Degree Of Freedom Oscillator Research Articles

High-efficiency nonlinear dynamic analysis for joint interfaces with Newton–Raphson iteration process

Nonlinear dynamic analysis of the assembled structures involves the complex nonlinearity of the joint interfaces. By combining the multi-harmonic balance method with the Newton–Raphson iteration process, a high-efficiency nonlinear dynamic analysis method is proposed to analyze the steady-state dynamic responses influenced by the hysteresis nonlinearity of the joint interfaces. The proposed method adopts a gradient vector based on the chain derivation of the continuous function to iterate the steady-state dynamic responses instead of the Jacobian matrix in each iteration process, and a 2-order interpolation function is chosen to compensate the continuity of the discrete harmonic coefficients. The proposed method is verified by a comparison with the Jacobian matrix-based Newton iteration method and the direct numerical integration by using a single degree-of-freedom (DOF) system, and the effects of the excitation amplitudes are also investigated. Based on the generalized mode superposition method, the complex lap-type bolted joint beam structure is reduced into a series of single DOF oscillators with coupling nonlinear dynamic behaviors of the joint interfaces. The proposed method is applied to calculate the frequency response function (FRF) and verified by a comparison with the prediction of the Jacobian matrix-based iteration method. The results show that the solutions of the proposed method agree well with the results predicted by the direct numerical integration and Jacobian matrix-based iteration in literature. Larger excitation amplitude will induce the loss of the joint stiffness and degrade the nonlinearity effects. The predicted FRFs of the bolted joint beam system agree well with the results iterated by the Jacobian matrix-based method, and indicate a better convergence performance and a higher efficiency with much less computational cost.

Simulation of Power Generation from Vibration of Railway Track

Ethiopia has a small percentage of its population accessing electricity with some power utilized by its transportation sector; there are frequent power outages that slow down economic activities such as trade and railway transportation. From this previous statement, it is clear that the demand for energy is significantly increasing and there is too much dependability on the current energy source. With the development of sustainable energy technology in the world today, this problem can be alleviated, while utilizing component vibration in railway infrastructure. The aim of this study was to estimate energy generation potential from the vibration of a railway track induced by a passing train using a two degree of freedom oscillator. Data was collected; energy harvesting system was modeled and simulated. Based on the result, the system produces enough energy to power communication and signaling equipment on the track. The optimum parameters: mass of 1 kg, a spring stiffness of 6 N/m and a damping coefficient of 4 Ns/m of the energy harvester were determined using sensitivity analysis. The cumulative mechanical power harvested by the inter-city-125 train at a speed of 195 km/h was calculated to be 224.56 W. Therefore, the energy demand of the communication and signaling equipment of the train track can be achieved using vibration energy harvesting system by installing the number that meets the demand.

Experimental analysis of the vibration dissipation induced by granular materials included into a 1 degree of freedom oscillator

The use of granular material is particularly attractive to damp vibrations in hollow structures, due to grains relative motion that induce shocks and friction, interaction grains/structure and possible deformation of the grains. Therefore, granular dampers are of great interest for many application types.However, predictive modelling of the damping performance of a granular material remains particularly difficult. The objective of this experimental work is to evaluate the vibration damping performance of various granular material samples, when inserted in a vertically shaken rigid cavity. This is done by Lissajous’ representation of displacement and force time signals, which, applied to this type of system, is a methodological originality allowing a detailed analysis of the overall motion of the grain cluster. The main insights are the identification of links between certain control parameters of the granular material and the vibratory energy dissipated. From Lissajous’ representation post-processing, it is shown that dissipation increases linearly with grain mass; at given mass the dissipation increases with grain size; the use of a viscoelastic material notably increases the performance, particularly at low acceleration levels; that the roughness of surfaces in contact with the grains plays a secondary role. Finally, the set of parametric variations give a better physical understanding of the phenomena involved, which gives orientations for future modelling works.

Experimental Research on the Impact Environment of Multiplate Cabinet

As an installation and protection device for electrical and electronic components, a shipboard cabinet is a typical multiplate structure. In order to study the impact environment distribution laws of such structures, impact testing was carried out on a shipboard cabinet under four working conditions in this paper. In addition, the impact response characteristics of such a multiplate structure were determined by numerical simulation and theoretical analysis. The impact environments of some pivotal points in cabinet were measured and some laws of dynamic response were found. The impact environment of central position was more severe on a single plate because of the first vibration modal. For different plates, the responses were usually similar at low‐frequency band and a little different at high‐frequency band. The theoretical analysis of the single degree of freedom oscillator was carried out, and the sensitivity of the response to the different characteristic frequencies was discussed based on the shock spectrum theory. A new method of calculating the response at a special frequency was proposed and verified.

Evolution of the entanglement of the N00N-type of states in a coupled two cavity system via an adiabatic approximation

We study a system of two cavities each encapsulating a qubit and an oscillator degrees of freedom. An ultrastrong interaction between the qubit and the oscillator is assumed, and the photons are allowed to hop between the cavities. A partition of the time scale between the fast-moving oscillator and the slow moving qubit allows us to set up an adiabatic approximation procedure where we employ the delocalized degrees of freedom to diagonalize the Hamiltonian. The time evolution of the N00N-type initial states now furnishes, for instance, the reduced density matrix of a bipartite system of two qubits. For a macroscopic size of the N00N component of the initial state the sudden death of the entanglement between the qubits and its continued null value are prominently manifest as the information percolates to the qubits after long intervals. For the low photon numbers of the initial states the dynamics produces almost maximally entangled two-qubit states, which by utilizing the Hilbert–Schmidt distance between the density matrices, are observed to be nearly pure generalized Bell states.

A novel stochastic linearization framework for seismic demand estimation of hysteretic MDOF systems subject to linear response spectra

This paper proposes a novel computationally economical stochastic dynamics framework to estimate the peak inelastic response of yielding structures modelled as nonlinear multi degree-of-freedom (DOF) systems subject to a given linear response spectrum defined for different damping ratios. This is accomplished without undertaking nonlinear response history analyses (RHA) or, to this effect, constructing an ensemble of spectrally matched seismic accelerograms. The proposed approach relies on statistical linearization and enforces pertinent statistical conditions to decompose the inelastic d-DOF system into d linear single DOF oscillators with effective linear properties (ELPs): natural frequency and damping ratio. Each such oscillator is subject to a different stationary random process compatible with the excitation response spectrum with damping ratio equal to the oscillator effective critical damping ratio. This equality is achieved through a small number of iterations to a pre-specified tolerance, while peak inelastic response estimates for all DOFs of interest are obtained by utilization of the excitation response spectrum in conjunction with the ELPs. The applicability of the proposed framework is numerically illustrated using a 3-storey Bouc-Wen hysteretic frame structure exposed to the Eurocode 8 elastic response spectrum. Nonlinear RHA involving a large ensemble of non-stationary Eurocode 8 spectrum compatible accelerograms is conducted to assess the accuracy of the proposed approach in a Monte Carlo-based context. The novel feature of iterative matching between the excitation response spectrum damping ratio and the ELP damping ratio enforces the required compatibility in the damping properties of the effective linear oscillators and the imposed elastic response spectra. It is found that this latter feature reduces drastically the error of the estimates (i.e., by an order of magnitude) obtained by a non-iterative application of the framework.

On the response of some discrete and continuous oscillatory systems with pure cubic nonlinearity: Exact solutions

This study is concerned with oscillatory systems with pure cubic nonlinearity, which are systematically presented, starting from a one degree of freedom oscillator, then focusing on chains with two and multi degrees of freedom, and ending with elastic systems (systems having an infinite number degrees of freedom). A one degree of freedom pure cubic oscillator is given as a reference case for the other systems under consideration as it has the exact closed-form solution for its free response. Similar normal modes and exact solutions for free response are investigated in chains consisting of two discrete masses attached with pure cubic springs in two configurations: when anchored at both ends, and anchored at one end only. The former class exhibits a pitchfork bifurcation related to their similar normal modes, and the way how this bifurcation is created and destroyed is demonstrated in terms of stiffness ratios. These chains are then enhanced with additional masses and springs of the same stiffness, and their similar normal modes are determined (modal shapes, frequencies and the exact closed-form solution for time responses). Finally, longitudinal vibration of a bar with a pure cubic relationship between the axial stress and strain are examined analytically, obtaining the exact solution for the first mode shape of a clamped–clamped and clamped–free bar. Their first modes are compared with those obtained for nonlinear chains with multiple masses as well as with the well-known results for linear longitudinal vibration of the bars with the same boundary conditions.

The vibration transmissibility of a single degree of freedom oscillator with nonlinear fractional order damping

ABSTRACTMotivated by the theoretical analysis of the effects of nonlinear viscous damping on vibration isolation using the output frequency response function approach, the output frequency response function approach is employed to investigate the effects of the nonlinear fractional order damping on vibration isolation based on Volterra series in the frequency domain. First, the recursive algorithm which is proposed by Billings et al. is extended to deal with the system with fractional order terms. Then, the analytical relationships are established among the force transmissibility, nonlinear characteristic coefficients and fractional order parameters for the single degree of freedom oscillator. Consequently, the effects of the nonlinear system parameters on the force transmissibility are discussed in detail. The theoretical analysis reveals that the force transmissibility of the oscillator is suppressed due to the existence of the fractional order damping, but presents different effects on suppressing the force transmissibility of the oscillator over the frequency region by varying the fractional order parameters. Moreover, the fractional order parameters, which affect the force transmissibility, the bandwidth of the frequency region and the resonance frequency, can be used as designing parameters for vibration isolation systems. At last, numerical studies are presented to illustrate the theoretical results.

Squeezed Schrödinger kitten states of a qubit–oscillator system: Generation and quantum properties in the phase space

We study the evolution of the hybrid entangled squeezed states of the qubit–oscillator system in the strong coupling domain. Following the adiabatic approximation we obtain the reduced density matrices of the qubit and the oscillator degrees of freedom. The oscillator reduced density matrix is utilized to calculate the quasiprobability distributions such as the Sudarshan–Glauber diagonal P-representation, the Wigner W-distribution, and the nonnegative Husimi Q-function. The negativity associated with the W-distribution indicates the nonclassicality of the developing state. The existence of the multiple time scales induced by the interaction introduces certain features in the bipartite system. In the strong coupling regime the transient evolution to low entropy configurations reveals brief emergence of nearly pure squeezed Schrödinger kitten states that may be regarded as superposition of uniformly separated distinguishable squeezed coherent states. However, the quantum fluctuations with a short time period engender bifurcation and subsequent rejoining of these peaks in the phase space. The abovementioned doubling of the number of peaks increases the entropy to its near maximal value. Nonetheless, these states characterized by high entropy values, are endowed with a large negativity of the W-distribution that points towards their non-Gaussian behavior. This may be ascertained by the significantly large Hilbert–Schmidt distance between the oscillator state and an ensemble of most general statistical mixture of squeezed Gaussian states possessing nearly identical second order quadrature moments as that of the oscillator.

Active vibration control of an oscillator with two translational degrees of freedom using centrifugal forces created by two eccentrically rotating masses

An active mass damper for vibration control of an oscillator with two translational degrees of freedom is presented along with the corresponding closed-loop control algorithm. The damper consists of two eccentrically rotating masses. In a preferred mode of operation, the masses rotate in opposite directions with a mostly constant angular velocity about a single axis. The resulting force is a harmonic control force. Its direction is determined by the relative angular position of the masses. In previous research, a similar control algorithm for vibration control of a single degree of freedom oscillator has been proven to be effective. In this paper, the control algorithm is augmented such that it can be applied to an oscillator with two translational degrees of freedom. Various state variables are introduced and a feedback control algorithm is developed. The presented algorithm ensures that the rotational motion of the masses is smooth and that the control force has the required orientation. The algorithm is verified experimentally with a test setup for the damping of free vibrations and numerically for stochastically forced vibrations. Finally, the device is compared with a conventional active mass damper. It is shown that the power demand and the energy consumption of the presented device are smaller than those of the conventional active mass damper.

On Response of a Single-Degree-of-Freedom Oscillator with Constant Hysteretic Damping Under Arbitrary Excitation

The simple constant hysteretic damping model is known to be non-causal although it is used often in diverse branches of engineering. In this paper the response of a single degree of freedom oscillator having linear hysteretic damping under arbitrary force excitation has been studied after deriving the impulse response function of the system. Some shortcomings of the results available in literature have been pointed out. It has been shown that the damping model can be practically used for calculating the response of a physical system when the damping is small and the force has small duration.

Strong Electron-Phonon Interaction in A Suspended Carbon Nanotube

We have developed an approach to calculate the full counting statistics of the Holstein polaron dot in a strong coupling regime. Using a polaron tunneling approximation we have derived an Levitov-Lesovik formula for the cumulant generating function with an effective normalized transmission coefficient. The effect of electron-phonon interactions on the charge transport through a contacted molecule was studied. We have simplified to contain a single electronic level linearly coupled to a local Holstein Phonon i.e. a bosonic oscillator degree of freedom with a single frequency. The vibrational modes of a such a quantum dot were also studied. We found the description of transport through molecules contacted using mechanically controlled break junctions.

Dynamical thermalization of Bose-Einstein condensate in Bunimovich stadium

We study numerically the wave function evolution of a Bose-Einstein condensate in a Bunimovich stadium billiard being governed by the Gross-Pitaevskii equation. We show that for a moderate nonlinearity, above a certain threshold, there is emergence of dynamical thermalization which leads to the Bose-Einstein probability distribution over the linear eigenmodes of the stadium. This distribution is drastically different from the energy equipartition over oscillator degrees of freedom which would lead to the ultra-violet catastrophe. We argue that this interesting phenomenon can be studied in cold-atom experiments.

A 2 DOF vibration harvester for broadband and multifrequency harvesting using a single electro-magnetic transducer

The narrow bandwidth of resonant vibration energy harvesters has long been seen as a drawback to exploitation. The narrow bandwidth is necessitated by the requirement to sufficiently amplify small source vibrations, but results in devices vulnerable to changes in excitation frequency, de-tuning due to ageing of components, and also makes efficient harvesting from sources with multiple frequency components difficult. In this paper a harvester based on a 2 degree of freedom oscillator is presented that not only enjoys the wider bandwidth of the higher order system, but configures the electromagnetic transducer in such a way that it requires no more components than the transducer of a typical single degree of freedom harvester. Theoretical models of the harvester system predict a range of possible frequency response functions dependent on easily-adjusted electrical parameters. These predictions are validated with experimental results.

Novel spectral features of nanoelectromechanical systems.

Electron transport through a quantum dot or single molecule coupled to a quantum oscillator is studied by the Keldysh nonequilibrium Green's function formalism to obtain insight into the quantum dynamics of the electronic and oscillator degrees of freedom. We tune the electronic level of the quantum dot by a gate voltage, where the leads are kept at zero temperature. Due to the nonequilibrium distribution of the electrons in the quantum dot, the spectral function becomes a function of the gate voltage. Novel spectral features are identified for the ground and excited states of nanomechanical oscillators that can be used to enhance the measurement sensitivity.

Rain-Wind-Induced In-Plane and Out-of-Plane Vibrations of Stay Cables

A theoretical model of in-plane and out-of-plane rain-wind-induced vibrations (RWIVs) of stay cables is developed in this paper. The proposed scheme models the cable, akin to sectional models of a bridge deck, as a segmental rigid element with two degrees of freedom (DOF) (in plane and out of plane) and the rivulet as a single DOF oscillator. The interaction between the cable surface and the water rivulet is first analyzed by using a wetting theory, which suggests a combined application of linear and Coulomb damping forces between the rivulet and the cable. Based on the model, the response of the combined cable-rivulet system is numerically evaluated. The results show that the in-plane and out-of-plane DOF of the cable are coupled. For the equilibrium of the rivulet on the cable surface, the gravitational and aerodynamic forces on the rivulet have similar time-averaged values with opposite signs. The interaction between the cable surface and the rivulet leads to energy dissipation, which helps to stabilize the rivulet. Like galloping, RWIV appears to be triggered by a sudden drop in the mean aerodynamic force coefficient. However, it is not a divergent type of vibration because the rivulet oscillates on the cable surface and its equilibrium position changes with wind velocity. Finally, the turbulence effect on the RWIVs is investigated. As the longitudinal turbulence intensity approaches 15%, the RWIV abates concomitantly.

Nonequilibrium phases in hybrid arrays with flux qubits and nitrogen-vacancy centers

We propose a startling hybrid quantum architecture for simulating a localization-delocalization transition. The concept is based on an array of superconducting flux qubits which are coupled to a diamond crystal containing nitrogen-vacancy (NV) centers. The underlying description is a Jaynes-Cummings-lattice in the strong-coupling regime. However, in contrast to well-studied coupled cavity arrays the interaction between lattice sites is mediated here by the qubit rather than by the oscillator degrees of freedom. Nevertheless, we point out that a transition between a localized and a delocalized phase occurs in this system as well. We demonstrate the possibility of monitoring this transition in a non-equilibrium scenario, including decoherence effects. The proposed scheme allows the monitoring of localization-delocalization transitions in Jaynes-Cummings-lattices by use of currently available experimental technology. Contrary to cavity-coupled lattices, our proposed recourse to stylized qubit networks facilitates (i) to investigate localization-delocalization transitions in arbitrary dimensions and (ii) to tune the inter-site coupling in-situ.

Bifurcation analyses on the chatter vibrations of a turning process with state-dependent delay

Bifurcation analyses are performed using the methods of multiple scales and harmonic balance in order to investigate the chatter vibration characteristics of a nonlinear turning system with state-dependent delay. In this study, the tool of the turning system is modeled as a two degrees of freedom oscillator and both the nonlinear stiffness and the nonlinear cutting force are considered. Prior to performing the bifurcation analyses, the nonlinear cutting force is appropriately expanded using a Taylor series, and an eigenanalysis is performed on the linearized system to obtain the linear stability boundaries. The bifurcation analyses are then performed and examined to explain the effect of state-dependent delay on the small- and large-amplitude behavior of the turning system. Analytic results derived from this study are validated through direct comparison with numerical results.

Frequency shift and hysteresis suppression in contact-mode AFM using contact stiffness modulation

In this paper the frequency response shift and hysteresis suppression of contact-mode atomic force microscopy is investigated using parametric modulation of the contact stiffness. Based on the Hertzian contact theory, a lumped single degree of freedom oscillator is considered for modeling the cantilever dynamics contact- mode atomic force microscopy. We use the technique of direct partition of motion and the method of multiple scales to obtain, respectively, the slow dynamic and the corresponding slow flow of the system. As results, this study shows that the amplitude of the contact stiffness modulation has a significant effect on the frequency response. Specifically, increasing the amplitude of the stiffness modulation suppresses hysteresis, decreases the peak amplitude and produces shifts towards higher and lower frequencies.

TWO MODELS OF NONSMOOTH DYNAMICAL SYSTEMS

Two examples of nonsmooth systems are considered. The first one is a two degrees of freedom oscillator in the presence of a stop. A discontinuity appears when the system position reaches a critical value. The second example consists of coupled oscillators excited by dry friction. In this case, the discontinuity occurs when the system's velocities take a critical value. For both examples, the dynamical system can be partitioned into different configurations limited by a set of boundaries. Within each configuration, the dynamical model is linear and the close form solution is known. Periodic orbits, including several transitions between the various configurations of the system, are found in analytical form. The stability of these orbits is investigated by using the Poincaré map modeling.