Underdamped Modes Research Articles

Analytical solutions and classification of vesicle motion and deformation in shear flow: Uncovering new tank-treading modes.

Using a small deformation approach, a fractional ordinary differential system is proposed to investigate the motion and deformation of a vesicle in shear flow. Closed analytical expressions of the orientation angle and the ellipticity of the vesicle contour (shape deformation) are provided. Three different motions are identified, the classical tank-treading state, and two new types of motions, namely, the over-damped tank-treading mode, in which the vesicle's orientation angle ψ and its shape deformation R tend more slowly toward equilibrium, and the under-damped tank-treading mode, in which ψ oscillates all the time along the flow direction with decreasing amplitude, while R starts making a breathing motion and then tends to an attractive amplitude. The implications of our findings extend widely within the field of fluid dynamics, revealing the potential for further advancements and applications in understanding complex fluid systems.

Investigation on Tuning Power–Frequency Droop for Improved Grid–Forming Inverter and Synchronous Generator Transient Load Sharing

Poor transient load sharing has been observed during the parallel operation of synchronous generators (SGs) and droop-controlled grid-forming (GFM) inverter-based resources (IBRs) in islanded microgrids. This can result in overcurrent protections, causing the inverters to trip, which leads to the collapse of voltage and frequency. In this article, small-signal analysis supported by electromagnetic transients (EMTs) simulation is performed on a detailed model of a microgrid containing IBRs and an SG to identify underdamped modes and their sensitivities to the power–frequency (P-F) droop parameter. Further EMT simulation of a more complex microgrid model with multiple GFM IBRs and SGs is performed to determine whether trends identified from the modal analysis extend to cases where the number of SGs in operation may vary. The objective is to investigate the effectiveness of tuning this parameter to reduce the likelihood of overcurrents in response to load and generation disturbances. The work uncovers both fast and slow modes of concern. Primarily, the findings show that increasing the P-F droop value improves the damping of the slower mode while degrading the damping of the faster mode, leading to an upper limit on the damping improvement of the GFM IBR output current. In EMT simulations of the more complex microgrid, the greatest damping improvement still exhibits significant maximum overshoot. Furthermore, the optimal value of the P-F droop parameter is sensitive to the number of SGs in operation. The conclusion is that tuning the P-F droop parameter is not an effective means to avoid the overcurrent tripping of GFM IBRs during response to large load or generation disturbances when operating in parallel to SGs in islanded microgrids.

Driving force and nonequilibrium vibronic dynamics in charge separation of strongly bound electron–hole pairs

Electron-hole pairs in organic photovoltaics efficiently dissociate although their Coulomb-binding energy exceeds thermal energy at room temperature. The vibronic coupling of electronic states to structured vibrational environments containing multiple underdamped modes is thought to assist charge separation. However, non-perturbative simulations of such large, spatially extended, electronic-vibrational (vibronic) systems remain an unmet challenge which current methods bypass by considering effective one-dimensional Coulomb potentials or unstructured environments where the effect of underdamped modes is ignored. Here we address this challenge with a non-perturbative simulation tool and investigate the charge separation dynamics in one, two and three-dimensional donor-acceptor networks to identify under what conditions underdamped vibrational motion induces efficient long-range charge separation. The resulting comprehensive picture of ultrafast charge separation differentiates electronic or vibronic couplings mechanisms for a wide range of driving forces and identifies the role of entropic effects in extended systems. This provides a toolbox for the design of efficient charge separation pathways in artificial nanostructures.

Start-up behavior of oscillating heat pipe in grinding wheel under axial-rotation conditions

• Start-up timing behavior of AR-OHP is characterized and studied. • AR-OHP filled with acetone has a fast start-up speed followed by methanol & water. • Start-up mode turns from over-damped to under-damped/ transient mode with increase of cooling. • Centrifugal acceleration improves AR-OHP flow motion and increases start-up time. • Start-up mode becomes over-damped mode @ a =738m/s 2 and benefits temperature control. High efficiency grinding of difficult-to-machine materials can generate a good amount of heat in a short time. If the heat is not timely transferred, the accumulation of heat will cause a high grinding temperature and can be harmful to the machining quality. The oscillating heat pipes (OHP) are designed in the grinding wheel to directly transfer out grinding heat when axial rotating with the grinding wheel. To avoid sudden burnout of the workpiece in high efficiency grinding, the OHPs in the grinding wheel are supposed to dissipate a high heat flux in a very short time, therefore, such OHPs should start-up very quickly. The start-up speed of the OHP plays an important role in the grinding efficiency and quality, while the study on the start-up behavior of the OHP under axial-rotation is insufficient. Given that, the start-up process of the OHP under axial-rotation is here described and characterized, and the insurgence of the start-up is evaluated by the settling time and the rising time. Besides, the influences of the working fluids, cooling conditions, and centrifugal accelerations are studied. As expected for other kinds of OHP, also for the present OHP, the OHP filled with acetone has the shortest start-up time (20s), followed by the OHP filled with methanol (45s), while the water as the working fluid has the longest start-up time (46.5s). Along with the increase of cold air pressure, the settling and the rising time decreases significantly, e.g., for acetone, the settling time decreases by 59% - 87%. The centrifugal acceleration promotes the OHP flow motion and heat transfer, in turn, increases the start-up time. Besides, the start-up behavior turns from under-damped mode to over-damped mode, which has no temperature overshoot and is beneficial to the temperature control for thermal management in axial-rotation scenarios.

Dynamic behavior and damping enhancement of cable with negative stiffness inerter damper

To improve the control performance of stay cables, a novel negative stiffness inerter damper (NSID) that consists of a negative stiffness device in parallel with an eddy current inerter damper is proposed in this study. The novelty of the NSID is that the damper can utilize the damping enhancement of the negative stiffness and inerter for cable vibration control, simultaneously. A further novelty is to carefully explore the modal behavior of the cable based on optimum parameters of the NSID. On this basis, the damping enhancement principle and conditions of the NSID for cable vibration control are demonstrated. In addition, the closed-form equation of the cable modal damping ratio and the design equation of the NSID for multi-mode cable vibration control are derived. And the control efficacy of the NSID is evaluated through the damping enhancement of the design cable modes. The results show that the modal behavior in the nth cable mode can be divided into three distinct regimes according to the negative stiffness and the corresponding optimum inertance, which are underdamped mode, subcritical damped mode, and the (n − 1)th underdamped mode. In the first regime, the inerter and negative stiffness have similar effects on enhancing damping ratio of the cable and reducing the optimum damping coefficient of the NSID. In the second and third regimes, the mode crossover phenomenon is observed, and the actual damping ratio of the nth cable mode decreases as the negative stiffness or inertance increases. The NSID performs better than the VD, VID, and NSD in mitigating multi-mode cable vibrations. Increasing negative stiffness can further improve the control performance and reduce the optimum inertance and damping coefficient of the NSID.

Underdamped longitudinal soft modes in ionic crystallites-lattice and charge motions observed by ultrafast x-ray diffraction.

Soft modes in crystals are lattice vibrations with frequencies that decrease and eventually vanish as the temperature approaches a critical point, e.g., a structural change due to a phase transition. In ionic para- or ferroelectric materials, the frequency decrease is connected with a diverging electric susceptibility and, for infrared active modes, a strong increase in oscillator strength. The traditional picture describes soft modes as overdamped transverse optical phonons of a hybrid vibrational-electronic character. In this context, potassium dihydrogen phosphate (KH2PO4, KDP) has been studied for decades as a prototypical material with, however, inconclusive results regarding the soft modes in its para- and ferroelectric phase. There are conflicting assignments of soft-mode frequencies and damping parameters. We report the first observation of a longitudinal underdamped soft mode in paraelectric KDP. Upon impulsive femtosecond Raman excitation of coherent low-frequency phonons in the electronic ground state of KDP crystallites, transient powder diffraction patterns are recorded with femtosecond hard x-ray pulses. Electron density maps derived from the x-ray data reveal oscillatory charge relocations over interatomic distances, much larger than the sub-picometer nuclear displacements, a direct hallmark of soft-mode behavior. The strongly underdamped character of the soft mode manifests in charge oscillations persisting for more than 10 ps. The soft-mode frequency decreases from 0.55 THz at T = 295 K to 0.39 THz at T = 175 K. An analysis of the Raman excitation conditions in crystallites and the weak damping demonstrate a longitudinal character. Our results extend soft-mode physics well beyond the traditional picture and pave the way for an atomic-level characterization of soft modes.

Temperature-Dependent Low-Frequency Modes in the Active Site of Bovine Carbonic Anhydrase II Probed by 2D-IR Spectroscopy.

Enzyme catalysis achieves tremendous rate accelerations. Enzyme reaction centers provide a constraint geometry that preferentially binds an activated form of the substrate and thus lowers the energy barrier. However, this transition state picture neglects the flexibility of proteins and its role in enzymatic catalysis. Especially for proton transfer reactions, it has been suggested that motions of the protein modulate the donor-acceptor distance and prepare a tunneling-ready state. We report the detection of frequency fluctuations of an azide anion (N3-) bound in the active site of the protein carbonic anhydrase II, where a low-frequency mode of the protein has been proposed to facilitate proton transfer over two water molecules during the catalyzed reaction. 2D-IR spectroscopy resolves an underdamped low-frequency mode at about 1 THz (30 cm-1). We find its frequency to be viscosity- and temperature-dependent and to decrease by 6 cm-1 between 230 and 320 K, reporting the softening of the mode's potential.

Piecewise constant response of underdamped oscillators through suppression of overshoots and undershoots in aerospace, civil, and mechanical systems

When a single-degree-of-freedom underdamped system is subjected to a step function force in order that it eventually acquires a constant displacement, its response shows undershoots and overshoots. Since a step function force is difficult, if not impossible, to apply to many aerospace, civil, and mechanical engineering systems because of their inertias, this paper looks at simple forces that can be generated from a practical standpoint and are not instantaneously applied, that cause an underdamped oscillator to acquire a constant displacement without any overshoots and/or undershoots. These forces are ramped up (or down) over a short duration of time and held constant thereafter. A preliminary approach to the development of such force–time histories is presented by using a force given by a polynomial in time. Open-loop optimal control is next considered, and then closed-loop optimal control. The optimal control problem does not fall within the standard rubric of terminal control and a new approach for doing this is developed. These ideas are then woven into the development of a methodology that allows an undamped/underdamped single degree of freedom system to track a desired piecewise constant displacement time-history using forces that do not need to be instantaneously applied and that generate rippleless responses with no overshoots/undershoots. The methodology is also applicable to classically damped multi-degree-of-freedom systems with underdamped modes of vibration.

Low-Frequency Protein Motions Coupled to Catalytic Sites.

This review examines low-frequency vibrational modes of proteins and their coupling to enzyme catalytic sites. That protein motions are critical to enzyme function is clear, but the kinds of motions present in proteins and how they are involved in function remain unclear. Several models of enzyme-catalyzed reaction suggest that protein dynamics may be involved in the chemical step of the catalyzed reaction, but the evidence in support of such models is indirect. Spectroscopic studies of low-frequency protein vibrations consistently show that there are underdamped modes of the protein with frequencies in the tens of wavenumbers where overdamped behavior would be expected. Recent studies even show that such underdamped vibrations modulate enzyme active sites. These observations suggest that increasingly sophisticated spectroscopic methods will be able to unravel the link between low-frequency protein vibrations and enzyme function.

Intramolecular vibrations enhance the\xa0quantum efficiency of excitonic energy transfer

We study the impact of underdamped intramolecular vibrational modes on the efficiency of the excitation energy transfer in a dimer in which each state is coupled to its own underdamped vibrational mode and, in addition, to a continuous background of environmental modes. For this, we use the numerically exact hierarchy equation of motion approach. We determine the quantum yield and the transfer time in dependence of the vibronic coupling strength, and in dependence of the damping of the incoherent background. Moreover, we tune the vibrational frequencies out of resonance with the excitonic energy gap. We show that the quantum yield is enhanced by up to 10% when the vibrational frequency of the donor is larger than at the acceptor. The vibronic energy eigenstates of the acceptor acquire then an increased density of states, which leads to a higher occupation probability of the acceptor in thermal equilibrium. We can conclude that an underdamped vibrational mode which is weakly coupled to the dimer fuels a faster transfer of excitation energy, illustrating that long-lived vibrations can, in principle, enhance energy transfer, without involving long-lived electronic coherence.

Start-up timing behavior of single-loop oscillating heat pipes based on the second-order dynamic model

Oscillating heat pipes have the potential to combine with the grinding wheel to enhance heat transfer in grinding process to avoid thermal damage of workpiece and grinding wheel. Since the time of grinding process is short, it raises a requirement for a good start-up behavior of oscillating heat pipes inside grinding wheel. In order to understand the start-up behavior, a new method is proposed based on the second-order dynamic theory to represent and evaluate start-up process. Several parameters (i.e., rising and settling time, percent-overshoot) are defined to value quantitatively the start-up process. Accordingly, three start-up modes are discovered with respect to the evaporator temperature, which are over-damped, under-damped and transient start-up modes. The start-up behavior is discussed with both pure fluids and nanofluids. Results show that as the heating power increases, the start-up speed enhances, and over-damped start-up mode develops to under-damped or transition start-up mode. The start-up speed of DI water is faster at low heating power, while acetone and nano-diamond solution without dispersant show faster start-up speed at high heating power. This methodology will help to compare the start-up timing of the OHP experiments in literatures, providing a more robust way to analyze start-up data.

Dissipation-Assisted Matrix Product Factorization.

Charge and energy transfer in biological and synthetic organic materials are strongly influenced by the coupling of electronic states to a highly structured dissipative environment. Nonperturbative simulations of these systems require a substantial computational effort, and current methods can only be applied to large systems if environmental structures are severely coarse grained. Time evolution methods based on tensor networks are fundamentally limited by the times that can be reached due to the buildup of entanglement in time, which quickly increases the size of the tensor representation, i.e., the bond dimension. In this Letter, we introduce a dissipation-assisted matrix product factorization (DAMPF) method that combines a tensor network representation of the vibronic state within a pseudomode description of the environment where a continuous bosonic environment is mapped into a few harmonic oscillators under Lindblad damping. This framework is particularly suitable for a tensor network representation, since damping suppresses the entanglement growth among oscillators and significantly reduces the bond dimension required to achieve a desired accuracy. We show that dissipation removes the "time-wall" limitation of existing methods, enabling the long-time simulation of large vibronic systems consisting of 10-50 sites coupled to 100-1000 underdamped modes in total and for a wide range of parameter regimes. For these reasons, we believe that our formalism will facilitate the investigation of spatially extended systems with applications to quantum biology, organic photovoltaics, and quantum thermodynamics.

Damped interfacial oscillation of a particle-embedded viscous drop

This article describes how a drop with an embedded particle exhibits interfacial waves with transient decay due to the interplay between capillary and viscous effects. To reveal the damped oscillation of the system properly, the deformation and pressure fields inside the domain are described in terms of complete sets of basis functions. Such representation leads to a matrix formulation which enforces no-slip condition at the solid-liquid interface and ensures correct discontinuity in normal stress due to surface tension at the drop periphery. The resulting characteristic equation involving the natural frequencies and the decay constants is solved numerically to determine these quantities. The matrix expression implies a block-diagonalized structure with two uncoupled blocks corresponding to two distinctly different dynamics. The first of these is related to pure rotational velocities on spherical surfaces which monotonically decay in time without any fluctuation in the absence of any peripheral deformation. By contrast, the second block is associated with the undulation in shape. Due to the restoring features of surface tension, the latter can exhibit underdamped oscillatory modes, if the capillary number Ca is below a critical value. However, even these waves would become overdamped if the critical number is exceeded. These values of Ca for a few most relevant modes are plotted in this paper as functions of particle-to-drop size ratio. Also, the natural frequencies for the underdamped cases as well as the damping constants for all considered modes are presented for different size ratios and capillary numbers. The findings are verified by matching the computed results to a novel boundary layer theory under low capillary number limit. Under the limiting condition, both sets of independent calculations yield the same decay constants and natural frequencies providing mutual validations.

Effect of Underdamped Vibration on Excitation Energy Transfer: Direct Comparison between Two Different Partitioning Schemes

We study excitation energy transfer (EET) in a model three-site system with a mixed-quantum classical dynamics method, by focusing on the effect of an underdamped vibration. We construct two types of models where the underdamped vibration mode is included either in the quantum subsystem or in the classical bath. We show that the two models yield practically equivalent results despite the different depictions of the vibration. In particular, both models consistently demonstrate accelerations of population relaxation induced by quasi-resonant vibration. This indicates that intricate features of EET dynamics that have been frequently ascribed to the quantal nature of vibrations, such as vibronic mixing, can be successfully reproduced by using physically equivalent but classically described bath modes. The mechanism behind the observed quantum-classical correspondence is proposed. We also systematically examine how the structure of the spectating continuum phonon modes affects the vibronic resonance and observe that phonon modes with different time scales influence the resonance in different manners.

Pole-placement Predictive Functional Control for under-damped systems with real numbers algebra

This paper presents the new algorithm of PP-PFC (Pole-placement Predictive Functional Control) for stable, linear under-damped higher-order processes. It is shown that while conventional PFC aims to get first-order exponential behavior, this is not always straightforward with significant under-damped modes and hence a pole-placement PFC algorithm is proposed which can be tuned more precisely to achieve the desired dynamics, but exploits complex number algebra and linear combinations in order to deliver guarantees of stability and performance. Nevertheless, practical implementation is easier by avoiding complex number algebra and hence a modified formulation of the PP-PFC algorithm is also presented which utilises just real numbers while retaining the key attributes of simple algebra, coding and tuning. The potential advantages are demonstrated with numerical examples and real-time control of a laboratory plant.

Experimental Study on Complex Modes of an End-Damped Continuous Beam

The complex modes of an end-damped cantilevered beam are studied as an experimental example of a nonmodally damped continuous system. An eddy-current damper is applied, for its noncontact and linear properties, to the end of the beam, and is then characterized to obtain the effective damping coefficient. The state-variable modal decomposition (SVMD) is applied to extract the modes from the impact responses in the cantilevered beam experiments. Characteristics of the mode shapes and modal damping are examined for various values of the end-damper damping coefficient. The modal frequencies and mode shapes obtained from the experiments have a good consistency with the results of the finite element model. The variation of the modal damping ratio and modal nonsynchronicity with varying end-damper damping coefficient also follow the prediction of the model. Over the range of damping coefficients studied in the experiments, we observe a maximum damping ratio in the lowest underdamped mode, which correlates with the maximum modal nonsynchronicity. Complex orthogonal decomposition (COD) is applied in comparison to the modal identification results obtained from SVMD.

Tracking the coherent generation of polaron pairs in conjugated polymers.

The optical excitation of organic semiconductors not only generates charge-neutral electron-hole pairs (excitons), but also charge-separated polaron pairs with high yield. The microscopic mechanisms underlying this charge separation have been debated for many years. Here we use ultrafast two-dimensional electronic spectroscopy to study the dynamics of polaron pair formation in a prototypical polymer thin film on a sub-20-fs time scale. We observe multi-period peak oscillations persisting for up to about 1 ps as distinct signatures of vibronic quantum coherence at room temperature. The measured two-dimensional spectra show pronounced peak splittings revealing that the elementary optical excitations of this polymer are hybridized exciton-polaron-pairs, strongly coupled to a dominant underdamped vibrational mode. Coherent vibronic coupling induces ultrafast polaron pair formation, accelerates the charge separation dynamics and makes it insensitive to disorder. These findings open up new perspectives for tailoring light-to-current conversion in organic materials.

Broadband Dielectric Spectroscopy on Lysozyme in the Sub-Gigahertz to Terahertz Frequency Regions: Effects of Hydration and Thermal Excitation.

We have performed dielectric spectral measurements of lysozyme in a solid state to understand the effects of hydration and thermal excitation on the low-frequency dynamics of protein. Dielectric measurements were performed under changing hydration conditions at room temperature in the frequency region of 0.5 GHz to 1.8 THz. We also studied the temperature dependence (83 to 293 K) of the complex dielectric spectra in the THz frequency region (0.3 THz to 1.8 THz). Spectral analyses were performed using model functions for the complex dielectric constant. To reproduce the spectra, we found that two relaxational modes and two underdamped modes are necessary together with an ionic conductivity term in the model function. At room temperature, the two relaxational modes have relaxation times of ∼20 ps and ∼100 ps. The faster component has a major spectral intensity and is suggested to be due to coupled water-protein motion. The two underdamped modes are necessary to reproduce the temperature dependence of the spectra in the THz region satisfactorily. The protein dynamical transition is a well-known behavior in the neutron-scattering experiment for proteins, where the atomic mean-square displacement shows a sudden change in the temperature dependence at approximately 200 K, when the samples are hydrated. A similar behavior has also been observed in the temperature dependence of the absorption spectra of protein in the THz frequency region. From our broadband dielectric spectroscopic measurements, we conclude that the increase in the spectral intensities in the THz region at approximately 200 K is due to a spectral blue-shift of the fast relaxational mode.

On overdamping phenomena in gyroscopic systems composed of high-loss and lossless components

Using a Lagrangian framework, we study overdamping phenomena in gyroscopic systems composed of two components, one of which is highly lossy and the other is lossless. The losses are accounted by a Rayleigh dissipation function. As we have shown previously, for such a composite system, the modes split into two distinct classes, high-loss and low-loss, according to their dissipative behavior. A principal result of this paper is that for any such system, a rather universal phenomenon of selective overdamping occurs. Namely, first of all, the high-loss modes are all overdamped, i.e., non-oscillatory, as are an equal number of low-loss modes. Second of all, the rest of the low-loss modes remain oscillatory (i.e., the underdamped modes), each with an extremely high quality factor (Q-factor) that actually increases as the loss of the lossy component increases. We prove that selective overdamping is a generic phenomenon in Lagrangian systems with gyroscopic forces and gives an analysis of the overdamping phenomena in such systems. Moreover, using perturbation theory, we derive explicit formulas for upper bound estimates on the amount of loss required in the lossy component of the composite system for the selective overdamping to occur in the generic case and give Q-factor estimates for the underdamped modes. Central to the analysis is the introduction of the notion of a “dual” Lagrangian system and this yields significant improvements on some results on modal dichotomy and overdamping. The effectiveness of the theory developed here is demonstrated by applying it to an electric circuit with a gyrator element and a high-loss resistor.

Completeness Verification of Complex Response Spectrum Method for Underdamped and Overdamped Multiple-Support Systems Regarding the Decoupled Damping as Mathematical Parameter without Physical Meaning

The completeness of the complex response spectrum method for both formally underdamped and overdamped modes is theoretically proved, and the physical meanings of the decoupled modes as well as involved parameters are recognized and clarified in this paper. For the system with relatively large non-classical damping, the eigenvalue pairs generated by the complex mode decomposition method are real and the so-called modal damping ratios are larger than unity. In this paper, we firstly clarified that the decoupled modes are virtual and the so-called modal frequency and damping ratio are mathematical parameters that have no physical meaning. Then, the completeness of the complex response spectrum method for both formally underdamped and overdamped modes is rigorously proved by allowing the “damping frequency” to be an imaginary number. For the virtually overdamped modes, Duhamel integral involved in the calculation for formally underdamped modes automatically convert to hyperbolic Duhamel integral. A numerical example taken from the published literature is given to verify this method. Structural responses for the system with coupled damping under multi-support seismic excitations are further analyzed and numerical results indicate the accuracy of complex response spectrum method.