This paper systematically analyzes linear oscillators, e.g., spring-mass-damper systems or RLC-circuits that are controlled by an extension of a phase-locked loop (PLL). These systems are often used in measurement applications where the stability and dynamics directly influence the measurement quality. Therefore, a description of the control loop in terms of phase signals is sought. However, the classical oscillator turns into a highly nonlinear system when it is formulated in amplitude/phase-variables of its input and output signals. Up to now, there were made either ab-initio assumptions of slowly varying parameters or trial-and-error designs. The novel approach proposed in this paper derives a universally valid description in state space form which enables the use of standard methods of nonlinear system theory. Using this description, the stability of phase controlled oscillators is analyzed by means of Lyapunov functions. A linearization is applied in order to effectively design the controller and optimize the closed-loop dynamics. Simulations with the original nonlinear systems are conducted to justify the linear approach. Thereby, two application scenarios are under consideration: Tracking of the desired target value (target phase shift) and resonance tracking (changes of the system parameters). It is found that including the phase dynamics of the oscillator significantly improves the description of the closed-loop behavior. Finally, the results are validated experimentally for an application measuring the viscosity of fluids.
This paper presents an innovative method to achieve the simultaneous phase control of multiple resonance frequencies of a linear multi-degree-of-freedom system using only one sensor/actuator pair. Each frequency is manipulated independently by means of a PLL-based control loop which comprises a digital averaging phase detector that combines the two most crucial tasks, namely the phase shift measurement and the frequency separation. The properties of the controller are designed individually for each mode using a linearized model of the control system. The method is applicable to all kinds of oscillators where frequencies near the structure’s natural frequencies are to be controlled. To validate the results experimentally, the controller is implemented on a digital signal processor (DSP) and applied to a torsional oscillator. Investigating two different damping conditions, the simultaneous control of five resonance frequencies of the oscillator illustrates the effectiveness and stability of the multiple frequency tracking. The method is able to significantly improve the accuracy and versatility of sensor applications. As an example a method is presented that enables the direct determination of the modal damping by using two frequencies corresponding to one single vibration mode.
This paper presents the extension of the classical torsional resonance viscometer towards the characterization of linear viscoelastic fluids. By simultaneously tracking multiple resonance frequencies, shear viscosity and elasticity can be monitored at multiple discrete frequencies at the same time. The proposed method is applied to the well-established sensor design of a simple rod-like structure, hence enabling a robust and versatile measurement system that can be used in different applications, such as in-line measurement or hand-held devices. In order to simultaneously control different resonance frequencies, multiple independent phase-locked loops are used, of which each subsystem is responsible for one natural vibration. An analytical description of the relationship between measurement parameters and physical fluid parameters is presented that is valid both for Newtonian and linear viscoelastic fluids. Consequently, calibration of the sensor is possible using Newtonian fluids only. To demonstrate the sensor, viscoelastic polymer and surfactant solutions are investigated at five frequencies between 2 kHz and 20 kHz. Additionally, the test fluids are characterized by means of a classical rotational rheometer (low frequency range) and diffusing-wave spectroscopy (high frequency range). The comparison of all methods shows very good agreement and validates the application of the sensor as resonant rheometer for viscoelastic fluids.
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