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Two Marks euestions and Answers What is mean by Frequenry and frequency ratio? Frequency is number of times the motion repeated in the same sense or altematively. It is the number of cycles made in one second (cps). It i" ar"o as Hertz (H2) named after the '' "xp."s"ed inventor of the tenn. The circular frequency ro in mits of sec-i i" giro by 2n;.

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2.

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What is ttre formula for free vibration response? The corresponding equation under fiee vrhrations can be obtained by substituting the right hand side ofequation as zero. This gives

mu+Cu+Ku=0

3.

VYhat are

4.

Effect on Human Sensitivity. Effect on Structural Damage What is mean by theory of vibration?

the effects ofvibration?

i.

ii.

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vibration is the motion of a particle or a body or a system of concentrated bodies having been displaced form a position ofequilibrium, upp"a.ing as an oscillation.

was recogdzed.in mechanicar systems fust and hence the study of vibrations . u]Yb' ^,, into fell the heading ..Mechanical Vibraiions,, as e*f,

5.

Define damping.

Damping

is a

uJo*

OZOO years ago.

of dissipation in a vibrating system. The dissipating . .elergy of the frictional form or vircou. forrr. In the former case, it is c6 ed dry friction or column damping and in the ratter case it is carled viscous damping. Danrping in a structural system generally assumed to be of viscous tlpe for mathematical convenience, viscous

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mechanism may be

damped force (Fd) is prcportionar to the velocity 1u; or a vibrating body. The corstant proportionality is called the damping constant (C). Its units are NS/m.

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6.

What do you mean by Dynamic Response? The Dynamic mav be d;fin1! simpry as

time varying. Dynamic load is therefore any load magnitude, direction uotr, iir, time. The structurar response (i.e., resulting dispracements and stresses) to a".d.ynamic load is also time varying or dynamic in nature. Hence

which varies

in its

it

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is called dynamic response.

s. AZHAOAP.SAMy rlrt.Tech, MfSTE 455T. PROFESSOR Deportment of Civil

Engg.

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What is mean by free vibration? A structure is said to be undergoing free vibrations if the exciting force that carred the vibration is no longer present and the oscillating structure is purely under influence of its own inertia or mass(m) and stiftress (k), Free vibration can be set in by giving an initial displacement or by giving an initial velocity (by strfting with a hammer) to the structure d an appropriate location on it.

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What is meant by F orced vibrations? Forced vibrations me produced in a structure when it is acted upon by the continuous prcsence of an e*ernal oscillating force acting on it. The structure under forced vibration nomally responds at the frequency ratio, i.e. (fin/ft) where ftn is the frequency ofexcitation and f,? is the Ddural frequency of the structure. a short note on

If 'u'

og

Amplitude. It is the maximum response of the vibrating body from its mean positiorr Ampiitude is genemlly associated with direction - v€rtical, horizontal, eE. It can be expressed in the form of displacement (u), velocity (ri) or acceleration (i): In the case of simple harmonic motiorl these terms are related though the frequency of oscilation (0. is displacemetrt amplitude, theD

:

:

itis

measured

in terrns of acceleration

tas

10. De{ine Resonance.

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Veloclty (*) 2zr f .u Acceleration (ri) QnD. Qt) : hr2 f2u When acceleration is used as a measure of vibration due to gravity, g (9.81 m/sec2).

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This phenomenon is characterized by the build -up area of large amplitudes of any given structural system and aq such , it has a significance in the design of dynamically loaded structures. Resonanoe should be avoided under all circumstances, whenever a structure is acted upon by a steady state oscillating force (i.e., frn is constant).

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The presence of damping, however, limits the amplitudes at resonance. This shows the importance of damping in controlling the vibrations of structures. According to IS 1893 -1975Indian standard code of practice on Earthquake resistant design of structures, following values of damping me recommended for design purposes.

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11. What is mean by Degrees of freedom? The number of degrees of freedom of system equals the minimum number co-ordinates necessary to defrne the configuration ofthe system.

o

f independent

12. Define static force. A push or pull

or a load or many loads on any.system creates static displacement or on whethff it is a lumped system or a continues system; there is no

deflection depending excitation and hence there is no vibration.

motion. Vibration is periodic motion; the simplest form of periodic motion is simple harmonic. More complex forms of periodic motion may be considered to be composed of a number of simple harmonics of various amplitudes and frequencies as specified in Fourier series 13. Write q

slort

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nof,e on simple Harmonic

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l,\.Tech, ASSI. PROFESSOR Departhent o{ Civil Enqg. S. AZHAG AR.s AMY

14. What is rhe response for impulsive load or Shock loads? Impulsive load is that which acts for a relatively short dLrdion. Examples are impact of a hammer on its foundation. Damping is not importanr in coryuting response to impulsive loads since the maximum response occurs in a very short time before darnfrng foices can absorb much energy from the structr.rre. Tberefore, onlv the ,,rrl*mJ,ed reslronse to irnpulsive loads wifi be considered. 15. Write a short note on single degree of Aeedom (SIX)F) systems. At any instant of time, tbe .mtioo of rhis srrstem cen be denoted by single co_ ordinate (x in this case). It is represented by a rigid rne
po t.in

,

tk

ircIuded

written as

FI + ID + FS:P(O (Inerti (Damp (EIas (Aop Force force) force forc This gives

og s

rnx+Cx+Kx=P

x, x, x respectively denote the displacement, velocity and acceleration ofthe systern p (t) is the time dependent force acting on the mass. The above equation ,"p."r"rr, the equation of motion ofthe single degree freedom system subjected to forced vibrations. 16. Define Clcle.

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tas .bl

The movement of a particre or body from the mean to its exreme position in the diectiorl then to the mean and then another exr"-" position and back to the mean is called a Cycle of vibration Cycles per second are the unit Hz.

17. Write short notes or DrAlembert's principle.

Write the mathematicat equation for springs in parallel and springs in Springs in parallel k": k, k,

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18.

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According to Newton,s law F: ma The above equation is in the form ofan equation ofmotion of force equilibrium in which the sum of the number of f,orce terms equal zero. Henie if an imaginary force wrri"i is equar to ma were applied to system in the direction opposite to the acceleration, the system could then be considered to be in equilibrium under the actibn of real force F and the imaeinary force ma. This ma is known as inertia force and the position of equilibrium is called dynamic i3.1Y_3*" equ rDnum. D'Alembert's principre which state rhat a sysrem may be in dynbmic equiribrium by adding to the extemal forces, an imaginary force, which is commonly knr*r, *,fr"l""*'^]L""*'

k.

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is called equivalent stiffness of the svstem

s. AZHAaAq.s.AMy iril.Iech, r$fSTE ASSL PROFESSOR Deportmant of Civil Engg.

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t

series

Springs in series

1:l*1

k. kr

k2

Define logarithmic decrement method. Logarilhmic decremeht is defined as the natural logarithmic value of the iatio oftwo adjacent peak values of displacemont in free vibration. It is a dimensionless paramet6r. It is denoted by a symbol d

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19.

short notes on Ilalf-power Bandwidth method. Bandwidth is the difference between two fiequencies corresponding to the same amplitude. Frequency response.curve is used to define the half-power bandwidtl. In which, the damping ratio is determined from the frequencies at which tie response a:nplitude is reduced 1rly'2 times the maximum amplitude or resonant amplitude. 20. Write

Magnification factor. Magnificati,on factor is defined as the ratio of dynamic displacement at any time to the displacement produced by static application ofload. 21. Define

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22. Whet is the difrerence betweer a static and dynamic force?

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ln a static problem, load is constant with respect to time and the dynamic problem is the time varying in nature. Because bottr loading and its responses varies with respect to fime Sratic problem has only one response that is displacement. But the dynamic problem has mainly three responses such as displacement, velocity and acceleration. 23. Define

critical damping.

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Critical damping is defined as the minimum amormt of damping for which the systern will not vibrate when disturbed initially, but it will retum tot the equilibrium posifion. This will result in non-periodic motion that is simple decay. The displacement decays to a negligible level after one nature period T.

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24. List out the types of damping.

(1)

Viscous Damping, (2) Coulomb Damping, (3) Structural Damping (4) Active

Damping, (5) Passive Damping.

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25. What is meant by

danping ratio?

The ratio of the actual damping to the criticai damping coefficient is called as damping ratioIt is denoted by a symbol p and it is dimensionless quantity. It ca be written as

p = c/cc

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Visit : Civildatas.blogspot.in AZHAGAPSAMY /tA,Tech, MISTE ASST. PROFESSOR Deportment of Civil engg.

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t{kso FiBbn Now the whole of kinetic energy is converted into elastic energy and this elastic energy again brings the ball to the equilibrium, position. In this way, vibratory inotion is repeated ioaen iiery-aoa exchange of energy takes place. This motion which repeats itself after certain interval called vibrarion. "f

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27. V,lhat arc the main causes of vibration? The main causes ofvibration are: l' unbalanced centrifugal forcc in the syste,r due to faulty design and poor manutbct,ring. 2. Elastic nature of system 3. Extemal excitation applied on the system 4' winds may cduse vibration of cerim system such as electricity lines, terephr:ne rines etc.

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Vibration causes excessive and unpleasant stresses in the rotating system. Vibration causes rapid wear and tear ofmaphine parts such as gears and bearings. Vibration causes loosening ofparts from the machine. Due to vibrations locomotive.can leave the track causing accident or heavy loss. Earthquakes are the cause of vibration because of whiih bu dings ,r*"*res (like bridges) 1nay collapse. "ri". Proper readings ofinstruments cannot be taken because ofheavy vjbrations.

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28. lYhat are the disadvantages of effects of vibratiol? Disadvantages harm_firl effects vibration:

-i

Resonance may_take prace if the frequency of system causing rarge amplitudes of vibration

Bridges

ex.itutro, mui"i"" J*,rr" **"r frequency of thereby resulting in failure of system,

S.AZHAGARSAT y lil.Tech, itrtISTE ASST. PROFESSOR Deportment of Civit Engg.

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29. How can you eliminate/reduce unnecessarT vibrations? . Unwanted vibrations can be reduced by: 1. Removing extemal excitation if possible. 2. Using shock absorbers. And Dynamic absoibers. 3. : P.roper balancing ofrotating parts. 4. Removing manufacturing defects and material in homogeneities. 5. Resting the system on proper vibration isolators. 30. What arc thc advantages ofvibration? 1. Musical Instrum€nts like guitar. 2. In study of earthquake for geological reasons. 3. Vibration is usefirl for vibration testing equipments. 4. Propagation of sound is due to viblations. 5. Vibratory conveyors are basbd on concept ofvibration.

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Pen&rlum clocks are based on the irinciple ofvibration.

31. DeIine the following: Periodic motion: A motion which repeats itself after certain interval of time is called periodic motion.

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2)

Time Period: it is time taker to complete one cycle. ' 3) Frequcnty: No's of cycles in one sec. Units : H

4) Amplitude:

Ir{aximrmr displacement

Amplitlde.

5) Naturel freqncnry:

of a vibrating

body

from mean position is called

6)

tas

When there is no extemal force applied on the system and it is given a slight disptacemeat the body vibrates. These vibrations are called free vibrations and

frequency offree vibration is called Natural frequency. Fuadamental mode of vibration: Frmdamental mode ofvibration is a mode

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32. Ctassify diftrent types of vibrations. Types of Vibrations

I. Free and Forced To and fio motion of the sygtem when disarbed initially without any extemal force acting on it are called free vibrations. e.g. pendulum. To and fro motions of the system under the influence of extemal force are called forced v.ibrations. E.g. Bell, Earthquake.

II. Damped and Undamped vibrations

Darnped vibrations are those in which amplitude vibrations are real and me alsb called transient vibrations.

of vibration

decreases

with time. These

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Undamped vibrations are those in which amplitude of vibration remains constant. In ideal system there would be no damping and so amplitude ofvibration is steady. PREPARED BY

Visit : Civildatas.blogspot.in S.AMY A'l.Iech, lrlISTE S. AZH AGAR A55T. PROFESSOR Deportment of Civil engg.

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33. What is DrAIembert,s principle? D'Alembert's princiPle that if the r€sultant fdce -force is zero. then the body will states he in static equilibrium Inertia force acting on the body is givcl by

F,

:

diEg

on a body along with the inertia

rn a, where

,r

= ma66 of tlre body and Iinear acceleaticn of ttre ege of rc Assuming that the resultant force acring on body is F, then the body

a:

sill tc i *,ra*q,--

;1

F+Fr:0 rying fuce of lte bo@ IG is z.trng 4ry163 d,.d&1.tbh61r i is afu i amu &ain- rb ,#g tusc is aig ha r-f hrrig s th Hy tu l{ * +L* S t d,L d bfrd -h ftdcfyirrbfrfi8. 6e

po t.in

Cud&

og s

ff-

tu2:2

,zi + Kr = O (D,Abmberds pr;ripte)

34. Classify different types of damping.

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other.

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Types of Damping [. Viscous 2. Coulomb 3. Structural 4. Non-linear, Slip or interfacial damping 35. What is meant by viscous damping? when the system is arlowed to vibrate in viscous medium the damping is caned viscous viscosity is the property ofthe fluid by virtue of lvhich it offers resistance to moment of one over the 36. What is the importance of critical damping? out of the three modes the vibmting body which has been displaced

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from its mean position would come to state of rest in smallest possiule time without overshooting i.e. without executing oscillation about mean position in critical damping mode. So critical damping is. used for practical applications in large g,ns so that after firing the retuming to original position in minimum time without vibrating and ready for next firing without delay' lf damping provided is overdamped or underdamped,, then there wrll be delay. This properry is also design of an instrument.

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I

relative to a motionless

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ratio of an the time domain. The logarithmic decrernent is the natural log of the ratio of the two successive peaks:

PREPARED BY

S. AZHAGAF.SAIAY tiil.Tech, i ISTE A55T. PROFESSOR DePortmeni of Civil En99. Visit : Civildatas.blogspot.in

9

EI(AMPIE 2.1 A unss of ore kg is suspeaded by a ryring hvitg a stiftess of mas i* dirplrrced dowar*ad &on irs quiUmurn (a) fruuim of uodq d 6o rystom (b) Ndrr.! fregacy of rbe jricren (c) Ttc rcspoosc 0f &E syseD c a ftrncthn of me (d) Tdd cE Ej, of Se sysrern.

fi:

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dr=ltC

r = 6{tr Nhr {a)

ltc quafgr of mb

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b

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li+6(Ilr=O freguacy is giveD b,

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Toral eoergy = 0.0j Nlm

S. A)$\EA\SNi) )i -T ec}, Nrf.SrE ASST. PROFESSOR Deporiment of Civil Engg.

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: a,(/*,r,'vl, j/i

-caslre
..

*

.- .1 ffr-+t.- -^d?1 (rr)

:r:/*uL

AT' I 47

,*g^!r,ry)

@.

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og sp ot. in ,id\ft*,j

+ c$

a( S+rr) r,!&

dt

-

i o

7re.

e*{f,,ar*

(]

u$

a

.

, i lru

$.:1,re^-"

1{_.

urU

'r' I EfflstLi 9'"4d-

Ci

i

$,"

vil

1

ag{, Ytu -brrk

da

a.

:

t,.h

tas .bl

>egan*l

tot*av-&*<

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z5

:

t'..;

og sp ot. in

vil

:,:'

Ci

!.a

da

i

i'::r

ry:+"a

tas .bl

t*

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F(sd,"G{

pcdod

i. givu

Trs @fft6d. of nofioa nrhom,

oo"

,! = bgJf

by,

.,t

r=1-!=o.rz"

l6

=

h{6)=37i

trd'ls

gs p

fl sed

= eS csr+

ot. in

4

^-?.#1"

tr

6 + 2,lt ?he rsimm rthcit], of r srsu! b gilEi by

Gm

ia=.Sg.r=Lllxni uianm dcrdioa

=7).46arle b i n= #. =LllxTl7z

.bl o

Tho

of a r).$rar

=D9#6rrl*

Phr$

aurB

.=h-'[f]

tas

-,lzxn11 :*.1 2s J

=7139Yff

r'{im

is .t = 4

&(q,r

= 125

lril

t f) = 2l I *e{3?.n + l5}

Ci vil da

E$&htr of

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27

ai

DI(Aildtl,,E A rr€fiirlrl c:6tc 3 m hry tai.a arua of 4 a rynighr of j0 kN. wbsr f,,i[ b he p",i"d-cross-M *o*f n"qp.."y

8=

2,1

x

ddt

l06l(8/cE2.

J *..* t "f frilJ

il

Sotrr$orrl Giya der',ik:

rl=4m2

ot. in

r=n)kll rr, 50x ld .=;:lJ_=505.818

=?-t x 16 *9tu1 , AE 4x2-txtff .=7=--E-=nnokC/ca

E

S[&ss

= 7I)0 x

x tS ftm

* lk l6.fftrnf q={;={ffi

*gaocy,

= 367

radifs

-2a

Na!{m! }e*iod,

*.?

= o.l?

I

s

I

ergnffiy

/ =;

= 5'E4

rlz (or) cpo

:SaMalr,. Givca

dctails:

i;

.bl o

ffi*1u*,e$8tr,ffik"1ffi*

tas r=fr-=s.or

Stuic Wc

ftfkcfion

{,

honr thal

I N,h*

'

nz

=;f

=9,31x1s-int=9^BI

EXfA}l?t E 25 A camitey€r beam _3 m toao o** uB ufimil Fiod tl. D€iod u*a and oa,*ir,"q,#,i ,*ljj l:** :yry*-:-"e*

",",-,

61, L

PREPARED BY

rri6rc*s $r

0n*

f*

uffi

r

I lrl/mr = 1000 !{/p r'l= I kg

' * -r#,.tff=31'62ratue

l'limial frcqnoncy

Ci vil da

I

= e*Of

gs p

lee

$l

;T

ErEr

joo

tg Is upper €r4 r/iE t $ iTTfdkffif;f# ilHffi ar

**{-T-}

b osrivabntti

r *-.i-_J

Fjc{rn ?,rl

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28

t\

**pr* 3x2.1xId (3mf

hftit.-{rp,bce,

x 1300

IF lz.vtxtd =rh={=m',

ftcgcocy

= 7l.37

rtds

(r

f =9;lla "2E

I.falud p.dod

7=l=3I=o.zr, TD.,

2f,

A cantil*ar beam AS of hDS&Lk{tackd{or{trhg}d*srBr.r,a, Gquadotr of m!{ioc rtrd {b) tr qppsrig er tb

dprn is ICgs"aIe {l) Frm es

fu

s.b l

of swfioo.

og sp

E:t{nr!r,E

"*

981 lrVqm

ot.

I*.trl

x

in

= 308 rSlcm = 3SJ 2-97 x td l.l/cm

*

Hgs,i 1r2

S&rinr.' (a) E+uion of motion duo

o Sc

tpp€Bd mrss lts is

da ta

$tiftns

TU8 gi&ecr

fr b actiry pfiaud

to

t

.', eqoirahm spriBE 3lifut8s & = tr +

O=X='# t

9El

=V+r

3w.+kE =**?-

Ci vil

Tls ffer*miel quation of motisn is mi- -k.x

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21

ot.

in

*-ffi1=o

da ta

s.b l

og sp

ffiAlrlil, 17 M b e{ esileJl of 6! g !|r dron h E!ilE a*h:!t&

&lptri

tffi* f!. aq q?ated sr.rb[} is shoflr ia 3lEsc ZJd Abo Ariqgs ll ad { al iD p.Etlcl ESrtedd fficts rtr = *r + lz aXX) + 2{IX) = x 40d)'t{/m

&lo

htu

.qtiy.hsr

E@ralom

sp*iag is

cffics*

pedbl

;

fs

Ci vil

;

S. AZHAOARSAI^y .rirt.Tech, MISTE AssT. PROFESSOR Deporthent of Civil Engg.

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3o

rBlhZ{{

fi:

ffm

'.:'ffi={

Nsrural ft$queEj!

ro

= 26.t16 rad/s

ot

.bl og

I =*=a-il*tz 7n

sp

;

ot.

in

q={&=B

Nh'

Ci

vil

da

tas

EXAMPLB aB Cossid.E ihe sysEm sho$s iB Figtrc 2.15. lf *r = ffi tqrn, *4* 1500 lrtrh, *r, S80 N/* aad Ia = *3:r 50d tia sJEeD hac * tmd Eeqcacy of l0 I:l&

AZHAGAR'ATAY M.Tech, AATSTE A55T. PROFESSoR DePqrtment of Civil Engg'

s.

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tu tb re

tr

in ot. sp og l,'lc

E$rfilaer{

.bl

rqEE

!+

fu

s

ryfi{,

1666.67 |,1/m

!+

o\EZf',ti}rEgA,

ah = 6z*8:l rarf*

fF

@-=.1* Y,'l

vil da

BIr

tas

r:!9L 'br

*- Lffitrl

,'l=-+E.---+ (dx

;?.6.J2*

w

Ci

:g+

.+ @l=m

s. AZHA6AS,5.AiAY ilA.Tech, iAfsT€ ASST. PROFESSOR Deportment of Civil Engg. Visit : Civildatas.blogspot.in

32-

ko

={+&. =4xfd +6xtd = f0 x

ld

Nrtr

.bl o

gs p

€o t@ r los tltn*

ot. in

3g hr a d*cr tqn, th. *ifhaa.is 4=?=

4*f*fOEfUrr

tas

t

F$r. at,

Ci vil da

BS

4 * tLfi x lS

N/m.

= 4956 rad/s

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3z

po t.in

tas .bl

og s

il

two sprirgr

tr

vil

Tb

da

fiSull a{8

Aiym

ty

and

Ci

:+

&

frur L19. . ..BxZlxld x 0.833 rran-.-,,-..'....'_!

-

000r

= 8+ kdsn a]lc io strr€s, EE equrvslEnt siifr€ss of th ths co[iForqd

fE=J*J=a*..1 E4rfl) *. * =

& &

45.65 Wcm 45.65 x 981

= 0..lrl8

x

I#

N/c{tr

Narural frequcncy al, =

=

PREPARED BY

12.22 .rad/s

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455T.

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rying

k

ot. in

a

(iii) Frequency: No's of cycles in one sec. Units: H (iv) Amplitude: Ma.ximum displacement of a vibrating body from mean position is called

.bl o

gs p

Amplitude. (v) Natural fr'equenry: whe, there is no extemal force applied on the system and it is given a slight displacement the body vibrates. These vibrations are called free vibrations and Aequency offree vibration is called Natiral frequency. '(vi) Fundamental mode of vibration: Fundamental mode ofvibration is a mode (vii) Degree of freedom:

r- SegbdegEe

tas

drhcGdoln-'

2. T!*o

da!*ce

dtpedgn'

Ci vil da

.

3. Ttlree

degte

oflteedom

The minimum no's of co-ordinates required to specifi motion of a system at any instant is called degree of freedom. (viii) Simple trIarmonic Motion (S.ILM,): The motion of a body "to" and "fro" about a fixed point is called S.H.M. S.FLM.. is a periodic motion and it. is firnction of ';Sine" or *Cosine". Acceleration

of S.H.M. is proportiirnal to displacement from

the mean position and is directed

towards the centre.

Difdeien tiate

wj.t. 'Y

dilferentiate

w

r =Asinarl x t,

Aot ccso.lt

i

-A

s.t-' t'

=

rpg sin t,llt

lAcc. *

iI

[u, =rJ PREPARED BY

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ot. in

.bl o

gs p

9. Explain different methods ofvibration analysis? Different methods ofvibration analysis are: Energr method: According to this method total energy of the system remains constant i.e. sum ofkinetic energy and potential energy always remains coostant.

*:5. = | P.E

1

=; #

*c

tas

K,B+P.E"

*i,

Ci vil da

l**r*|*2""

FIs. 1.4

PREPARED BY

Dlff€rcErgating

w'i.f,

r ,rr*

i"r K* j I(r

',i+ '

Let

. Puttins in (I) we get

=0 =0

x - A sinarl i - - Aarzsln arl

i

=

-a2x

rnx(-rr2r)+Kr =0

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34

ot. in

temt is

.bl o

gs p

, f;n1l*f-l*'

H$ t.5

tas

:

Accordiag to this mettod the sum of. forces and moments acting on the syStem is zero e:rtemal fqce is applied on the system. Consider fig. I ' ar3 +c,+xlt =0 (If rro gcsrna! furtn ls appL:ed)

Ci vil da

mi tci +Xx =F

PREPARED BY

no

(If ecdend'fqce Pb applied)

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L

if

Z7

9. Classify

-

differenf types ofvibrations. Ans. Types of Vibmtions

rTo and

ao motion of the sys[em when dishrbed initially without any extemal force actiag on fio motions of the system under the influence ofextemal force are called forced vibrations. E.g. Bell, Earthquake. IL Linear and Non-linear vibrations . Liiear vibrations: The linear vibrations are those which obey law of superimposition. If ar and a2 are the solutions ofa difflercntial equation, the, al + o2 5tror16 ulso be the solution. arc called free vibratioos. e.g. pendurum. To and

og sp ot. in

it

i +ci +k =fr{0 ,ni+ti+k=fr{r) , mi+g,itkx-Fr{r)+fz{4

a

Non-linear vibrations: when ampritude of vibrations tends towards large varue, then vibrations become non-rinear in nature. They do not obey raw of superimposition. iil. Damped

and Undamped vibrations

II. Damped and Undamped vibralions Damped vibrations are those in which ampritude of vibration

decreases

tas .bl

vibrations are real and are also called transient vibrations.

with time.

These

da

undamped vibrations are those in which ampriiude of vibration remains consiant. In ideal system there would be no damping and so amplitude ofvibration is steady. 1V. Determiiistic and Random vibrations (Non_Deterministic).

vil

Deterministic vibrations are trat whose externar excitation is known or can be detemrined whereas Random vibrations are those whose extemar excitation cannot be determined. e. g. Earthquake

Ci

V. Longitudinal, Transverse and Torsional yibrations

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56

lO Ew cru we make a system vibrate in one of its natural node?

'

Wh

a system is displaced slightly from its equilibrium position and allowed to vibrate then these are called free vibrations and the system is said to vibrate m its natural mode without any exterml force impressed on it. 4

.11. What do

ot. in

you mean by undamped free vibrations? If the body vibrates with intemal forces and no exterilal force is included, it is further during vib,rations if there is no loss of energy due to friction or resistance, it is known as

.12. What is

gs p

D'Alembert's Principle? D'Alembert's principle states that ifthe resultant force acting on a body along with the inertia force is zero, then the body will be in static equilibrium. Inertia force acting on the bbdy is given by F, = rrl a, where, ar = ErasF df the body. and a = *near agreleration of tlre ctrtke sf mass

equilfuium

.bl o

Assuming that &e resultant force acting on body is F, then the body will be in static

if

F+F, =0

tas

Consid€r fig. the spring force of the body Ik is acting upwards and acceleration of the body i is acting in downward direction. The accelerating force iS acting downward so iirerlia force is actirg upwards, so the body is

Ci vil da

M static equilibrium rmder'the action of the two fories Kx and mi. Mathematically it can be writlen as

lx" t..

l'*

Frig, 2.,:t

4X

PREPARED BY

+K*:

0

.

(tfahrnberfs principte)

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5.

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3n

in ot.

#-# m.+53*E

0er2*

l.T ?-r

\ rr

og

r''-

sp

,stiffness of thA spring can be celcul,eUd as

.',- *tbk = ErZ0

N/ni

.bl

.lr

,,

Tftneperio{fft.*

aec.

tas

rE

t-*rF

7r *szs= %r^E

Ci

vil

da

So

3ec.

_ = zzli

', E5t/?

tTZ -s.Sr xo.ol -

O.22 sec.

PREPARED BY

ASSI.

S. AZHA6,AR SAMY PROFESSOR

rtrt. Tech, rtilrSTE Depdrtheni of Civil Engg.

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:Nlm-3?, = 1m0x931

ld

x

N/m

ot.

*

in

Siiflhos(*):

fn=*F^ -1

tn

bt ln = L878llz

Find the natural frequency of the system shown in Fig.

fu

-

dQo !$eF-&

-ts

=,1500

lrlAq lrl =

1o

ks;

lLEe +ings ae in paallell tteir equivalent sfess can be calculated

Ci vil

Sincc

t1

s.b l

Gh

da ta

16.

og sp

Natural frequency of spring-mass system in vertical position is given by

1.-4* b+ ts=1m0+'1500+ 1500 ={00N/m

Na|rlt kqelry'f, 'fo

PREPARED BY

1E

Ztllrl

1lffi-

urtl

to

=3-7Al1z^

aZHiEARSAAY il.Tech. rllISTE A55T. PROFESSOR Depariment ol Civil engg. S.

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L

suspendbd from a Q. 18. A mass is. snspended 3 spring system as showr in figure 2,13, Determine lhe trrtural of .the system. fe = Srrrp r\l/m; &

-

&s

-

0000

fv,[l;'a'-

ZE

tg

Srnce qpa:ag k2 and k3 are corulected in parallel l so their equivalent k3. fuain k and kl are connected in serieq so the equivaleat ke i,

in

k is t- given as k = + gir"ok - -

1.

ot.

k' ,\ kz+k. \ k" =1+1=--]--++ 11 + +

The natural frequency

l

og

1

lt<-

Lzlm

Jn

sp

8000+8fr)0 5000 ----.1.k, =3s0e.52N/m -

l3soe52 2zI 25

tas

.bl

I = 1 96172

Ci

vil da

fb. at3

PREPARED BY

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_

f

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&g

t

I

etcsr sfioI,rr in fi3$r€ 214

& =r&g#e&=lrr€{r@ *f = Slq6f/cnt f. * *s = oi lqfcn

E'*f""t f* " rr€hrat q , fu trid *r r'c ir selis

stild rtt*fE$w

Alt*

SPrtngP

..

erluivelenl

trL

|*ft"!

l. ,rrd *s arc in pwal!*l equiealert *titrers, *.,

again &r, and

.'-

i.,

are

6ek quivelest

= r.+n

Fr!

t. =0657kt'/dt

.

.'.

$ g'-

=4-q'q stiftu* t'171 'l =

Springs

{reqte"cy of

ot. in

;

(:onsk *l< 't

-t,i&-

O.S*OS-r f67<*.

inparalkl-

fuq

l,

=

kr+ &,=0.517+

-f, -

1.0

!.66? x

.bl o

-7&ZWr:a!i*,

at.

gs p

Q. 19.

1f P7-

t h-ffixlm' IF ',r{r,toz"r w/'gsr

f.s'briderlt-g? . DatiE .

is

tas

W = a.1{ N

fu reslmce offered by a body to thi motion of a vibratory

system-

Ib rsirc cly be applied to liquid or solid internally or extemally at the start of the vih'y min tte amplitude of vibration is maximum wkij6es on decreasing with time. depends upon the amount of damping-

Ci vil da

Tb rae of &creing mplitude

ffircnt

types of damPingTypes of Damping

2. Clesd$

5. Viscous 6. Coulomb 7. Structural 8. Non-line'ar,

Slip or interfacial damping

1. Viscous damping: When the system is allowed to vibrate in viscous medium 'the damping is called viscous Viscosity is the property of the fluid by virtue of which it offers resistance to moment of one over the other. S.AZHAGARS,AMY rr .Tech, A rsTE A55T. PROFESSOR DePortment ol Civil Engg.

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15

The force F required to maintain the velocity x ofplatc is given by:

Fls 3.t

.

F

= ni"

A+

6,ie*rons law of visc6r.g)

area ol plate

r _. thickness of fluid film

coefficient of dynamic viscocity-

The force F can also be wriften

F

=

as:

ci

:

Where c is called viscous damping coefficient from

ttA

and

...(2)

(2),

au.per are cylinder, piston and viscous fluid.

.bl

vir"J*

(l)

da

tas

The main components of

-.-o)

og s

I -

*

po t.in

Marirfg Phb V=

Ci

vil

The damping resistance depends upon pressure difference on both sides olpiston in viscous medium' The cleararrce is left between pirt"" -d wdls. More rle. clearance more will be the velocity of piston and ress wilr be the varue of";li;;;; vi""o*

PREPARED BY

ou.ping

Ftst 33

YrDcr

"oefficient.

a.inS

s-Az{r6arsl|IyLTGGI-rr.srEDcfrr-'* of O: Er.

ASST_ PROfESSQ

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4+

Equation of Motion

r,!.i+c.t +Kf,=0

Errt$t disi?ad in vistous damping per rycle (AE) AF. =raoA2

:

specific damping capacity

= max. K.E. of sysem

og sp ot. in

B

g = $;*no*'f E

2' Coulomb DamPitrg: When a body is allowed to slide over the other body the surface of o offers resistance to the movement ofglod over it. This resisting force is callcd force offriction.

F!,€e (eFlisrt)

Flgr

t

4

FtcR11

.

F = p Rp; where p =

tas .bl

Coelficient of friction Some ofthe energy is wasted in friction and amplitude ofvibrations goes on decreasihg. Such type of damping is called coulomb damping. 3. Structural danrping: This type of tlamping tniscs because of iltemrolecular friction beti- thc molecules of structure which opposes its movement. The magnitude of this damping is very small as compared to other damping. Elastic materials during loading and unloading from a loop or sfess strain curve known as hysteresis loop. The area of thii loop gives the amount of energy dissipated in one cycle during vibratiorts. This is also called hysteresis damping.

da

The energy loss per cycle is given as;

83

Ci

vil

St

E = xldA2

* t tl A

MEP.NED BY

FEr A5

Amplitude of vi$ration dimensionless damping factor and .is i€laEd to property of rnaerial represents shape, size and property of matrrlal

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46

Ifenerry

dissipated is treated equar to energy dissipated by viscous damping then; E = BA2

=ttcr,lA2 B

'

'

:

B. _ f--x

I " -*"*-, r'1"*,.:tu I

lt(r)

snl

F = ---'-7-.

zl0

og sp

The amplifude decay is ofexponential nature.

ot.

The damping'force, F

in

,ta)

s.b l

4' Linear & non-linear or IntBrfacial damping: Machine elements are connected th.gugh various joints and microscopic slip occurs over the joints of machine elements which usdisspoint of energy when machine elements are in contact with fluctuating load. The energy dissipated per cycle depends upon coefficient of fiiction, pressure at contacting surfac-e and amplitude of vibration. There is aa optimum value of contact pressure at which energi dissipated is maximum for different amplitudes.

da ta

Zton"on*,^*

6, Differentiate between Coloumb and Viscous damping.

Ci vil

I ' In case of viscous damping ratio of any two successive amplitudes is constaot qaereas in coulomb damping difference between ,*o ,u"""r.iu. amplitudes is constant. 2' ln viscous damping enverope of the maximize in displacement-time plot is aD curve here as in coulomb darnping envelope ",ryone,,ial of maximize of dispracement-time prot is a *aigtt line3' In case of viscous damping the body once disturbed and from equ,iuirm pcition wilr come to rest in equilibrium position although it make theoretically infinite time bao so whqcas in case of coulomb damping the body may finaliy come to res in equilibri,rn positin or m fis!,r,ccd position depending upon initial amplitude and amount of ftiction present

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I{I5TE

4SST PROFESSOR Department of Civil Eryg. Visit : Civildatas.blogspot.in il '.

4b

in sp ot.

s.b log

20. A cantilever beam of negligible rnass is loaded with mass 4m' at the free end. Find the natural frequeucy of the mass m'.

Flg. Z.r5

Ar.

Deflectim of catilever beam loaded at one end can be given as

cbflecticn

dtbtfue

Siftess of beam

FP

=;,

can be calculated as

3E I F Flr3Er:15-

da ta

Force t-m'=

Gdc;rftdminfuwde-fed

Aee vibration is given as

rri +lr =0

vil Ci

r

3EI z7+-r=O IJ

PREPARED BY

t

t

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