f 2 f n where m is the total mass of the object. Expression (6a) is plotted in Figure 8 for several values of damping ( ).

Transcription

1 F o F o / k A = = 6 k n r n n n RESONANCE It is seen in Figure 7 that displaement and stress levels tend to build up greatly when the oring requeny oinides with the natural requeny, the buildup being restrained only by damping. This ondition is known as RESONANCE. In many ases, the ored vibration is aused by an unbalaned rotating mass, suh as the rotor o an eletrial motor. The degree o unbalane an be expressed as distane e between the C.G. o the rotor and its axis o rotation. The vertial omponent o the entriugal ore generated by the unbalaned rotor mass M is F.. = M e sin t = M e sin t, 7 where is angular speed o rotation in rad/ se and is the number o revolutions per seond. In ase o vibration exitation by the unbalaned rotor, ombining o 6 and 7 results in A F o /k... = T1 T e h n i a l S e t i o n A = Me n k 1 + r n Figure / n Amplitude-Frequeny Charateristis o Massive Blok Motion in Figure 6 Me / = = Me n, 6a k 1 m + n n 1 n + n where m is the total mass o the objet. Expression 6a is plotted in Figure 8 or several values o damping.. Vibration Isolation Although VIBRATION ISOLATION is a very large area o vibration ontrol, there are two most widely used tehniques o vibration isolation: Redution o transmission o vibratory or shok ores rom the objet, in whih these ores are generated, to the base; and Redution o transmission o vibratory motions o the base to the work area o vibration-sensitive objets. These tehniques are similar, but also quite dierent. They both deal with TRANSMISSIBILITY or TRANSMISSION RATIO. There are several transmission ratios. Usually these reer to the ratios o the maximum values o the transmitted ore or displaement to the maximum values o the applied ore or the ored motion. The important diretion o transmission is rom the objet to the base or the ore isolation, or rom the base to the objet or the motion isolation. T1-9

2 A M e m Frequeny Ratio, / n Figure 8.1 Vibration Isolation o Vibration-Produing Produts Figure 9 shows a simpliied single-degree-o-reedom model o a vibration isolation system. While in models in Figure 1 and Figure, the base oundation is shown as having ininite mass, in Figure 9 model the oundation has a inite mass m. I the ore Ft = F o sin t is generated in the objet mass m, the ore transmissibility F rom the objet to the oundation is equal to the motion transmissibility x rom the oundation to the objet and is expressed as 1 + F = x = F m = x 1 = n. 8 F o x m + m 1 + n n This expression or m = is plotted in Figure whih shows that "isolation" o the ore soure or the ondition o F < 1 develops at requenies greater than = 1.1 n and ast improving with urther inreasing o the requeny ratio / n. The maximum transmissibility ours at the resonane when the requeny ratio / n = 1. At resonane = n, the transmissibility at not very high damping is expressed as m m + m F max = x max. 9 While inreasing o damping is beneiial at and around the resonane, the isolation at high requenies deteriorates with inreasing damping. This eet must be onsidered in designing the isolation system or a given appliation. Still, a reasonable inrease o damping is important sine it makes the system more robust i subjeted to inevitable spurious exitations. Also, the higher damping improves behavior o the system i the objet generates ores in a broad requeny range; e.g., as unbalaned motors generating ontinuously hanging exitation requeny during its aeleration phase. It should be onsidered that the transmissibility urves in Figure are plotted or visous damping in the isolators. Damping in elastomeri and wire-mesh or able elements is dierent, so-alled hystereti damping. This latter type o damping does not aet the preresonane and the resonane behavior o the system, but demonstrate only a minimum deterioration o the isolation at high requenies even or highly-damped isolators more in [1]. r = Amplitude-Frequeny Charateristis o Massive Blok Motion in Figure 6 Exited by an Unbalaned Rotor Figure 9 k F t m m x 1 x Dynami Model o a Basi Vibration Isolation System T1 T e h n i a l S e t i o n T1-

3 As mentioned beore, the goal o vibration isolation o vibration-sensitive objets rom the base vibration is to redue relative vibratory displaements in the work zone. Transmissibility o the base motion into the relative vibrations θ = x 1 x is or any value o m : θ rel = = n. x n 1 + n Expression is plotted in Figure 11. It is lear that transmissibility o low requeny as ompared with the natural requeny oundation vibrations into the relative vibrations is very small sine at low requenies the motions are very slow and the objet is moving ollowing the vibrating oundation. ISOLATION EFFICIENCY Isolation is the perent o vibration ore that is not transmitted through the vibration mounts and whih improves with inreasing requeny ratio. Isolation eiieny o 81.1% orresponding to a requeny ratio o., is generally adequate as shown in Table. Figure 1, the basi vibration hart, gives stati deletion vs. requeny and % o vibration isolation 1 - F. It is useul or seletion o vibration isolators/mounts and or alulations see Setion 11. A more omplete treatment o this ase o vibration isolation, onsidering more omplex and more realisti several degrees o reedom models is given in [1]. Table : Frequeny Ratio VIBRATION ABSORPTION Vibration Absorption, Perent resonane Results Attained exellent exellent very good good air poor none worse than with no mountings Absolute Transmissibility, Relative Transmissibility, rel = = r Frequeny Ratio, / n Figure Fore/Motion Transmissibility in Figure 9 System =.1.6 r =. 1.6 =. r =.7 r r = = 1.6 r =.1 T1 T e h n i a l S e t i o n. T Frequeny Ratio, / n Figure 11 Transmissibility o Vibratory Base Motion to Relative Vibratory Motion in the Work Zone

4 VIBRATION FREQUENCY Hz STATIC DEFLECTION INCHES RESONANCE NATURAL FREQUENCY ISOLATION EFFICIENCY % STATIC DEFLECTION CM T1 T e h n i a l S e t i o n REGION OF AMPLIFICATION Figure VIBRATION FREQUENCY CYCLES PER MINUTE Vibration Frequeny vs Stati Deletion o Isolators vs Isolation Eiieny. Vibration Isolation o Vibration-Sensitive Objets Sine, or this group o objets, the relative vibrations in the work zone are determined by dynami harateristis o the objet itsel, a model in Figure 1 should be onsidered. Floor oundation vibration x 1 = x sin t is transmitted through vibration isolators stiness k v, damping oeiient v to rame/bed o the objet mass M B ausing its vibrations x B = x B sin t. The work zone o the objet is between the rame/bed and its "upper unit", mass M u e.g., tool head o a mahine tool or illumination unit o a photo-lithography tool. Stiness k m and damping oeiient m desribe strutural dynami harateristis o the objet, whose strutural natural requeny is m = 1 k m M u + M B. 11 M u M B T1-1

5 Aordingly, transmissibility o the vibratory motion o the oundation into the work zone an be expressed as a produt o transmissibility x o the oundation motion X 1 to the rame motion rom expression 7 where x = X 1, x 1 = X B ; and m = M B and transmissibility o the rame motion X B to the relative motion X rel in the work zone rel rom expression 8 where x = X B, θ = X rel, and n = m rom expression 11. This operation is illustrated in Figure 1. In Figure 1, the plot a is maximum intensity a o o loor vibration displaements amplitudes ompounded rom numerous on-site measurements. It is shown in [1] that or a majority o manuaturing plants a o. m in the - Hz range and is muh smaller outside o this range or vertial loor vibrations, and a o. m in the - Hz range and muh smaller outside o this range or horizontal loor vibrations. For high preision ailities, the levels o allowable loor vibrations are reommended by BBN plots in Figure 1. The next plot b in Figure 1 illustrates transmissibility rom the loor to the objet rame or three ases: a - the objet installed on rigid mounts e.g., jak mounts or rigid isolator mounts; b - the objet installed on soter, isolating mounts lower n with the same degree o damping height o the resonane peak as the mounts in a; - the same n as in b, but greater damping. The third plot illustrates transmissibility rom the rame o the objet into its work zone; m is the strutural natural requeny o the objet. The bottom plot shows the produt o the previous three plots. An installation is onsidered suessul i the vibration amplitude in the work zone does not exeed the allowable amplitude o. It an be seen that a rigid installation results in two peaks o the relative vibration amplitude, whih oten exeed the tolerane. Both peaks are redued by using sot isolator mounts: the seond one due to redued transmissibility at high requenies per expression 6, and the irst one due to lower sensitivity o the objet struture to lower resonane requeny o the objet on soter isolating mounts. It is lear, that inreasing damping also results in redued relative vibrations. Aordingly, the requirement or an adequate vibration isolation o a vibration-sensitive objet is ormulated not as a required upper limit o the natural requeny n, but as a required upper limit o the "Isolation Criterion" Φ, Φ = n. 1a X rel X Figure 1 X a o K m X B M u M B C m K v, C v Floor a Maximum Intensity a o o Floor Vibrating b Two-Mass Dynami Model or Vibration Sensitivity o Preison Objet n n1 b Transmissibility rom Floor to Objet Frame a T1 T e h n i a l S e t i o n The magnitude o this riterion an be alulated i vibration sensitivity o the objet in the requeny range o interest is measured and its tolerane is assigned, see [1]. The objet is properly isolated i o Φ <, 1b X Transmissibility From Objet Frame to Work Zone m where o is the maximum tolerated vibratory displaement in the work zone o the objet, X is the maximum amplitude o loor vibration with requeny ; is the transmissibility into the work zone at requeny ratio o relative vibration amplitude in the work zone to amplitude o the objet rame vibration at requeny. Aording to this riterion widely validated by pratial appliations, stiness o isolators or a given installation an be inreased usually, a very desirable eature i the isolators have higher damping. o b v v1 m d Resultant Transmissibility Produt o a, b & Figure 1 Model o Vibration Transmission rom Floor to Work Zone a T1-1

6 Thus, while vibration isolation o the ore-produing objets requires reduing natural requeny in aordane with nomogram in Figure 1, isolation o a vibration-sensitive objet an be suessul even when some part o the system is at resonane, provided that the natural requeny o the isolation system and its damping are properly seleted. Vibration isolation in the latter ase is greatly simpliied i strutural stiness and strutural natural requeny o the vibration-sensitive objet are enhaned. Veloity Level db re 1 miro-inh/se m m 1 m. m Figure 1. Shok Isolation The inormation in this setion has been taken rom [] with permission o the publisher. It is oten neessary to determine the eetiveness o a shok isolator as well as the magnitude o the aeleration experiened by elements o the proteted equipment. Figure 16, similar to Figure 1, desribes the system experiening a veloity shok as illustrated by the displaement-time urves o Figure 17. The displaement o equipment y supported by isolators and subjeted to a veloity shok V is expressed by the ollowing equation: y = V 1 1 sin y t 1 y where y = system 1 k y m y Workshop ISO Oie ISO Residential day ISO Operating theatre ISO VC - A miro-inhes/se VC - B miro-inhes/se VC - C mir-inhes/se VC - D miro-inhes/se Figure 16 m z m y Support. m.1 m.6 m VC - E 1 miro-inhes/se.1 m. m One-third Otave Band Center Frequeny Hz is the natural requeny, Hz, o the elasti onsisting o hassis m y and isolator k y. Double dierentiation o equation 1 yields the aeleration experiened by the equipment hassis during shok. This is designated the transmitted aeleration and is expressed as: k z k y BBN Vibration Criteria VC or Installation o Preision Equipment Shemati Representation o Equipment, Comprised o Chassis my and Element mzkz, Mounted Upon Isolator ky. z y s Rms Veloity, miro-inh/se T1 T e h n i a l S e t i o n y = y V 1 The units o aeleration y, are linear distane inhes, m, et per seond per seond. This equation an be expressed another way, using more onvenient engineering units, as: y Transmitted Shok = = π y V = y V, g where: V = shok veloity hange, in/se. y = natural requeny o isolator, Hz. y / g = maximum aeleration experiened by hassis, expressed as a dimensionless multiple o the aeleration due to gravity. Thus, the maximum aeleration o the hassis during shok, is diretly proportional to the magnitude o the veloity hange and to the natural requeny o the isolator. Figure 18 is a graphi representation o the maximum transmitted aeleration omputed rom Equation 1. T1-1 Figure 17 t Displaement-Time Curves or Support, Chassis, and Element o Equipment Inelasti Impat

7 The maximum aeleration experiened by the hassis o the mounted equipment, as indiated in Figure 18, should not be onused with the maximum aeleration experiened by various elements o the equipment. The latter is equal to the produt o the maximum hassis aeleration y and the ampliiation ator A, whih is deined as the ratio o the maximum aeleration o the element z to the maximum aeleration o the hassis y and is given by: A = z 16 y In the absene o damping, A is a untion only o the element's natural requeny z and the isolator's natural requeny y. For an undamped system, shok transmissibility T s is related to the ampliiation ator A as ollows: T s = A 17 y z where shok transmissibility T s is the ratio o the maximum aeleration o the mass element, m z, to the maximum aeleration o the same element whih would our i the isolator's spring onstant, k y, were ininitely rigid. Using values or the ampliiation ator A as determined in [], and plotted or a range o values o damping ratio, shok transmissibility an be determined or a damped system as shown in Figure 19. The damping between m z and m y is assumed to be onstant at one perent ritial damping =.6. However, wide variations in the degree o damping have little eet on the results. Figure gives the ampliiation ator A or the system shown in Figure 16 when the support experienes veloity shok as illustrated in Figure 17. The ator A is the ratio o the maximum aeleration o mass m z to the maximum aeleration o mass m y. Shok transmissibility Ts Maximum deletion o isolator sy, in., dotted lines in/se in/se Figure 18. Natural requeny o isolator, Hz 8 6 Maximum Aeleration o Chassis my and Maximum Deletion o Linear Isolator ky Shown in Figure 16, When Support Experienes Veloity Shok as Illustrated in Figure 17. Damping ratio or isolator Damping ratio or element m z k z = Maximum transmitted aeleration y/g, solid lines Maximum Transmitted Shok T1 T e h n i a l S e t i o n Ratio Natural requeny o element Natural requeny o isolator z y Figure 19 Shok Transmissibility or System Shown in Figure 1, When Subjeted to Veloity Shok as illustrated in Figure 17 []. T1-1

8 Ampliiation ator A Damping ratio or isolator Damping ratio or element m z k z = Figure Ratio Natural requeny o element Natural requeny o isolator..1 Shok Motion o Base Base Suddenly Stops or Aelerates The time history o the sudden aeleration proess o the base in Figure. 1a is shown in Figure 1b. The analytial results taken rom [] are also appliable to the objet equipment unit dropping rom a height onto a hard surae. z y Ampliiation Fator or System Shown in Figure 16 When Subjeted to Veloity Shok as Illustrated in Figure 17 T1 T e h n i a l S e t i o n x Objet Vibration Isolator Base k W Damping onstant y V t y = t y = V t Veloity hange o base Time a System b Motion o Base Figure 1 Vibration Isolation System or Objet W a Subjeted to Shok Motion o Base with Time History b I: V = sudden veloity hange o base, in/se or m/se / r = / = damping ratio where is log derement n = undamped natural requeny o system, Hz g = gravitational onstant, 86 in/se = 9.81 m/se d max = max. isolator deletion, measured rom equilibrium position, in. or m d st = stati isolator deletion = W/k, in. or m a max = maximum aeleration o objet, in/se or m/se then, or / r. or 1., d max = a max = n 1 / r 18 d st g g T1-16

9 Figure illustrates Equation 18. When the damping is small, maximum ore transmitted to equipment is very nearly kd max. V g ξ =. = 1.6 ξ = = ξ =.1 = d max d st = a max g.. Sudden Impat on Equipment [] Impulse Time t o a FORCE TIME CURVE OF AN IMPULSE Objet Vibration Isolator Base Figure k W I Shok Eet at Dierent Damping Values b SYSTEM Sudden impulse large ore F o ating over very short time t o : I = F o t o. x Damping onstant T1 T e h n i a l S e t i o n Figure Vibration Isolation System o Objet W b Subjeted to Sudden Impat on the Objet with Time History a Sudden impat, or a sharp blow is haraterized by a large ore F o ating or a short period o time t o as shown in Figure a. For pratial purposes, suddenness is taken to mean that t o is small in omparison with the natural period o vibration o the system in Figure b. The impulse, I, is deined as the area under the ore-time urve; i.e., I = F o t o lb-se or kg m/se 19 Appliation o impulse I results in a sudden downward veloity V o the objet, V = Ig/W. The maximum isolator deletion and the maximum aeleration o the objet an be obtained by substituting V into Equation 18.. NONLINEARITIES The equations previously given or transmissibilty Setion.1 make ertain assumptions whih may not always be valid. For example, it is assumed that the damping is visous or linear resistane to relative motion is proportional to the relative veloity. The assumption greatly simpliies the analysis. However, the damping provided by wire mesh is a ombination o loalized ritional losses by individual wires and hysteresis in the ushion itsel. Damping in elastomeri materials has similar harateristis. In pratial terms, this means that the damping is a untion o displaement in addition to veloity, and the terms desribing the damping in the equations o motion are nonlinear. At resonane, where the displaement is large, the damping is high. In the isolation band, where displaement is small, the damping is negligible. This ondition gives the best o both worlds as damping is only desirable under resonane onditions. Thus, the idealized urves in Figure are on the onservative side sine they show deterioration o isolation in the high requeny ater resonane range. T1-17