108 S. Peter et al. excitation is troublesome especially in the case of strong nonlinearity where a jump in the Frequency Response Function (FRF) occurs. Close to resonance, even for small disturbances, the jump can occur prematurely. Also the tuning, which is usually done manually, is time consuming and the estimation of an excitation level driving the system in a nonlinear range requires some a priori knowledge about the system or several trial and error runs. Additionally, by using the time frequency analysis of a free decay, sophisticated signal processing such as wavelet transforms [15] is required and some inaccuracy might be induced due to transient effects. This paper presents a novel method for phase resonance testing to overcome these practical issues by an automated procedure using a PLL. The PLL is originally an analogue circuit used in radio technology in the 1930s [16] and is nowadays widely used in electronics applications like radios, TVs or smart phones [17]. The general idea of PLL concepts is to generate a harmonic signal with a frequency which is tuned based on the phase difference to a reference signal. There are many different designs of PLLs which are all essentially nonlinear oscillators generating a harmonic output dependent on this phase difference. The design of the PLL has to be adapted to its application which can be the tuning of digital or analogue signals originating from linear or nonlinear systems [18, 19]. Yet, only few attempts have been made to use these concepts for the measurement of mechanical structures. Most of them remain theoretical and the examples are mostly numerical ones [20]. There have been attempts to use PLLs in nonlinear micro systems modeled with one DOF [21] but the applicability for testing macro scale mechanical systems or continuous structures is mostly unclear. A publication by Mojrzisch [22] showed the usefulness of the PLL experimentally for the measurement of nonlinear FRFs of a macro scale single DOF Duffing type system by phase sweeping. However, to the authors knowledge there have no attempts been made to use the PLL for tracking of backbone curves of nonlinear continuous structures and exploit its potential for the measurement of NNMs. The paper is organized as follows. In Sect. 11.2 some basics of phase resonance testing for nonlinear structures are briefly reviewed. In the subsequent Sect. 11.3 some more specific aspects of phase resonance method using the PLL including some design aspects of the PLL used within this paper are explained. In Sect. 11.4 a numerical example is used to illustrate the method and highlight some of its characteristics. The numerical example is followed by an experimental demonstration of its functionality in Sect. 11.5. The paper closes with a conclusion and some aspects of future work in Sect. 11.6. 11.2 Nonlinear Modal Analysis Using the Phase Resonance Method A general mechanical system with conservative nonlinearities can be described in a spatially discretized form by the differential equation MRxCDPxCKxCFnl.x/ DFexc.t/; (11.1) where Mdenotes the mass matrix, D the viscous damping matrix, K the linear stiffness matrix and Fnl.x/ represents a vector of nonlinear, conservative forces. The vector of external excitation is represented Fexc. In contrast, the NNMs of an autonomous conservative system are governed by a differential equation of the form MRxCKxCFnl.x/ D0: (11.2) For numerical systems periodic solutions of this differential equation can be obtained in a straightforward way using shooting or the HBM. These periodic solutions provide the NNMs of the system and can for example be visualized in Frequency Energy Plots (FEPs) [4]. Due to its high efficiency and its filtering property, which is particularly interesting in conjunction with measurements, the numerical NNM calculations throughout this paper are obtained by the HBM. The details of the numerical method are described in a previous publication [23]. The experimental realization of periodic motions of systems which motions are governed by differential Eq. (11.2) is more difficult as undamped systems cannot be realized practically as there are always sources of material damping or damping in interfaces of coupled structures. Hence, the forced and damped system described by Eq. (11.1) has to be considered as a representation of real systems. For lightly damped structures the assumption of proportional damping can provide a reasonable approximation. For the realization of a NNM motion of the underlying conservative system, the non-conservative system is sought to behave like a system described by Eq. (11.2). By comparing Eqs. (11.2) and (11.1) it can be seen that theoretically the forcing should exactly balance out the damping for all points of the structure and for all times i.e. DPx.t/ DFexc.t/; 8t: (11.3)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==