# Exercise 11 ISOLATION OF MACHINE VIBRATIONS

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2 Rubber as a vibro-isolatig material Rubber is widely used as a visco-elastic isolatig elemet. I compariso with steel, rubber has high iteral dampig, a ability of soud absorptio ad very good shape elasticity. The stiffess coefficiet of the rubber vibro-isolator depeds o its hardess ad chages i a o-liear way with a icrease i the static load. The stiffess coefficiet is defied as: d Q N k = d y m, (11.1) where: Q static load of vibro-isolator, y deflectio of vibro-isolator. 600 Q [N] Q P 1 Q s P y st y y Fig No-liear characteristic curve of rubber Figure 11.1 shows the so-called hard characteristic curve. I this case, the k value icreases whe the loadig elarges. If the static load does ot exceed a certai value Q s, (poit P s i Fig. 11.1), the characteristic curve is liear. I this loadig rage, the stiffess coefficiet k has a costat value. If the static load elarges, for example to the value Q 1 (poit P 1 ), the stiffess coefficiet icreases i a o-liear way. The slope i the poit P 1 determies the k value i this loadig rage.

3 For the isolatio purposes, vibratio dampig properties of rubber are very importat. They cause the absorptio of vibratio eergy ad facilitate a getle passage through resoace. Depedig o the rubber type, the value of dimesioless dampig coefficiet h/α is i the rage from 0.0 to 0.1 (where: h viscous dampig coefficiet referred to the mass i [rad s -1 ], α - atural frequecy i [rad s -1 ]). Cotrary to steel, rubber does ot corrode ad has high fatigue resistace. This does ot apply uder low temperatures, because rubber pads may be used i temperatures from -30 o to +75 o C..3. Trasfer factor A oe-degree-of-freedom vibratig system is cosidered. The excitig harmoic force F sit acts o the mass m (Fig. 11.). Betwee the vibratig mass ad its foudatio, there is a sprig with the stiffess coefficiet k ad a viscous dashpot with the dampig coefficiet c. m F si t x k c Fig Model of the vibratig system The equatio of motio of the cosidered model ca be expressed i the followig form: m& x + c x& + k x = F si t. (11.) After some trasformatios, equatio of motio (11.) takes the form: & x&+ h x& + x = q si t, (11.3) where: c k F = h; = ; = q. (11.) m m m The particular solutio to Eq. (11.3) ca be predicted as: x = Asi( t ϕ), (11.5) where the amplitude of displacemet A is expressed by: q, (11.6) A= h ( ) + ad the phase shift agle ϕ betwee excitatio ad displacemet is determied by the followig depedece: h ϕ = arc tg. (11.7) The force which acts o the foudatio cosists of a force i the sprig ad a force i the damper. The force i the sprig is as follows: S = kx = kasi( t ϕ). (11.8) The force i the damper is equal to: R = cx& = ca cos( t ϕ). (11.9) The maximal total force actig o the foudatio is expressed as:

4 P max = S + R = ( ka) + ( ca). (11.10) max max Substitutig Eq. (11.) ad Eq. (11.6) ito Eq. (11.10), oe receives: P max h F 1+ = (11.11) h 1 + The ratio of the maximal total force actig o the foudatio P max to the excitig force amplitude F is called the trasfer factor ad will be deoted by ν i further cosideratios. Usig Eq. (11.11), the trasfer factor ca be determied i the form: ν h 1+ Pmax = = (11.1) F 1 h + Figure 11.3 shows plots of the trasfer factor ν versus the excitig frequecy γ = / (the ratio of the dimesioal excitig frequecy to the atural frequecy ) for differet values of the dimesioless dampig coefficiet h/. Depedig o the rubber type, the value of the dimesioless dampig coefficiet h/ is i the rage from 0.0 to 0.1. To receive the trasfer factor plot value of about 0.1 o the basis of Fig for these dampig values, the dimesioless excitig frequecy γ = / should be higher tha 3. 5 ν h / = 0 3 h / = 0. 1 h / = 0. h / = / Fig Plot of the trasfer factor

5 .. Example of the isolator selectio for active vibro-isolatio o the basis its static characteristics I this umerical example, a system cosistig of a fa ad a drivig motor is cosidered. The total mass of the system is m 1 = 00 kg. The rotatioal speed of the motor is = 150 rev/mi. The system is mouted i a supportig structure of the mass m = 100 kg. The foudatio should be isolated from vertical vibratios of the system by meas of four rubber isolators. After the isolatio, the trasfer factor value should ot be higher tha 0.1. Solutio The excitig frequecy is equal to: π = = [rad/s] (a) 60 The gravitatioal force actig o oe isolator is as follows: m1 + m Q = g = [N] (b) It is assumed that rubber isolators will work i the liear part of their static characteristics (Fig. 11.1). Two kids of M00-type isolators deoted by A ad B, depedig o their hardess, have bee selected to isolate the foudatio of the system. The static characteristics of M00-type isolators are show i Fig Q [N] Hardess B Max force Q = 00 N Hardess A Max force Q = 1300 N Fig Static characteristics of M00-type isolators From the characteristic curve of the M00-type isolator of the hardess A, oe ca read the static deflectio uder the Q load. This deflectio is y st = 5.89 mm. The, oe ca determie the value of the atural frequecy of such a isolator: k mg g 9.81 = = = = 0.81 [rad/s] = (c) m y m y st st The ratio of the excitig frequecy to the atural frequecy is equal to: γ = = = 3.7 (d) y [mm] 1

6 Still, oe ought to check the trasfer factor value for the limitig value of the h/ ratio of rubber isolators. ν = h Measuremet device lim h + lim = 1+ 0,1 3,7 ( 1 3,7 ) + 0,1 3,7 = 0,097 < 0,1 A scheme of the measuremet device is show i Fig Loudspeaker device 1 is mouted o its foudatio, represeted by rigid plate 3. Oe should isolate the foudatio from vertical vibratios of the loudspeaker device by meas of four rubber isolators. Plate 3 is coected with the buildig costructio by meas of four supportig pads of the kow stiffess. Vibratios of loudspeaker device 1 ad plate 3 are measured with piezoelectric sesors 5 ad 6, respectively. The sesors are coected with a set of vibratio meters ad arrow-bad aalyser 7. (e) y x Fig Scheme of the measuremet device. Course of the exercise The active isolatio of loudspeaker device vibratios should be performed. After the isolatio, the trasfer factor betwee the maximal total force actig o the plate ad the excitig force amplitude should ot be higher tha 0.1. The followig operatios are to be performed: 1) mout rigidly the loudspeaker device o the plate modellig the foudatio, ) tur o the loudspeaker device, 3) measure the vibratio frequecy ad the acceleratio amplitude alog the vertical directio, ) switch off the loudspeaker device, 5) calculate the amplitude of the excitig force, 6) o the basis of the example, select four rubber isolators, 7) calculate the value of the theoretical trasfer factor, 8) locate the selected isolators i the loudspeaker device support, 9) tur o the device, 10) measure the vibratio frequecy ad the acceleratio amplitude of the plate alog the vertical directio, 11) switch off the device, 1) calculate values of the force actig of the plate ad the trasfer factor.

7 5. Laboratory report should cotai: 1) Aim of the exercise. ) Experimetal ad calculatio results preseted i the followig table: Table 1 Loudspeaker device mouted rigidly o the plate Vibratio frequecy: f = [Hz] Agular excitig frequecy: =... [rad/s] Acceleratio amplitude alog the vertical directio: A y =... [mm/s ] Excitig force amplitude: F y = [N]. 3) Isolator selectio - calculatio results. The gravitatioal force actig o oe isolator: Q =...[N]. The selected isolator type: The static deflectio uder the Q load: y st =... [mm]. The atural frequecy of the isolator: ν =... [rad/s]. The ratio of the excitig frequecy to the atural frequecy: γ = / =... The trasfer factor value for the limitig value of the ratio of rubber isolators (h / = 0.1): ν =.... ) Experimetal ad calculatio results preseted i the followig table: Table Loudspeaker device supported o isolators Vibratio frequecy: f = [Hz] Agular excitig frequecy: =..... [rad/s] Vibratio amplitude of the plate alog the y directio: A y = [mm/s ] Force actig o the plate: F y = [N] Trasfer factor value: ν y =... [-] 5) Coclusios.

8 Refereces 1. Goliński J.: Wibroizolacja maszy i urządzeń, WNT Warszawa Parszewski Z.: Drgaia i dyamika maszy, WNT Warszawa Iformatio o isolatig materials of the Swedish compay Trelleborg AB.. PN-91/N Admissible values of acceleratios of vibratios which act o the huma body. 5. Rao S.S.: Mechical Vibratios, 3rd ed., Pretice Hall, NY, 1995.

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