Dynamic Practical Lab Report Cantilever Beam

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Introduction Introduction

Cantilever beam is used to find the modulus of elascity of a thin film because that measurement Cantilever beam is used to find the modulus of elascity of a thin film because that measurement of a bulk materials is easier compared to thin film b

of a bulk materials is easier compared to thin film b y showing the analysis of the frequency y showing the analysis of the frequency ofof vibration of cantilever beam. One end

vibration of cantilever beam. One end of cantilever beam is fixed while the other of cantilever beam is fixed while the other end is free.end is free. Free vibration of cantilever beam with natural frequenc

Free vibration of cantilever beam with natural frequenc y is starting with by initial displacementy is starting with by initial displacement height to the cantilever beam with h

height to the cantilever beam with hοο=3mm. !nd is displaced by "#$mm%#=3mm. !nd is displaced by "#$mm%#

&mm%mm%&mm%$mm' from initial height. (he beam will deflect to the curve when

&mm%mm%&mm%$mm' from initial height. (he beam will deflect to the curve when load isload is removed by particular height measured from the meter rule. (he larger the load the larger the removed by particular height measured from the meter rule. (he larger the load the larger the deflection. !ft

deflection. !fter the free vibration finished% we have er the free vibration finished% we have conducted the experiment bconducted the experiment by 3mm%&mmy 3mm%&mm and with viscous damper in water which connected to the cantilever beam.

and with viscous damper in water which connected to the cantilever beam. )amping is present in this experiment. )amping vibration means

)amping is present in this experiment. )amping vibration means that energy have lost from thethat energy have lost from the system and finally vibration stops% which the amplitude of vibration decreases gradua

system and finally vibration stops% which the amplitude of vibration decreases gradua lly from thelly from the cantilever beam.

cantilever beam.

(he cantilever beam is left to vibrate with no

(he cantilever beam is left to vibrate with no external force in free vibration. *uch vexternal force in free vibration. *uch vibration willibration will not stops unless force being applied.

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Figure 1 Figure 1 The simple cantileve

The simple cantilever beam shown in Fr beam shown in Figure 1 can be moigure 1 can be modeled as adeled as a mass-spring system where the governi

mass-spring system where the governing equation of ng equation of motion is given bymotion is given by m m x x´´ =-kx =-kx oror x x´´  ωωnn 2 2 x = ! x = ! n

n is known as the natural circular frequency of the system and is given byis known as the natural circular frequency of the system and is given by

n n

=

=¿

¿

ω ω_¿_¿ mm k k

+quation "$' is a homogeneous second#order

+quation "$' is a homogeneous second#order equation linear differential equation% has theequation linear differential equation% has the following general solution,

following general solution,

x = x =

´´

x x ω

ω_n_n

((

00

))

sinsinωωnnt t - x"' - x"' coscosωωnnt t _."$./'_."$./'

(he natural period of the oscillation is established from

(he natural period of the oscillation is established from ωωnnτ τ ==22π π _or_or

τ

=

22_π_π mm

k

k ."$.3'."$.3' (he natural frequency of the s

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f f _n_n ₌₌ 11 τ τ = = 1 1 2 2_π_π

√√

k k m m ..."$.0'..."$.0'

Viscously damped Vibration Viscously damped Vibration

"very mechanical system possesses some inherent degree of friction#

"very mechanical system possesses some inherent degree of friction# whichwhich dissipates

dissipates

mechanical energy$ %recise mathematical models of the dissipative friction mechanical energy$ %recise mathematical models of the dissipative friction forces are usually

forces are usually

complex$ &iscous damping force can be expressed by complex$ &iscous damping force can be expressed by

F

F _d_d _=c_=c x x_´´ _."$.&'_."$.&'

(he equation of motion of a

(he equation of motion of a free#damped vibration system is given asfree#damped vibration system is given as mm x x´´++cc ´ ´ x x++kxkx==0.0. (he (he

general solution is given as general solution is given as

ξξ22

−

11

−

ξξ

+

√ √

¿

ξξ22

−

11

−

ξξ

−

√ √

¿

ωωnnt t

¿

x x

=

A A11ee ¿ ¿ ..."$.1' ..."$.1' (he radicand " (he radicand " ξξ 2 2

−

11

¿

_{may be positive% negative or 2ero% giving rise to three categories of}_{may be positive% negative or 2ero% giving rise to three categories of}

damped motion,

damped motion, ξξ

>

11 "over#damped%figure /'% "over#damped%figure /'% ξξ

=

11 "critically damped% Figure /' and"critically damped% Figure /' and ξξ<<11

"under damped% Figure 3'. "under damped% Figure 3'.

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Figure 2 Figure 2

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Figure 3 Figure 3

(he frequency of damped vibration

(he frequency of damped vibration ωωdd

=

√ √

11

−

ξξ

2 2

ω

ω_n_n _{."$.'}_{."$.'} Natural frequency of a Cantilever Beam

Natural frequency of a Cantilever Beam

Figure 4 Figure 4

(he maximum deflection of the cantilever bea

(he maximum deflection of the cantilever beam under a concentrated end m under a concentrated end force 4 is given byforce 4 is given by y

y_max_max

=

P P LL

3 3 3 3_EI_EI

=

P P k k !"1!1#$!"1!1#$

(herefore the stiffness of the beam is given by k= (herefore the stiffness of the beam is given by k=

3 3_EI_EI L L33 "$.$$' "$.$$' 5here 5here

6= length of the beam 6= length of the beam

7=moment of inertia% for rectangular area% 7 = 7=moment of inertia% for rectangular area% 7 = bb hh

3 3

12 12

b= width of the beam b= width of the beam h=height of the beam h=height of the beam

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+= modulus of

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%b&ective' %b&ective' (art 1

(art 1, (o investigate the natural frequency of a , (o investigate the natural frequency of a natural frequency of a cantilever beamnatural frequency of a cantilever beam (art 2

(art 2, (o find out relationship between both unda, (o find out relationship between both undamped and damped free vibration motion mped and damped free vibration motion of aof a cantilever beam.

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)et*odology )et*odology (rocedure (rocedure

1 'omputer and the strain recorder is switched on$ 1 'omputer and the strain recorder is switched on$

($ )train recorder application is started software by double click on the ($ )train recorder application is started software by double click on the *+'1!,"ng.

*+'1!,"ng. shortcut

shortcut icon icon on on the the computer computer desktop$desktop$ /$ The strain

/$ The strain recorrecorder and the der and the recorrecorder application software is refer to theder application software is refer to the operational

operational manual manual for for the the operationoperation

,$The viscous damper is removed if it is attached

,$The viscous damper is removed if it is attached to the beam$to the beam$ 0$ The beam#

0$ The beam# y y maxmaxrefer to Figure ,2 by -(! mm# -10 mm# -1! mm# -0 mm# isrefer to Figure ,2 by -(! mm# -10 mm# -1! mm# -0 mm# is

displaced and hold displaced and hold

! mm# 0 mm# 1! mm# 10 mm and (! mm and record the strain recorder ! mm# 0 mm# 1! mm# 10 mm and (! mm and record the strain recorder reading for each

reading for each

displacement value manually from the *3umerical 4onitor. screen of the displacement value manually from the *3umerical 4onitor. screen of the application

application software$ software$

5$The relationship of the displacement is obtainedof the free end of 5$The relationship of the displacement is obtainedof the free end of thethe beam2 and the strain

beam2 and the strain recor

recorder reading by plotting der reading by plotting an appropriate graph using an appropriate graph using a spreadsheet$a spreadsheet$ 6$ The beam is displaced by /! mm and leave the beam to vibrate on its 6$ The beam is displaced by /! mm and leave the beam to vibrate on its own$ ecord the strain

own$ ecord the strain recor

recorder reading by clicking on the *%lay. and *)top. der reading by clicking on the *%lay. and *)top. button$button$

7$ etrieve the recorded 8le by clicking on the *ead 9):. button$ 7$ etrieve the recorded 8le by clicking on the *ead 9):. button$ ;$ The graph of

;$ The graph of the beam displacement versus the time#the beam displacement versus the time# t t is is plotted$plotted$ 1!$ The experiment is repeated by using beam displacement of 0! mm$ 1!$ The experiment is repeated by using beam displacement of 0! mm$ 11$ The viscous damper is connected$ )teps

11$ The viscous damper is connected$ )teps 6 and 1! is 6 and 1! is repeated by usingrepeated by using beam displacement of /! mm and 0! mm# respectively$

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+esults' +esults'

(art1' ,train recorder reading for eac* displacement value (art1' ,train recorder reading for eac* displacement value 7nitial 7nitial displacement"cm displacement"cm '' 6engthen from 6engthen from initial initial displacement"mm displacement"mm '' *train *train "$'"mm' "$'"mm' 3 3--!!-- ..22## //2233## 3 344!!-- ..11-- //1100## 3 344!!## ..11## //1111## 3 333!!-- ..-- //## 3 333!!## ## ## 3 322!!-- //-- ## 3 322!!## //11## 1122## 3 311!!-- //11-- 1100## 3 311!!## //22## 2233## --//!! --((!! --11!! !! 11!! ((!! //!! -/!! -/!! -(!! -(!! -1!! -1!! ! ! 1!! 1!! (!! (!! /!! /!! -(/! -(/! -17! -17! -11! -11! -5! -5! ! ! 5! 5! 1(! 1(! 17! 17! (/! (/!

'hart Title

)trainmm2 )trainmm2

lengthen from initial displacementmm2

+esult for Free vibration for 3#mm' +esult for Free vibration for 3#mm'

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! ! (( ,, 55 77 11!! 11(( ! ! ( ( , , 5 5 7 7 1! 1! 1( 1(

e

sult

for free vib

rat

ion for

/!mm

Result for Free vibration for 50mm Result for Free vibration for 50mm

! ! (( ,, 55 77 11!! 11(( ! ! ( ( , , 5 5 7 7 1! 1! 1( 1(

e

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Results for damped vibration in water for 30mm Results for damped vibration in water for 30mm

! ! (( ,, 55 77 11!! 11(( ! ! ( ( , , 5 5 7 7 1! 1! 1( 1(

esult for damped vibration in water /!mm

+esults for damped vibration in ater for -#mm +esults for damped vibration in ater for -#mm

! ! (( ,, 55 77 11!! 11(( ! ! ( ( , , 5 5 7 7 1! 1! 1( 1(

e

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iscussion iscussion

For theoretical natural frequency# f

For theoretical natural frequency# f nn22ththeoeo calcuation# according to dampedcalcuation# according to damped and undamped experiment# the following

and undamped experiment# the following datas were needed$datas were needed$ 1$ 4odulus of

1$ 4odulus of elasticity of aluminium"2 = 6!<%aelasticity of aluminium"2 = 6!<%a ($ +imension of the

($ +imension of the cantilever beam = ;(5mm x 1;mm x 5mmcantilever beam = ;(5mm x 1;mm x 5mm /$ 4ass of

/$ 4ass of the cantilever beam = (;0gthe cantilever beam = (;0g ,$ 4ass of the damper = 1((g

,$ 4ass of the damper = 1((g

:y substituting the data above into the equation# we obtained# :y substituting the data above into the equation# we obtained#

4oment of inertia# = 4oment of inertia# = $/$/ :: bh bh = = $/$/ :: '' 3 3 $ $ x x 1 1 '" '" 3 3 $ $ x x "$ "$ −− −− = /$,( x = /$,( x 10 10−−1010_m_m44

)ti>ness of cantilever beam# k= )ti>ness of cantilever beam# k= ::

3 3 L L EI EI k k

=

33

((

7070××1010 9 9

)(

3.423.42_×_×1010−−1010

))

((

926926_×_×1010−−33

))

= ;!$,0 = ;!$,0 NmNm − −11

(he equivalent mass of cantivalent beam%

(he equivalent mass of cantivalent beam% mmeqeq

=

=¿

¿

_""

33 33 140 140

¿

¿((

mm

))

m m_eq_eq

=

=¿

¿

_"" 3333 140 140

¿

((

295295××1010 − −33

))

_{kg=.8kg}_{kg=.8kg}

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;atural frequency% ;atural frequency% f f _n_n = = 1 1 2 2_π_π mmeqeq k k = = ₂₂11_π_π

√√

90.45 90.45 Nm Nm − −11 0.07 0.07 = &.8/<2 = &.8/<2 'o

'ompmpararining g wiwith th ththe e frfreqequeuencncy y obobtatainined ed in in eexpxpererimimenentt 30mm 30mm withwithoutout

damping

damping#f #f nn22expexp ==

τ τ $ $ = ?@ = ?@

"xperimental valuereferred from graph2= "xperimental valuereferred from graph2=

1 1 T T 22−−11

=

11 316 316

−(−

328328

))

=

1.5521.552 Hz Hz %ercentage errorA=

%ercentage errorA=

((

theortheoretca!etca! "a!#e_{theoretca!}_theor"a!#e_{etca! "a!#e}

−

ex$ermeex$ermenta!_"a!#e nta! "a!#e"a!#e

))

××100100

= = 5.72 5.72_Hz_Hz

−

1.5521.552_Hz_Hz 5.72 5.72 ××100100 =6($;A =6($;A 'o

'ompmpararining g wiwith th ththe e frfreqequeuencncy y obobtatainined ed in in eexpxpererimimenentt 50mm 50mm withwithoutout

damping damping# f # f nn2 =2 = 1 1 T T == 1 1

−

344344

−

298298

=

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= = 5.72 5.72

−

1.561.56

¿

5.72 5.72××100100 =8/.8=8/.8

In vibration with damping(both for 30mm

In vibration with damping(both for 30mm# f # f nn2=2= 1 1 T T

=

1 1 248 248

−

246246

=

%ercentage errorA=

((

−

))

××100100

= =

((

5.72 5.72

−

0.500.50

))

5.72 5.72 ××100100 =91%=91%

In vibration with damping(both for 50mm

In vibration with damping(both for 50mmf f nn2=2= 1 1 T T

=

1 1 372 372

−

324324

=

0.020.02 %ercentage errorA=

%ercentage errorA=

((

−

))

××100100

5.72

−

0.020.02 5.72

5.72 ××100100 =!5%=!5%

" #al$ulation for damped period " #al$ulation for damped period

τ τ = ( = (ΠΠ k k m m τ τ = ( = (ΠΠ 0& 0& .. 3 3 $3/ $3/ ..   ./>3 ./>3 = = τ τ s s

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#al$ulation for damped fre&uen$' #al$ulation for damped fre&uen$'

f f nn== τ τ $ $ = = Π Π / / $ $ m m k k = = $3/ $3/ ..   0& 0& .. 3 3 / / $ $ Π Π = = 3.0&<23.0&<2

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7n

7n the the experiment%we experiment%we have have use use the the apparatus apparatus and and materials materials available available such such as as explainedexplained above.(he theorectical natural frequency o

above.(he theorectical natural frequency of free vibration at 3mm is &.8/<2.!nd for thef free vibration at 3mm is &.8/<2.!nd for the experimental? value is $.&&/<2 which percentage error is 8/. which shows that the experimental? value is $.&&/<2 which percentage error is 8/. which shows that the experiment have a result which not

experiment have a result which not accurate.(his is greatly determine by the graph. accurate.(his is greatly determine by the graph. 7t is believe7t is believe that the reason why the data

that the reason why the data not accurate is as our human not accurate is as our human error while doing the experiment.error while doing the experiment. 5hen carry out the experiment% our hand is not removed from the table and hence making the 5hen carry out the experiment% our hand is not removed from the table and hence making the vibrations of the cantilever beam.Furthermore% there have

vibrations of the cantilever beam.Furthermore% there have parallax error occur as ourparallax error occur as our

measurement is not correct as the meter rule which use to measure the length of cantilever beam measurement is not correct as the meter rule which use to measure the length of cantilever beam being displaced is not cling on it with almost correct of certain length. *econdly

being displaced is not cling on it with almost correct of certain length. *econdly%the results of the%the results of the free vibration average data should be

free vibration average data should be recorded in order to have recorded in order to have a nice distribution graph. !ftera nice distribution graph. !fter the damper is added in

the damper is added in the water% the vibration become more unaccurate as the percenthe water% the vibration become more unaccurate as the percen tage errortage error show $ for free vibration of 3mm in water

show $ for free vibration of 3mm in water with damper and />& for &with damper and />& for &mm for vibrationmm for vibration under water with damper.

under water with damper. *train

*train gauge gauge is is mounted mounted on on cantilever cantilever beam beam to to act act asasresistance strain gages and to measureresistance strain gages and to measure forces% moments% and the deformations of structures and materials. (wo of the strain gauge is mounted on forces% moments% and the deformations of structures and materials. (wo of the strain gauge is mounted on top of the beam and

top of the beam and two mounted below the beam. (he stress of the two mounted below the beam. (he stress of the surface of bending beam can besurface of bending beam can be

calculated as following formula

calculated as following formula E E

=

% % ϵ ϵ (hus(hus ϵ ϵ

=

1 1 k k & & m' m' ' ' &&1010 − −33 ' ' m' m'

!s the length of the strain gauge

!s the length of the strain gauge increase from one end to anoincrease from one end to another end thus the strain decreasesther end thus the strain decreases from the bending cantilever beam.

from the bending cantilever beam.

Conclusion Conclusion

!s a conclusion% the cantilever beam

!s a conclusion% the cantilever beam is a device to measure the is a device to measure the vibration of thin film as the bulkvibration of thin film as the bulk product is really hard measured by it. (he measurement which carried out by cantilever beam product is really hard measured by it. (he measurement which carried out by cantilever beam should be very cautious while doing the experiment as the graph showing will depends on the should be very cautious while doing the experiment as the graph showing will depends on the amplitude of vibration.*o that the external forces should be

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+eference +eference 1$ 1$ *ttp'!me!unm!edu56almot*cantilever7lab!pdf *ttp'!me!unm!edu56almot*cantilever7lab!pdf 2$ 2$ *ttp'iitg!vlab!co!in8sub92:brc*91;-:sim91#0#:cnt91*ttp'iitg!vlab!co!in8sub92:brc*91;-:sim91#0#:cnt91