Absence Of Damping Research Articles

The effect of damping on rigid block motion under harmonic forcing

The effect of damping on the response of a rigid block to harmonic excitation is considered. It is shown that the structure seen in the absence of damping is preserved in this case. Multiple solutions, asymmetric responses, period- and impact-doubling cascades still occur, albeit in different parts of parameter space. Explicit expressions are given for the stability boundaries. It is shown that, at certain values of the forcing frequency, the stability boundaries possess a local maximum as a function of the damping.

Quantum coherence and classical chaos in a pulsed parametric oscillator with a Kerr nonlinearity.

We consider a parametric amplifier driven by a periodically pulsed pump field inside a cavity containing a Kerr nonlinearity. The dynamics of the device is modeled as a kicked nonlinear system. The pulsed parametric amplifier constitutes the kick. In between kicks the dynamics is determined by the Kerr nonlinearity and damping. In the absence of damping, a classical description of the device exhibits a rich phase-space structure including fixed points of multiple period and chaos. We contrast the classical behavior of the mean intensity with that predicted by quantum dynamics. The mean photon number inside the cavity is shown to undergo regular collapse and revival in the regular region of the phase space and irregular revivals in the chaotic region. When damping is included, the quantum recurrences are rapidly suppressed, and the classical behavior is restored. In this case a stable steady state is possible. The damping represents the effect of photon-number measurements on the system. We also discuss the photon statistics in the steady state.

Modelling of dynamic response of pavements to impact loading

This contribution presents a discrete layer model for studying the elastostatic response of a pavement to falling weight deflectometer impact loading. The displacement field is represented by a Fourier-Bessel series expansion in the radial direction and piecewise linear variation in displacement in the vertical. Explicit time marching has been adopted to follow the history of a response. The results show that, although dynamic and static deflection bowls are different, little amplification in displacement due to inertia occurs directly under the load, even in the absence of damping. The results also suggest that measured displacements farther away from the load should not be used for backcalculation, if static analysis is used to interpret FWD generated data.

Dynamic response of some dissipative systems by means of functions of matrices

A certain class of linear, viscously damped dynamical systems, the matrix coefficients of which satisfy a certain commutativity condition, are known to exhibit the same normal modes as the ones that exist in the absence of damping. These systems are said to have classical normal modes. In the present study, such systems are shown to be particular cases of dynamical systems that possess a zero commutator, which form a subset of separable systems. The response of the latter is investigated by means of functions of matrices, and its connection with the classical eigenfunction expansion approach is established. The paper continues the author's previous work where functions of matrices were used to (a) develop an algorithm for the numerical integration of the equations of motion for such systems, and (b) to establish their conservation laws.

The motion of a superconducting loop in an homogeneous magnetic field: the harmonic oscillator equation in an unfamiliar setting

If a conducting loop is allowed to fall under the influence of gravity in an inhomogeneous magnetic field, Faraday's law and Newton's second law together lead to differential equation for the vertical velocity of the loop. A simple case is analysed in which this equation turns out to be just the familiar second-order differential equation for damped harmonic motion. The same equation applies no matter what the resistance of the loop, but the resulting motion of a superconducting loop is dramatically different from that of a loop with small but non-zero resistance: the magnetic flux through a superconducting loop is time independent, and the result is that in the absence of damping, both the vertical position of the loop and the current in the loop oscillate indefinitely, with a frequency determined by the properties of the loop and the strength of the magnetic field.

Time-dependent collisional approximations for two-level systems.

We derive two types of perturbative approximation for the response of collisionally broadened two-level atoms to a time-dependent field. The perturbative approximations for the inversion and polarization are valid to any desired order in the derivatives of the field. Comparison with a numerical solution of the full density-matrix equations of motion demonstrates that the new approximations significantly improve upon the adiabatic-following and collisional steady-state approximations for time-dependent pulses in the appropriate damping regime. Application of the approximations to pulse propagation, sideband instability, and population pulsations is discussed. The collisional steady-state and adiabatic-following approximations find extensive use in many areas of nonlinear optics. The deficiencies of these approximations are well known: The collisional steady-state approxirnation is valid only for monochromatic (cw) radiation and the adiabatic-following approximation is valid only in the absence of damping and only if the product of the detuning and a characteristic time is much larger than one. In spite of their manifest faults, the collisional steady-state and adiabatic-following approximations are often exceptionally good approximations, in the appropriate damping regime, for the local response of a vapor of two-level atoms to a slowly varying field. However, timedependent nonlinear processes modify a propagating field zuch that it varies less slowly and the neglected portions of the atomic response become more important. ' Therefore improved approximations for the damped atomic response to a time-dependent field are desirable. In this paper we present two improved perturbative approximations for the inversion and polarization of collisionally broadened two-level atoms induced by a time-dependent field. The adiabatic-following approximation for two-level systems was derived phenomenologically by Grischkowsky, who used it to explain certain experimental results of near-resonance laser-pulse propagation. Motivated by the work of Grischkowsky, Crisp provided an analytic derivation for the adiabatic-following approximation and obtained a quadrature for the error. The essential feature of Crisp's derivation is the expansion of the kernel of the integral representation of the off-diagonal density-matrix element p2, in a Fourier series with integration proceeding term by term. Both the adiabatic-following approximation and the collisional steady-state approximation were obtained in first order. Both Crisp and Grischkowsky used the adiabatic inversion to derive a correction to the adiabatic-following approximation for the polarization. We have shown how Crisp's method can be extended to obtain corrections to the adiabatic inversion and to improve corrections to the polarization in the absence of damping. In this paper we show how the integrated Fourier series can be used to obtain two types of perturbative approximation for the inversion and polarization of collisionally broadened twolevel systems that are valid to any desired order in the derivatives of the field. The two types of perturbative approximation to the inversion are (i) a formal solution of the rate equation for the inversion which is obtained from the density-matrix equations for the populations; and (ii) an approximation obtained using the approximation (p22 — p») —I — 4p2, P, 2, valid for weak damping. The approximations for the polarization are obtained by substituting either (i) or (ii) for the inversion in the integrated-Fourier-series expansion of p2, . The approximation obtained using the first type of approximate inversion converges most rapidly for strong damping and reduces to the collisional steady-state approximation when the magnitude and phase of the field are constant. The approximation obtained using the second type of approxirnate inversion converges most rapidly for weak damping and in first order reduces to the adiabaticfollowing approximation for zero damping. Comparison with a numerical solution of the density-matrix equations demonstrates that retention of higher-order derivatives by retaining more terms in the integrated Fourier series results in a significant improvement in accuracy for a time-dependent field. However, Rabi oscillations, which can be an important component of the nonlinear response, ' cannot be obtained by finite-order approxi

Observation of the Temporal Decoupling Effect on the Macroscopic Quantum Tunneling of a Josephson Junction

We have measured the Macroscopic Quantum Tunneling (M.Q.T.) rate out of the zero voltage state of a Josephson junction shunted by a delay line terminated by a resistor. As the length of the line is increased in situ, thereby increasing the delay with which the resistor damps the junction, the M.Q.T. rate increases until it almost recovers its value in the absence of damping: This constitutes the temporal decoupling effect. We compare our results to the theoretical predictions.

Homoclinic chaos in the rf superconducting quantum-interference device

We consider a simple model of the flux in a rf superconducting quantum-interference device (SQUID) ring subjected to an external periodic magnetic field. The dynamic equation describing the flux response of the SQUID is solved analytically in the absence of damping and external driving terms. We then introduce these terms as small perturbations, and construct, for this system, the Melnikov function, the zeros of which indicate the onset of homoclinic behavior. For the parameter values under consideration, excellent agreement is obtained between our theoretical predictions and numerical calculations of the stable and unstable (i.e., time-reversed) solution manifolds. A chaotic attractor is shown to appear somewhat above the homoclinic threshold.

Free Vibration of a Large-Amplitude Deflected Plate—Reexamination by the Dynamical Systems Theory

For a simply supported large-amplitude deflected plate, Fourier expansion of displacement reduces the nonlinear plate equation to a system of infinitely coupled modal equations. To close off this system, we have suppressed all but the four lowest-order symmetric modes. In the absence of damping and forcing, the four-mode truncation can be recasted into a Hamiltonian of 4 DOF. Hence, the free vibration of nonlinear plate can be investigated by the standard technique of Hamiltonian systems. It has been found that subsystems of 2 DOF are practically stable in that the invariant tori remain on a smooth surface up to total energy of 1000, at which modal displacements can be 40 times the plate thickness. On the other hand, the trajectory of 4 DOF system develops chaos at a much lower energy value of 76, corresponding to modal displacements twice the plate thickness. This has been evidenced by many spikes in the power spectral density of displacement time-series and an erratic pattern that modal energy components cut through an energy sphere.

Reconstruction of one-dimensional inhomogeneities in elastic modulus and density using acoustic dimensional resonances

A method is described for quantitatively reconstructing a spatial inhomogeneity along a one-dimensional structure from measurements of its resonant frequencies (fundamental and higher modes). A relationship between the shifts in the resonant frequencies of the structure due to a material inhomogeneity (computed relative to the frequencies of the homogeneous state) and the coefficients in a Fourier expansion of the inhomogeneity, which holds to the first order in the material perturbation, is derived. If a number of successive resonant frequencies are excited and measured, the unknown inhomogeneity may then be reconstructed by Fourier summation. The material inhomogeneity recovered by this technique is the sum of the fractional changes in elastic modulus and density. For simplicity, the analysis is carried out in one dimension in the absence of damping. Compared to pulse-echo methods, the advantages of the dimensional resonance method are that it can image slowly varying impedance variations, can utilize narrow bandwidth detection, has its signal enhanced by the resonant Q, and generally utilizes lower frequencies where problems of attenuation and scattering are less serious.

Conservation laws for dissipative systems possessing classical normal modes

Linear, viscously damped dynamical systems whose matrix coefficients satisfy a certain commutativity condition are known to exhibit the same normal modes as the ones associated with the same system in the absence of damping. Such dissipative systems are said to possess classical normal modes. In the present study, it is shown that the original equation of motion in the displacemnt vector which exhibits velocity coupling can be transformed into an equation where the velocity term is no longer present, but whose matrix coefficients are time-dependent. A commutativity condition further reduces the resulting equation into one with constant coefficients. Global and modal energy conservation laws are constructed for this equation governing the undamped position vector, and the results are expressed in terms of the original damped physical vector. As a by-product of the analysis, independent first integrals equal in number to the degrees of freedom of the system are obtained.

A perturbation technique for gyroscopic systems with small internal and external damping

The response of linear damped gyroscopic systems can be obtained by means of techniques of linear systems theory, which involves the computation of the transition matrix. The response is in terms of complex quantities, which is likely to cause computational difficulties as the order of the system increases. In the absence of damping, it is possible to derive the response of a linear gyroscopic system with relative ease by working with real quantities alone. When damping is small, one can use a perturbation approach to produce the response by regarding the undamped gyroscopic system as the unperturbed system. In a previous paper, a perturbation analysis was used to derive the response of a gyroscopic system with small internal damping. This paper extends the approach to the case of external damping, which is characterized not only by symmetric coefficients multiplying velocities but also by skew symmetric coefficients multiplying displacements, where the latter terms are known as circulatory. A numerical example is presented.

Gravitational radiation from a solid-crust neutron star

The measured period derivative1–3 for the millisecond pulsar PSR1937 + 214 is Ṗ ≃ 1.05×10−19, whereas the present upper limit5 for PSR1953 + 296 is Ṗ<4.2×10−17. For electromagnetic braking, the combination of rapid rotation with such small period derivatives requires a small (∼108 G) magnetic field. A natural association between rapid rotation and small magnetic field is provided by accretion spin-up scenarios7–9. Here we address the analogous problem for gravitational radiation. Balancing gravitational radiation power against the rate of rotational energy loss, we find that if the effective triaxiality eeff exceeds emax∼10−9, gravitational radiation alone would spin a millisecond pulsar down at a faster rate than the observed Ṗ. We show that eeff<<emax, so that gravitational radiation is not likely to play a major role in the evolution of millisecond pulsars even in the absence of damping. Damping times for precession of the solid crust are estimated to be <2×l03 yr.

Instability and wave over-reflection in stably stratified shear flow

We reexamine the related problems of instability of parallel shear flows and over-reflection of internal waves at a critical level, concentrating on the stratified case. Our primary aim is to delineate the specific aspects of a flow that permit overreflection and instability. A related and partly realized aim is to develop a mechanistic ‘picture’ of how over-reflection and instability work. In the course of this study we have also uncovered some new results concerning the instability of stratified shear flows – showing how regions of enhanced static stability and enhanced damping can destabilize otherwise stable flows.For the scattering of steady plane waves, we show that, of the conditions found by Lindzen & Tung (1978), in the unstratified case, only the existence of wave-propagation regions above and below the critical level is always necessary for over-reflection (at least in the absence of damping), although a trapping region around the critical level and a reflecting surface bounding the upper wave region may play crucial roles in some cases. Our results suggest that the role of the upper wave region may be to allow a wave flux through the critical level. Moreover, we show numerically that the effect of an upper wave region can be mimicked by a region of localized damping which leads to over-reflection as well.We also consider an initial-value problem, using numerical methods. When a wave is incident on the incident level, the reflection and transmission coefficients grow smoothly to their final values. The rate of growth depends on the flow parameters, but there is some evidence to suggest there is a characteristic timescale involved that depends only on the shear (and not on wave travel time). This fits a mechanistic picture of over-reflection and instability that we describe, in which the essential part is a kinematic interaction between wave and mean flow at the critical level, depending only on shear.

A unified theory of a class of mode conversion problems

Many radio-frequency heating methods involve conversion of an incoming wave to another mode which only propagates within the plasma and is ultimately damped. A method previously used to consider electron cyclotron heating by the O mode is extended to other mode conversion phenomena.From the properties of the dispersion relation in the neighborhood of the mode conversion point, differential equations for the mode amplitudes are constructed, in a well-defined way, which give energy conservation in the absence of damping. An analytic solution gives the transmission and conversion coefficients in terms of parameters defining the local behavior of the dispersion relation. The technique is applied to a number of problems, showing that they can all be solved by elementary algebraic manipulations of the local dispersion relation. The phenomenon of electron cyclotron emission also falls rather naturally into this mode conversion picture. It is also suggested that the method of dealing with cyclotron emission may have some relevance to the more general problem of energy transport across a magnetic field. The technique, therefore, brings together into a unified theory a number of problems which have been treated by diverse, and usually more complicated, mathematical techniques in the existing literature.

Barotropic Wave Propagation and Instability, and Atmospheric Teleconnection Patterns

A global barotropic model, linearized about the 300 mb climatological mean January flow, is perturbed by applying a series of localized forcings distributed throughout the tropics and subtropics. Structures which resemble the observed “Pacific/North American” and “East Atlantic” teleconnection patterns noted by Wallace and Gutzler (1981) tend to recur in the responses. Similar patterns are found to result from the dispersion of isolated initial perturbations placed at a variety of locations in the tropics and midlatitudes. It is shown that these structures are related to the most rapidly growing mode associated with barotropic instability of the zonally-varying climatological basic state. In the absence of damping, this mode has an e-folding time of about a week and a period close to 50 days. In localized regions the instantaneous growth rates can be competitive with those of baroclinic instability. These episodes of rapid local barotropic growth are interspersed with intervals in which the local perturbation relaxes as energy disperses throughout the hemisphere. Some of the less unstable modes exhibit a similar structure and time evolution, while others are spatially fixed and grow exponentially in time, without any periodic modulation. The dominant processes by which the growing disturbances extract kinetic energy from the basic state can be described in terms of two conversion terms: −u′v′∂ū/∂y and (v′2−u′2)∂ū/∂x, where u and v are the velocity components in the zonal (x) and meridional (y) directions, respectively, primes refer to the perturbations, and overbars to the basic state. In the fastest growing mode the dominant process contributing to the growth is found to be the second term, which is largest in the Pacific jet exit region where u′2 ≫ v′2. Nonlinear initial value calculations for weak damping rates exhibit strong low-frequency oscillations similar in structure and period to those of the most unstable mode. Nonlinear forced solutions show some sensitivity to the polarity of the forcing and they tend to show a larger response in the time average over the first month of the integration than in longer term means. Sensitivity to model resolution, dissipation, and choice of basic state is illustrated by means of selected experiments. On the basis of these results and recent work by Hoskins et al. (1983), it is suggested that much of the low-frequency variability of the Northern Hemisphere wintertime general circulation is associated with disturbances which derive their energy from the basic state through barotropic instability.

Dynamic Behavior of a Moving Belt Supported on Elastic Foundation

The dynamic behavior of a belt moving on an elastic foundation and supported on two pulleys at the ends is investigated. The problem is formulated to include the nonlinear terms arising from large amplitude oscillations as well as material damping and the variation in tension along the belt. The differential equation of motion is solved employing numerical techniques, and the spatial response variations with time are presented graphically for different belt velocities. These results indicate that in the absence of damping, the system is unstable for any belt velocity larger than the wave velocity in the belt material. The results are useful in investigating the stability of large continuous conveyor systems supported on elastic foundations.

Theory of Second‐Harmonic Generation by Soft Ultrasonic Shear Waves in Transforming A‐15 Compounds

AbstractA theory of second‐harmonic generation by pulsed “soft” ultrasonic shear waves in a transforming A‐15 compound, A3B, is presented which accounts for the generation observed in V3Si by Testardi. It is shown that this phenomenon, which is forbidden on the basis of static symmetry considerations, is possible under conditions of quasi‐static distortion of the crystal by the free A‐atom oscillations excited by the front of the driving‐wave pulse. Under these conditions not only should the driving‐wave‐amplitude dependence of the second‐harmonic waves be cubic at low drive levels and linear at high drive levels, as observed by Testardi, but the generated waves should also be amplitude modulated by a sine curve (in the absence of damping) of a period equal to the free‐oscillation period of the A atoms. This latter effect can thus be used to measure the soft‐phonon frequency as a function of temperature and the phonon dispersion at very low wave numbers.

Oscillating two-stream instability with pump of finite extent

In one spatial dimension, the effects of finite pump length L on the oscillating two-stream instability are considered. In the absence of damping, a purely growing instability for very small pump extent is found. Under certain conditions, the finite system exhibits faster growth rates than the corresponding infinite system.

A Theorem on the Free Vibration of Damped Systems

A damped linear system may oscillate freely (with a decaying motion) in “modes” [1]. Certain sufficient conditions have been found [2-4] under which those modes are the principal modes defined in the absence of damping. A simple condition is established in this paper that is both necessary and sufficient for this to be the case. It admits other, hitherto undisclosed, conditions under which this great simplification can validly be adopted.