EXPERIMENT OF SIMPLE VIBRATION

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Elvin Maxwell
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1 EXPERIMENT OF SIMPLE VIBRATION. PURPOSE The purpose of the experimet is to show free vibratio ad damped vibratio o a system havig oe degree of freedom ad to ivestigate the relatioship betwee the basic vibratio parameters.. THEORY.. Free Vibratios Cosider a system icludig a body of mass m movig alog oly the vertical directio, which is supported by a sprig of stiffess coefficiet k ad a egligible mass (Figure a). Figure. A system havig oe degree of freedom Let the mass m be give a dowward displacemet from the equilibrium positio ad the released. At some time t, the mass will be at a distace x from the equilibrium positio. The et force o the mass is the sprig force (-kx) which will ted to restore it to its equilibrium positio. Therefore, the equatio of the motio by Newto s secod law (Figure b). d x m kx dt or () d x x 0 () dt

2 where k (3) m Solutio of equatio VIII. with the iitial coditios x ( t 0) 0 (4a) ad dx ( t 0) 0 dt is x t) x cos t (5) ( 0 The motio ca be see i figure () (4b) T Figure The motio of free vibratio of sprig mass system f is give as The period T of the motio is determied from Figure. From equatio frequecy k f (6a) T m or f where g s mg s (7) k (6b)

3 .. Damped Vibratios Cosider a system icludig a body of mass m hug by a sprig of stiffess coefficiet k ad egligible mass. The mass is also attached by a perforated pisto to a oilcylider whose resistace is proportioal to the velocity of the mass (Figure 3a). Figure 3. Oil-cylider with a perforated pisto. Let the mass m be give a dowward displacemet from the equilibrium positio ad the released. To determie the equatio of motio, a free body diagram of the mass at a distace x from the equilibrium positio at some time t is show i figure 3b. Net forces o the mass are the dampig force (-Cdx/dt) which will ted to restore its equilibrium positio, the sprig force ad the iertia force. C is the dampig coefficiet of the oil. From the Newto s secod law d x m kx C dt or dx dt (8) d x dx r x 0 (9) dt dt where k C C ad r (0) m m km r is called the dampig factor (ratio) i equatio 9 3

4 Solutio of the equatio of motio is depedet o the dampig ratio. If r>, a operiodic movemet which approaches the equilibrium positio is obtaied (Figure 4). r> x( t) ( r r ) t ( r r ) t Ae Be () A ad B are costats which are depedet o the iitial coditios of motio. If r=, the motio correspodig to the critical dampig ratio is approachig the equilibrium positio with time. r= x r t ( t) ( A Bt) e () If r<, a motio which has a period T ad a decreasig amplitude with time t is obtaied. r< x t e r t ( ) ( Acos r t Bsi r t (3) Figure 4. The motio of damped free vibratio of sprig-mass system The followig relatio is foud betwee two cosecutive wave widths, as ca be see from equatio 3. x rt xe (4) The logarithmic decremet is defied as follows to calculate the dampig ratio. x log (5a) e x where x: th. maximum amplitude durig vibratio; x+: is the ext maximum. 4

5 If we cosider the drop i amplitude i successive cycles the the log decremet is give by x log (5b) 0 e x where x0: vibratio amplitude of iitial cycle; x: vibratio amplitude of. cycle. Figure5. Damped vibratios with dampig ratio less tha. r t (6) Vibratio period T is T d r where ωd is the damped atural frequecy. Note that time iterval betwee two cosecutive peaks i the graphic is equal to period, T. Because of this, time t betwee x ad x cycles must be equal to period T. So t=t. The dampig ratio is calculated as follows (7) r r t (8) r or / r (9) ( / ) for small dampig we have r log x e x r. So the dampig ratio, (0) 5

3. EXPERIMENTAL APPARATUS The experimetal rig basically cosists of a frame free to move vertically o

The movable frame is attached to the fixed frame. A recordig pe is also attached to the fixed frame.

to the fixed frame. The mass of the movig frame is approximately.7 kg.

Various sprigs, whose sprig costats are to be determied experimetally, will be available.

6 3. EXPERIMENTAL APPARATUS The experimetal rig basically cosists of a frame free to move vertically o roller guides, removable weights ad a drawig pe attached to frame. The movable frame is attached to the fixed frame. A recordig pe is also attached to the fixed frame. A paper rollig system drive by a electric motor ad a perforated pisto i a oil-cylider are also attached to the fixed frame. The mass of the movig frame is approximately.7 kg. There are some mass blocks that will be added o the movable frame durig experimet. Various sprigs, whose sprig costats are to be determied experimetally, will be available. The drawig pe records the sprig vibratios o the paper strip movig at a speed 0.0 m/s. (a) Experimetal rig (b) Differet sprigs ad removable weights (c) Plotter (d) Frame (e) Dashpot Figure. Experimetal rig system 6

respectively. These data are goig to be used to calculate the sprig coefficiet.

7 4. EXPERIMENTS 4.. Free Vibratio Tests Below steps are performed for each oe of the sprigs: a) Elogatio values are measured ad recorded from a referece poit while the frame is empty ad carryig, ad 3 kg masses respectively. These data are goig to be used to calculate the sprig coefficiet. b) Vibratio amplitude ad frequecy are determied o the recordig surface while the frame is empty, ad while it has a,, 3 kg mass respectively. Amplitude [mm] Time distace [mm] Figure. A example vibratio amplitude plottig while F=5.7 N 4.. Damped Vibratio Tests Below steps are repeated for each sprig after the oil-cylider is charged with oil ad the pisto is coected with the frame. a) The portable frame is released from a certai height ad the amplitude-time curves are determied o the recordig surface whe the frame is empty ad whe additioal masses are attached respectively. Also for each mass added, tests are performed agai i the /4 tur of the adjustig ut of pistos holes. Amplitude [mm] Time distace [mm] Figure. A example vibratio amplitude plottig from damped vibratio tests b) Attach a suitable mass o the frame ad record the motio after the frame is released from a iitial displacemet. 7

8 5. GRAPHS a) The relatios betwee the force ad elogatio are read o the scaled paper by usig displacemet values resultig from static loads. The sprig coefficiets are calculated from the curve for each oe of the sprigs. b) The experimetal period ad frequecy values are determied from the vibratio plottig of each mass-sprig system. Also the theoretic period ad frequecy values also will be calculated usig Equatio 6 a, b for the same mass-sprig systems. Usig these results, the whole data are recorded o the Table. Table. Sample table for free vibratio test. Sprig coefficiet k (kn/m) Mass m (kg) Period (s) Frequecy (Hz) Test Theory Test Theory Static displacemet (cm) Below graphics will be obtaied from the experimetal data tabulated above. i) Plot chage i frequecy as a fuctio of mass for each sprig.(test ad theory) ii) Plot the chage i frequecy as a fuctio sprig costat for each sprig. iii) Plot the curve with the static displacemets o the abscissa ad the frequecies o the ordiate axes. c) The velocities, which correspod to the each pisto-hole-adjustig ut positios, are calculated usig the vibratio plottig of each mass-sprig-dashpot system. These values ca be recorded i Table. Table. Sample table for experimetal data Adjustig ut positio Closed /4 tur /4 tur 3/4 tur 4/4 tur Force Pisto velocities (cm/s) Force Adjustig ut positio Closed /4 tur /4 tur 3/4 tur 4/4 tur Dampig coefficiet of oil-cylider C 8

9 From the experimetal data ad calculated data i the tables, below graphics will be obtaied: ) Chage i pisto velocity as a fuctio of the adjustig ut positio ) Chace i force with respect to pisto velocity. 3) Chage i the dampig coefficiet of oil-cylider with respect to the adjustig ut positio. 4) Calculate the dampig ratio of ay mass-sprig system. 6. OBSERVATIONS AND DISCUSSIONS Observatios ad results from figures i sectio 5.Graphs will be discussed. While discussig, it will be useful to look for aswers to the followig questios: a) What is the relatioship betwee the force ad displacemet? b) Frequecy of free vibratio chages with respect to the mass ad sprig coefficiet. Compare the frequecies determied from the theory with those determied experimetally. Which error sources exits? What are the other differeces betwee theoretical ad experimetal results? How are they explaied? What is the relatioship betwee frequecy ad static displacemet? Does it give the same relatio regardless of the mass ad sprig chose? c) What is the chagig i the pisto velocity with respect to the adjustig ut positio? How are the force ad velocity related? Is it i accordace with the assumptios used i the motio? Which differeces are observed betwee the theory ad experimet? What are the error sources ad how ca they be decreased? What are your observatios o the damped-free vibratio experimet? d) Search other techiques to measure vibratios. e) What are the advatage ad disadvatages of the vibratios of mechaical parts? Explai briefly. 9

EXAMPLE CALCULATIONS Mass & elogatio data are i the Figure ad Table. (.7.7).9,8 k 43 N (7 0).0 Figure. mass versus x relatio System s Total Mass m (kg).7.7 3.

5 m k Obtaiig Period & Frequecy (Experimetally) (9 6).0 Amplitude & time plottig for free vibratio experimet is i the Figure. Figure. F=5.7 N ad m=.

10 EXAMPLE CALCULATIONS Mass & elogatio data are i the Figure ad Table. (.7.7).9,8 k 43 N (7 0).0 Figure. mass versus x relatio System s Total Mass m (kg) Elogatio (mm) (3.7.7).9,8 (6 7).0 (4. 3.7).9, m k 090 N k 3 m N 3 m i i k N mea m k Obtaiig Period & Frequecy (Experimetally) (9 6).0 Amplitude & time plottig for free vibratio experimet is i the Figure. Figure. F=5.7 N ad m=.7 kg T f Experimet Experimet 7 0 T 0.35 sec Experimet.86[ Hz] Obtaiig Period & Frequecy (Theoretically) k f Hz T m f T.7 59 T 0.8sec f Obtaiig Static Displacemet Δs (mm) s mg.7 * m k. 04 0

Experimetally calculatig the dampig ratio r Dampig ratio r is calculated experimetally usig amplitudes of the cosecutive two peaks ad the formula of r log x e x vibratio; x+ is the ext maximum.

Figure m=.7 kg ad full closed positio of the whole holes, k=090 N/m. We ca use C r. We eed to determie C before to calculate.

11 Experimetally calculatig the dampig ratio r Dampig ratio r is calculated experimetally usig amplitudes of the cosecutive two peaks ad the formula of r log x e x vibratio; x+ is the ext maximum. where r: dampig ratio, x: th. maximum amplitude durig 5 r l 0,043.5 Figure m=.7 kg ad full ope positio of the whole holes. To calculate the dampig ratio r of a over damped vibratig system. Figure m=.7 kg ad full closed positio of the whole holes, k=090 N/m. We ca use C r. We eed to determie C before to calculate. For this, we ca km use C mg kx v, where, m=mass, k=sprig coefficiet, v=pisto velocity. Before to calculate, we eed to also determie pisto velocity VPisto first. X Pisto V m Pisto 6.984*0 s t 63/ 0 mg kx.7*9.8090*0.05 C v Pisto 6.984*0 C 479 So; r km 090*.7 Ns m

FREE VIBRATION RESPONSE OF A SYSTEM WITH COULOMB DAMPING

Mechaical Vibratios FREE VIBRATION RESPONSE OF A SYSTEM WITH COULOMB DAMPING A commo dampig mechaism occurrig i machies is caused by slidig frictio or dry frictio ad is called Coulomb dampig. Coulomb dampig