Document 99a28YK6Rr376VwkRyk14wKVe

214 CHAPTER 14 1965 Guide And Data Book $aDnd:C<Snt^ol, 215 fig. 22 .... Vertical Forces Acting on a Vibrating Madune Hie flexibility of the machine support is best expressed in terms of its deflection under the total supported static load: ' (18) token ,- d,-- static deflection of support, inches. W a share of weight of machine including any rigid base, rest- (; ing on support, pounds. - ' 4, vertical stiffness of the support, pounds per inch.". . fig. 24 .... Effect of Mount Flexibility of Support on Vibrational Displacement at 1000 RPM where D = displacement amplitude for negligible damping, inches. U = unbalance, ounce-inches. Resonance occurs when the inertia force is equal in magni tude to the spring force: The inevitable flexibility of the machine support.allows.a certain amount of vibrational motion in response to any dis turbing forces, such as the centrifugal force due to-imbalance. This means that the machine is displaced from its equilibrium position, alternately, up-and down as-shown in Fig.-22. Re versal at the top and bottom of each cycle requires substantial acceleration. This gives rise-to an inertia reaction'which .is proportional to the mass of the machine and the square of the speed: The force halmw- diagrams of fig. 23'show'that'the inertia force is always opposed to the spring force exerted by the supports. Thus, apart from the relatively small'damping force, the difference between the spring force and inertia force always has to be such as to balance the disturbing force: Since both spring force mid inertia force are proportional to displacement; as shown by the formulas in' Fig. 22, displace^ meat can be calculated from Equation 18 and the condition of force balance: KRPM\* /1 \*i U /RPM\* sr) -(i)]-nx(tst) !19> - At resonance, the displacement of the machine is limited only by the rate at which energy is dissipated by damping in the supporting structure or machine* mounts. Fig. 24 shows curves of relative displacement for. a speed of 1000 rpm as a function of the static deflection for various amounts of damp-, ing and various amounts of machine weight. It should be noted that damping has a significant effect on displacement only near resonance, and that a static deflection of as little as it in. will produce resonance at 1000 - When the static deflection is much greater than that which produces resonance, the vibrational, displacement of the machine levels off and becomes independent of static deflect tion. Displacement for this condition can be calculated from Equation 19 by neglecting the 1/d, term, and isalso seen to be independent of speed: U / 188 \* for <L 3 l^rrl (21) Motion of a mashing on sufficiently flexible mounts is masscontrolled. Displacement for a given amount of imbalance can -only be reduced by adding mass to the machine, as illustrated in Fig. 24. The conditions for mass control as well as resonance are shown in Fig. 25. When the static deflection is much less than that which produces resonance, the motion is stiffness-controlled. Dis- placement for a given unbalance and speed is only a function :,of the stiffness of the support: (S) stiffness controclco (S) ** controclcd fig. 23 .... Balance of Forces Acting on a Vibrating Machine 16 V 188 / V*/ ,, E f*K\' /*\ " 16 V 188 / \w) 1/ 188 V r ** s 5 \rpm) (22) - Practical, limits for Mounting Madiines Rigidly Rigid (stiffness-controlled) mounting of a marhine has some ' merit as the vibrational motion of the machine can be kept to .. a minimum. In practice, the condition for ridigity shown" in Fig. 25 can be fulfilled only when the machine is relatively light, and'the speed is relatively low. Fig. 25 .... Condition of Resonance and Approximate Regions in which Vibrational Motion is Mass:Ccintrolled and Stiffness-Controlled - The weight of the machine has to-be considered from two-" standpoints: ' - 1. The heavier the machine, the more difficult it becomes to meet the condition of Fig. 25 for a rigid mounting, because of.the inevitable-flexibilities of the supports. 2. If a'machine is installed on or suspended from a floor or roof slab, the natural frequency of the slab may be modified by more than 10 percent if the machine weight exceeds 10 percent of.the effective weight of the slab. Also, the natural.frequency of .the machine will be affected by the dynamic propa-ties of the Structure. The two resonant speeds, which should Be avoided by at least 50 percent, can be found from Fig. 26. The effective uoar weight can be taken as the weight of that part of the >* under the projected area of the machine. For reciprocating machinery, rigid mounting is particularly to be avoided because some of the disturbing forces at mul tiples of the rotational speed would fall into the resonant region. fig. 26 .... Resonant Speeds for Machine Mounted on Flexible Floor that equation, displacements under actual operating condi tions are usually much larger and can be estimated better from Fig. 27." During start up and shut down of a ma^Hin* still larger displacements will occur, at least momentarily, as the speed goes through resonance. Practical Considerations in Mounting Machines flexibly : , ... When a high speed machine is installed on an upper floor <5r other structure of low rigidity, resonance can best be avoided (L. P^^ting the machine on mounts which are sufficiently flexible to meet the condition of Equation 21. Even though jwge turbo-machines may appear to.be rigidly mounted, the wer part of the foundation is generally HpaignpH to provide we necessary amount Of flexibility.71 . . . j_/]enble marhiftg mounts are properly, called vibration iso- * rather than vibration eliminators or absorbers, because "*ey confine the vibrations to .the mnrhino by allowing it to j^PWid freely to the .disturbing.forces. Equation 21 may be ~ed: to estimate, the minimum amount of .displacement..Due various simplifying assumptions made in.the;derivation of fig. 27 . ... Practical Range of Vibrational-Motion .for Machines;Mounted on Vibration Isolators11