Nonlocal Damping Research Articles

Non-local finite element analysis of damped beams

Non-local viscoelastic beam models are used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local damping models the internal force of the non-local model is obtained as weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two-node beam elements. However, for non-local damping, nodes remote from the element do have an effect on the energy expressions, and hence on the damping matrix. The expressions of these direct and cross damping matrices may be obtained explicitly for some common spatial kernel functions and Euler–Bernoulli beam theory. Alternatively numerical integration may be applied to obtain solutions. Examples are given where the eigenvalues are compared to the exact solution for a pinned–pinned beam to demonstrate the convergence of the finite element method. The results for beams with other boundary conditions are used to demonstrate the versatility of the finite element technique.

Intrinsic Spin Relaxation Processes in Metallic Magnetic Multilayers

Spin relaxation processes in metallic magnetic nanostructures are reviewed. First a brief review of the phenomenology of magnetic damping is presented using the Landau Lifshitz Gilbert (LLG) equations of motion. It is shown that the Gilbert damping in bulk metallic layers is caused by the spin orbit interaction and itinerant character of 3d and 4s-p electrons. Spin dynamics in magnetic nanostructures acquires an additional nonlocal damping. This means that a part of the magnetic damping is not given by the local Gilbert damping but arises from the proximity to other layers. Spin pumping and spin sink concepts will be introduced and used to describe the interface nonlocal Gilbert damping in magnetic multilayers. The modified LLG equation of motion in magnetic multilayers will be introduced and tested against the ferromagnetic resonance (FMR) data around the accidental crossover of FMR fields. The spin pumping theory will be compared to the early theories introduced in the 1970s for the interpretation of transmission electron spin resonance (TESR) measurements across ferromagnet/normal metal sandwiches.

Optimal Nonlocal and Asymmetric Structural Damping Using Regenerative Force Actuation Networks

A regenerative force actuation (RFA) network consists of multiple electromechanical forcing devices distributed throughout a structural system and actuated in such a way as to reduce the response of the structure when it is subjected to an excitation. The associated electronics of the devices are connected together such that they are capable of sharing electrical power with each other. This makes it possible for some devices to extract mechanical energy from the structure while others reinject a portion of that energy back into the structure at other locations. The forcing capability of an RFA network is constrained by the requirement that the total network must always dissipate energy. As such, it differs from fully active control devices in that its operation requires only a small amount of external power. Furthermore, its power-sharing capability gives it a forcing versatility beyond that attainable with semiactive and traditional passive damping systems. In this paper, RFA networks are analyzed in the context of their ability to apply supplemental linear structural damping, taking into account dissipation due to electrical resistances and viscous damping associated with the actuators. It is shown that these systems can be used to produce nonlocal damping (i.e., damping forces between distant degrees of freedom) and asymmetric damping matrices. By comparison, semiactive and passive devices can only impose local damping forces. The more generalized linear damping capabilities of RFA networks are shown to yield significant improvements in linear-quadratic optimal performance in stationary response. Examples are given in which a RFA network is used in various configurations to reduce the stationary response of the three-story shear structure to stochastic base excitation.

Modal Analysis of Nonviscously Damped Beams

Linear dynamics of Euler–Bernoulli beams with nonviscous nonlocal damping is considered. It is assumed that the damping force at a given point in the beam depends on the past history of velocities at different points via convolution integrals over exponentially decaying kernel functions. Conventional viscous and viscoelastic damping models can be obtained as special cases of this general damping model. The equation of motion of the beam with such a general damping model results in a linear partial integro-differential equation. Exact closed-form equations of the natural frequencies and mode shapes of the beam are derived. Numerical examples are provided to illustrate the new results.

Giant nonlocal damping by spin-wave emission: Micromagnetic simulations

A micromagnetic simulation scheme is employed to evaluate the damping processes occurring in all-optical pump-probe experiments. We reveal that small wavelength and high-energy, nanometer-sized domains form during the first stage of the relaxation process. The emission of spin waves propagating out of the excitation spot area dominates the energy dissipation process. This process becomes dominant for small excitation spot areas and increases the macroscopically observed damping constant $\ensuremath{\alpha}$ drastically. We find that the damping time can be increased by a factor of more than 100, depending on the geometry, and must be considered in the interpretation of all-optical pump probe experiments with spot diameters less than $1\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$.

A Galerkin method for distributed systems with non-local damping

In this paper, a non-local damping model including time and spatial hysteresis effects is used for the dynamic analysis of structures consisting of Euler–Bernoulli beams and Kirchoff plates. Unlike ordinary local damping models, the damping force in a non-local model is obtained as a weighted average of the velocity field over the spatial domain, determined by a kernel function based on distance measures. The resulting equation of motion for the beam or plate structures is an integro-partial-differential equation, rather than the partial-differential equation obtained for a local damping model. Approximate solutions for the complex eigenvalues and modes with non-local damping are obtained using the Galerkin method. Numerical examples demonstrate the efficiency of the proposed method for beam and plate structures with simple boundary conditions, for non-local and non-viscous damping models, and different kernel functions.

Spin-transfer-induced magnetic excitation: The role of spin-pumping induced damping

Spin-transfer-induced magnetic excitation in large magnetic field applied perpendicular to the thin film junction surface reveals both a current threshold Ic and a voltage threshold. The current threshold follows the Slonczewski-type of magnetic field dependence [J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996)]. The voltage step at Ic is ΔV which appears to scale with the applied field with a prefactor of the order of 2μB∕e, suggesting a threshold to magnetic excitation. Furthermore, experimentally it is observed that ΔV≈IcδR, where δR is the magnetoreistance between the parallel and the antiparallel states. This apparent coincidence can be unified when one includes the effect of spin-pumping-related nonlocal damping. The spin-pump damping relates magnetic instability threshold Ic to δR, producing (dIc∕dH)δR that is about 2μB∕e, explaining the origin of the coincidence.

Structural control with regenerative force actuation networks

A Regenerative Force Actuation (RFA) Network consists of multiple electromechanical forcing devices distributed throughout a structural system and actuated in such a way as to reduce the response of the structure when subject to an excitation. The associated electronics of the devices are connected together such that they are capable of electrical with each other. This makes it possible for some devices to extract mechanical energy from the structure, while others re-inject a portion of that energy back into the structure at other locations. The forcing capability of an RFA network is constrained only by the requirement that in the aggregate the total network must always dissipate energy. The electromechanical currents generated by RFA networks must be controlled to create the desired structural forces. This control is facilitated by the alternation of a multitude of power-electronic transistor switches in the electrical network. In this study, a sliding-mode switching controller is proposed for realizing zero-error force command tracking. It is shown that parameter uncertainty is a critical issue for force commands which require the network to operate near its optimum transmissive efficiency. RFA networks can be used to create velocity-proportional damping forces in structures. However, unlike traditional structural damping, RFA networks have the ability to create non-local and asymmetric damping forces. It is shown that this more generalized damping capability can lead to significant improvements in the forced response of a structure, as compared with traditional linear damping. RFA networks may also be used for feedback control. In this context, the forcing capability of the RFA network is constrained by its physical limitations. In this study, a systematic method of nonlinear control design called Damping-Reference control is proposed, which guarantees a certain level of quadratic performance for the structural response. Variants of the control law synthesis are proposed for quadratic regulation, stochastic control, and H[infinity] control contexts. These ideas are illustrated in the context of earthquake engineering through a simulation example, involving a three-story structure with a two-actuator RFA network installed. In this example, it is shown that the power sharing nature of the RFA network has a significant influence on the response.

Role of dynamic exchange coupling in magnetic relaxations of metallic multilayer films (invited)

The relaxation processes were investigated by ferromagnetic resonance (FMR) using magnetic single, Au/Fe/GaAs(001), and double layer, Au/Fe/Au/Fe/GaAs(001), structures prepared by molecular beam epitaxy. These structures provided an excellent opportunity to investigate nonlocal damping which is caused by spin transport across a nonmagnetic spacer. In the double layer structures thin Fe layers F1 were separated from a second thick Fe layer F2 by a Au(001), normal metal spacer. The interface magnetic anisotropies separated the FMR fields of F1 and F2 by a big margin which allowed us to investigate FMR in F1 while F2 had a negligible angle of precession. The main result is that the ultrathin Fe films in magnetic double layers acquire a nonlocal interface Gilbert damping. Several mechanisms have been put forward to explain the nonlocal damping. A brief review of each mechanism will be presented. They will be compared with the experimental results allowing one to critically assess their applicability and strength. It will be shown that the precessing layers act as spin pumps and spin sinks. This concept was tested by investigating the FMR linewidth around an accidental crossover of the resonance fields for the layers F1 and F2.

Role of spin current in magnetic relaxations of metallic multilayer films

In our recent studies the relaxation processes were investigated by ferromagnetic resonance (FMR) using magnetic single, Au/Fe/GaAs(0 0 1), and double layer, Au/Fe/Au/Fe/GaAs(0 0 1), structures prepared by molecular beam epitaxy (MBE). Single and double layer structures were grown on (4×6) reconstructed GaAs(0 0 1) templates. The samples have high crystalline quality with the magnetic properties corresponding to bulk Fe, only modified by interface anisotropies. The magnetic relaxation in these structures is well described by Gilbert damping with no extrinsic contributions to the FMR line width. These structures provide an excellent opportunity to investigate non-local damping which is caused by spin transport across a non-magnetic spacer. In the double layer structures thin Fe layers were separated from a second thick Fe layer of 40 monolayer (ML) thickness by a 40 ML thick Au(0 0 1) spacer. The main result of these studies is that the ultrathin Fe films in magnetic double layers acquire an additional Gilbert damping. The additional FMR line width followed an inverse dependence on the Fe film thickness which is expected for interface damping. The spin pump model will be discussed and it will be shown that it is closely related to the dynamic part of interlayer exchange coupling.

Magnetic relaxations in metallic multilayers

The intrinsic damping mechanism in metals caused by incoherent scattering of itinerant electron-hole pair excitations by phonons and magnons will be reviewed. The unique features of magnetic relaxations in multilayers were studied by ferromagnetic resonance (FMR) using magnetic single, Au–Fe–GaAs(001), and double layer Au–Fe–Au–Fe–GaAs(001) structures prepared by molecular beam epitaxy. The magnetic relaxation in single-layer films is described by the Gilbert damping with no extrinsic contributions to the FMR linewidth. These films provided an excellent opportunity to investigate nonlocal damping. The main result of these studies is that ultrathin Fe films in magnetic double layers acquire an additional interface Gilbert damping. This is in agreement with recent predictions of nonlocal interface damping which is based on the transport of spin angular momentum between the ferromagnetic layers. Measurements of the nonlocal Gilbert damping offer a possibility to carry out quantitative studies of the relaxation torques caused by nonlocal spin momentum transfer. Numerical simulations of magnetization reversal and stationary precession for an applied perpendicular current in Au–Fe–Au–Fe–GaAs(001) multilayers will be shown.

Role of Spin Momentum Current in Magnetic Non-Local Damping of Ultrathin Film Structures

Non-local damping was investigated by Ferromagnetic Resonance (FMR) using ultrathin magnetic single and double layer structures prepared by Molecular Beam Epitaxy (MBE). The double layer structures show magnetic damping which is caused by spin transport across a normal metal spacer (N). In double layer structures a thin Fe layer, F1, was separated from a thick Fe layer, F2, by a Au(001) spacer. The interface magnetic anisotropies separated the FMR fields of F1 and F2 by a big margin allowing one to investigate FMR in F1 while F2 had a negligible angle of precession, and vice versa. The Fe films in magnetic double layers acquire non-local interface Gilbert damping. It will be shown that the precessing magnetic moments act as spin pumps and spin sinks. This concept was tested by investigating the FMR linewidth around an accidental crossover of the resonance fields for the layers F1 and F2. There is another possible mechanism for non-local damping which is based on a “breathing Fermi surface” of the spacer. The temperature dependence of the non-local damping indicates that this mechanism is weak in Au spacers. Surprisingly the Au spacer acts as an additional impedance for the spin pump mechanism. Finally, it will be shown that electron-electron correlations in a Pd spacer can lead to a significant enhancement of the non-local damping.

Wave Number Downshift in Modulated Wavetrain through a Nonlinear Damping Effect

The time evolution of a nonlinear modulated wavetrain is investigated using the Dysthe equation with an additional nonlocal nonlinear damping term. It is found that the additional damping term has no significant effect on qualitative features of the modulational instability of the Stokes wave. Numerical results show that for appropriate values of the coefficient of the damping term, it is practically effective only near the crest of envelope, and causes the wave number downshift. These results suggested that two factors are essential to cause the downshift. The first is the enough nonlinearity to produce the asymmetry in spectral distribution. The second is the nonlinear dissipation that affects especially the higher components only when the wavetrain is strongly modulated.

Theory and three-dimensional simulation of light filamentation in laser-produced plasma

A desire to interpret recent experiments on filamentation with and without beam-smoothing techniques led to the development of a three-dimensional fluid model that includes the effects of nonlocal electron transport and kinetic ion damping of the acoustic waves. The damping of the electron-temperature perturbations that drive thermal filamentation by nonlocal electron conduction, valid in the diffusive limit, is supplemented in the present model by electron Landau damping in the collisionless limit when the wavelength of the perturbation is much less than the electron–ion scattering mean-free path. In this collisionless limit, Landau damping of the ‘‘temperature’’ fluctuations makes ponderomotive forces universally more important than thermal forces. Simulations in plasmas of current interest illustrate the relative importance of thermal and ponderomotive forces for strongly modulated laser beams. Although thermal forces may initiate filamentation, the most intense filaments are associated with ponderomotive forces. The present simulations of filamentation model well the density perturbations observed in experiments [Young et al., Phys. Rev. Lett. 61, 2336 (1988)]. In addition, a simple criterion is obtained analytically and supported by simulations for stabilization of filamentation by laser beam-smoothing techniques such as induced spatial incoherence and random phase plates [Eq. (1)].

Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics

Fluidlike models have long been used to develop qualitative understanding of the drift-wave class of instabilities (such as the ion temperature gradient mode and various trapped-particle modes) which are prime candidates for explaining anomalous transport in plasmas. Here, the fluid approach is improved by developing fairly realistic models of kinetic effects, such as Landau damping and gyroradius orbit averaging, which strongly affect both the linear mode properties and the resulting nonlinear turbulence. Central to this work is a simple but effective fluid model [Phys. Rev. Lett. 64, 3019 (1990)] of the collisionless phase mixing responsible for Landau damping (and inverse Landau damping). This model is based on a nonlocal damping term with a damping rate ∼ vt‖k∥‖ in the closure approximation for the nth velocity space moment of the distribution function f, resulting in an n-pole approximation of the plasma dispersion function Z. Alternatively, this closure approximation is linearly exact (and therefore physically realizable) for a particular f0 which is close to Maxwellian. ‘‘Gyrofluid’’ equations (conservation laws for the guiding-center density n, momentum mnu∥, and parallel and perpendicular pressures p∥ and p⊥) are derived by taking moments of the gyrokinetic equation in guiding-center coordinates rather than particle coordinates. This naturally yields nonlinear gyroradius terms and an important gyroaveraging of the shear. The gyroradius effects in the Bessel functions are modeled with robust Padé-like approximations. These new fluid models of phase mixing and Landau damping are being applied by others to a broad range of applications outside of drift-wave turbulence, including strong Langmuir turbulence, laser–plasma interactions, and the α-driven toroidicity-induced Alfvén eigenmode (TAE) instability.

Non-local damping of coupled helicon-phonon modes

The dispersion equation for a helicon-phonon interaction has been derived for the case when non-local effects are significant. Numerical studies have been carried out for potassium. The nature of the variation of the damping of the four modes with the magnetic field strength has been studied. The results confirm that in the interaction region, mode conversion takes place and the waves are of a hybrid nature.

Nonlocal Damping of Helicon Waves

The nonlocal damping of helicon waves propagating in the simple metals is discussed. An approximate dispersion relation is presented which includes the effects of both normal collision damping and nonlocal damping. Two types of experiments designed to measure the damping in sodium and potassium are described. The first type involves measurements of the standing-wave resonances in thin-plate sodium samples. This technique is useful in the regime $\mathrm{ql}<5$, where $q$ is the helicon wave vector and $l$ is the carrier mean free path. The second type involves transmission experiments using both sodium and potassium. These experiments span the range $4\ensuremath{\lesssim}\mathrm{ql}\ensuremath{\lesssim}30$. The experimental results are in close agreement with the theoretical predictions based on the free-electron model.

Nonlocal Damping of Helicon Waves

Nonlocal Damping of Helicon Waves