Linear Mechanical Oscillator Research Articles

On the Synchronization of Parallel Power Converters via Emulation of Linear Mechanical Oscillators

On the Synchronization of Parallel Power Converters via Emulation of Linear Mechanical Oscillators

Self-balancing characteristics of three self-synchronous linear motion mechanical oscillators

How to restrain the unbalance caused by the manufacturing and installation of parts during the working process of the rotating oscillators is rarely addressed, yet it is very important for evaluating the performance of rotating machines. This paper treats a system comprising three linear motion mechanical oscillators (LMMOs) fixed to a rigid body and clarifies the self-balancing characteristics of the system in the over-resonance state. An electromechanically coupled dynamic model of LMMO is constructed using Lagrange’s energy approach. The small parameter average method is used to derive the synchronization conditions among the LMMOs, and the reasonable coupling torque parameter range that the system should be satisfied. The motion stability is analyzed using a linearized matrix of the torque perturbation equation describing the system, and the stability criteria of synchronous linear motion are provided. Subsequently, the necessary conditions are found for implementing self-balancing among the three LMMOs to eliminate the unbalanced vibration in the system, and the influences of the structural parameters on self-balancing behavior are investigated. The results demonstrate that the structural parameters and phase differences among the three LMMOs can positively reduce the unbalanced vibration of the system, when the necessary condition for the vibrating displacement response nearing to be zero is fully satisfied. This research can provide theoretical guidance for the vibrating isolation of rotating machinery body with several linear motion vibrating sources installed on a common oscillating platform.

Broadband coherent multidimensional variational measurement

Standard Quantum Limit (SQL) of a classical mechanical force detection results from quantum back action perturbing evolution of a mechanical system. In this paper we show that usage of a multidimensional optical transducer may enable a broadband quantum back action evading measurement. We study a corresponding technique for measurement of a resonant signal force acting on a linear mechanical oscillator coupled to an optical system with three optical modes with separation nearly equal to the mechanical frequency. The measurement is performed by optical pumping of the central optical mode and measuring the light escaping the two other modes. By detecting optimal quadrature components of the optical modes and post-processing the measurement results we are able to exclude the back action in a broad frequency band and characterize the force with sensitivity better than SQL. We discuss how proposed scheme relates to multidimensional system containing quantum-mechanics-free subsystems (QMFS) which can evade the SQL using idea of so called "negative mass".

Hierarchy of quantum non-Gaussian conservative motion

Mechanical quantum systems embedded in an external nonlinear potential currently offer the first deep excursion into quantum non-Gaussian motion. The Gaussian statistics of the motion of a linear mechanical quantum system, characterised by its mass and a linear-and-quadratic potential, possess a limited capacity to reduce noise in nonlinear variables. This limitation induces thresholds for noise reduction in nonlinear variables beyond which linear mechanical oscillators cannot pass. Squeezing below the thresholds for such variables is relevant for the implementation of nonlinear mechanical devices, such as sensors, processors or engines. First however, quantum non-Gaussian conservative motion must be identified in experiments with diverse nonlinear potentials. For this purpose, we provide sufficient criteria for quantum non-Gaussian motional states in conservative systems based on the observation of squeezing in nonlinear variables. We further extend these criteria to a hierarchy able to recognise the quantum non-Gaussian motion induced via diverse nonlinear potentials through their various capacities to produce nonlinear squeezing.

Dynamics and Control of a Magnetic Transducer Array Using Multi-Physics Models and Artificial Neural Networks.

A linear mechanical oscillator is non-linearly coupled with an electromagnet and its driving circuit through a magnetic field. The resulting non-linear dynamics are investigated using magnetic circuit approximations without major loss of accuracy and in the interest of brevity. Different computational approaches to simulate the setup in terms of dynamical system response and design parameters optimization are pursued. A current source operating in baseband without modulation directly feeds the electromagnet, which consists commonly of a solenoid and a horseshoe-shaped core. The electromagnet is then magnetically coupled to a mass made of soft magnetic material and attached to a spring with damping. The non-linear system is described by a linearized steady-space representation while is examined for controllability and observability. A controller using a pole placement approach is built to stabilize the element. Drawing upon the fact that coupling works both ways, enabling estimation of the mass position and velocity (state variables) by processing the induced voltage across the electromagnet, a state observer is constructed. Accurate and fast tracking of the state variables, along with the possibility of driving more than one module from the same source using modulation, proves the applicability of the electro-magneto-mechanical transducer for sensor applications. Next, a three-layer feed-forward artificial neural network (ANN) system equivalent was trained using the non-linear plant-linear controller-linear observer configuration. Simulations to investigate the robustness of the system with respect to different equilibrium points and input currents were carried out. The ANN proved robust with respect to position accuracy.

Broadband dichromatic variational measurement

Standard Quantum Limit (SQL) of a classical mechanical force detection results from quantum back action impinged by the meter on a probe mechanical transducer perturbed by the force of interest. In this paper we introduce a technique of continuous \vy{broadband} back action avoiding measurements for the case of a resonant signal force acting on a linear mechanical oscillator supporting one of mirrors of an optical Michelson-Sagnac Interferometer (MSI). The interferometer with the movable mirror is an opto-mechanical transducer able to support polychromatic probe field. The method involves a dichromatic optical probe resonant with the MSI modes and having frequency separation equal to the mechanical frequency. We show that analyzing each of the harmonics of the probe reflected from the mechanical system separately and postprocessing the measurement results allows excluding the back action in a broad frequency band and measuring the force with sensitivity better than SQL.

An experimentally validated piezoelectric nonlinear energy sink for wideband vibration attenuation

Various researchers have investigated the behavior of a linear mechanical oscillator coupled to a nonlinear mechanical attachment that has essential stiffness nonlinearity. Under certain conditions, the essentially nonlinear attachment acts as a nonlinear energy sink (NES) and one-way energy transfer from the main structure to the nonlinear attachment can be achieved. An important characteristic of an essentially nonlinear attachment is that it does not posses any preferential resonance frequency, resulting in increased robustness against detuning, thereby enabling frequency-wise wideband performance. This work presents an experimentally validated piezoelectric-based NES for wideband vibration attenuation. The electrical circuit consists of a negative capacitance shunt (introduced for cancelling the piezoelectric capacitance) combined in series with a nonlinear capacitance of cubic order that is realized using operational amplifiers. Design and practical implementation of the NES shunt circuit are discussed in detail. The performance of the piezoelectric NES to attenuate vibrations over a wide range of frequencies is numerically simulated and experimentally validated for a cantilever in the absence and presence of tip mass attachments.

Vibration control using nonlinear damped coupling

In this paper, a dynamical system, which consists of two linear mechanical oscillators, coupled with a nonlinear damping device is considered. First, the dynamic equations are derived, then, an analytical method such as harmonic balance method, is applied to obtain the response to a harmonic base excitation. The response of the system depends on the excitation characteristics. A parametric study is carried out based on different base excitation amplitudes, frequencies, and different nonlinear damping values and the response of the system is fully described. For validation, time domain simulations are carried out to obtain the nonlinear response of the coupled system.

Experimental and Numerical Validation of Digital, Electromechanical, Parametrically Excited Amplifiers

A parametric amplifier having a tunable, dual-frequency pumping signal and a controlled cubic stiffness term is realized and investigated experimentally. This device can be tuned to amplify a desired, single frequency weak signal, well below resonance. The transition between a previously described theoretical model and a working prototype requires an additional effort in several areas: modeling, design, calibration, identification, verification, and adjustment of the theoretical model. The present paper describes these necessary steps and analyzes the results. Tunability is achieved here by adding a digitally controlled feedback, driving a linear mechanical oscillator with an electromechanical actuator. The main advantage of the present approach stems from the separation of the controlled parametric and nonlinear feedback terms which are linked to the resonating element. This separation allows for the realization of feedback in an electronic form where a digital implementation adds further advantages as the feedback coefficients can be tuned in situ. This arrangement benefits from the mechanical resonance of a structure and from the ability to set the parametric excitation such that it accommodates sinusoidal input signals over a wide range of frequencies. The importance of an in situ identification phase is made clear in this work, as the precise setting of model and feedback parameters was shown to be crucial for successful application of the amplifier. A detailed model-identification effort is described throughout this paper. It has been shown through identification that the approach is robust despite some modeling uncertainties and imperfections.

Essentially nonlinear piezoelectric shunt circuits applied to mistuned bladed disks

An essentially nonlinear piezoelectric shunt circuit is proposed for the practical realization of nonlinear energy sink, and then applied to a mistuned bladed disk for blade vibration reduction. First, the global dynamics of a single degree-of-freedom linear mechanical oscillator, coupled to an essentially nonlinear shunted piezoelectric attachment, is studied. Under certain conditions, the nonlinear targeted energy transfer, i.e. a fast, passive energy transfer from the mechanical oscillator to the nonlinear attachment is observed. A numerical method, referred to as the variable-coefficient harmonic balance method, is developed to calculate quasi-periodic responses arising in the electromechanical system under harmonic forcing. Characterized by the nonexistence of a resonance frequency, the essentially nonlinear shunt circuit is able to work robustly over a broad frequency band with a smaller inductance requirement compared with the linear resonant shunt circuit. The application of piezoelectric shunt damping to simplified blade–disk structures is then taken into consideration. Shunted piezoelectrics are attached onto the disk surface in our damping strategy in order to reduce blade vibrations. Essential nonlinearity is also introduced into the piezoelectric shunted bladed disk system. Since the piezoelectric-based nonlinear energy sink is not a priori tuned to any specific frequency, a sound damping performance is achieved when blades become inevitably mistuned.

Simulation of the linear mechanical oscillator on GPU

In this short paper we are describing some methods for solving stochastic differential equations which appears in mechanics. We are concerned with solutions of mechanical oscillators which are subject to random excitation. We are primarly interested in solving stochastic differential equations excited with white noise. We rewrite the random mechanical oscillator equation in the form suitable for applying numerical procedures for solving it. Finally, we give some design notes on the methods used to implement numerical simulation of the graphical processing units.

On the stiffness-switching methods for generating self-excited oscillations in simple mechanical systems

The present article proposes stiffness-switching methods for generating artificial self-excited oscillation in simple mechanical and micromechanical systems and theoretically investigates the feasibility of the methods. The basic method proposed involves a stiffness reduction mechanism which operates in an on–off fashion depending on the measured states of the system. The efficacy of the method is first studied in case of a single-degree-of-freedom linear mechanical oscillator and then extended to an essentially nonlinear micromechanical resonator. Analytical and numerical investigations reveal that the stiffness-switching method can be effectively utilized for generating and controlling self-excited oscillations in simple mechanical structures. However, the amplitude of oscillation obtained by the basic method involving only the stiffness reduction mechanism is found to be extremely sensitive to the uncertainties in the inherent damping of the structure. An alternative switching scheme is also proposed to alleviate this shortcoming.

Optomechanical stochastic resonance in a macroscopic torsion oscillator

Linear mechanical oscillators have been applied to measure very small forces, mostly with the help of noise suppression. In contrast, adding noise to nonlinear oscillators can improve the measurement conditions. Here, this effect of stochastic resonance is demonstrated in a macroscopic torsion oscillator, for an optomechanical nonlinear potential. The signal output is enhanced for a subthreshold electronic signal. This nonlinear oscillator serves as a model system for the enhancement of signal-to-noise ratio in high precision optomechanical experiments.

New Approach for the Analysis of Damped Vibrations of Fractional Oscillators

The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a definition of the fractional derivative enables one to analyse approximately vibratory regimes of the oscillator without considering the drift of its position of equilibrium. The assumption of small fractional derivative terms allows one to use the method of multiple time scales whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and nonlinear elastic terms is possible to be carried out. The interrelationship of the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when orders of the small fractional derivative term and of the cubic nonlinearity entering in the oscillator's constitutive equation coincide.

New Approach for the Analysis of Damped Vibrations of Fractional Oscillators

The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a definition of the fractional derivative enables one to analyse approximately vibratory regimes of the oscillator without considering the drift of its position of equilibrium. The assumption of small fractional derivative terms allows one to use the method of multiple time scales whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and nonlinear elastic terms is possible to be carried out. The interrelationship of the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when orders of the small fractional derivative term and of the cubic nonlinearity entering in the oscillator's constitutive equation coincide.

Synchronization in a ring of mutually coupled electromechanical devices

We study the synchronization in a ring of mutually coupled electromechanical devices. Each device consists of an electrical Duffing oscillator coupled magnetically with a linear mechanical oscillator. By varying the coupling coefficient, we find the ranges for cluster and complete synchronization, either in the regular state or in the chaotic one. Effects of this coupling parameter show various types of bifurcation sequences.

Vibration analysis and bifurcations in the self-sustained electromechanical system with multiple functions

We consider in this paper the dynamics of the self-sustained electromechanical system with multiple functions, consisting of an electrical Rayleigh–Duffing oscillator, magnetically coupled with linear mechanical oscillators. The averaging and the harmonic balance method are used to find the amplitudes of the oscillatory states respectively in the autonomous and nonautonomous cases, and analyze the condition in which the quenching of self-sustained oscillations appears. The influence of system parameters as well as the number of linear mechanical oscillators on the bifurcations in the response of this electromechanical system is investigated. Various bifurcation structures, the stability chart and the variation of the Lyapunov exponent are obtained, using numerical simulations of the equations of motion.

Nonlinear Dynamics of Mechanical Systems Under Complicated Forcing

This paper considers the problem of simple linear mechanical oscillators forced by complicated forcing functions, such as those induced by fluids or humans. Such problems arise in civil engineering in the design of bridges, grandstands and towers. The paper questions whether low dimensional models have much to contribute to the understanding of such problems, given the complicated possibilities of high-dimensional forcing.

Nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions

This paper deals with the nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions, described by an electrical Duffing oscillator magnetically coupled to linear mechanical oscillators. Firstly, the amplitudes of the sub- and super-harmonic oscillations for the resonant states are obtained and discussed using the multiple time scales method. The equations of motion are solved numerically using the Runge–Kutta algorithm. It is found that chaotic and periodic orbit coexist in the electromechanical system depending on the set of initial conditions. Secondly, the problem of synchronization dynamics of two coupled electromechanical systems both in the regular and chaotic states is also investigated, and estimation of the coupling coefficient under which synchronization and no-synchronization take place is made.

General scaling law for stiffness measurement of small bodies with applications to the atomic force microscope

A general scaling law connecting the stiffness and dissipative properties of a linear mechanical oscillator immersed in a viscous fluid is derived. This enables the noninvasive experimental determination of the stiffness of small elastic bodies of arbitrary shape by measuring their resonant frequency and quality factor in fluid (typically air). In so doing, we elucidate the physical basis of the method of Sader et al. [Rev. Sci. Instrum. 70, 3967 (1999)] for determining the stiffness of rectangular atomic force microscope cantilevers, and discuss its applicability. The validity of the derived general technique is demonstrated by calibrating atomic force microscope cantilevers with complex geometries, and its implications to small bodies in general are discussed.