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

Highlights

• Response controlled stepped sine test¨ is proposed for experimental modal analysis of nonlinear systems.

• Modal parameters of nonlinear structures are identified as functions of modal amplitude.

• Identified modal parameters are used to synthesize FRFs for untested harmonic forcing scenarios.

• ¨Harmonic force surface¨ approach is proposed for measuring unstable branches of frequency response curves.

• The method is validated using a numerical example, a benchmark structure and a real missile structure.

Abstract

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.

Introduction

Structural dynamics community witnessed important developments in the field of nonlinear system identification over the last two decades [1], [2]. Despite the significant progress in the state-of-the-art, there are still difficult problems such as joint nonlinearities (multiple and discrete) or geometric (i.e. continuously distributed) nonlinearities due to large deformations. Since there are generally several joints in engineering structures, it would be very difficult, if not impossible, to identify each joint nonlinearity separately. On the other hand, in case of geometric nonlinearities, the concept of discrete nonlinear elements cannot be used. Consequently, the right identification strategy in these complex problems is to quantify the resultant effect of all nonlinearities, instead of focusing on individual nonlinear elements. Fortunately, the concept of nonlinear normal modes (NNMs), which dates back to Rosenberg [3], [4] (1960s), provides a rigorous theoretical framework to study the overall effect of several nonlinearities in a structure. Analogous to linear normal modes, Rosenberg defined an NNM as a vibration in unison of the system, i.e. synchronous periodic motion. Later, in 1979, Szemplinska-Stupnicka proposed a novel technique called the single nonlinear mode method [5] to study the effect of nonlinearities on the resonant vibrations of multi-degree-of-freedom (MDOF) systems. The method showed that if the modes are well separated and no pronounced modal coupling occurs in the energy range of interest, near-resonant frequency responses of a nonlinear system can be represented accurately by a single NNM and its corresponding natural frequency which are functions of modal amplitude. This pioneering work of Szemplinska-Stupnicka led to the development of various nonlinear modal identification techniques. For example, in the early 1990s, Setio et al. [6] proposed to extract nonlinear modal parameters by minimizing the error function between frequency responses measured by force-controlled sine-sweep testing and analytical frequency responses represented by superposition of NNMs, which gives satisfactory results if the modes are well separated and no internal resonances occur. The method was validated on a real cantilever beam supported at its free end by a string which creates cubic stiffness effect. Later, in the early 2000s, Gibert [7] proposed a similar modal identification algorithm based on minimization of the error function between measured and analytical frequency responses and showed that superposition of NNMs gives satisfactory results for the synthesis of frequency responses of the Ecole-Central-de-Lyon (ECL) benchmark [8] over the frequency range including the first three elastic modes.

Early 2000s witnessed other promising nonlinear modal identification strategies. For example, Göge et al. [9] proposed a novel method called the identification of nonlinearity by time-series-based linearity plots (INTL) which is based on the application of the restoring force surface (RFS) method [10] in modal space. The test strategy used in the INTL approach is the normal mode force appropriation which is also known as phase resonance testing, where the nonlinear system is harmonically excited at its resonance frequency by means of an excitation force pattern appropriated to a single mode of interest. Nonlinear modal stiffness and damping parameters can then be determined by curve fitting to the measured nonlinear restoring force in modal space. The application of the method was demonstrated on a real space structure. Platten et al. [11] also proposed another RFS-based technique called the nonlinear resonant decay method (NLRDM). This method differs from the INTL approach in that it can take also the cross-coupling of nonlinear modes into account by virtue of its excitation strategy which consists of a burst sine at the resonance frequency of the mode of interest. The NLRDM technique was validated on a wing-like structure with hardening stiffness at the pylon connections [12] and on a real transport aircraft [13].

The major difference between the new generation techniques developed in the last decade and the afore-mentioned ones is that in the new generation algorithms, computational effort is minimized at the cost of experimental effort. Most of the recently developed nonlinear modal identification methods are inspired by phase resonance testing approach and they focus on direct parameter estimation, which reduces computational effort considerably. For example, in the method proposed by Peeters et al. [14] in 2011, the phase lag quadrature criterion was generalized to nonlinear structures in order to locate a single NNM during experiment. Once the NNM appropriation is achieved, the frequency-energy dependence of that nonlinear mode can be determined by applying time-frequency analysis to the free decay response data. The proposed methodology was demonstrated on numerical examples [14] and on a real nonlinear beam structure [15]. However, an important drawback of this early version of the nonlinear phase resonance testing was the manual tuning process of the phase lag between response and excitation. By virtue of the recently proposed control algorithms [16], [17], the tuning of the phase lag was automated throughout the entire NNM backbone curve. The control technique proposed by Peter and Leine in [16] is called phase-locked-loop (PLL) control which provides a robust and fast way of tracing out back-bone curves as well as of stabilizing the unstable branches of near-resonant frequency response curves. The PLL control strategy, which outputs zeroth and higher harmonics of nonlinear modes as well as the fundamental harmonic, was validated on a benchmark beam structure in [16] and on a circular plate, a chinese gong and a piezoelectric cantilever beam in [18]. In accordance with the single nonlinear mode assumption [5], synthesis of the near-resonant frequency response curve of a nonlinear cantilever beam from a single NNM measured by PLL technique was also demonstrated in [19]. Another interesting application of the PLL technique is the identification of the nonlinear dissipation at a bolted joint [20]. The control approach proposed by Renson et al. in [17], called control-based continuation (CBC), is similar to the PLL method, and also enables backbone curve identification of nonlinear structures. The application of the CBC method was demonstrated on a real single-degree-of-freedom (SDOF) oscillator in [17].

All of the nonlinear modal identification techniques mentioned above have their own advantages and limitations. Early identification methods [6], [7] which rely on simple frequency response data measured by standard force-controlled sine sweep testing involve considerable computational cost. On the other hand, INLT and NLRDM require considerable effort both in computation and experiment, and they are applicable to weakly nonlinear systems. In nonlinear phase resonance testing method [14], the computational effort is considerable reduced, but the manual tuning of the phase lag between response and excitation process requires careful and time consuming experimentation. Although recently developed PLL and CBC control strategies automated the tuning of the phase lag as well as the determination of the backbone curve, they cannot make use of the available standard equipment. Furthermore, although experimental extraction of natural frequencies and deflection shapes at resonance is straightforward in the methods based on phase resonance testing approach, determination of nonlinear modal damping is still an important issue.

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. The method is based on the single nonlinear mode assumption of Szemplinska-Stupnicka [5], where near-resonant frequency responses of a nonlinear system can be represented accurately by a single NNM and its corresponding natural frequency which are functions of a single modal amplitude. Accordingly, the proposed method hypothesizes that if the displacement (equivalently modal) amplitude is kept constant with the RCT strategy during modal testing, measured frequency response functions (FRFs) come out in quasi-linear form. That makes it possible to use standard linear modal analysis tools to extract all modal parameters as functions of modal amplitude. These identified modal parameters can then be employed to synthesize near-resonant frequency response curves including unstable branches, if there is any, for various untested harmonic forcing scenarios. Furthermore, in case of using multiple sensors during RCT, NNMs can also be experimentally extracted from identified modal constants. Therefore, the contribution of the proposed method is threefold. Firstly, it relies on standard controllers (available in commercial modal testing hardware and driven by commercial software) which makes it very attractive especially for industrial applications. Secondly, identification of modal damping and mass normalization of NNMs, which are necessary for the prediction of frequency responses of untested harmonic forcing scenarios, is straightforward with the proposed method, by applying linear modal analysis methods available in commercial software packages to measured constant-response FRFs of nonlinear structures. Finally, the proposed approach provides two different ways of determining near-resonant frequency response curves for untested constant-amplitude harmonic forcing scenarios; either computationally by using the nonlinear modal parameters identified during RCT, or experimentally by directly extracting isocurves of constant-amplitude forcing from the measured Harmonic Force Surface (HFS), a new concept proposed in this paper. It should be noted that both approaches are capable of determining unstable branches of frequency response curves, which may occur in strongly nonlinear systems. Theoretically, both approaches must give identical results, which constitutes a self-validation measure for the proposed method.

The paper is organized as follows. In Section 2, the theoretical background of the proposed approach is given. Subsequently, in Section 3 the proposed experimental methodology used to identify nonlinear modal parameters, and two different ways of determining near-resonant frequency responses corresponding to constant-amplitude harmonic forcing are explained in detail. Section 4 is dedicated to the numerical validation of the proposed approach with a lumped MDOF system with strong conservative nonlinearity. In Section 5, the method is applied on a real cantilever beam supported at its free end by thin metal strips which create cubic stiffness effect due to geometric nonlinearity, and also on a real missile structure which exhibits considerable nonlinear damping due to bolted joints. Finally, conclusions are discussed in Section 6.