Showing papers on "Parametric oscillator published in 2003"

Vibration control of active structures : an introduction

Abstract: Preface to the third edition.- Preface to the second edition.- Preface to the first edition.- 1 Introduction.- 1.1 Active versus passive.- 1.2 Vibration suppression.- 1.3 Smart materials and structures.- 1.4 Control strategies.- 1.4.1 Feedback.- 1.4.2 Feedforward.- 1.5 The various steps of the design.- 1.6 Plant description, error and control budget.- 1.7 Readership and Organization of the book.- 1.8 References.- 1.9 Problems.- 2 Some concepts in structural dynamics.- 2.1 Introduction.- 2.2 Equation of motion of a discrete system.- 2.3 Vibration modes.- 2.4 Modal decomposition.- 2.4.1 Structure without rigid body modes.- 2.4.2 Dynamic flexibility matrix.- 2.4.3 Structure with rigid body modes.- 2.4.4 Example.- 2.5 Collocated control system.- 2.5.1 Transmission zeros and constrained system.- 2.6 Continuous structures.- 2.7 Guyan reduction.- 2.8 Craig-Bampton reduction.- 2.9 References.- 2.10 Problems.- 3 Electromagnetic and piezoelectric transducers.- 3.1 Introduction.- 3.2 Voice coil transducer.- 3.2.1 Proof-mass actuator.- 3.2.2 Geophone.- 3.3 General electromechanical transducer.- 3.3.1 Constitutive equations.- 3.3.2 Self-sensing.- 3.4 Reaction wheels and gyrostabilizers.- 3.5 Smart materials.- 3.6 Piezoelectric transducer.- 3.6.1 Constitutive relations of a discrete transducer.- 3.6.2 Interpretation of k2.- 3.6.3 Admittance of the piezoelectric transducer.- 3.7 References.- 3.8 Problems.- 4 Piezoelectric beam, plate and truss.- 4.1 Piezoelectric material.- 4.1.1 Constitutive relations.- 4.1.2 Coenergy density function.- 4.2 Hamilton's principle.- 4.3 Piezoelectric beam actuator.- 4.3.1 Hamilton's principle.- 4.3.2 Piezoelectric loads.- 4.4 Laminar sensor.- 4.4.1 Current and charge amplifiers.- 4.4.2 Distributed sensor output.- 4.4.3 Charge amplifier dynamics.- 4.5 Spatial modalfilters.- 4.5.1 Modal actuator.- 4.5.2 Modal sensor.- 4.6 Active beam with collocated actuator-sensor.- 4.6.1 Frequency response function.- 4.6.2 Pole-zero pattern.- 4.6.3 Modal truncation.- 4.7 Admittance of a beam with a piezoelectric patch.- 4.8 Piezoelectric laminate.- 4.8.1 Two dimensional constitutive equations.- 4.8.2 Kirchhoff theory.- 4.8.3 Stiffness matrix of a multi-layer elastic laminate.- 4.8.4 Multi-layer laminate with a piezoelectric layer.- 4.8.5 Equivalent piezoelectric loads.- 4.8.6 Sensor output.- 4.8.7 Beam model vs. plate model.- 4.8.8 Additional remarks.- 4.9 Active truss.- 4.9.1 Open-loop transfer function.- 4.9.2 Admittance function.- 4.10 Finite element formulation.- 4.11 References.- 4.12 Problems.- 5 Passive damping with piezoelectric transducers.- 5.1 Introduction.- 5.2 Resistive shunting.- 5.3 Inductive shunting.- 5.4 Switched shunt.- 5.4.1 Equivalent damping ratio.- 5.5 References.- 5.6 Problems.- 6 Collocated versus non-collocated control.- 6.1 Introduction.- 6.2 Pole-zero flipping.- 6.3 The two-mass problem.- 6.3.1 Collocated control.- 6.3.2 Non-collocated control.- 6.4 Notch filter.- 6.5 Effect of pole-zero flipping on the Bode plots.- 6.6 Nearly collocated control system.- 6.7 Non-collocated control systems.- 6.8 The role of damping.- 6.9 References.- 6.10 Problems ..- 7 Active damping with collocated system.- 7.1 Introduction.- 7.2 Lead control.- 7.3 Direct velocity feedback (DVF).- 7.4 Positive Position Feedback (PPF).- 7.5 Integral Force Feedback(IFF).- 7.6 Duality between the Lead and the IFF controllers.- 7.6.1 Root-locus of a single mode.- 7.6.2 Open-loop poles and zeros.- 7.7 Actuator and sensor dynamics.- 7.8 Decentralized control with collocated pairs.- 7.8.1 Cross talk.- 7.8.2 Force actuator and displacement sensor.- 7.8.3 Displacement actuator and force sensor.- 7.9 References.- 7.10 Problems.- 8 Vibration isolation.- 8.1 Introduction.- 8.2 Relaxation isolator.- 8.2.1 Electromagnetic realization.- 8.3 Active isolation.- 8.3.1 Sky-hook damper.- 8.3.2 Integral Force Feedback.- 8.4 Flexible body.- 8.4.1 Free-free beam with isolator.- 8.5 Payload isolation in spacecraft.- 8.5.1 Interaction isolator/attitude control.- 8.5.2 Gough-Stewart platform.- 8.6 Six-axis isolator.- 8.6.1 Relaxation isolator.- 8.6.2 Integral Force Feedback.- 8.6.3 Spherical joints, modal spread.- 8.7 Active vs. passive.- 8.8 Car suspension.- 8.9 References.- 8.10 Problems.- 9 State space approach.- 9.1 Introduction.- 9.2 State space description.- 9.2.1 Single degree of freedom oscillator.- 9.2.2 Flexible structure.- 9.2.3 Inverted pendulum.- 9.3 System transfer function.- 9.3.1 Poles and zeros.- 9.4 Pole placement by state feedback.- 9.4.1 Example: oscillator.- 9.5 Linear Quadratic Regulator.- 9.5.1 Symmetric root locus.- 9.5.2 Inverted pendulum.- 9.6 Observer design.- 9.7 Kalman Filter.- 9.7.1 Inverted pendulum.- 9.8 Reduced order observer.- 9.8.1 Oscillator.- 9.8.2 Inverted pendulum.- 9.9 Separation principle.- 9.10 Transfer function of the compensator.- 9.10.1 The two-mass problem.- 9.11 References.- 9.12 Problems.- 10 Analysis and synthesis in the frequency domain.- 10.1 Gain and phase margins.- 10.2 Nyquist criterion.- 10.2.1 Cauchy's principle.- 10.2.2 Nyquist stability criterion.- 10.3 Nichols chart.- 10.4 Feedback specification for SISO systems.- 10.4.1 Sensitivity.- 10.4.2 Tracking error.- 10.4.3 Performance specification.- 10.4.4 Unstructured uncertainty.- 10.4.5 Robust performance and robust stability.- 10.5 Bode gain-phase relationships.- 10.6 The Bode Ideal Cutoff.- 10.7 Non-minimum phase systems.- 10.8 Usual compensators.- 10.8.1 System type.- 10.8.2 Lead compensator.- 10.8.3 PI compensator.- 10.8.4 Lag compensator.- 10.8.5 PID compensator.- 10.9 Multivariable systems.- 10.9.1 Performance specification.- 10.9.2 Small gain theorem.- 10.9.3 Stability robustness tests.- 10.9.4 Residual dynamics.- 10.10References.- 10.11Problems.- 11 Optimal control.- 11.1 Introduction.- 11.2 Quadratic integral.- 11.3 Deterministic LQR.- 11.4 Stochastic response to a white noise.- 11.4.1 Remark.- 11.5 Stochastic LQR.- 11.6 Asymptotic behavior of the closed-loop.- 11.7 Prescribed degree of stability.- 11.8 Gain and phase margins of the LQR.- 11.9 Full state observer.- 11.9.1 Covariance of the reconstruction error.- 11.10Kalman-Bucy Filter (KBF).- 11.11Linear Quadratic Gaussian (LQG).- 11.12Duality.- 11.13Spillover.- 11.13.1Spillover reduction.- 11.14Loop Transfer Recovery (LTR).- 11.15Integral control with state feedback.- 11.16Frequency shaping.- 11.16.1Frequency-shaped cost functionals.- 11.16.2Noise model ..- 11.17References.- 11.18Problems.- 12 Controllability and Observability.- 12.1 Introduction.- 12.1.1 Definitions.- 12.2 Controllability and observability matrices.- 12.3 Examples.- 12.3.1 Cart with two inverted pendulums.- 12.3.2 Double inverted pendulum.- 12.3.3 Two d.o.f. oscillator.- 12.4 State transformation.- 12.4.1 Control canonical form.- 12.4.2 Left and right eigenvectors.- 12.4.3 Diagonal form.- 12.5 PBH test.- 12.6 Residues.- 12.7 Example.- 12.8 Sensitivity.- 12.9 Controllability and observability Gramians.- 12.10Internally balanced coordinates.- 12.11Model reduction.- 12.11.1Transfer equivalent realization.- 12.11.2Internally balanced realization.- 12.11.3Example.- 12.12References.- 12.13Problems.- 13 Stability.- 13.1 Introduction.- 13.1.1 Phase portrait.- 13.2 Linear systems.- 13.2.1 Routh-Hurwitz criterion.- 13.3 Lyapunov's direct method.- 13.3.1 Introductory example.- 13.3.2 Stability theorem.- 13.3.3 Asymptotic stability theorem.- 13.3.4 Lasalle's theorem.- 13.3.5 Geometric interpretation.- 13.3.6 Instability theorem.- 13.4 Lyapunov functions for linear systems.- 13.5 Lyapunov's indirect method ..- 13.6 An application to controller design.- 13.7 Energy absorbing controls.- 13.8 References.- 13.9 Problems.- 14 Applications.- 14.1 Digital implementation.- 14.1.1 Sampling, aliasing and prefiltering.- 14.1.2 Zero-order hold, computational delay.- 14.1.3 Quantization.- 14.1.4 Discretization of a continuous controller.- 14.2 Active damping of a truss structure.- 14.2.1 Actuator placement.- 14.2.2 Implementation, experimental results.- 14.3 Active damping generic interface.- 14.3.1 Active damping.- 14.3.2 Experiment.- 14.3.3 Pointing and position control.- 14.4 Active damping of a plate.- 14.4.1 Control design.- 14.5 Active damping of a stiff beam.- 14.5.1 System design.- 14.6 The HAC/LAC strategy.- 14.6.1 Wide-band position control.- 14.6.2 Compensator design.- 14.6.3 Results.- 14.7 Vibroacoustics: Volume displacement sensors.- 14.7.1 QWSIS sensor.- 14.7.2 Discrete array sensor.- 14.7.3 Spatial aliasing.- 14.7.4 Distributed sensor.- 14.8 References.- 14.9 Problems.- 5 Tendon Control of Cable Structures.- 15.1 Introduction.- 15.2 Tendon control of strings and cables.- 15.3 Active damping strategy.- 15.4 Basic Experiment.- 15.5 Linear theory of decentralized active damping.- 15.6 Guyed truss experiment.- 15.7 Micro Precision Interferometer testbed.- 15.8 Free floating truss experiment.- 15.9 Application to cable-stayed bridges.- 15.10Laboratory experiment.- 15.11Control of parametric resonance.- 15.12Large scale experiment.- 15.13 References.- 16 Active Control of Large Telescopes.- 16.1 Introduction.- 16.2 Adaptive optics.- 16.3 Active optics.- 16.3.1 Monolithic primary mirror.- 16.3.2 Segmented primary mirror.- 16.4 SVD controller.- 16.4.1 Loop shaping of the SVD controller.- 16.5 Dynamics of a segmented mirror.- 16.6 Control-structure interaction.- 16.6.1 Multiplicative uncertainty.- 16.6.2 Additive uncertainty.- 16.6.3 Discussion.- 16.7 References.- 17 Semi-active control.- 17.1 Introduction.- 17.2 Magneto-rheological fluids.- 17.3 MR devices.- 17.4 Semi-active suspension.- 17.4.1 Semi-active devices.- 17.5 Narrow-band disturbance.- 17.5.1 Quarter-car semi-active suspension.- 17.6 References.- 17.7 Problems.- Bibliography.- Index.

Parametric resonance in quantum field theory.

Abstract: We present the first study of parametric resonance in quantum field theory from a complete next-to-leading order calculation in a 1/N expansion of the two-particle irreducible effective action, which includes scattering and memory effects. We present a complete numerical solution for an O(N)-symmetric scalar theory and provide an approximate analytic description of the nonlinear dynamics in the entire amplification range. We find that the classical resonant amplification at early times is followed by a collective amplification regime with explosive particle production in a broad momentum range, which is not accessible in a leading-order calculation.

Non-linear resonance in nearly geodesic motion in low-mass X-ray binaries

Abstract: We have explored the ideas that parametric resonance affects nearly geodesic motion around a black hole or a neutron star, and that it may be relevant to the high frequency (twin) quasi-periodic oscillations occurring in some low-mass X-ray binaries. We have assumed the particles or fluid elements of an accretion disc to be subject to an isotropic perturbation of a hypothetical but rather general form. We find that the parametric resonance is indeed excited close to the radius where epicyclic frequencies of radial and meridional oscillations are in a 2:3 ratio. The location and frequencies of the highest amplitude excitation vary with the strength of the perturbation. These results agree with actual frequency ratios of twin kHz QPOs that have been reported in some black hole candidates, and they may be consistent also with correlation of the twin peaks in Sco X-1.

Non-Linear Resonance in Nearly Geodesic Motion in Low-Mass X-Ray Binaries

Abstract: We have explored the ideas that parametric resonance affects nearly geodesic motion around a black hole or a neutron star, and that it may be relevant to the high-frequency (twin) quasi-periodic oscillations that occur in some low-mass X-ray binaries. We have assumed the particles or uid elements of an accretion disc to be subject to an isotropic perturbation having a hypothetical but rather general form. We nd that the parametric resonance is indeed excited close to the radius where epicyclic frequencies of the radial and meridional oscillations are in a 2 : 3 ratio. The location and frequencies of the highest amplitude excitation vary with the strength of the perturbation. These results agree with actual frequency ratios of twin kHz QPOs that have been reported in some black hole candidates, and they may be consistent also with correlation of the twin peaks in Sco X-1.

Record performance of parametric amplifier constructed with highly nonlinear fibre

Abstract: Record performance from a two-pump parametric amplifier constructed with highly nonlinear fibre is reported. Equalised continuous-wave parametric gain of 40 dB was achieved over 33.8 nm without any gain-flattening elements. Maximal difference between signal and idler powers was measured to be 1 dB within the equalised gain band.

A method to derive an equation for the oscillation frequency of a ring oscillator

Abstract: A new method for deriving an equation for the oscillation frequency of a ring oscillator is proposed. The method is general enough to be used for a variety of types of delay stages. Furthermore, it provides a framework to include various parasitic and secondary effects. The method is used to derive an equation for a common ring oscillator topology. The validity of the method and the resulting equation have been verified through simulation. The oscillation frequencies predicted by the proposed method are more accurate than existing equations and account for more secondary effects.

Passive harmonic switch mixer

Abstract: A passive harmonic switch mixer is shown that is immune to self mixing of the local oscillator greatly reducing leakage noise, pulling noise, and flicker noise when used in a direct conversion receiver or direct conversion transmitter circuit. The passive harmonic switch mixermixes an input signal received on an input port with an in-phase oscillator signal and a quadrature-phase oscillator signal and outputs an output signal on an output port. Because the quadrature-phase oscillator signal is the in-phase oscillator signal phase shifted by 90 °, the passive harmonic switch mixer operates with a local oscillator running at half the frequency of the carrier frequency of an RF signal. Additionally, because the passive harmonic switch mixer has no active components, the DC current passing through each switch device is reduced and the associated flicker noise of the mixer is also greatly reduced.

Broadband and flat parametric amplifiers with a multisection dispersion-tailored nonlinear fiber arrangement

Abstract: We describe a simple scheme to allow for the achievement of flat gain over ultrabroad bands with a single-pump fiber-optic parametric amplifier operating in the zero-dispersion wavelength region. The proposed method, based on a multisection dispersion-tailored in-line nonlinear fiber arrangement, is demonstrated by both modulational instability theory and numerical simulations of the nonlinear Schrodinger equation. The results show that the design can be adjusted to generate gain bands that exceed either 100 nm with a ripple of less than 0.2 dB and for a pump power of only 500 mW, or even 200 nm when a pump power of 5 W is used. In addition, the robustness of this gain-flattening technique has been numerically checked against random fluctuations of the zero-dispersion wavelength in each of the fiber sections.

Mechanical domain coupled mode parametric resonance and amplification in a torsional mode micro electro mechanical oscillator

Abstract: In this paper, we experimentally demonstrate non-degenerate parametric resonance in a torsional micro electro mechanical (MEM) oscillator with two interacting mechanical modes of oscillation. The parametric oscillation results from a displacement-dependent electrostatic force generated in both the modes of oscillation. There is a decoupling of the input and the output frequencies in this mode of operation where the system responds at the primary natural frequency when driven at the sum of the first two mechanical modes. This can be implemented in many commonly used MEM oscillator configurations including the cantilever beam geometry. In this mode of operation, single oscillator non-degenerate parametric amplifiers as well as resonant mode sensors based on the frequency selectivity of parametric resonance can be implemented.

Tuning the dynamic behavior of parametric resonance in a micromechanical oscillator

Abstract: We describe how to significantly change the dynamic behavior of parametric resonance in a micromechanical oscillator. By varying the voltage amplitude of applied electrical signal, the frequency response of the first order parametric resonance changes dramatically. We attribute this variation to the tuning of effective cubic stiffness of the oscillator, which is a contribution of both structural and electrical cubic stiffness. This phenomenon is well explained by the first-order perturbation analysis of nonlinear Mathieu equation.

An Nth-harmonic oscillator using an N-push coupled oscillator array with voltage-clamping circuits

Abstract: The push-push oscillator is commonly used for implementing a second-harmonic oscillator. By combining two out-of-phase oscillators, their fundamental frequency components are canceled and the second-harmonic components are enhanced. This structure can be extended to triple-push, quadruple-push and hence N-push harmonic oscillators. From the oscillator injection-locking phenomenon, the relative phase between coupled oscillators can be controlled by the oscillator free-running frequency. As the output phase-shifted version signals are properly shaped and combined, the desired harmonic components are constructively added and lower-order harmonic components are canceled. This structure can be viewed as the general case of push-push oscillators. Since the output power is combined in a passive circuit, it does not suffer from the power limit of the output device in the cascade structure. The desired harmonic component can be selected by tuning the relative phase of the coupled oscillators and the conductive angle of the voltage-clamping circuit. Second-harmonic, third-harmonic and fourth-harmonic oscillators are designed and verified experimentally.

Continuous-wave operation of a single-frequency optical parametric oscillator at 4–5 μm based on periodically poled LiNbO 3

Abstract: We present a cw, Nd:YAG-pumped singly resonant single-frequency narrow-linewidth high-power optical parametric oscillator with idler tuning from 3.7 to 4.7 μm . In this spectral range the absorption of the idler wave in the LiNbO3 crystal is significant, causing the oscillation threshold to increase with a subsequent decrease in output power from 1.2 W at 3.9 μm to 120 mW at 4.7 μm . The optical parametric oscillator’s cavity was stabilized and mode-hop tuned with a rotatable solid etalon but with a subsequent reduction in idler power of as much as 50%. We demonstrated the usefulness for spectroscopy by recording the photoacoustic spectrum of a strong CO2 absorption, using a 24-GHz continuous idler scan.

Understanding mechanical domain parametric resonance in microcantilevers

Abstract: In this paper we present a mathematical model for the dynamics of an electrostatically actuated micro-cantilever. For the common case of cantilevers excited by a periodic voltage, we show that the underlying linearized dynamics are those of a periodic system described by a Mathieu equation. We present experimental results that confirm the validity of the model, and in particular illustrate that parametric resonance phenomena occur in capacitively actuated micro-cantilevers. The combined parametric/harmonic mode of operation is investigated as well and experimental data are provided.

Synchronously pumped CdSe optical parametric oscillator in the 9–10 μm region

Abstract: Continuous mode-locked operation of a singly resonant, synchronously pumped optical parametric oscillator (SPOPO) based on CdSe has produced idler output tuned over the range of 9.1–9.7 μm , the longest wavelength generated so far to our knowledge from a SPOPO. Average idler powers as high as ~70 mW are generated in the crystal. Tandem pumping with a diffraction-grating-tuned parametric oscillator in periodically poled lithium niobate provides a convenient and agile means of tuning the noncritically phase-matched CdSe device. The absence of any detrimental thermal effects in the CdSe crystal suggests that significant further power scaling should be possible, with idler tuning ranges extendable to cover 8–12μm .

Parametric Dwarf Spheroidal Tidal Interaction

Abstract: The time-dependent tidal interaction of the Local Group dwarf spheroidal (dSph) galaxies with the Milky Way (MW) can fundamentally affect their dynamical properties. The model developed here extends earlier numerical descriptions of dSph-MW tidal interactions. We explore the dynamical evolution of dSph systems in circular or elliptical MW orbits in the framework of a parametric oscillator. An analytic model is developed and compared with more general numerical solutions and N-body simulation experiments.

Polariton parametric amplifier pump dynamics in the coherent regime.

Abstract: We studied the pump coherent dynamics in a II-VI microcavity parametric amplifier, using angle-resolved four-wave mixing. The polariton parametric amplification is found to result in a strong quenching and saturation of the pump coherence lifetime above the threshold. For the polariton scattering processes that remain below the amplification threshold, we find an angle-dependent collision broadening associated with the efficiency of the polariton scattering towards the excitonic reservoir.

A 2GHz voltage tunable FBAR oscillator

Abstract: This paper describes the design and measured performance of a low-noise varactor tuned oscillator based on a film bulk acoustic resonator (FBAR) at 2 GHz. Using varactor tuning, this oscillator demonstrated a 2.5 MHz frequency tuning range at 1985 MHz with a phase noise of -112 dBc/Hz at 10 kHz from the carrier. This represents the first example of a low noise Si-bipolar FBAR tunable oscillator.

A MOS capacitor-based discrete-time parametric amplifier with 1.2 V output swing and 3/spl mu/W power dissipation

Abstract: Experimental results are reported for an amplifier based on a little-known effect in MOS capacitors: a voltage sampled on the gate rises when the channel charge is pulled out. It is demonstrated that this phenomenon can be used to provide micro-power, low-gain, and low-noise signal amplification.

Correspondence between theq-deformed harmonic oscillator and finite range potentials

Abstract: The q-deformed harmonic oscillator is revisited in connection with the spectrum of finite range potentials. It is found that the finite series expansion of the canonical variables, the q-deformed phase space coordinate and momentum variables, generates a local momentum dependent interaction. It turns out that the resulting spectrum exhibits features of the spectrum of a finite range potential, added to the low-lying harmonic oscillator behavior. Thus, the otherwise unbounded spectrum of the harmonic oscillator behaves, in the q-deformed version, as the spectrum of a finite range potential subject to specific boundary conditions.

Design and characterization of an all-cryogenic low phase-noise sapphire K-band oscillator for satellite communication

Abstract: An all-cryogenic oscillator consisting of a frequency-tunable sapphire resonator, a high-temperature superconducting filter and a pseudomorphic high electron-mobility transistor amplifier was designed for the K-band frequency range. The high quality factor of the resonator above 1 000 000 and the low amplifier phase noise of approximately -133 dBc/Hz at a frequency offset of 1 kHz from the carrier, gave oscillator phase-noise values superior to quartz-stabilized oscillators at the same carrier frequency for offset frequencies higher than 100 Hz. In addition to low phase noise, the oscillator possesses mechanical and electrical frequency tunability. We have implemented a two-step electrical tuning arrangement consisting of a varactor phase shifter integrated within the amplifier circuit (fine tuning by 5 kHz) and a dielectric plunger moved by a piezomechanical transducer inside the resonator housing (coarse tuning by 50 kHz). This tuning range is sufficient for phase locking and for electronic compensation of temperature drifts occurring during operation of the device employing a miniaturized closed-cycle Stirling-type cryocooler.

Generation of vortex beams by an image-rotating optical parametric oscillator

Abstract: We generate optical vortex beams in a nanosecond optical parametric oscillator based on an image-rotating resonator. This efficient new method of vortex generation should be adaptable to pulsed or continuous lasers.

Transformer coupled quadrature tuned oscillator

Abstract: First and second oscillator cells include tank circuits that may be tuned to a desired resonant frequency using varactors as variable reactance devices in a phase-shifting network. First and second transformers are connected to the oscillator cells to control the varactors.

Stability analysis of conducting jets under ac radial electric fields for arbitrary viscosity

Abstract: A temporal linear modal stability analysis is presented for conducting viscous liquid jets flowing with nonzero velocity relative to an ambient gas and subjected to an ac radial electric field. Parametric resonance between natural dc frequencies and the frequency (or multiple) of the imposed ac field eventually leads to destabilization of the jet for perturbations with wave numbers in the stable domain. In this way, it is possible to obtain drops of smaller size. The main result is the extension of the stability analysis to liquids of arbitrary viscosity using a dynamical approach, instead of previous variational models valid for slightly viscous liquids. The effect of the outer gas in relative motion is taken into account in the framework of currently available semiempirical theories. A brief discussion of the dispersion relation for dc fields is included as the natural starting point for the discussion of the ac case. Use of the 1-D averaged model for axisymmetric perturbations, an alternative to the 3-...