respectively. These nonlinear modal parameters are functions of a single parameter; the modal amplitude
.
It was shown that if the modal amplitude is kept constant during stepped-sine testing, the measured constant-response FRFs turn out to be quasi-linear. In the case of a single input sine testing, the modal amplitude can be kept constant by just keeping the displacement amplitude of the driving point constant. In References [
13,
16,
18], it was shown that nonlinear modes can be isolated with acceptable accuracy by just using single-point single harmonic excitation under the condition that no internal resonance occurs. On the other hand, the RCT method can theoretically be employed by using multi-input sine testing, which requires careful tuning of the amplitude ratios of excitation signals to keep the modal amplitude at a constant level [
16].
Due to the popularity of accelerometer in modal testing, it was preferably used as the control sensor in the applications of the proposed method. Accordingly, a constant displacement amplitude condition at the driving point over the frequency range of interest was achieved in an indirect way by feeding the closed-loop controller with an appropriate user-defined acceleration profile, which is supported by standard modal testing software (e.g., LMS Test Lab).
The HFS concept was successfully used for the validation of constant-force FRFs synthesized from identified nonlinear modal parameters [
16]. The focus of the current paper is demonstrating the performance of HFS in experimentally extracting the backbone curves of strongly nonlinear systems by emphasizing its ability to accurately extract the turning points of frequency response curves with unstable branches.
As shown in the left column of the flow chart given in
Figure 2, HFS is constructed from the harmonic force spectra of the driving point measured at different constant displacement amplitude levels by using linear interpolation. A frequency response curve corresponding to a specific constant force level, which will include any existing unstable branch, is simply determined by picking up points of the HFS corresponding to that force level.
In this paper, it is proposed to determine the backbone curves of nonlinear systems by picking up resonance peaks of nonlinear frequency response curves extracted from HFS at various different constant force levels. Picking up resonance peaks requires smoothly identified turning points, which is an important issue for the state-of-the-art experimental continuation techniques. The successful application of HFS in extracting backbone curves of strongly nonlinear systems is demonstrated with numerical and experimental case studies in the subsequent sections.
5. Conclusions
Conventional constant-force sine testing is not suitable for accurately measuring turning points of frequency response curves of nonlinear structures exhibiting strong conservative nonlinearity due to the jump phenomenon. Consequently, the accuracy of backbone curves determined by constant-force testing is always questionable, which led to the development of advanced experimental continuation algorithms to extract backbone curves in the last decade. Although the current state-of-the-art provides promising control algorithms, it cannot directly make use of available modal testing equipment. This paper proposes an alternative approach that relies on standard equipment to identify backbone curves of strongly nonlinear systems directly from experimental measurements by using the Response-Controlled stepped-sine Testing (RCT) and the Harmonic Force Surface (HFS) concept recently proposed by the authors. These methods can be applied to systems where there are several nonlinearities at unknown locations. However, these techniques are not applicable to systems where internal resonances occur. So, the method is applicable to systems with strong nonlinearity in the sense that they exhibit unstable branches in the frequency response curves obtained with constant-force testing, as illustrated in the numerical and experimental case studies given in this paper, but not for nonlinearities which give rise to internal resonance.
In the proposed method, harmonic excitation force spectra measured at several different constant displacement amplitudes are collected into a smooth HFS by using linear interpolation. Next, constant-force frequency response curves with smooth turning points and unstable branches (if there is any) are extracted by cutting the HFS with constant force planes. Finally, the backbone curve is determined by connecting the resonance peaks of these frequency response curves. Points which correspond to the same resonance frequency on the backbone curves that belong to different locations build-up the NNM of the structure under test. The proposed method is numerically validated on a 5 DOF lumped system which exhibits strong conservative nonlinearity due to cubic stiffnesses. Furthermore, the method is successfully applied on a cantilever beam with a nonlinear connection 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.