Mass-spring Oscillator Research Articles

Computational method of underwater acoustic radiation from a spherical shell coupled with nonlinear systems

During the continuous research and practical application of nonlinear vibration isolation systems, new problems arise for the vibro-acoustic calculation of underwater targets. The present frequency-domain analysis method can only deal with linear problems, so the acoustic radiation calculation of underwater targets coupled with nonlinear systems can only be performed in the time domain. When it comes to the real-time calculation, the huge computation cost will definitely bring challenges to the practical application. To solve this problem, a new calculation method is established for the underwater vibro-acoustic problems of structures coupled with nonlinear systems, taking elastic spherical shells as research objects in this paper. This method realizes the time-frequency domain conversion of the fluid-structure coupled vibration equation of an elastic spherical shell through the Fourier transform. The explicit difference method is further used to calculate the vibration response of the spherical shell, while the Newmark method is used to calculate the vibration response of the internal nonlinear system. An efficient integrated calculation of the two coupling parts is then achieved by considering boundary conditions at the connection point. Without the time-domain acoustic field solution, this method is able to calculate the vibration response of both the internal nonlinear system and the spherical shell, as well as the radiated acoustic power generated by the spherical shell vibration in real time. The calculation complexity and efficiency are radically improved. The nonlinear system discussed in this paper is a nonlinear mass-spring oscillator system installed at the bottom of a spherical shell. Finally, the correctness of the method described in this paper is verified through specific numerical examples, and a series of calculations and analyzes are carried out to explore the influence of nonlinear effects on the acoustic radiation of underwater structures. The calculation method described in this article is not limited to spherical shells, and is still applicable to general underwater targets.

Influence of End Wall Clearance on Guide Vane Self-Excited Vibrations at Small Openings during Pump Mode’s Starting Up Process of a Reversible Pump Turbine

Unstable guide vane torsional mode self-excited vibrations that occur at small guide vane openings during the transient operations with pump flow, such as the starting and closing of the pump mode, are considered to have potentially severe consequences, such as guide vane slippage or damage to the link and lever mechanism. Related site tests have indicated that the end wall clearance of a guide vane may have important influences on torsional mode self-excited vibrations. In this paper, numerical investigations, which were based on computational fluid dynamics (CFD) with a single degree of freedom (1DOF) mass-spring oscillator, were carried out on a prototype high-head reversible pump turbine. The results showed that the guide vane self-excited vibrations are unstable under steady-state conditions and during the pump mode’s starting up process for cases with small end wall clearances. In addition, the critical conditions of self-excitation instability under steady-state conditions have larger safety margins than those during the pump mode’s starting up process. After further discussion, it was concluded that increasing the end wall clearance to suppress unstable guide vane self-excited vibration is unreliable due to the complexity and randomness of the initial vibration excitations.

Multiple spring-mass oscillator for wide bandwidth piezoelectric energy harvester

This work proposed a design of unique wideband piezoelectric energy harvester. The solution toward broadband response was achieved by using multiple spring-mass oscillators integrated with piezoelectric energy harvester array. The harvester was demonstrated to harvest continuous power from vibration source under various frequencies in the range of 0-250 Hz. It is shown that the stable output power of the harvester over a wide range of source frequency was achieved. The maximum output voltage and power of each beam were 4, 3.6, 9.8, 10 V and 0.01, 0.07, 0.45, 0.52 mW for 15, 30, 45 gm of the proof masses and the superimposed case, respectively. The resonant frequencies of the harvester were in the low ambient vibration frequency range. Expanded in resonance frequency was achieved in this new design harvester.

Broadband Nonreciprocal Acoustic Scattering Using a Loudspeaker with Asymmetric Feedback

This paper describes an acoustic system enabling us to achieve asymmetric transmission over a broad frequency range. The basic elements of the system are a moving coil loudspeaker placed along the axis of a duct, and a microphone flush mounted next to the front side of the loudspeaker. The loudspeaker acts both as a mass-spring oscillator through which incident waves are reflected and transmitted, and as a sound re-emitter controlled by the microphone through a feedback loop. It is shown that the reciprocity of the resulting two port is broken, and that it can be used to design a broadband asymmetric wave transmitter. The experimental results show that there exists a frequency band ranging from about 300 Hz up to 1500 Hz, where an almost perfect transmission in the forward direction is observed whereas a weak (i.e., of about 10%) transmission is achieved in the reverse direction. Owing to the broadbandness of this efficient and compact nonreciprocal scatterer, it is also shown through its response to incident pulses that it can be used as a sound trap.

Measurement of a spring force constant using sound, door alarm, and soundcard oscilloscope

The increasing or decreasing magnetic field of a permanent magnet—attached to the bottom of the mass hanger of a spring-mass oscillator—would turn ON or OFF, respectively, the switch of a door alarm system. The sound pulses—now voltage pulses as a result of conversion by the built-in microphone of a PC—were recorded by the PC-based oscilloscope as a function of time and on the basis of these data, the period of oscillations was obtained; thus, the spring constant. We obtained (29.6 ± 0.1) N m−1 which is a good match to the accepted value of 30.0 N m−1.

A coupled closed-form/Doubly Asymptotic Approximation approach for the response of orthotropic plates subjected to an underwater explosion

ABSTRACT A closed-form analytical solution procedure is proposed to solve the coupled fluid-structure interaction (FSI) equations that involve the first-order Doubly Asymptotic Approximation (DAA1) formulation. An efficient method comprised of the nonstandard finite difference (NSFD) scheme is applied. First of all, analytical equations are developed to analyze the response of a rigid mass-spring oscillator in an air-backed condition when it is subjected to a plane shock exponential wave coming from a far-field underwater explosion. After validating the results with LS-DYNA/USA (DAA1), these equations are extended to determine the response of a two-dimensional, simply-supported rectangular plate, and then tested on the isotropic and orthotropic plates within a small deflection regime. Parametric studies are also performed by varying the load decay times, peak pressures, as well as the aspect ratios of the plate. Finally, the advantages and limitations of the proposed formulae are exposed along with suggestions for the future work.

Demonstration to Show Resonant Oscillations of a Simple Pendulum Coupled to a Mass-Spring Oscillator

Resonance is a widely observed physical phenomenon, occurring when an oscillator experiences a driving force whose frequency closely matches one of its own natural frequencies. Due to its broad applicability in physics, resonance is a familiar topic in the physics teaching literature. Previous studies include resonant energy transfer between similar mechanical systems such as coupled pendulums, resonant oscillations of wine glasses driven by sound waves, and resonance in AC circuits. Moreover, special drivers have been constructed to produce resonant oscillations of a variety of mechanical systems.

The optimal displacement control for fractional incommensurate mass-spring oscillators by Shifted Legendre polynomials

Based on the Riemann–Liouville fractional derivative, an optimal displacement control strategy for fractional incommensurate mass-spring oscillators has been presented. According to the calculus of variations and the Lagrange multiplier technique, the optimality conditions for the given problem are obtained. Using Shifted Legendre polynomials, the problem of solving differential equations is transformed into the problem of solving a set of algebraic equations. The validity, high efficiency and applicability of the proposed method are demonstrated through the simulation results.

Non-reciprocal Rayleigh wave propagation in space–time modulated surface

Research on non-reciprocal propagation of waves is of great significance in the field of photonic and phononic crystals for realizing flexible one-way propagation devices with potential engineering applications. Here, non-reciprocal Rayleigh waves are investigated in a continuous two-dimensional (2D) semi-infinite medium bound with an array of space–time modulated spring–mass oscillators. The involved modulation is a wave-like perturbation of the surface of the continuous medium that breaks time-reversal symmetry and reciprocity. To characterize the propagation of Rayleigh waves in such a complex 2D medium with continuous and discrete interface, an analytical study is performed to obtain dispersion-engineered bandgaps by adopting the asymptotic method and coupled mode theory, which is also validated by numerical simulation. Specifically, the non-reciprocal transmission of Rayleigh waves with one-way mode conversion is illustrated, and various relevant physical quantities, including conversion length and band gap size, are quantitatively estimated. This work sheds light on versatile control of Rayleigh wave propagation ranging from sensing and evaluation of engineering structures to guided wave-based damage detection techniques.

On the dynamics of a novel energy harvester to convert the energy of the magnetic noise into electrical power

Abstract Magnetic and mechanical noise in the frequency range of MHz are available in the environment and could be harvested for human convenience. This manuscript focuses on the dynamical behavior of hybrid magneto-mechano-electric (MME) energy harvesters to convert the energy of the magnetic noise and mechanical noise to electrical power using a composite energy scavenging structure. The proposed apparatus is composed of a piezoelectric (PZT-5A) layered beam on which a magnetostrictive material (Metglas-2605SC) is deposited. Once the device is exposed to external magnetic noise, Metglas-2605SC undergoes mechanical strain, and as a result, the mechanical strain is converted to electrical potential difference throughout the PZT-5A layer. In the present manuscript, the energy harvesting device is modeled as a cantilever beam, and the equations of motion are derived using Newton’s second law. The governing equations of motion, along with the output electrical potential difference equation are then discretized and numerically integrated over time, the frequency response curves for deflection, harvested power, and voltage are determined, and the effect of governing parameters on the output power is investigated. It is concluded that in the absence of mechanical damping, the response resembles that of a damped mass-spring oscillator confirming the energy consumption throughout the output circuit. In addition, as the external load resistance increases up to a particular value (164k Ω ), the attenuation rate of the response amplitude, and accordingly, the harvested power also increases. Beyond that particular value, the collected energy decreases by further increasing the load resistance. The results revealed that between two successive natural frequencies, there exists an anti-resonance region, where the response amplitude dramatically drops, and the operating area of the energy harvester needs to be kept well away from this zone in the frequency domain. The analytical results are verified by presenting a finite element simulation of the cantilever energy harvesting model, in which the distribution of stress and harvested voltage are determined.

Design of a Magnetic Interaction-Based Vibration Absorber for Continuous Beam

Classical absorber for vibration suppression of a continuous structure is constructed as a spring-mass oscillator, which only provides coupling force to suppress the vibration of primary structure. In this study, absorber beam is introduced and coupled on the continuous primary beam with magnetic interaction. Thus, the magnetic interaction and coupling bending moment affect the responses of primary beam. Based on the model of the system and Galerkin truncation, the natural frequencies for different magnetic parameters are obtained, which demonstrates that the fundamental frequency can be reduced to zero and the vibration of primary beam can be suppressed in a wide frequency band. Considering the vibration suppression on frequency band, we propose two criteria to evaluate the vibration suppression effect: one is the width of band for vibration suppression and the other is the width for vibration absorption. The two criteria not only show the vibration reduction effect but also correspond to different vibration suppression mechanism. Due to the advantages of zero fundamental frequency induced by the proposed magnetic interaction coupling and wide vibration suppression frequency band, utilizing absorber beam in vibration suppression of continuous structure has potential applications for flexible aim in the fields of manufacturing and aerospace.

The Identification of Period Doubling in a Nonlinear Two-Degree-of-Freedom Electromagnetic Vibrational Energy Harvester

Common vibrational energy harvesters are generally based on a linear mass-spring oscillator model, and these typically show narrow bandwidth and high resonant frequency at small scales. To overcome these problems, a two-degree-of-freedom nonlinear velocity-amplified energy harvester has been developed. The device comprises two masses, relatively oscillating one inside the other between four sets of magnetic springs. The magnetic springs introduce nonlinear effects, such as period doubling, that can be exploited to enhance the output power and bandwidth of the harvester. This article studies the dynamics of the harvester when a key geometrical parameter (the height of the device) is varied. For large height values, a significant increase of the output power in regions far from the resonant frequency of the device is observed, and the associated period-doubling effect was verified through high-speed imaging. It is demonstrated that nonlinear effects can be used to enhance the bandwidth of the device in order to harvest energy in regions far from resonance.

Passage through a resonance for a mechanical system, having time-varying parameters and possessing a single trapped mode. The principal term of the resonant solution

We consider a forced oscillation and passage through resonance for an infinite-length system, having time-varying parameters and possessing a single trapped mode. The system is a string, lying on the Winkler foundation and equipped with a discrete linear mass-spring oscillator of time-varying stiffness. We obtain the principal term of the asymptotic expansion for the resonant solution describing the motion of the inclusion (i.e., the mass-spring oscillator). The obtained result was verified by independent numerical calculations based on solution of the corresponding partial differential equation by means of the method of finite differences. The comparison demonstrates a good agreement in a neighbourhood of the instant of resonance.

Driven Series RLC Circuit and Resonance: A Graphic Approach to Energy

Resonance is an important, and rather dramatic, phenomenon in physics. The mechanical case of a mass-spring oscillator is generally the first example that students encounter in introductory courses. In electromagnetism, the resonance in driven RLC circuits is presented in analogy with mechanics, since the mathematical equations describing them are similar. In this work, the resonance phenomenon is approached by means of the graphs of the power as a function of time, something unusual in textbooks on electricity. The maximum absorption of energy by the circuit at resonance is emphasized, using the mathematical solutions for this oscillation. The meanings of power delivered by the external source and energy absorbed by the system are highlighted by means of both the physical interpretation of the equation for P(t) and the graphs of the power vs. time.

Study of heat conduction in three-layered structure—a sandwich model

One-dimensional sandwich model, a three-layered structure - segmented thermoelectric material, is studied based on the mass-spring oscillator structure which effectively allows (resists) heat and can work as a heat conductor (heat barrier). The temperature profile, heat flux, interface thermal resistance, thermal conductivity and figure of merit of such sandwich structures are studied by varying the mass of an oscillator of the sandwich segment. It is seen that the heat transfer can be tuned by changing the mass of the middle segment of the sandwich model. When mass of an oscillator in middle layer is less (larger) than the mass of an oscillator of outer layers, the material behaves as a heat barrier (good heat conductor). It is shown that the proper mass asymmetry improved the figure of merit of the segmented thermoelectric material up to 244% in comparison to the single thermoelectric material.

Liquid Sloshing in a Rigid Cylindrical Tank Equipped with a Rigid Annular Baffle and on Soil Foundation

Fluid sloshing in a rigid circular cylindrical tank is investigated; the tank is resting on soil foundation and is excited by horizontal seismic accelerations. A rigid annular baffle is connected to the inner wall of the storage tank to reduce liquid sloshing. By using the fluid subdomain method, the convective velocity potential is derived. An equivalent model with mass-spring oscillators is proposed to describe the sloshing motions of the contained liquid. Then, by means of the least square method, a complex polynomial fraction is employed to fit the dynamic impedance of the soil. A nested lumped parameter model is established to model the effect of the soil foundation. The substructure method allows to obtain the soil–tank–liquid coupled model. The equation of motion of the coupled system is solved by the Newmark-[Formula: see text] method. A comparison between the present sloshing results and those published in the literature shows an excellent agreement. The effects of the soil parameters, the baffle position and its size on the dynamic behavior of the soil–tank–liquid system are discussed in detail. The results demonstrate that the soil properties and the baffle parameters can have a remarkable influence on liquid sloshing. The novelty of this research is that an analytical model for the soil–tank–liquid–baffle coupled system is derived for the first time and it allows to study the dynamics and sloshing response of the system with accuracy and low computational cost.

A receptance-based method for frequency assignment via coupling of subsystems

This paper presents a theoretical study of the frequency assignment problem of a coupled system via structural modification of one of its subsystems. It deals with the issue in which the available modifications are not simple; for example, they are not point masses, grounded springs, or spring-mass oscillators. The proposed technique is derived based on receptance coupling technique and formulated as an optimization problem. Only a few receptances at the connection ends of each subsystem are required in the structural modification process. The applicability of the technique is demonstrated on a simulated rotor system. The results show that both bending natural frequencies and torsional natural frequencies can be assigned using a modifiable joint, either separately or simultaneously. In addition, an extension is made on a previously proposed torsional receptance measurement technique to estimate the rotational receptance in bending. Numerical simulations suggest that the extended technique is able to produce accurate estimations and thus is appropriate for this frequency assignment problem of concern.

Tuning bandgaps in metastructured beams: numerical and experimental study

Tunable metastructures (including phononic crystals and metamaterials) have the unique advantage that one can change the operating frequency and acoustic wave characteristics as needed. In this paper, the bandgap characteristics and their controllability of a metastructured beam with mass-spring oscillators and under an axial force are investigated in depth both by the finite element method and by experiment. The experimental and numerical results indicate that there is one local resonance (LR) bandgap and multiple Bragg scattering (BS) bandgaps. The width and position of each bandgap can be tuned effectively by adjusting the axial force, lattice constant, and spring stiffness, and a super wide pseudo-gap can be obtained under suitable conditions. By integrating different mass-spring oscillators into one metastructured beam, the bandgap width can be broadened and pseudogap- like characteristics can be achieved. By changing the number of different oscillators, the propagating distance of elastic waves in the beam can also be controlled. It is further revealed that point defects have a large influence on the BS bandgaps but little effect on the LR bandgap. The present work provides an important reference for the optimal design of adjustable high-performance metastructures.

A Frequency Up-Converted Hybrid Energy Harvester Using Transverse Impact-Driven Piezoelectric Bimorph for Human-Limb Motion.

Energy harvesting from human-body-induced motion is mostly challenging due to the low-frequency, high-amplitude nature of the motion, which makes the use of conventional cantilevered spring-mass oscillators unrealizable. Frequency up-conversion by mechanical impact is an effective way to overcome the challenge. However, direct impact on the transducer element (especially, piezoelectric) increases the risk of damaging it and raises questions on the reliability of the energy harvester. In order to overcome this shortcoming, we proposed a transverse mechanical impact driven frequency up-converted hybrid energy harvester for human-limb motion. It utilizes the integration of both piezoelectric and electromagnetic transducers in a given size that allows more energy to be harvested from a single mechanical motion, which, in turn, further improves the power density. While excited by human-limb motion, a freely-movable non-magnetic sphere exerts transverse impact by periodically sliding over a seismic mass attached to a double-clamped piezoelectric bimorph beam. This allows the beam to vibrate at its resonant frequency and generates power by means of the piezoelectric effect. A magnet attached to the beam also takes part in generating power by inducing voltage in a coil adjacent to it. A mathematical model has been developed and experimentally corroborated. At a periodic limb-motion of 5.2 Hz, maximum 93 µW and 61 µW average powers (overall 8 µW·cm−3 average power density) were generated by the piezoelectric and the electromagnetic transducers, respectively. Moreover, the prototype successfully demonstrated the application of low-power electronics via suitable AC-DC converters.

Two-degree-of-freedom velocity-amplified vibrational energy harvester for human motion applications

Conventional vibrational energy harvesters (VEHs) are generally based on a linear mass-spring oscillator model that features narrow bandwidth and high resonant frequencies at small scales. To overcome these limitations, a two-degree-of-freedom (2-Dof) velocity-amplified VEH was developed. The harvester comprises two masses, relatively oscillating one inside the other, between four sets of magnetic springs. Impacts between the two masses are allowed, and momentum is transferred from the larger outer mass to the smaller inner mass, thereby providing velocity amplification. Electromagnetic transduction was chosen because it can be easily implemented in a velocity-amplified VEH. The harvester was tested with harmonic excitation of different amplitudes and two peaks of similar heights were observed at arms = 0.6 g, resulting in a −3 dB bandwidth of 10 Hz. The VEH was also tested under human motion and at a running speed of 10 km/h the harvester generated P = 0.44 mW, a power level that could be accumulated in a storage medium over time and used for powering wireless sensor nodes.