Chapter 11: Vibration Isolation of the Source [Part I]

Transcription

1 Chapter : Vibration Isolation of the Source [Part I] Eaple 3.4 Consider the achine arrangeent illustrated in figure 3.. An electric otor is elastically ounted, by way of identical isolators, to a - thick steel plate. When the otor is driven, its rotating parts generate a vertically-oriented, sinusoidal eciting force between the achine and the joists. Calculate the ratio between the total force acting on the foundation with and without the vibration isolators. Carry out the calculations at low frequencies under the assuption that the electric otor, when operating, generates a vertical haronic eciting force with circular frequency and aplitude Fˆ. The ass of the otor is kg, and each isolator s cople stiffness (see chapter 5, section 5..5) is (. +.i) 4 N/. F stör a) b) c) F stör 4F F Single isolator 4 4F F Figure 3. a) Electric otor elastically ounted to a large steel plate via four vibration isolators. b) Siplified odel of the syste in a. c) The syste in b represented by its separated subsystes. [] Solution Assue that the ecitation frequency is so low that: (i) the otor can be considered a rigid body; (ii) the foundation can be regarded as rigid; and, (iii) each isolator can be described as an ideal assless spring. Assue, additionally, that the otors otions are strongly doinated by sall-aplitude vertical translations. In these circustances, the single degree-of-freedo syste in figure 3. b is a useful odel to describe the proble.

2 With isolators Starting with the syste in figure 3. c, the equation of otion can be constructed for the ass, as well as Hooke s law for spring Thus, d F 4F ec (3.46) dt where 4F is the total force acting on the foundation, i.e.,. the force transitted through all four isolators, and F ( ) (3.47) Assue a sinusoidal, cople-valued displaceent i t e ˆ and eliinate using both of the relations given above. Then, the force on the foundation, noralized by the eciting force, is 4F F ec 4 where is the achine s so-called ounting resonance, i.e., the resonance frequency of the achine ass on the copliance of the isolators. Note that the first ter in the equation only applies to achines with four ounting points. For achines ounted at n points, the ter 4/ should be replaced by n /. (3.48) Without isolators For the case of no isolators, it becoes evident upon reflection that the force on the foundation is equal to F ec. The desired ratio between the force with and without isolators is therefore F F u If an insertion loss is defined on the basis of that ratio, one obtains (3.49) D IL log 4 log Y Y log Y I (3.5) in which Y and Y I are the achine s and the isolator s respective obilities. Note that that forula even applies to cases of shielding isolation.

3 The insertion loss, in that case, has several characteristic properties; see figure 3.. First of all, no isolation is obtained for ecitation frequencies far below the ounting resonance f corresponding to. Secondly, the insertion loss takes on large, negative values for ecitation frequencies near the ounting resonance; there, the force on the foundation is aplified rather than reduced. Finally, large, positive insertion losses are obtained for ecitation frequencies well above the ounting resonance. The increase in the insertion loss asyptotically approaches 4 db/decade, i.e., 4 db for each increase of the ecitation frequency by a factor of. The isolation sees, therefore, to be very effective above the ounting frequency. Unfortunately, that effectiveness is largely the result of the grossly siplified odel. In reality, the increase in insertion loss is interrupted above, say, f. D IL [db] f Frequency [Hz] Figure 3. The insertion loss for a rigid body ounted elastically to a rigid foundation. The ounting resonance is tuned to f =.8 Hz. Note the deep trough in the insertion loss at the ounting resonance, and its negative values elsewhere at low frequencies. The vibration isolation syste is therefore counterproductive at low frequencies; above all, it is essential that the ecitation frequency not fall in the vicinity of the achine s ounting resonance frequency. []

4 A very iportant conclusion fro eaple 3.4 is that the vibration isolators ust be designed to prevent the coincidence of the achine s ounting frequency with any iportant ecitation frequency. Moreover, it is clear that a positive effect is obtained fro the isolators at frequencies above the ounting frequency. The iplication is that as low as possible a ounting resonance frequency ust be sought. In practice, achine ounting is often designed so that the ounting resonance frequency falls in the - Hz band. Fleible foundation As the ecitation frequency increases, the deforation of the foundation due to the ecitation force soon becoes too large to ignore. A odel in which the foundation is fleible ust then be used. A nuber of different odels with differing characteristics are available for this situation. If, for eaple, the foundation is a syste of joists with considerable diensions, an infinite plate odel ight be used to describe the otions of the foundation. If the foundation ehibits a resonance, then a ass-daper syste can be used as a first approiation to describe its behavior. Eaple 3.5 Consider the achine ounting situation of eaple 3.4. Assue that an infinite plate would be a valid odel of the foundation response. Calculate the ratio between the total force on the foundation with and without isolators. Solution Assue that the deforation of the foundation is the sae at all four achine feet. Additionally, conditions (i) and (iii) fro eaple hold,

5 a) F r b) F s Single isolator F 4F 4 4F F Figure 3.3 Siple odel of a achine ounted to a fleible foundation The equation of otion, Hooke s law, and the obility of a plate yield the following syste of equations: Eliinate and d dt F ec 4F F ( ), i ) Yplate4 ( F. 4F F with ec ( i) Y plate i Y i 4 YI Y Y Y Y Without isolators, the force on the foundation can be deterined by ecluding the second of the equations fro the syste given above, and setting equal to. The syste then has the solution I plate (3.5) F Y without 4 i Fec i Yplate Y Yplate (3.5) The insertion loss is therefore D IL Y Y Y I plate log (3.53) Y Y plate Forula (3.53) can be shown to even apply to the shielding isolation case. The obility of a c thick, very large steel plate is calculated fro

6 3 3( ) = 78 kg/ =.3 5 Y.6 /Ns. plate 4h E E. N/ h =. Putting values into the epression for the insertion loss leads to the graph in figure 9. That figure also shows the corresponding results for a rigid foundation. Apparently, the fleible foundation affects the insertion loss in two bands: at the ounting resonance; and, at frequencies over about 5 Hz. At the ounting resonance frequency, the insertion loss increases, due to the ability of the infinite plate to act as an energy sink. Above 5 Hz, the fleible foundation provides a significantly lower insertion loss than the rigid foundation. Here, the isolation obtained is largely deterined by the ratio of the isolator obility to the obility of the plate. At high frequencies, the insertion loss now asyptotically approaches a db per decade rate of increase, instead of the 4 db per decade obtained earlier. D IL [db] 8 Rigid foundation 6 4 Copliant foundation - -4 f Frequency [Hz] Figure 3.4 Insertion loss for a rigid body elastically ounted to an infinite steel plate. Copared to an ideal, rigid foundation, the aplification peak at the ounting resonance frequency is reduced and the rate of increase of the insertion loss falls off. That latter effect is due to the diinished obility or ipedance gap between the isolator and the foundation.

7 Wave propagation in the isolator When the ecitation frequency has increased so uch that the deforation field in the isolator is a wave otion, the ideal spring odel becoes less and less tenable. Depending on the isolator design, different odels for wave propagation in the isolator ay be appropriate. In eaple -, an eaple is given of a siple wave propagation odel of the isolator. Eaple 3.6 Consider the achine ounting situation illustrated in eaple 3.4. Assue that the isolator is a circular cylindrical bar undergoing priarily aial deforations. For aial deforation, the otion in the isolator is built up of longitudinal waves. In order to perit a direct coparison between the eaples, every isolator is assued to have a length of.5 and a cross-sectional area of.5. The isolator aterial is assued to have a density of 5 kg/ and a cople E-odulus.( + i.) MPa. For these input values, the isolator stiffness at low frequencies atches that used in eaples 3.4 and 3.5 above. Solution Block one end of the isolator. By calculating the ratio of the force on the blocked end to the ecited displaceent response at the free end, frequency-dependent dynaic isolator stiffness can be calculated. If the stiffness in the result fro eaple 3.4 is replaced with this dynaic stiffness, an insertion loss accounting for longitudinal wave propagation is obtained for the isolator. The result can be directly copared to those obtained earlier; see figure 3.5.

8 DIL [db] 8 6 Copliant Rigid Eftergivligt underlag Oeftergivligt underlag 4 64 Hz 3 Hz Wave propagation Vågutbredning i isolatorn h l - -4 f Frekvens [Hz] Frequency [Hz] Figure 3.5 Insertion losses of isolators. A jutaposition of all the results fro eaples 3.4 to 3.6. In real vibration isolation, these results are further odified by the achine s dynaic behavior. Resonances in the achine, for eaple, negatively ipact the insertion loss. That is because the difference in ipedance between the achine and the isolator is reduced at a achine resonance. For cases in which there is longitudinal wave propagation in the isolators, they becoe very stiff at certain specific frequencies. That is due to the interaction between waves propagating in opposite directions. At these frequencies, the isolators no longer function as copliant eleents. We have frequencies at which little isolation is obtained. In the eaple given above, these critical frequencies lie at 64 Hz, Hz, etc. Figure - clearly shows that the first such critical frequency is an upper bound, above which the isolation insertion loss no longer has an unbroken rising trend. Above that frequency, the average isolation reains about constant. In the eaple given above, the insertion loss above, say, 5 Hz, is in the vicinity of 4 db, ecept at the critical frequencies. Eaple 3.4 clearly shows that wave propagation in the isolator brings the trend of increasing insertion loss to an end after a certain point.

9 Deforable achine In eaples 3.4 to 3.6, it was assued that the achine oves along a coordinate direction as a rigid point ass. The insertion loss is then very low at ecitation frequencies near the ounting frequency. If several of the achine s si rigid body degrees-of-freedo are taken into account by the odel, several ounting resonance frequencies are then ehibited. In the ost general case, we therefore have critical frequencies at si different ounting resonances. In fact, every real achine also ehibits internal resonances at certain frequencies. Typically, the first resonance frequency of a copact achine with a -kg ass, e.g., a sall internal cobustion engine, falls in the Hz - 5 Hz range. If the achine is coposed of fleibly attached sections, the first resonance can of course lie at even lower frequencies. The possibility of wave propagation in the achine also affects the isolation insertion loss obtained fro elastic ounting. That depends, after all, on the relative stiffnesses of the achine and the isolator. If the achine stiffness varies significantly, due to resonances and antiresonances, then even the insertion loss will vary. On average, the isolation perforance is degraded above the achine s first resonance frequency. Figure 3.6 shows the insertion loss of a siple syste consisting of a achine with internal resonances. The foundation is rigid and the isolator is the sae as in eaple. D IL [db] Rigid achine structure Machine structure with internal resonances f Frequency [Hz] 5 ( +, i) MN/ F et 5 kg 5 kg 5 kg kg Machine with Internal Resonances Figure 3.6 Insertion loss for the achine ounting situation of eaple 3.4, but with internal resonances in the achine. The right side of the figure shows a echanical odel of the achine, and the input data used.

10 In the eaple, the achine has resonances at 85, 45 and 55 Hz, and antiresonances at 6, 5 and 495 Hz. Figure 3.6 shows that, at the resonance frequencies, at which the achine is copliant, there are insertion loss inia, i.e., frequencies at which the isolation is poor. On the other hand, etra isolation is obtained at the anti-resonances, at which the achine is very stiff. For increasing ecitation frequency in the region above the first achine internal resonance, the average insertion loss falls off gradually. That effect is due to the ever saller unsprung ass that takes part in the otions of the point(s) of contact with the isolator(s). Fundaentals of Sound and Vibrations by KTH Sweden [], this book is used under IITR-KTH MOU for course developent.

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