Data processing: measuring – calibrating – or testing – Measurement system in a specific environment – Mechanical measurement system
Reexamination Certificate
1997-10-16
2002-08-06
Shah, Kamini (Department: 2857)
Data processing: measuring, calibrating, or testing
Measurement system in a specific environment
Mechanical measurement system
C324S166000
Reexamination Certificate
active
06430516
ABSTRACT:
FIELD OF THE INVENTION
The present invention provides: (1) The exact characteristic equation of vibrations of a circular cylindrical shell; (2) The characteristic equation is good for any linear viscoelastic materials including elastic solids and non-Newtonian fluids; (3) Computer programs to generate data for the vibrational wave spectrum analysis; (4) The physical and geometrical dimensions of the shafts can be determined between two critical frequencies.
BACKGROUND OF THE INVENTION
High speed rotating shafts are one of the most important parts of machines that are used for the transformation of other energies into kinetic energy. They are used in turbines, rotors, motors, pumps and flywheels that are the most vital components of space shuttles, jetliners, and all other vehicles operated in space, air, land and in hydrospace. Thus far, the design of low and moderate speed rotating shafts has been very well known and successfully applied and used in our daily machines. However, because of the need for higher performance of the machines in our modem times, special care must be given to design high speed rotating shafts. The mechanical structural failure of the shaft may lead to a disaster of losing human lives in airplanes because fragments of the broken shaft carry high kinetic energy that could destroy airplane components in their paths. If this is allowed to happen, the airplane will be shattered into pieces.
To solve the problem, two aspects have been followed: one by means of experimental tests; the other by means of theoretical analysis. They are complimentary to each other. The former provides a practical solution of the problem. It is more expensive than using the latter on account of the choices of physical and geometrical parameters of various modem materials of the shafts for different purposes. The latter provides a systematic guide for the tests to choose the appropriate material for the shaft once the analysis is completed and perfected with a fully developed computing program.
From the theoretical aspect, the proposed solution of the problem has been attempted by means of three methods: 1. particle dynamics; 2. structural dynamics; and 3. elastodynamics. Historically, the particle dynamics approach to solve the rotary dynamic problems can be traced back to two papers in 1895 by Dunkerly, S., “On the Whirling and Vibration of Shafts,” Phil. Trans. Roy Soc. A., Vol. 185, pp. 269-360, and by Foppl, A., “Das Problem der Laval'schen Turbinewlle,” Civilinggenieur 41, pp. 332-342. The interests of those problems have been extended and continued by many others in the academic and industrial organizations. Most recently, the paper by Crandall, S. H., “The Physical Nature of Rotor Instability Mechanisms” in “Rotor Dynamical Instability” (M. L. Adams, ed.) ASME Special Publication, AMD-Vol. 55, pp. 1-18 (1983), provided simple physical explanations of several instability mechanisms. They are whirling due to Coriolis acceleration; internal damping in the rotor; flow about the rotor; and internal flow within a rotor. It is well known that the dynamics of particles approach involves the vibration of masses between springs. Similarly, the vibration of continuous structures are also usual topics in the field of structural mechanics. A combination of both, and the efforts of many as indicated by the many papers by Myklestad, N.O., “A New Method of Calculating Natural Modes of Uncoupled Bending Vibration of Airplane Wings and Other Types of Beams” J. Aero. Sci., pp.153-162 (April 1944), Prohl, M. A., “A General Method for Calculating Critical Speeds of Flexible Rotors.” Trans. ASME A-142 (September 1945), Pestel, E. C., and Leckie, F. A., “Matrix Methods in Elastodynamics” McGraw-Hill, N.Y. (1963) and by Thomson, W. T., “Matrix Solution for the Vibration of Nonuniform Beams.” J. Appl'd Mech., pp.337-339 (September 1950) and “Vibration Theory and Application,” Prentice-Hall, Englewood Cliff, N.J. (1965), brings about a method to solve many complicated structural problems.
The most important element of the structural dynamics approach is the transfer matrix of the structural system. As can be seen from the books by Thomson and by Pestel and Leckie, the transfer matrix can be derived from the governing differential equations of a physical problem. A great deal of these have been done for the beam theories on torsional, axially compressional and transversal vibrations. However, the transfer matrix for the exact theory of elastodynamics and thermoelastodynamics is comparatively unknown. This was so because general solutions of the governing equations of elastodynamics were either too tedious or uninformative for any practical uses. One of these examples is the well known Pochhammer and Love solution of the flexural vibration of an elastic rod. See, Pochhammer, L., “Ueber die Fortflanzungsgeschwindigkeiten Schwinggungen in ein Unbegrenzten Isotropen Kreiszylinder” J. Fur Math., Vol. 81, pp. 324-336 (1876), and Love A. E. H., “A Treatise on the Mathematical Theory of Elasticity” Cambridge University Press, Fourth Edition, pp. 287-292 (1927). The solution was originated by Pochhammer in 1876 and made independently by Chree in 1886. The original Pochhammer-Chree solution of the problem was very tedious. Concise and modem systematic solution of the same problem together with the wave spectra analysis were separately provided by Gazis in 1959, Gazis, D. C., “Three-Dimensional Investigation of the Propagation of Waves in Hollow Circular Cylinders I: Analytical Foundation, and II: Numerical Results, J. Ac. Soc. Amer., Vol 31, pp. 568-578, by Greenspon in 1960, Greenspon, J. E., “Vibration of a Thick-walled Cylindrical Shell—Comparison of the Exact Theory with Approximate Theories.”, J. Ac. Soc. Amer., Vol 32, pp. 571-578, and by Wong in 1967, Wong, P. K., “On the Unified General Solution of Linear Wave Motions of Thermoelastodynamics and Hydrodynamics with Practical Examples” Transaction of ASME, Journal of Applied Mechanics, Vol 34, pp. 879-887 (December 1967) and Vol. 35, pp 847 (December 1968). The wave spectra were extended for the entire class of linear viscoelastic materials for solids and shells by Wong in 1970, Wong, P. K., “Waves in Viscous Fluids, Elastic Solids, and Viscoelastic Materials” Ph.D. Dissertation, Department of Aeronautics and Astronautics, Stanford University, Stanford, Calif. (1970).
SUMMARY OF THE INVENTION
The invention is based upon elastodynamnic methods; and certain details can be traced from Wong, P. K., “On the Unified General Solution of Linear Wave Motions of Thermoelastodynamics and Hydrodynamics with Practical Examples.” Transaction of ASME, Journal of Applied Mechanics, Vol 34, pp. 879-887 (December 1967) and Vol. 35, pp 847 (December 1968).
The invention illustrates two main features different from other approaches to solve the design of high speed rotating shafts: (1) since the general solutions of the governing equations of elastodynamics and thermoelastodynamics are shown by Wong in 1967, 1968 and 1970, the derivation of transfer matrices for the exact theories is therefore possible; and (2) it can be shown that the solutions are also useful for practical design purposes. These can be demonstrated in a practical example which can be solved both from the lumped mass technique and from the elastodynamics theory.
It is known that the lumped mass technique is a combination of particle dynamics and structural mechanics approaches. The comparison of lumped mass techniques with elastodynamics is discussed below and in connection with accompanying
FIGS. 1-3
. Consider, for example, an elastic solid rotating shaft of mass m, density &dgr;, Young's modulus E and area moment of inertia I=&pgr;r
4
/4. The shaft is simply supported by two bearings as shown in
FIG. 1
for a homogeneous circular cylindrical rotating rod and its equivalence being replaced by a massless shaft with its equivalent mass m=&pgr;r
2
l&dgr; concentrating at the center of the rod as shown in FIG.
2
. The natural frequency of transversal vibration of the system
Shah Kamini
Wong Po Kee
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