ABSTRACT
The The expe experim rimen entte tteste sted d two two type typess of vibr vibrati ation on syste system m which which are undamped undamped fre freee vibration and damped free vibration . Free vibration system is a system where there is no external forces subjected to the system after the initial force or disturbance is exerted which causes the system to vibrate on its own.If no energy is lost or dissipated in friction or other resistance during oscillation, the vibration is known as undamped vibration. If any energy is lost in this way wa y, however, it is called damped vibration. In many physical systems, the amount a mount of damping is so small that it can be disregarded for most engineering purpose. However, consideratio consideration n of damping damping becomes becomes extremely extremely important important in analysing analysing vibratory systems near resonance.
The system set up for this experiment is the basic cantilever beam with a spring attached on the top of the beam. For undamped free vibration system, the lengths of the lever arm, for hanging the spring of the cantilever beam were set differently for for total of times and ! different stiffness of the spring. The different in length and spring stiffness could result in different behaviour of the vibration system. The experimental value of natural fre"uency obtained will be different according to the changes of the set up.
#s for damped free vibration system, the spring stiffness and lever arm length for spring is constant.Three different length of lever arm length for damper was set differently. $ith ith damp damper er attach attached ed on the the syst system, em, the the ener energy gy of the the syste system m will will is lost lost durin during g the the oscill oscillatio ation n differ different ently ly accordin according g to the length. length. The damped fre"uency fre"uency for the ! tests tests of damped free vibration system could be analysed and thus, it might help to illustrate the effect of damping on the system. For the design engineer, it is very crucial for them to consider the mechan mechanical ical vibrati vibration on of a structu structure re to avoid avoid from from any catastro catastrophi phicc failur failures. es. This This would would happen if the structure is excited at its resonance fre"uency% the damping is low and excessive of vibrations.
OBJECTIVES
The objective of this experiment is to& '
(ompar (omparee the the theor theoreti etical cal natu natural ral fre" fre"uen uency cy for for vario various us cons constan tants ts and and leve leverr arms arms with with the values obtained by measurement.
•
Illu Illust stra rate te the the eff effec ectt of dam dampi ping ng on on the the deca decay y beha behavi vior or..
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THEORY
PROCEDURE
DATA COECTIO! A
"ree And Undamped O#$i%%ation Tab%e& Geometric Dimension of the Beam
en't( of Beam) *mm+ ,ei'(t of Beam)
)**
)**
)**
)+*
)+*
.-
.-
.-
.-
.-
m *-'+
2
ever Arm) a *mm+ Sprin' Con#tant) $ *!.mm+
B
-*
**
**
!**
!**
*./
*./
.//
.//
!.*0
"ree And Damped O#$i%%ation Tab%e& Geometric Dimension of the Beam
en't( of Beam) *mm+ ,ei'(t of Beam)
.-
.-
.-
**
!**
*
*./
*./
*./
m *-'+ ever Arm) b *mm+ Sprin' Con#tant) $ *!.mm+
RESUTS A "ree And Undamped O#$i%%ation Tab%e & Data for Free & Damped Oscillation Analysis
1efer 2raph in the appendix.
3pring (onstant, c 4 *./ 56mm 7ever #rm, a 4 -* mm Theoretical 5atural Fre"uency, f * 4 .)+ H8 (alculated 5atural Fre"uency, f 4 .** H8
3
+
!
/
1efer graph in the appendix
3pring (onstant, c 4 *./56mm 7ever #rm, a 4 ** mm Theoretical 5atural Fre"uency, f * 4 /./* H8 (alculated 5atural Fre"uency, f 4 .** H8
1efer graph in the appendix
3pring (onstant, c 4 .// 56mm 7ever #rm, a 4 ** mm Theoretical 5atural Fre"uency, f * 4 .)- H8 (alculated 5atural Fre"uency, f 4 .** H8
1efer graph in the appendix
1efer graph in the appendix
3pring (onstant, c 4 .// 56mm 7ever #rm, a 4 !** mm Theoretical 5atural Fre"uency, f * 4 !.!- H8 (alculated 5atural Fre"uency, f 4 !.!! H8 3pring (onstant, c 4 !.*0 56mm 7ever #rm, a 4 !** mm Theoretical 5atural Fre"uency, f * 4 /.0! H8 (alculated 5atural Fre"uency, f 4 .** H8
B "REE A!D DA/PED OSCIATIO! Tab%e 0& Data for Free & Damped Oscillation Analysis
1efer 2raph in the appendix.
+
1efer graph in the appendix
!
1efer graph in the appendix
7ever arm, b 4 ** mm 9egree of 9amping, 9 4 Theoretical 9amped Fre"uency, 7ever arm, b 4 !** mm 9egree of 9amping, 9 4 Theoretical 9am ed Fre uenc 7ever arm, b 4 *mm 9egree of 9amping, 9 4 Theoretical 9am ed Fre uenc
ωd
4
ωd
4
ωd
4
CACUATIO!S A "ree And Undamped O#$i%%ation Tab%e & Percentage of Error of Undamped Frequency "re1uen$ie# of Undamped O#$i%%ation E2periment
E2perimenta% Data
Period) T *#+ *.+
"re1uen$3) f *H4+ .**
+
*.+
.**
T(eoreti$a% "re1uen$3) f *H4+
Per$enta'e of Error *5+
.)+
+.
/./*
!.-/ 4
!
*.+
.**
.)-
!.0
/
*.!
!.!!
!.!-
*.0
*.+
.**
/.0!
./+
SA/PE CACUATIO! *E6PERI/E!T 7+
Theoretical 5atural Fre"uency&
√
2
1 ❑ 3c a ƒ = 2 π m L2 ˳
ƒ = ˳
√(
(
)(
1 ❑ 3 1440 0.3 2 π
1.68
)
2
2
) [ ( 0.72 ) ]
= 3.36 Hz
(alculated 5atural Fre"uency& ¿
f 0=
¿
f 0=
1
T 1 0.3
=3.33 Hz
:ercentage of ;rror&
Percentage error ( ) =
Percentageof error ( ) =
Theoretical − Experimental Theoretical
3.36 −3.33 × 100= 0.89 3.36
B "ree And Damped O#$i%%ation Tab%e & Percentage of Error of Damped Frequency
E2periment
E2perimenta% Damped ωd "re1uen$3)
!./+
+
!./!
!
!./+
T(eoreti$a% Damped ωd "re1uen$3)
Per$enta'e of Error *5+
5
T exp1 =T exp2=T exp3=0.2 s
DISCUSSIO!
A "ree And Undamped
For the Free and Force
√
2
1 3 ca = ƒ also can be known from the formula of natural fre"uency, 2 π m L2 ˳
where the lever
arm, a increases the natural fre"uency also will increased. #part from that, from experiment + and ! there were similarity in length of lever arm but different in spring coefficient. From here it can be seen that when the stiffness of spring increases, the natural fre"uency as like natural angular fre"uency also will increase. It can be proven from the formula where the higher the stiffness of the spring, the higher the natural fre"uency happened. 3o for the calculation there was a percentage different between the theoretical calculated values on natural fre"uency and experimental values. Here we can conclude that our result of experiment had effected by some disturbance. #s the instruments being so sensitive with the vibration, the wind factor in the lab might be the one of the disturbance. The little amount of the wind can affect the result. =oreover, the springs also not in good condition due to overuse and this affect its constant so the result will be not so accurate. The beam also might not place hori8ontally completely so that the spring will not completely in vertical state. This will causes the imbalance condition and then effect the oscillation of the cantilever beam. B "ree And Damped
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>ased on this experiment, the same springs and the damper had been used. 3o, the spring coefficient was same in each experiment and the numbers of experiments were ! in total. Here, the position of damper on the lever arm was being manipulated. Then, the varieties in position of damper on the lever arm affected the fre"uency of the oscillations. If the viscous damper placed further from the fixed mount, then the period of oscillation will be shorter as the result the less fre"uency happened. #s the length of lever arm increases, the degree of damping also increases. The graph showed the behaviour of free vibration with viscous damper in cantilever beam. From the graph, it shown that the shorter the length of lever arm the longer the period for the system to reach stability state.. It can be concluded that the rate of losing or absorbing energy is higher when the distance between viscous damper and fixed mount is farther. In this experiment there were a few errors which are human error where the force from student on the lever arm during experiments is not same in every experiment .3econd is the mass of viscous damper affecting the oscillations. #nd the third error is the wind factor. #nd the last, the external vibration nearer to the experiment.#ll of this factor of course affected the result of this experiment.
CO!CUSIO!
$e successfully conducted the experiment by following the steps provided in the manual aided by 2#. In conclusion,free vibrations are oscillations where the total energy stays the same over time. This means that the amplitude of the vibration stays the same.Forced vibrations occur when the object is forced to vibrate at a particular fre"uency by a periodic input of force. ?bjects which are free to vibrate will have one or more natural fre"uency at which they vibrate. If an object is being forced to vibrate at its natural fre"uency, resonance will occur and large amplitude vibrations will be observed. The theoretical natural fre"uencies are compared to experimentally obtained data of damped and undamped free vibrations. :ercentages of error are shown. #lso, the effect of damping on the decay behaviour of the vibration is shown through the aid of graphs and calculations.The results are recorded for further calculation. #ll the objectives of the experiment are met.
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APPE!DI6
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