Dynamic Practical Lab Report Cantilever Beam

Introduction

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 y showing the analysis of the frequency of vibration of cantilever beam. One end of cantilever beam is fixed while the other end is free. Free vibration of cantilever beam with natural frequenc y is starting with by initial displacement height to the cantilever beam with hο=3mm. !nd is displaced by "#$mm%# &mm%mm%&mm%$mm' from initial height. (he beam will deflect to the curve when load is removed by particular height measured from the meter rule. (he larger the load the larger the deflection. !fter !fter the free vibration finished% we have conducted the experiment by b y 3mm%&mm and with viscous damper in water which connected to the cantilever beam.  )amping is present in this experiment. )amping vibration means that energy have lost from the system and finally vibration stops% which the amplitude of vibration decreases gradua lly from the cantilever beam. (he cantilever beam is left to vibrate with no external force in free vibration. *uch vibration v ibration will not stops unless force being applied.

Figure 1

 The simple cantilever cantilever beam shown in Figure Figure 1 can be modeled modeled as a mass-spring system where the governing governing equation of motion is given by m  x´ =-kx n

2

or

´  ωn  x

x=!

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

n =¿ ω¿

k  m

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

x=

 x´ ( 0 )sin ω n t   - x"' ωn

cos ω n t 

 

."$./'

(he natural period of the oscillation is established from

τ =2 π 

m k   

(he natural frequency of the system s ystem is

."$.3'

ω n τ =2 π   or

1

f  n = τ   =

√

 k    2 π  m 1

..."$.0'

Viscously damped Vibration

"very mechanical system possesses some inherent degree of friction# which dissipates mechanical energy$ %recise mathematical models of the dissipative friction forces are usually complex$ &iscous damping force can be expressed by

 F d =c  x´  

."$.&'

 x + kx =0.  (he (he equation of motion of a free#damped vibration system is given as m x´ + c  ´ 2

ξ −1 −ξ +√ ¿

¿ ξ −1 general solution is given as −ξ −√ ¿ ωn t    ..."$.1' ¿ ¿ ¿ ¿  x = A e 2

1

2

−1 ¿  may be positive% negative or 2ero% giving rise to three categories of damped motion, ξ > 1  "over#damped%figure /'% ξ =1 "critically damped% Figure /' and (he radicand " ξ

ξ < 1  "under damped% Figure 3'.

Figure 2

Figure 3

(he frequency of damped vibration ω d= √ 1− ξ ω n   2

."$.'

Natural frequency of a Cantilever Beam

Figure 4

(he maximum deflection of the cantilever beam bea m under a concentrated end force 4 is given by 3

 P L  P  y max =  = !"1!1#$ k  3 EI 

3 EI 

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

3

 L

5here 6= length of the beam 3

7=moment of inertia% for rectangular area% 7 =  b= width of the beam h=height of the beam

bh

12

  "$.$$'

+= modulus of elasticity% elasticity% for aluminium% += 894a

%b&ective' (art 1, (o investigate investigate the natural frequency of a natural frequency of a cantilever beam

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

)et*odology (rocedure

1 'omputer and the strain recorder is switched on$ ($ )train recorder application is started software by double click on the *+'1!,"ng. shortcut icon on the computer desktop$ /$ The strain recorder recorder and the recorder recorder application software is refer to the operational manual for

the operation

,$The viscous damper is removed if it is attached to the beam$ 0$ The beam# y max max refer to Figure ,2 by -(! mm# -10 mm# -1! mm# -0 mm# is displaced and hold ! mm# 0 mm# 1! mm# 10 mm and (! mm and record the strain recorder reading for each displacement value manually from the *3umerical 4onitor. screen of the application software$ 5$The relationship of the displacement is obtainedof the free end of the beam2 and the strain recorder recorder reading by plotting an appropriate graph using a spreadsheet$ 6$ The beam is displaced by /! mm and leave the beam to vibrate on its own$ ecord the strain recorder recorder reading by clicking on the *%lay. and *)top. button$ 7$ etrieve the recorded 8le by clicking on the *ead 9):. button$ ;$ The graph of the beam displacement versus the time# t  is plotted$ 1!$ The experiment is repeated by using beam displacement of 0! mm$ 11$ The viscous damper is connected$ )teps 6 and 1! is repeated by using beam displacement of /! mm and 0! mm# respectively$

+esults' (art1' ,train recorder reading for eac* displacement value

7nitial

6engthen from

*train

displacement"cm

initial

"$'"mm'

'

displacement"mm '

3-!34!34!# 33!33!# 32!32!# 31!31!#

.2# .1.1# .# //1# /1/2#

/23# /10# /11# /# # # 12# 10# 23#

'hart Title (/! /!! 17! (!! 1(! 5! 1!!

lengthen from initial displacementmm2

! ! -5! -/! -(! -1! ! 1 ! (! /! -11! -1!! -17! -(/! -(!! -/!!

)trainmm2

+esult for Free vibration for 3#mm'

esult e sult for free vibrat vibration ion for /!mm 1( 1! 7 5 , ( !

!

(

,

5

7

1!

1(

Result for Free vibration for 50mm

esult e sult for free vibrat vibration ion for 0!mm 1( 1! 7 5 , ( !

!

(

,

5

7

1!

1(

Results for damped vibration in water for 30mm

esult for damped vibration in water /!mm 1( 1! 7 5 , ( !

!

(

,

5

7

1!

1(

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

esult e sult for dampe damped d vibrat vibration ion in w ater 0!mm 1( 1! 7 5 , ( !

!

(

,

5

7

1!

1(

iscussion

For theoretical natural frequency# f n2theo theo calcuation# according to damped and undamped experiment# the following datas were needed$ 1$ 4odulus of elasticity of aluminium"2 = 6!