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Substituting the first natural frequency back into the equation of motion solution gives the result of the first natural frequency: (k1 + k2 − ω21m1)C11 − k2C21 = 0............................................... 105) and ( ) −k2C11 + k2 − ω21m2 C21 = 0. 106) The ratio of C11 to C21, known as the mode shape, for the first natural frequency is: Print Search Contents Home Chapter 2 Bookmarks Help I-28 Petroleum Engineering Handbook—Vol. I λ1 = C11 C21 = k2 k1 + k2 + ω21m1 k2 − ω21m2 = k2 ;......................................
2 Partial Differential Equations. PDEs are an extension of the concept of ordinary differential equations (ODEs). Unlike an ODE, which depends on only one independent variable, a PDE depends on two or more independent variables. In the previous example, Eq. 6 depends on two independent variables: one space dimension (x) and time (t). The order of Eq. 6 de- Print Search Chapter 1 Home Chapter 3 Bookmarks Help Chapter 2—Mathematics of Fluid Flow I-47 pends on the form of concentration and flux. The order of a PDE is the order of the highest derivative that appears in the equation.
K. (1980). 21. Mal, A. : Deformation of Elastic Solids, Prentice-Hall, Englewood Cliffs, New Jersey (1991). 22. : Vibrations and Waves, Butterworth, London (1963). 23. : Wave Propagation in Petroleum Engineering: Modern Applications to Drillstring Vibrations, Measurement-While-Drilling, Swab-Surge, and Geophysics, Gulf Publishing, Houston (1994). 24. : Wave Propagation, McGraw-Hill Book Co. , New York City (1973). 25. , New York City (1989) 26. , San Diego, California (1989). 27. : The FFT: Fundamentals and Concepts, Prentice-Hall, Englewood Cliffs, New Jersey (1985).