This book equips the reader to understand every important aspect of the dynamics of rotating machines. Will the vibration be large? What influences machine stability? How can the vibration be reduced? Which sorts of rotor vibration are the worst? The book develops this understanding initially using extremely simple models for each phenomenon, in which (at most) four equations capture the behavior. More detailed models are then developed based on finite element analysis, to enable the accurate simulation of the relevant phenomena for real machines. Analysis software (in MATLAB) is associated with this book, and novices to rotordynamics can expect to make good predictions of critical speeds and rotating mode shapes within days. The book is structured more as a learning guide than as a reference tome and provides readers with more than 100 worked examples and more than 100 problems and solutions.

/pdf/dynamics-of-rotating-machines-499re2kizd.pdf

Dynamics of rotating machines

Identification of chatter free cutting conditions, the chatter stability lobes, requires a measurement of the frequency response function (FRF) of each tool mounted on the spindle. This paper presents a method of assembling known dynamics of the spindle–tool holder with an analytically modeled end mill using the receptance coupling technique. The classical receptance technique is enhanced by proposing a method of identifying the end mill–spindle/tool holder joint dynamics, which include both translational and rotational degrees of freedom. The method requires measurement of FRFs with impact tests applied on the spindle–tool holder assembly and blank calibration cylinders attached to the spindle. The spindle and tool holder characteristics are completely identified from the two experiments, and used for the mathematical prediction of FRF for end mills with arbitrary dimensions. The proposed method is experimentally proven and verified in cutting tests.

Receptance coupling for end mills

Piezoelectric Actuators and Motors: Materials, Designs, and Applications

This paper is a review of the remarkable progress that has been made during the past few decades in design, modeling, and fabrication of micromachined resonators. Although micro-resonators have come a long way since their early days of development, they are yet to fulfill the rightful vision of their pervasive use across a wide variety of applications. This is partially due to the complexities associated with the physics that limit their performance, the intricacies involved in the processes that are used in their manufacturing, and the trade-offs in using different transduction mechanisms for their implementation. This work is intended to offer a brief introduction to all such details with references to the most influential contributions in the field for those interested in a deeper understanding of the material.

Micromachined Resonators: A Review

This paper describes a procedure for identifying mass and stiffness imperfections present in a microring gyroscope. In general mass and stiffness imperfections will be present as a result of fabrication variances and will contribute to mistuning of the gyroscope. Both mass and stiffness imperfections are treated as perturbations of the ideal isotropic mass and stiffness. The mass and stiffness perturbations are combined to form a structural imperfection, which is shown to be determinable from Nyquist plots of the frequency response functions. An analysis of electrostatic "tuning" is presented and demonstrates that the nonlinear negative spring component produced from a particular arrangement of capacitive electrodes is capable of removing such structural imperfections. If uncorrected the structural imperfections would otherwise result in anisoinertia and anisoelasticity. A comparison between the theoretically predicted tuning voltage and the experimentally derived value is made and shown to be in agreement. The major advantage of electrostatic tuning is that it may be implemented on-chip during operation thus providing active tuning. This is in contrast to the trimming techniques employing laser ablation or focussed ion beam, which may only be done prepackaging.

/pdf/electrostatic-correction-of-structural-imperfections-present-22c84zq5bx.pdf

Electrostatic correction of structural imperfections present in a microring gyroscope

Thermoelastic damping of the in-plane vibration of thin silicon rings

A simple theory for the analysis and correction of frequency splitting in slightly imperfect rings

A three axis sensor including a substantially planar vibrator resonator (2) having a substantially ring or hoop like structure, drive means (4) for causing the resonator (2) to vibrate in an in plane cos2θ vibration mode, carrier mode pick-off means (5) for sensing movement of the resonator in response to said drive means, pick-off means (6) for sensing in plane sin2υ resonator motion induced by rotation about the z-axis, drive means (7) for nulling said motion, pick-off means (8) for sensing out of plane sin3υ resonator motion induced by rotation about the x-axis, drive means (9) for nulling said motion, pick-off means (10) for sensing out of plane cos3υ resonator motion induced by rotation about the y-axis, drive means (11) for nulling said motion, and support means (9) for flexibly supporting the resonator, wherein the support means comprises only L support beams, where L≠2Jx3K and J=0,1 or 2 and K=0 or 1 with L>24.

Vibratory gyroscopic rate sensor

Multi-mode trimming of imperfect rings

Zener's model for thermo-elastic loss, when applied to uniform beams undergoing flexural vibrations, gives theoretical predictions of mechanical Q-factor that often agree well with experimental measurements. The use of silicon ring resonators in MEMS devices is now becoming increasingly common. This paper considers the application of Zener's theory to thin, circular rings and presents a simple expression for the Q-factor associated with in-plane flexural modes of vibration. The theoretical predictions are shown to be in good agreement with experimental measurements for a practically relevant range of ring sizes. The relationships between ring dimensions, ambient temperature and Q-factor are explored.

http://iopscience.iop.org/article/10.1088/0960-1317/14/9/019/pdf

C.H.J. Fox

Papers

Thermoelastic damping of the in-plane vibration of thin silicon rings

A simple theory for the analysis and correction of frequency splitting in slightly imperfect rings

Vibratory gyroscopic rate sensor

Multi-mode trimming of imperfect rings

A preliminary investigation of thermo-elastic damping in silicon rings