tailieunhanh - Harris' Shock and Vibration Handbook Part 4

Tham khảo tài liệu 'harris' shock and vibration handbook part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | VIBRATION OF SYSTEMS HAVING DISTRIBUTED MASS AND ELASTICITY Rayleigh s Method. An accurate approximation to the fundamental natural frequency of this system can be found by using Rayleigh s method. The motion of the mass can be expressed as um u0 sin at. If it is assumed that the deflection u at each section of the rod is proportional to its distance from the fixed end u u0 x l sin ant. Using this relation in the appropriate equation from Table the strain energy V of the rod at maximum deflection is SE f du 2 SE fl Y seuo2 V 2 ị 11 dx 2 IT dx -i The maximum kinetic energy T of the rod is . Sy f Sy x y _ Sv 2l T 2g V dx 2g au I J dx 2g a u0 t The maximum kinetic energy of the mass is Tm m2an2u02 2. Equating the total maximum kinetic energy T Tm to the maximum strain energy V gives a f SE y 2 an l m2 m where mi Syl g is the mass of the rod. Letting SE l k an k_n y M m 3 This formula is included in Table . The other formulas in that table are also based on analyses by the Rayleigh method. EXAMPLE . The natural frequency of a simple mass-spring system for which the weight of the spring is equal to the weight of the mass is to be calculated and compared to the result obtained by using Eq. . SOLUTION. For m1 m2 l the lowest root of Eq. is an m k m2 m1 an _ m2 Using the approximate equation a. I y m2 1 1 3 y m2 LATERAL VIBRATION OF STRAIGHT BEAMS Natural Frequencies from Nomograph. For many practical purposes the natural frequencies of uniform beams of steel aluminum and magnesium can be determined with sufficient accuracy by the use of the nomograph Fig. . This nomograph applies to many conditions of support and several types of load. Figure indicates the procedure for using the nomograph. Classical Solution. In the derivation of the necessary equation use is made of the relation d 2y dx2 M CHAPTER SEVEN FIGURE Nomograph for determining fundamental natural frequencies of beams. From the point on .