Characterization of Nonlinearities in a Structure Using Nonlinear Modal Testing Methods 169 Fig. 1 Phase-locked loop (PLL) schematic 3 Phase-Locked Loop A phase-locked loop (PLL) is a control system where a specific phase relationship exists between its output and input signal. It follows from the corollary that two signals starting from the same reference will have same frequency only when the phase between them is zero. Conversely for dynamic systems, the same principle is used to obtain a phase lag quadrature relationship between the excitation and displacement; the frequency where this quadrature is observed can be assumed to satisfy one or more conditions for a normal mode (linear or nonlinear) as described in the previous section. Traditional sine-sweep type approaches to nonlinear testing lack any such control loops that provide monitoring and control of input and output relationships. In its simplest form a PLL consists of the components as shown in Fig. 1. The phase detector is used to separate the phase information between the reference p(t) and output s(t) signals. In theory, this is achieved by multiplying the input and output signals for single frequency sinusoidal type signals, i.e., q(t) =p(t) * s(t). The resulting DC component of signal can be used to extract the phase information between the two signals. A loop filter, comprising of low pass filter and variations of a PID controller for stability controls, is generally used to extract the phase information. A voltage-controlled-oscillator (VCO) is used to generate a signal that has the required characteristics based on the output of the loop filter. The output of the VCO may be amplified as required to obtain the desired voltage level (excitation level). Detailed descriptions of these components are available in [16]. In nonlinear modal testing, it is generally assumed that the influence of higher harmonics is negligible and only the fundamental frequency is considered for tuning the PLL. The PLL can hence be operated to obtain a phase quadrature for the fundamental harmonic of the excitation frequency. For systems whose fundamental frequency varies with excitation amplitude, the loop circuit will ensure a signal is generated such that a phase quadrature is obtained between excitation and displacement as soon as the excitation levels are altered. The PLL operation can alternately be executed using any software platform [19] that is typically used to operate and acquire data in experimental vibration analysis. Some advantages of using a software PLL (sPLL) are highlighted in the following sections. 4 Software Phase-Locked Loop PLL circuits described above typically operate on a per-wave basis. These variations over longer time durations may be used to determine performance and operation of the PLL circuit. In any experimental analysis of dynamic systems and vibrations, averaging is most crucial in ensuring confidence of data acquired. Moreover, the above-described traditional PLL requires physical components or its equivalent to obtain a nonlinear mode for a given system. The process also involves careful and experienced tuning of the loop filter using additional PID controllers. Alternately, software phase-locked loop (sPLL) provides many advantages and offers more fidelity from a vibration measurement perspective against employing a traditional PLL. The sPLL method used in this paper uses MATLAB [22] and its available features as described in the further sections. A VXI mainframe is used for data acquisition and control of the modal shaker through a power amplifier. The complete schematic for operation of the sPLL is given in Fig. 2. Consistent with [16], two assumptions are made for nonlinear modal testing using sPLL, viz. (1) The fundamental harmonic is used for phase computations and effect of higher harmonics is negligible and (2) The RMS of input and output quantities is mostly dominated by the signal at the fundamental frequency and hence is used for force and response computations. A linear experimental modal analysis is performed on the test system to determine the mode that is considered for investigation. Once the linear modal frequency is determined, a steady sine wave is generated at a frequency very close to this linear modal frequency. At extremely low amplitudes of vibration, the steady sine wave excitation will ideally be in quadrature with the response at the same frequency as the experimentally determined linear modal frequency. To obtain a backbone curve, the response amplitude and frequency of quadrature are essential. The excitation levels can be stepped up either using voltage control of the mainframe or using the modal shaker power amplifier. For nonlinear systems of approximately cubic hardening type nonlinearity, the nonlinear modal frequency increases with increase in vibration levels.
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