Degenerate Vibration Modes Research Articles

Double-Gate Vectorial Frequency Control in Piezoresistive Nanowire Electromechanical Devices

Vacuum-gap and orthogonally aligned double-gate geometry is employed in a nanowire-based electromechanical resonator device to independently control two nearly degenerate orthogonal vibration modes. In the device, piezoresistance is the dominant mechanism of motion-induced conductance variation, which provides an efficient scheme for the self-transduction of vibrational motion into an electric signal. Two simultaneously applied gate voltages induce an opposite combined effect on the frequency shift between two vibration modes, which is well explained by model calculations. Using the opposite double-gate effect, we demonstrate vectorial control of two mode frequencies, which is not possible with single-gate geometry.

Eigenmode operation of piezoelectric resonant gyroscopes

The theory of eigenmode operation of Coriolis vibratory gyroscopes and its implementation on a thin-film piezoelectric gyroscope is presented. It is shown analytically that the modal alignment of resonant gyroscopes can be achieved by applying a rotation transformation to the actuation and sensing directions regardless of the transduction mechanism. This technique is especially suitable for mode matching of piezoelectric gyroscopes, obviating the need for narrow capacitive gaps or DC polarization voltages. It can also be applied for mode matching of devices that require sophisticated electrode arrangements for modal alignment, such as electrostatic pitch and roll gyroscopes with slanted electrodes utilized for out-of-plane quadrature cancellation. Gyroscopic operation of a 3.15 MHz AlN-on-Si annulus resonator that utilizes a pair of high-Q degenerate in-plane vibration modes is demonstrated. Modal alignment of the piezoelectric gyroscope is accomplished through virtual alignment of the excitation and readout electrodes to the natural direction of vibration mode shapes in the presence of fabrication nonidealities. Controlled displacement feedback of the gyroscope drive signal is implemented to achieve frequency matching of the two gyroscopic modes. The piezoelectric gyroscope shows a mode-matched operation bandwidth of ~250 Hz, which is one of the largest open-loop bandwidth values reported for a mode-matched MEMS gyroscope, a small motional resistance of ~1300 Ω owing to efficient piezoelectric transduction, and a scale factor of 1.57 nA/°/s for operation at atmospheric pressure, which greatly relaxes packaging requirements. Eigenmode operation results in an ~35 dB reduction in the quadrature error at the resonance frequency. The measured angle random walk of the device is 0.86°/√h with a bias instability of 125°/h limited by the excess noise of the discrete electronics.

On the phenomenon of natural-frequency splitting of a guitar string caused by a magnetic field

A magnet can affect the vibration of the metal strings in a musical instrument. We show that the magnetic field from a magnetic pickup can cause a frequency splitting of a metal guitar string’s normally degenerate transverse vibration modes, leading to a beat note in the resulting sound. The amount of frequency splitting induced by the magnet depends on the product of the induced magnetization in the ferromagnetic string, and the magnet’s spatial gradient at the position of the string, and is on the order of a few Hz. We apply free vibration theory to the string to obtain an eigenvalue problem, which we solve using a distributed-transfer function method. This method accurately predicts the natural frequencies of the vibrating guitar string in a non-uniform magnetic field. Videos of a high-E string vibrating with and without a nearby magnet will be shown.

Natural-frequency splitting of a guitar string caused by a non-uniform magnetic field.

A magnet can affect the vibration of the metal strings in a musical instrument. Here it is shown that a magnetic field can cause splitting of a metal guitar string's normally degenerate transverse vibration modes, leading to a beat note in the resulting sound. A distributed-spring model accurately predicts the natural frequencies of a vibrating guitar string in the non-uniform magnetic field.

Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope.

Parametric amplification, resulting from intentionally varying a parameter in a resonator at twice its resonant frequency, has been successfully employed to increase the sensitivity of many micro- and nano-scale sensors. Here, we introduce the concept of self-induced parametric amplification, which arises naturally from nonlinear elastic coupling between the degenerate vibration modes in a micromechanical disk-resonator, and is not externally applied. The device functions as a gyroscope wherein angular rotation is detected from Coriolis coupling of elastic vibration energy from a driven vibration mode into a second degenerate sensing mode. While nonlinear elasticity in silicon resonators is extremely weak, in this high quality-factor device, ppm-level nonlinear elastic effects result in an order-of-magnitude increase in the observed sensitivity to Coriolis force relative to linear theory. Perfect degeneracy of the primary and secondary vibration modes is achieved through electrostatic frequency tuning, which also enables the phase and frequency of the parametric coupling to be varied, and we show that the resulting phase and frequency dependence of the amplification follow the theory of parametric resonance. We expect that this phenomenon will be useful for both fundamental studies of dynamic systems with low dissipation and for increasing signal-to-noise ratio in practical applications such as gyroscopes.

Correlation of structural distortion with magnetic properties in electron-doped Ca0.9R0.1MnO3 perovskites (R=rare-earth)

A series of electron-doped orthorhombic-perovskite manganites Ca0.9R0.1MnO3 (R=La, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, and Yb) are synthesized for a systematic study of their crystal structure and magnetic properties. The structural distortions, in terms of the average Mn–O–Mn bond angle θMn–O–Mn and Mn–O bond length dMn–O, are characterized as a function of the A-site ionic size. Two degenerate vibration modes Q2 and Q3 are used for describing the bond length splitting and the evolution of the octahedral-site distortion. With R3+ doping, the magnetization increases markedly at low temperatures, which can be attributed to the formation of ferromagnetic clusters in the antiferromagnetic matrix. Both low temperature magnetization and paramagnetic susceptibility vary with the radius of R3+ ion and enhanced ferromagnetic domain is found in Ca0.9Ho0.1MnO3. The Néel temperature TN, varying from 100 to 116 K, is strongly dependent on the crystal structural distortions and can be well described as functions of three structural parameters θMn–O–Mn, dMn–O, and A-site cation size variance σ2. The best size matching between Dy3+ and Ca2+ leads to the highest TN in Ca0.9Dy0.1MnO3.

A cylindrical traveling wave ultrasonic motor using longitudinal and bending composite transducer

A cylindrical type traveling wave ultrasonic motor using longitudinal and bening composite transducer was proposed in this paper. A composite transducer is attached to the cylinder on its outer surface to excite two degenerate flexural vibration modes spatially and temporally orthogonal to each other in the cylinder. In this new design, a single transducer can excite a flexural traveling wave in the cylinder. Two cone type rotors are pressed in contact to the borders of the inner surface of teeth by means of a spring and nut system. The working principle of the motor was analyzed. The resonance frequencies of the two vibration modes of stator were tuned to be close to one another using finite element method. A prototype motor was fabricated and tested. Typical output of the prototype is no-load speed of 281 rpm and maximum torque of 1.2 N m at an exciting voltage of 200 V rms.

Actuating Mechanism and Design of a Cylindrical Traveling Wave Ultrasonic Motor Using Cantilever Type Composite Transducer

BackgroundUltrasonic motors (USM) are based on the concept of driving the rotor by a mechanical vibration excited on the stator via piezoelectric effect. USM exhibit merits such as simple structure, quick response, quiet operation, self-locking when power off, nonelectromagnetic radiation and higher position accuracy.Principal FindingsA cylindrical type traveling wave ultrasonic motor using cantilever type composite transducer was proposed in this paper. There are two cantilevers on the outside surface of cylinder, four longitudinal PZT ceramics are set between the cantilevers, and four bending PZT ceramics are set on each outside surface of cantilevers. Two degenerate flexural vibration modes spatially and temporally orthogonal to each other in the cylinder are excited by the composite transducer. In this new design, a single transducer can excite a flexural traveling wave in the cylinder. Thus, elliptical motions are achieved on the teeth. The actuating mechanism of proposed motor was analyzed. The stator was designed with FEM. The two vibration modes of stator were degenerated. Transient analysis was developed to gain the vibration characteristic of stator, and results indicate the motion trajectories of nodes on the teeth are nearly ellipses.ConclusionsThe study results verify the feasibility of the proposed design. The wave excited in the cylinder isn't an ideal traveling wave, and the vibration amplitudes are inconsistent. The distortion of traveling wave is generated by the deformation of bending vibration mode of cylinder, which is caused by the coupling effect between the cylinder and transducer. Analysis results also prove that the objective motions of nodes on the teeth are three-dimensional vibrations. But, the vibration in axial direction is minute compared with the vibrations in circumferential and radial direction. The results of this paper can guide the development of this new type of motor.

Design and Analysis of a Cylindrical Traveling Wave Ultrasonic Motor Using Orthogonal Longitudinal Composite Transducer

A cylindrical type traveling wave ultrasonic motor using orthogonal longitudinal composite transducer is proposed. The composite transducer contains two longitudinal transducers, which are located orthogonal to each other. Each longitudinal transducer includes one exponential horn located on leading end, and the horns insect with each other at tip ends. The composite transducer is attached to the cylinder on its outer surface to excite two degenerate flexural vibration modes spatially and temporally orthogonal to each other in the cylinder. Thus, a flexural traveling wave in the cylinder could be excited. The working principle of proposed motor is analyzed. The stator is designed with FEM. The vibration modals of stator are degenerated. Transient analysis is developed to gain the vibration characteristics of stator, and results indicate that the motion trajectories of nodes on the teeth are ellipses. The results of this paper verify the theoretical feasibility and provide instructions for the development of proposed motor.

Design and Analysis of a Cylindrical Traveling Wave Ultrasonic Motor Using Composite Transducer

A cylindrical type traveling wave ultrasonic motor using composite transducer is proposed in this paper. A composite transducer is attached to the cylinder on its outer surface to excite two degenerate flexural vibration modes spatially and temporally orthogonal to each other in the cylinder. In this new design, a single transducer can excite the flexural traveling wave in the cylinder. Thus, an elliptical motion is achieved at the particle on the tooth. And the driving force is the frictional force between rotor and teeth. The working principle of proposed motor is analyzed. The cylinder and transducer are designed with FEM. The sensitive parameters of resonance frequencies of transducer and cylinder are gained with modal analysis. The resonance frequencies of two vibration modals of stator are degenerated, and the motion trajectories of nodes on the teeth are analyzed. The result of transient analysis shows that the trajectory of node on the tooth is nearly an ellipse. The results of this paper could guide the development of this new type of motor.

Intrinsic structural distortion in orthorhombic perovskite oxides

Octahedra in the orthorhombic perovskite structure are not rigid. The bond-length splitting at an octahedral site can be described by two degenerate vibration modes, the orthorhombic ${Q}_{2}$ and the tetragonal ${Q}_{3}$. A polar plot of $r=\ensuremath{\rho}={({Q}_{2}^{2}+{Q}_{2}^{2})}^{1/2}$ and $\ensuremath{\phi}={\text{tan}}^{\ensuremath{-}1}({Q}_{3}/{Q}_{2})$ has been made to map out the site distortion as a function of the rare-earth ionic size for three families, $R{\text{FeO}}_{3}$, $R{\text{VO}}_{3}$, and $R{\text{MnO}}_{3}$. The octahedral-site distortion deduced from $R{\text{FeO}}_{3}$ is found to be intrinsic to orthorhombic perovskites and to strongly bias orbital ordering and spin ordering in compounds that have the same structure.

Traveling Wave Ultrasonic Motor Using a Flexural Composite Transducer

This paper presents a new design for a traveling wave ultrasonic motor with a bolt-clamped Langevin transducer. A flexural composite transducer was attached to a ring around its circumference to excite two degenerate flexural vibration modes spatially and temporally orthogonal to each other in the ring for flexural traveling wave excitation. The prototype motor, composed of a 30-mm-diameter composite transducer attached along the outer edge of a 130-mm-diameter ring, was designed and tested. A maximum torque of 1.28 N·m with a maximum efficiency of 3.8% was achieved when driven at an applied voltage of approximately 200 Vrms and a preload of 172 N.

Traveling wave excitation in a flexural vibration ring by using a torsional-flexural composite transducer.

This paper presents a new mechanism for traveling wave excitation of the flexural vibration in a ring. A torsional-flexural composite transducer was used to excite two degenerate flexural vibration modes spatially and temporally orthogonal to each other in the ring. The proto-type, composed of a 32-mm diameter composite transducer attached along the outer edge of an 180-mm diameter aluminum annular ring, was designed, constructed, and tested. The ring was designed via finite element analysis to operate using a (0.8) transverse flexural traveling wave, and the motion was confirmed experimentally.

Effect of Pressing Force Applied to a Rotor on Disk-Type Ultrasonic Motor Driven by Self-Oscillation

In this paper the relationship between the pressing force applied to a rotor and the rotation characteristic of an ultrasonic motor driven by self-oscillation are discussed. The motor used here is an in-phase drive-type ultrasonic motor using two degenerate bending vibration modes of a disk. The picking-up electrical signal caused by self-oscillation is positively fed back into the piezoelectric ceramics for driving through an operational amplifier and a step-up transformer. The pressing force applied to a rotor was measured using a force gauge coupled to the shaft of the ultrasonic motor. As a result, it was confirmed that the selection of the picking-up position for the feedback signal is important for a stable starting and running of the disk-type ultrasonic motor driven by self-oscillation.

Rod-Type Ultrasonic Motor Using Two Degenerate Second Flexural Vibration Modes and Characteristic Consideration Using Its Equivalent Circuit Expression

This paper deals with a simplified equivalent circuit of an ultrasonic motor and its application to the estimation of motor characteristics. Investigation of this equivalent circuit is conducted from the viewpoint of systematization of research on ultrasonic motors. The equivalent circuit must follow the principle of motor operation and must be expressed simply. We have considered the characteristics of an ultrasonic motor using its equivalent circuit expression. We propose simplified equivalent circuits of a rod-type ultrasonic motor using the two degenerate second flexural vibration modes; moreover, we show that the maximum torque, the suitable load, and power transmission can be easily estimated. As a result, we found that the estimated maximum torque was sufficiently close to the measured one and that the power transmission could be estimated by correction of the measured value of internal resistance.

An ultrasonic micromotor using a bending cylindrical transducer based on PZT thin film

Ultrasonic motors have a good high-torque performance. In particular, it seems that cylinder-type ultrasonic motors are superior to disk- or ring-type ones in the sense of high-torque application. For miniaturization of the motor size, a simple structure of an ultrasonic transducer becomes effective. This paper presents an ultrasonic micromotor using a bending cylindrical transducer based on a lead zirconate titanate (PZT) thin film on the side wall of a small tube. The transducer of the ultrasonic motor consists of a short cylindrical titanium pipe, PZT thin film and electrodes. The titanium pipe is introduced as a base material on which to deposit the PZT thin film by a hydrothermal method. The thickness of the film is about 9 μm. The electrodes are divided into four parts in a circumferential direction to excite degenerate bending vibration modes with two r.f. driving electrical sources. A rotor loaded on the transducer turns in the intended direction. The maximum revolution speed is 295 rpm.

Stepping Drive of Self-Oscillation-Type Ultrasonic Motor

A stepping drive method that does not use an extra clock signal source for a self-oscillation-type ultrasonic motor is described. The motor used here is an in-phase drive-type ultrasonic motor using two degenerate bending vibration modes of a disk. The output electrical signals caused on a self-oscillation are used as a clock signal for the stepping drive. An angular displacement at every step is realized using an open-loop control system without a rotary encoder as an angular displacement sensor. In order to eliminate the deviation of angular displacement at every step, a multitooth rotor was used. As a result, stable stepping drive of an ultrasonic motor was realized.

Transient Response Characteristics of a Same-Phase Drive-Type Ultrasonic Motor

This paper deals with a same-phase drive-type ultrasonic motor, using two degenerate bending vibration modes of a disk. Characteristics of the motor are influenced considerably by the mechanisms for supporting a stator and pressing a rotor. By investigating these points experimentally, a motor with high efficiency characteristics was realized. Moreover, the driving method for improving the transient response of the motor is described.

Same phase drive-type ultrasonic motors using two degenerate bending vibration modes of a disk

Same-phase drive-type ultrasonic motors, using two degenerate bending vibration modes of a disk, are presented. The distinctive feature of the motor is in using the standing wave modes. The motor is not driven by two input signals with different phases, but by input signals with the same phase. Therefore, only one amplifier is sufficient to drive the motor. The experimental results have proved that the motor can yield stable operational characteristics at low speed and high torque.

Characteristics and a New Control Method of a Same-Phase Drive-Type Ultrasonic Motor

This paper deals with a same-phase drive-type ultrasonic motor, using two degenerate bending vibration modes of a disk. Characteristics of the motor driven at low input power are shown. Signal waves picked up from a rotor are also shown. It is found that these signals are closely related to revolution states of the motor. Moreover, a new control method for the motor is proposed and tried.