Pl2 4 by using holzers method for all m m and al l k

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Pl2-4 by using Holzer's method for all m, - m and al l k, - k. m 1 k, m 2 * 2 m *• ,, "/n r/� Figure Pll-4. 12-5 Repeat Prob. 12-4 when m 1 - m, m 2 - 2m, m3 - 3m, k1 - k, k 2 - k, and k, - 2 k. 12-6 Compare the equations of motion for the linear spring-mass system versus torsional system with same mass and stiffness distribution. Show that they are similar. 12-7 Determine the natural frequencies and mode shapes of the spring-mass system of Fig. P12- 7 by the Holzer method when all masses are equal and al l stiffnesses are equal. Figure Pll-7. 12-8 A fighter-plane wing is reduced to a series of disks and shafts for Holzer's analysis as shown in Fig. P12-8. Determine the first two natural frequencies for symmetric and antisymmetric torsional oscillations of the wings, and plot the torsional mode corresponding to each.
Chap. 12 Problems n l 2 3 4 5 6 J lb. in. sec.2 50 138 145 181 260 t x 140,00 0 0 =·�CD K lb. in./rad. 15 x HJ" 30 22 36 120 40" 7011 105" 145" 2 00" I I I I I I I I I I I I 4 I I I 3 2 I I I I I I I I I : Figure Pll-8. 345 12-9 Determine the natural modes of the simplified model of an airplane shown in Fig. Pl2-9 where M/m - n and the beam of length I is uniform. t m1 )---'----1 M r-�l-� m Figure Pll-9. 12-10 Using Myklestad's method, determine the natural frequencies and mode shapes of the two-lumped-mass cantilever beam of Fig. P12-10. Compare with previous results by using influence coefficients. Figire Pll-10. m m 0 0 I 12-11 Determine the first two natural frequencies and mode shapes of the three-mass cantilever of Fig. P12-11 Figure Pll-11. o / o-,--<�>-- , .. .., 12-12 Using Myklestad's method, determine the boundary equations for the simply supported beam of Fig. P12-12. Figure Pl 2-12. 2 3 12-13 The beam of Fig. P12-13 has been previously solved by the method of matrix iteration. Check that the boundary condition of zero deflection at the left end is satisfied for these natural frequencies when Myklestad's method is used. That is,
346 Numerical Procedures for Lumped Mass Systems Chap. 12 check the deflection for change in sign when frequencies above and below the natural freque�cy are used. 500kg 100kg c m �� () ����- o ! f- - -.1...- -- ! -.!...- -- �-- -j Figure 12-13. Figure 12-14. 12-14 Determine the flexure-torsion vibration for the system shown in Fig. P12-14. 12-15 Shown in Fig. P12-15 is a linear system with damping between mass 1 and 2. Carry out a computer analysis for numerical values assigned by the instructor, and determine the amplitude and phase of each mass at a specified frequency. Figure Pl2-15. 12-16 A torsional system with a torsional damper is shown in Fig. Pl2-16. Determine the torque-frequency curve for the system. g, = 104 T J2=100 Figure Pll-16.
Chap. 12 Problems 347 12-17 Determine the equivalent torsional system for the geared system shown in Fig. P12-l 7 and find its natural frequency. 3" dia. 611 dia. d z =2" 1. z= 3 0 .. J 1 = 1 0 lb-in.-sec 2 Figure Pll-17. Jz = 24 12-18 If the small and large gears of Prob. 12-17 have the inertias J' - 2, J" - 6, determine the equivalent single shaft system and establish the natural frequen- cies.