Mechanical Vibrations: Theory And Application T...

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Prerequisite: MECHENG 211, Math 216. (3 credits) Introduction to theory and practice of the finite element method. One-dimensional, two-dimensional and three dimensional elements is studied, including structural elements. Primary fields of applications are strength of materials (deformation and stress analysis) and dynamics and vibrations. Extensive use of commercial finite element software packages, through computer labs and graded assignments. Two hour lecture and one hour lab. (Course Profile)

Prerequisite: MECHENG 211, MECHENG 240, Math 216. (3 credits)Exact and approximate techniques for the analysis of problems in mechanical engineering including structures, vibrations, control systems, fluids, and design. Emphasis is on application. (Course Profile)

Prerequisite: MECHENG 211, Math 450. (3 credits) The general theory of a continuous medium. Kinematics of large motions and deformations; stress tensors; conservation of mass, momentum and energy; constitutive equations for elasticity, viscoelasticity and plasticity; applications to simple boundary value problems.

Prerequisite: MECHENG 320. (3 credits) Fundamental concepts and methods of fluid mechanics; inviscid flow and Bernoulli theorems; potential flow and its application; Navier-Stokes equations and constitutive theory; exact solutions of the Navier-Stokes equations; boundary layer theory; integral momentum methods; introduction to turbulence.

Prerequisite: MECHENG 360 or graduate standing. (3 credits)Geometrical representation of the dynamics of nonlinear systems. Stability and bifurcation theory for autonomous and periodically forced systems. Chaos and strange attractors. Introduction to pattern formation. Applications to various problems in rigid-body dynamics, flexible structural dynamics, fluid-structure interactions, fluid dynamics, and control of electromechanical systems.

Prerequisite: ME 461 or equivalent. (3 credits) Theoretical principles and practical techniques for controlling mechatronic systems are taught in the context of advanced manufacturing applications. Specifically, the electro-mechanical design/modeling, basic/advanced control, and real-time motion generation techniques for computer-controlled manufacturing machiens are studied. Hands-on labs and industrial case studies are used to re-enforce the course material.

Prerequisite: MECHENG 520. (3 credits) An introduction to the theory of hydrodynamic stability with applications to stability of thermal flows, rotating and curved flows, wallbounded and free shear flows. Development of the asymptotic theory of the Orr-Sommerfeld equation. Review of the fundamental concepts and current work in nonlinear theory of hydrodynamic stability.

Prerequisite: MECHENG 541. (3 credits) Large amplitude mechanical vibrations; phase-plane analysis and stability; global stability, theorems of Liapunov and Chetayev; asymptotic and perturbation methods of Lindstedt-Poincare, multiple scales, Krylov-Bogoliubov-Mitropolsky; external excitation, primary and secondary resonances; parametric excitation, Mathieu/Hill equations, Floquet theory; multi-degree of freedom systems and modal interaction.

The components of a mechanical filter are all directly analogous to the various elements found in electrical circuits. The mechanical elements obey mathematical functions which are identical to their corresponding electrical elements. This makes it possible to apply electrical network analysis and filter design methods to mechanical filters. Electrical theory has developed a large library of mathematical forms that produce useful filter frequency responses and the mechanical filter designer is able to make direct use of these. It is only necessary to set the mechanical components to appropriate values to produce a filter with an identical response to the electrical counterpart.

While the meaning of mechanical filter in this article is one that is used in an electromechanical role, it is possible to use a mechanical design to filter mechanical vibrations or sound waves (which are also essentially mechanical) directly. For example, filtering of audio frequency response in the design of loudspeaker cabinets can be achieved with mechanical components. In the electrical application, in addition to mechanical components which correspond to their electrical counterparts, transducers are needed to convert between the mechanical and electrical domains. A representative selection of the wide variety of component forms and topologies for mechanical filters are presented in this article.

The theory of mechanical filters was first applied to improving the mechanical parts of phonographs in the 1920s. By the 1950s mechanical filters were being manufactured as self-contained components for applications in radio transmitters and high-end receivers. The high "quality factor", Q, that mechanical resonators can attain, far higher than that of an all-electrical LC circuit, made possible the construction of mechanical filters with excellent selectivity. Good selectivity, being important in radio receivers, made such filters highly attractive. Contemporary researchers are working on microelectromechanical filters, the mechanical devices corresponding to electronic integrated circuits.

Mechanical filter design was developed by applying the discoveries made in electrical filter theory to mechanics. However, a very early example (1870s) of acoustic filtering was the "harmonic telegraph", which arose precisely because electrical resonance was poorly understood but mechanical resonance (in particular, acoustic resonance) was very familiar to engineers. This situation was not to last for long; electrical resonance had been known to science for some time before this, and it was not long before engineers started to produce all-electric designs for filters. In its time, though, the harmonic telegraph was of some importance. The idea was to combine several telegraph signals on one telegraph line by what would now be called frequency division multiplexing thus saving enormously on line installation costs. The key of each operator activated a vibrating electromechanical reed which converted this vibration into an electrical signal. Filtering at the receiving operator was achieved by a similar reed tuned to precisely the same frequency, which would only vibrate and produce a sound from transmissions by the operator with the identical tuning.[8][9]

It was not enough to just develop a mechanical analogy. This could be applied to problems that were entirely in the mechanical domain, but for mechanical filters with an electrical application it is necessary to include the transducer in the analogy as well. Poincaré in 1907 was the first to describe a transducer as a pair of linear algebraic equations relating electrical variables (voltage and current) to mechanical variables (force and velocity).[13] These equations can be expressed as a matrix relationship in much the same way as the z-parameters of a two-port network in electrical theory, to which this is entirely analogous:

Wegel, in 1921, was the first to express these equations in terms of mechanical impedance as well as electrical impedance. The element z 22 {\displaystyle z_{22}\,} is the open circuit mechanical impedance, that is, the impedance presented by the mechanical side of the transducer when no current is entering the electrical side. The element z 11 {\displaystyle z_{11}\,} , conversely, is the clamped electrical impedance, that is, the impedance presented to the electrical side when the mechanical side is clamped and prevented from moving (velocity is zero). The remaining two elements, z 21 {\displaystyle z_{21}\,} and z 12 {\displaystyle z_{12}\,} , describe the transducer forward and reverse transfer functions respectively. Once these ideas were in place, engineers were able to extend electrical theory into the mechanical domain and analyse an electromechanical system as a unified whole.[10][14]

Harrison used Campbell's image filter theory, which was the most advanced filter theory available at the time. In this theory, filter design is viewed essentially as an impedance matching problem.[17] More advanced filter theory was brought to bear on this problem by Norton in 1929 at Bell Labs. Norton followed the same general approach though he later described to Darlington the filter he designed as being "maximally flat".[1] Norton's mechanical design predates the paper by Butterworth who is usually credited as the first to describe the electronic maximally flat filter.[18] The equations Norton gives for his filter correspond to a singly terminated Butterworth filter, that is, one driven by an ideal voltage source with no impedance, whereas the form more usually given in texts is for the doubly terminated filter with resistors at both ends, making it hard to recognise the design for what it is.[19] Another unusual feature of Norton's filter design arises from the series capacitor, which represents the stiffness of the diaphragm. This is the only series capacitor in Norton's representation, and without it, the filter could be analysed as a low-pass prototype. Norton moves the capacitor out of the body of the filter to the input at the expense of introducing a transformer into the equivalent circuit (Norton's figure 4). Norton has used here the "turning round the L" impedance transform to achieve this.[20]

Modern mechanical filters for intermediate frequency (IF) applications were first investigated by Robert Adler of Zenith Electronics who built a 455 kHz filter in 1946.[23] The idea was taken up by Collins Radio Company who started the first volume production of mechanical filters from the 1950s onwards. These were originally designed for telephone frequency-division multiplex applications where there is commercial advantage in using high quality filters. Precision and steepness of the transition band leads to a reduced width of guard band, which in turn leads to the ability to squeeze more telephone channels into the same cable. This same feature is useful in radio transmitters for much the same reason. Mechanical filters quickly also found popularity in VHF/UHF radio IF stages of the high end radio sets (military, marine, amateur radio and the like) manufactured by Collins. They were favoured in the radio application because they could achieve much higher Q-factors than the equivalent LC filter. High Q allows filters to be designed which have high selectivity, important for distinguishing adjacent radio channels in receivers. They also had an advantage in stability over both LC filters and monolithic crystal filters. The most popular design for radio applications was torsional resonators because radio IF typically lies in the 100 to 500 kHz band.[24][25] 781b155fdc