Mechanic al Vibration s Mte-321l Tahir Tahir Hasan Institute of Mechatronics Mechatronics Engineering, Engineering, UET, Peshawar 12PWMT!1"# lass $ !% Instructor& 'ir (ai) Muha**a+
ab $ 1 aout of Mechanical Vibrations ab. Objectives: The basic ob/ecti0e ob/ecti0e of this lab is to gain gain nowle+ge about about +ierent +ierent in+s of aaratus that are resent in the Mechanical Vibrations lab. 4 brief +escrition an+ alication of each instru*ent is shown.
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Free Vibration Apparatus:
Vibrations are a *a/or art of the real worl+. There are 0ibrations in *an of the hsical heno*enon heno*enon we see for e5a*le a car engine, a hanging bri+ge or e0en our 0oice is trans*itte+ in the for* of 0ibrations. The free 0ibration aaratus is use+ to 0isuali)e free 0ibrations. It consists of a sring *ass sste* an+ a +a*er. The whole whole asse*bl asse*bl is calle+ calle+ a free free 0ibratio 0ibration n aaratus aaratus because no e5ternal force is alie+ +uring the 0ibration. The e5citation force is /ust ro0i+e+ once an+ then its resonse is lotte+. It can be use+ use+ to erfor erfor* * a nu*be nu*berr of e5er e5eri*e i*ents nts such as +eter +eter*in *ining ing the +a*ing in a sste*, in0estigating the nature of 0ibration, 6n+ing the +a*ing coe7cient an+ *an *ore.
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Forced Vibration Apparatus:
4 force+ 0ibration aaratus is the sa*e as a free 0ibration aaratus but it is use+ to +e*onstrate the force+ 0ibrations. It consists of a *otor an+ a sring *ass sste* that causes force+ oscillations of so*e fre8uenc, which is in the control of the user. 4 0ertical bar is attache+ to the *otor 0ia a sring. When the *otor rotates the 0ertical bar oscillates an+ ro+uces force+ oscillations. 4 +a*ing sste* is also resent in this aaratus. 9a*er is use+ as an energ absorbent in the 0ibrating sste*. The +a*ing of the 0ibration can also be 0arie+ using a nob on to of it.
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Torsional Vibration Apparatus:
This e5eri*ental e5eri*ental unit unit is use+ to e5lain torsion an+ torsional 0ibration in e5eri*ents. The core of the e5eri*ent is a *etal torsion bar. With the ai+ of chucs, *ass +iscs of 0aring inertia can be attache+ to the bar. It this wa it is ossible to asse*ble torsional 0ibrator sste*s with u to three *asses. Vibrations can be in+uce+ on the sste* using an electrical e5citer sulie+ fro* the control unit. 4n a+/ustable +a*er can be use+ to in:uence the 0ibrations. 'haft enco+ers on the chucs ro0i+e the a*litu+e of 0ibration as an electrical signal. The control control unit con+itions con+itions these these signals an+ an+ *aes the* a0ailable
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Static and Dynamic Balancing Apparatus:
The *ain ele*ent ele*ent of the bench bench to unit is is a s*ooth shaft to which four 0ariable unbalance weights can be attache+ at an angle or +istance. The rotor is suorte+ hori)ontall in ball bearings an+ is +ri0en b a see+controlle+ *otor. The see+ is *easure+ electronicall an+ shown on a +igital +isla. (or +eter*ination of the unbalance weight b *easuring the balance of *o*ents the +ri0ing belt can be re*o0e+. Using +ierent weights +ragging on the ulle +e6ne+ *o*ents can be e5erte+ to the shaft. The can be co*are+ with those cause+ b the unbalance weights when rotating. The transarent hoo+ re0ents contact with the rotating arts an+ ro0i+es a goo+ 0iew of the rotor. (astening of the suorting base with elastic ele*ents neutrali)es un+esirable 0ibrations.
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Inverted Pendulum Trainer:
In0erte+ en+ulu* sste* is a nonlinear unstable sste*, an i+eal e5eri*ent latfor* for teaching control theories an+ con+ucting 0arious
control e5eri*ents. Man abstract control concets, such as the stabilit an+ the controllabilit of a control sste*, can all be shown 0isuall through the in0erte+ en+ulu* sste*. In a++ition to e+ucational uroses, an in0erte+ en+ulu* is also a research area for *an researchers of *o+ern control theories. Through the continuous research on new was of controlling in0erte+ en+ulu*, researchers ha0e +e0eloe+ new control *etho+s, an+ al the* to the high tech areas such robotics , thans to the characteristics as aeronautical engineering and robotics, characteristics of the sste*, such as high-or+er, instabilit *ulti0ariables, non-linearit an+ strong couling.
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yroscope:
;roscoe is a +e0ice for *easuring or *aintaining *aintaining orientation, base+ on the rinciles of angular *o*entu*. Mechanical groscoes ticall co*rise a sinning wheel or +isc in which the a5le is free to assu*e an orientation. 4lthough the orientation of the sin a5is changes in resonse to an e5ternal tor8ue, the a*ount of change an+ the +irection of the change is less an+ in a +ierent +irection than it woul+ be if the +is were not sinning. 4lications of groscoes inclu+e inertial na0igation sste*s where *agnetic co*asses woul+ not wor
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De!ection o" Beam Apparatus:
The bench *ounte+ *ounte+ aaratus aaratus has a hea0 hea0 steel base with a 65e+ suort at one en+ an+ a *o0eable suort at the other. The suorts can be 6tte+ with nife e+ges or cla* lates one of which er*its hori)ontal *o0e*ent for an encastre bea*. 4 steel bea* an+ two loa+ hangers are sulie+ together with two +ial gauges for *easuring bea* +e:ections an+ sloes. This e8ui*ent e8ui*ent is art of a range +esigne+ +esigne+ to both +e*onstrate +e*onstrate an+ e5eri*entall e5eri*entall con6r* basic engineering rinciles.
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#$irling o" S$a"t Apparatus:
The *o+es of of oscillation an+ resonances resonances of rotors with continuous *ass +istribution can be clearl +e*onstrate+ using this unit. 9ue to the use of thin, elastic rotor shafts *a+e of high-strength steel, the oscillator heno*ena can be easil un+erstoo+. 4 range of shaft +ia*eters an+ the free choice of the bearing arrange*ent *ae it ossible to erfor* a wi+e 0ariet of e5eri*ents. 4+aters in the bearings co*ensate for +ierent +ia*eters, catch bearings li*it the a*litu+e of the oscillation. The free+o* of *o0e*ent of the rotor is assure+ b an elastic couling. 4 a0al rotor with +iscrete *ass +istribution can be asse*ble+ using a *ass +isc.
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%ompre$ensive Fly #$eel Apparatus:
4 :wheel is a rotating *echanical +e0ice that is use+ to store rotational energ. (lwheels ha0e a signi6cant *o*ent of inertia an+ thus resist changes in rotational see+. The a*ount of energ store+ in a :wheel is roortional to the s8uare of its rotational see+. Energ is transferre+ to a :wheel b aling tor8ue to it, thereb increasing its rotational see+, an+ hence its store+ energ. on0ersel, a :wheel releases store+ energ b aling tor8ue to a *echanical loa+, thereb +ecreasing the :wheel>s rotational see+.
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%entri"ugal Force Apparatus:
This aaratus aaratus is use+ to 0erif that centrifugal centrifugal force 0aries 0aries with the s8uare s8uare of the see+, the rotating *ass, an+ the ra+ius of gration. The bench-*ounte+ unit houses a see+-controlle+ *otor, which carries a hori)ontal asse*bl. This asse*bl co*rises a boo* which rotates. The see+ of rotation of the boo* is +islae+ on the front anel along with the +igital force rea+ing. 4 uni8ue feature is that all three 0ariables can be set an+ the centrifugal force +irectl rea+ fro* the +igital force +isla +is la..
Ot$er Apparatus: 'o*e other aaratus which are resent are *ass attache+ to sring aaratus, torsion 0ibration aaratus, connecting ro+s, co*ressional srings. 4ll of the aaratus are use+ to +e*onstrate the natural ti*e erio+ an+ fre8uenc of 0ibrations. ?alancing of *asses aaratus is resent which is use+ to +e*onstrate the heno*enon heno*enon of balancing of *asses base+ on the rincile of *o*entu* an+ 0ibrations.
ab $ !2 To T o 6n+ 6 n+ the @atu @atural ral (re8uen re8uenc c an+ a n+ ti*e erio+ of a Torsiona orsionall 'ste*. 's te*. Apparatus: • •
Torsional Torsional 'ste* 'ste* or
T$eory: 4 torsional sste* consists of a circular +is attache+ to a rigi+ suort 0ia a circular shaft. It is use+ to 0isuali)e the heno*enon of torsional 0ibrations. Torsional Torsional 0ibrations 0ibrations are are base+ on the the rincile of of olar *o*ents *o*ents of inertia. inertia. Parameters o" t$e torsional system: weight of the +is A 2B.% oun+s weight of chuc A 1.% oun+s length of shaft A 1C#.%c* +ia*eter of shaft A !."3c*
Procedure: • • •
•
•
•
•
Ensure that the aaratus is in e8uilibriu* osition. Deset the stowatch to )ero. Then al a coule on the the circular circular +is, but be be sure that that it is in the the safe li*its an+ will not +a*age the aaratus. 4s soon as ou release the +is start the sto watch. The +is will start an oscillator *otions. ount the nu*ber of oscillations oscillations an+ the ti*e taen taen to co*lete the oscillations. @ow bring the aaratus to rest again, an+ al a coule on the +is. @ow again count the oscillations an+ the ti*e taen to co*lete the oscillations. Deeat these stes 3 to C ti*es.
Observations and %alculations: &at$ematical Formul Formulation: ation: W n=
W n
√
K T I
A natural (re8uenc
K T =
GJ L
; A oungs *o+ulus F A G+232
@u*ber of oscillations
Ti*e
(re8uenc scillationsT i*e
2 + ¿ f 3
f ¿ 3 ¿
avg =¿ ¿ f ¿
Time period '
()" avg avg '
ab $ !3 9eter*ine the ti*e erio+ an+ natural fre8uenc of *ass attache+ to sring Apparatus: • •
Mass string sste* 'to watch
T$eory: Mass attache+ to sring is the si*lest for* of a en+ulu*. The string use+ is an ine5tensible one. When so*e e5citation is gi0en to the *ass in articular +irection it oscillates about its *ean osition. These oscillations are erio+ic an+ are calle+ as 0ibrations. The *ass string sste* can be oerate+ in three *o+es. In 6rst *o+e it
0ibrates fro* left to right. In the secon+ *o+e its 0ibrations are torsional in nature, while it the thir+ *o+e it 0ibrates in to an+ fro *anner.
Procedure: • • • • •
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Ensure that the aaratus is in its e8uilibriu* osition Deset the stowatch @ow +islace the aaratus towar+s an e5tre*e osition i.e. left or right. @ow as soon as ou release the aaratus start the sto watch. ount the nu*ber of oscillations it *aes an+ the ti*e taen to *ae those oscillations. @ow again bring the aaratus to rest an+ reeat the abo0e stes, 3 ti*es.
&ode *: To To oerate in *o+e *o+e 2 the *ass *ass is sub/ecte+ sub/ecte+ to a coule. coule. When the force is release+ the *ass starts to oscillate in a torsional *anner. The sa*e roce+ure roce+ure is is reeate+ i.e. noting the ti*e erio+ an+ the nu*ber nu*ber of oscillations. Deeat the roce+ure 3 ti*es to get an a0erage 0alue. • • •
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&ode +: The roce+ure roce+ure is sa*e but the e5citation e5citation this ti*e is in an e5tre*e e5tre*e osition osition towar+s the bac or forth . Ti*e an+ an+ oscillations oscillations are note+ note+ to calculate the ti*e erio+ erio+ an+ natural natural fre8uenc. •
•
Observations and %alculations %alculations:: •
s. no
Mo+e 1 @u*be Ti* r e f
1 2 3
•
Mo+e 2
(re8uenc cless ec
s. no
@u*be Ti* r e f
(re8uenc cless ec
1 2 3
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s. no
Mo+e 3 @u*be Ti* r e f
(re8uenc cless ec
1 2 3
ab $ !C To T o 6n+ 6 n+ the fre8uen fre8uenc c an+ a n+ +a*i + a*ing ng ratio in the case of free 0ibration. 0ibration.
Apparatus: • • • • •
(ree 0ibration aaratus 'to watch Masses ;rah aer Pencil
T$eory: (ree 0ibrations are the te of 0ibrations in which e5citation force is ro0i+e+ once an+ then re*o0e+. The 0ibrations +een+ uon the *ass an+ the sring constant. There is a +a*er also resent +ue to which there e5ists a logarith*ic +ecre*ent in the 0ibration an+ with the assage of ti*e the a*litu+e beco*es )ero.
Procedure: •
• •
•
•
• • •
•
ut a longitu+inal sheet of grah aer an+ wra it aroun+ the roller on the aaratus. Put a encil or a en to note the resonse of the 0ibration. @e5t a++ *asses to the aaratuss ro0i+e+ slots for *asses. n the +a*er bloc there is a screw which can be use+ to increase or +ecrease the +a*ing in the sste* @ow gi0e +e:ection to the sli+er of the aaratus, as soon as ou gi0e +e:ection start the *otor. 4t the sa*e ti*e start the sto watch to *easure the ti*e taen for oscillations, an+ count the nu*ber of oscillations also. When the a*litu+e of 0ibration beco*es )ero, sto the *otor. @ow a++ *ore weights an+ reeat the abo0e stes. Each ti*e ou see the resonse on the grah aer ou will notice a change in the a*litu+e of the 0ibration. This change change in a*litu+e a*litu+e is +ue to the the change of *asses *asses in the sste*, which can ro0e the fact that *asses eect the 0ibrations.
Observations and calculations: Desonse of the sste* is gi0en b the e8uation&
Where the e5onential ter* gi0es us with +ecre*ent in the a*litu+e. ogarith*ic +ecre*ent is &
ab $ !% ?alancing of unbalance+ *asses using balancing *achine. Apparatus: • •
?alancing *achine aaratus Masses
T$eory: The aaratus aaratus consists of of eccentric +is +is of nown *asses attache+ attache+ to a hori)ontal hori)ontal shaft. The shaft is connecte+ to outut of a *otor through a J0ee beltK. There are slots ro0i+es on the +iss to ass *asses to the*. In a balance+ con+ition there is a less 0ibration while in an unbalance+ con+ition there is a huge a*ount of 0ibration.
Procedure: •
•
• •
•
•
•
9isconnect the *otor fro* ower outlet to eli*inate the +anger of an electric shoc. @ow a++ *asses to the eccentric +iss, two *asses are a++e+ on a nown angle an+ the other two are a++e+ in unnown ositions at constant ra+ii +iss. The *asses are tightene+ tightene+ carefull carefull to a0oi+ a0oi+ sliage +uring +uring oeration. oeration. When the *otor is turne+ on the shaft start to rotate an+ there is a large a*litu+e of 0ibration +ue to unbalance+ *asses. If the *achine is oerate+ in balance+ con+ition there will be *ini*u* 0ibration. The angles of of two *asses are nown an+ two are are et unnown unnown an+ +eter*ine+. When the unnown angles are foun+ out the *asses are lace+ at those angles an+ it is obser0e+ that the a*litu+e of 0ibration now +ecreases to a *ini*u* le0el.
Observations and %alculations: Place
Mass <b.=
Da+iu s
4ngle
*r *r
*ros L
*r 'inL
Mo*e nt 4r* <5=
*r5o sL
*r5'i nL
ab $ ! To T o balan ba lance ce unbal u nbalanc ance+ e+ *asse * asses s using static an+ +na*ic balancing aaratus. Apparatus: • • •
'tatic an+ +na*ic balancing aaratus 4llen e Masses
T$eory: The *ain ele*ent ele*ent of the bench bench to unit is is a s*ooth shaft shaft to which which four 0ariable 0ariable unbalance weights can be attache+ at an angle or +istance. The rotor is suorte+ hori)ontall in ball bearings an+ is +ri0en b a see+-controlle+ *otor. The see+ is *easure+ electronicall an+ shown on a +igital +isla. (or +eter*ination of the unbalance weight b *easuring the balance of *o*ents the +ri0ing belt can be re*o0e+. Using +ierent weights +ragging on the ulle +e6ne+ *o*ents can be e5erte+ to the shaft. The can be co*are+ with those cause+ b the unbalance weights when rotating. The transarent hoo+ re0ents contact with the rotating arts an+ ro0i+es a goo+ 0iew of the rotor. (astening of the suorting base with elastic ele*ents neutrali)es un+esirable 0ibrations.
Procedure: • •
• •
• •
9isconnect the aaratus fro* ower sul. @ow re*o0e the transarent to, an+ on the shaft there are rectangular blocs in which *asses can be attache+. The rectangles rectangles can rotate rotate as well as the can sli+e sli+e on the shaft. shaft. To To *easure *easure the +istance +istance fro* the the *ean osition osition a 0ertical 0ertical scale is resent resent an+ the to *easure the angle fro* the *ean osition a gra+uate+ +is with angles is also resent. We can a+/ust the angular osition an+ the linear osition to our own +esire. Then weights are a++e+ a++e+ an+ are are fastene+ tightl tightl..
• • • • •
@ow the to is again lace+ on the aaratus an+ the *otor is starte+. The see+ of the *otor can be be seen on the +isla +isla unit. unit. We can also control the see+ of the *otor. In an unbalance+ con+ition there will be a 0ibration in the aaratus. This rocess rocess can be use+ to either 6n+ the osition osition of the unbalance+ unbalance+ *asses, or the unbalance+ *asses the*sel0es, if osition is nown.
Observations and calculations: Place
Mass
Da+iu s <**=
4ngle
*r *r
*ros L
*r 'inL
Mo*e nt 4r* <5=
*r5o sL
*r5'i nL
ab $!B To T o 6n+ 6 n+ the ti*e eri erio+ o+ an+ a n+ fre8uenc of an E+ge susen+e+ onnecting ro+ Apparatus: • •
E+ge susen+e+ connecting ro+ 'tan+
T$eory: The connecting connecting ro+ or con ro+ ro+ connects the iston to the cran or cranshaft. Together with the cran, the for* a si*le *echanis* that con0erts recirocating *otion into rotating *otion. It ser0es as the inter*e+iate con0ersion +e0ice between two tes of *otion.
Procedure: •
• • • • •
•
The connecting connecting ro+ is lace+ on the the stan+ s*aller s*aller en+ uwar+ uwar+ an+ larger larger en+ +ownwar+. The aaratus is brought brought into e8uilibriu* e8uilibriu* osition. osition. @ow a s*all angular +islace*ent is gi0en to the ro+ in either +irection. @ow start the stowatch to count the ti*e taen to co*lete the oscillations This rocess rocess is reeate+ reeate+ three three ti*es. @ow the ro+ is lace+ usi+e +own i.e. the larger +ia*eter hole is uwar+s an+ the other +ownwar+s. The sa*e roce+ure roce+ure is is reeate+ for this osition osition as well well an+ the oscillations oscillations an+ ti*e is note+ to calculate the fre8uenc.
Observations and %alculations %alculations:: '.n o
@u*ber Ti*e for for of oscillatio 0ibratio ns
(re8uen c
ns
'.n o
<'=
@u*ber Ti*e for for of oscillatio 0ibratio ns ns <'=
(re8uen c
ab $ !# 9eter*ination 9eter *ination of sring constant for a gi0en free 0ibration aaratus Apparatus: • • • •
(ree 0ibration aaratus ;rah aer Masses Pointer
T$eory: 'ring constant is +e6ne+ as the stiness in the sring. It is a *easure of restoring force the sring alies +ue to the +islace*ent. In the aaratus we lace *asses in the carriage, an+ also we gi0e so*e initial +islace*ent to the aaratus. There is stiness of the sring ele*ent that arises. is the sring constant, in @ewtons er *eter <@*=, an+ 5 is the +islace*ent of the sring fro* its e8uilibriu* osition. The sring constant, , is reresentati0e of how sti the sring is. The stier a sring is the *ore +i7cult is to stretch it.
Procedure: • • • •
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n a free 0ibration aaratus roll a grah aer on the roller. @ow lace so*e weight in the carriage. 4+/ust the +a*ing of the sste*, fro* the ro0i+e+ +a*er. @ow gi0e the carriage of the aaratus a slight +islace*ent. The aaratus will start 0ibration. 4t the sa*e ti*e start the *otor to recor+ the resonse of the 0ibrating sste*.
• •
•
4s soon as the a*litu+e beco*es )ero sto the *otor. 4++ *ore *ass to the sste* an+ again gi0e +e:ection an+ note the resonse of the sste* with the hel of a grah aer. This rocess rocess can be reeate+ an an ti*es with with the change of of *asses to ensure there is a change in force which results +islace*ent.
Observations and %alculations %alculations::
ab $ !" (in+ the natural ti*e erio+ an+ fre8uenc of a co*rehensi0e :wheel aaratus. Apparatus: • •
o*rehensi0e :wheel aaratus 'to watch
T$eory: 4 :wheel is a rotating *echanical +e0ice that is use+ to store rotational energ. (lwheels ha0e a signi6cant *o*ent of inertia an+ thus resist changes in rotational see+. The a*ount of energ store+ in a :wheel is roortional to the s8uare of its rotational see+. Energ is transferre+ to a :wheel b aling tor8ue to it, thereb increasing its rotational see+, an+ hence its store+ energ. on0ersel, a :wheel releases store+ energ b aling tor8ue to a *echanical loa+, thereb +ecreasing the :wheel>s rotational see+. o**on uses of a :wheel inclu+e, ro0i+ing continuous energ when the energ source is +iscontinuous. (or e5a*le, :wheels are use+ in recirocating engines because the energ source, tor8ue fro* the engine, is inter*ittent.
Procedure: • • •
• •
Mae sure that the :wheel aaratus is in an e8uilibriu* osition @ow reset the sto watch. ;i0e a slight +islace*ent to the :wheel an+ release it, an+ at the sa*e ti*e start the sto watch. @ote the nu*ber of oscillations an+ also the ti*e taen to co*lete it. @ow again bring the aaratus to rest an+ reeat the abo0e rocess three to four ti*es an+ *easure ti*e for a seci6c nu*ber of 0ibrations.
Observations and %alculations %alculations:: '. n o 1 2 3
@u*b er of Ti*e oscilla tions
(re8uenc scillation sti*e
ab $1! 9e*onstration an+ use of the (orce+ Vibration 4aratus. Apparatus: • •
(orce+ 0ibration aaratus ;rah aer
T$eory: The oscillator oscillator consists of a bea* *ounte+ *ounte+ in ball ball bearings at at one en+N a helical helical sring is hung on the other en+ of the bea*. The attach*ent of srings, the e5citer an+ a +a*er to a erforate+ anel er*its a wi+e range of +ierent set-us to be rero+uce+. Either an unbalance e5citer or a +islace*ent e5citer generate 0ibrationsN the fre8uenc of the 0ibrations can be a+/uste+ using the electronic control unit. The +islace*ent e5citer can be 6tte+ +irectl to the base of the sring. The 0ibrations can be +a*e+ using the a+/ustable 0iscosit +a*er. To recor+ 0ibration rocesses o0er ti*e, a *echanical +ru* lotter is inclu+e+.
Procedure: • • •
•
• • • •
• •
• •
4 grah aer is rolle+ on to the +ru* roller to recor+ the 0ibration. @ow the aaratus is connecte+ to a ower sul unit. There is a controller controller resent resent that controls controls the see+ of the *otor which which *eans it can control the fre8uenc of 0ibration. 4 linear +ashot is also resent, with a bolt to 0ar its +a*ing o0er a wi+e range. The +a*ing +a*ing is re+uce+ re+uce+ to a low 0alue 0alue an+ the *otor is turne+ turne+ on. We will see that the bea* starts to 0ibrate. @ow the see+ of the *otor is increase+. 4s the see+ of the *otor increases the fre8uenc of 0ibrations also increases. @ow for the secon+ ti*e the +a*ing 0alue is a+/uste+ to a *a5i*u* 0alue. 4gain the *otor is starte+ at the sa*e fre8uenc, now we can note that there is less 0ibration in the bea*. The osition of the +a*er +a*er can also be be 0arie+. It it is *ore close to the 0ibrating en+ there will be *ore +a*ing an+ less 0ibration, but on the other if it is far awa fro* the 0ibrating en+ the *agnitu+e of 0ibration will increase.
