c t,MT- II MULTIPLE DEGREE OFFREEI}OM SYSTEM
'flvo
fieedom system -- Norrnal modes of vibration - r.l*atural - Mode shapes .- Introduction to MDOF sy.stems - Decoupling of equations of degreo
of
frequencies motion - Concept of mode superposition (No derivatiors).
Two lvlorks Questions ond Ansyers 1. Define degrees of fieedom. The n-o. of indep€ndeht tlisplaccments required to define the d.isplaced positiom of all the masses relative to their original position is called the no. of degrees of freedom for d;,n:imic analysis.
2. Write
a
short note or matii:deflation technique-
Whenever the starting vector, tbe vector'iteration metbod yieltls the same lowest Eigen value. To otftain the next lowest valug the one aheady found must be suppressed. This is possible by selecting vector tbat is ortbogonal to the eigen values already foruid, or by modifying any arbitra-rily selectgd !4itial. vectgr fore oithogo;nal to alrg{f.'ev.aJeated vectors:
The Eigen vectors xL2 cor.npr+lgd br jterllon as in the previous eFmple x1 would . be orthogonal to the xll. The conesporlding frequency will !e higher *r* l,rt but fowel 3. :\Yrite tlie.e;amples of multi degrees of freedbm systein
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AZHAGARSAMY /t{-Tech, AAISTE ASST. PROFESSOR Deportmenl of Civil En99'
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M.Ethod
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cmtile us to cahrhte the lowest Eigen vahres one after anotler, Bdlhd riilds all tb Eigen vahes simultaneously. By a series of transformations of chssfual frrm of tbe maix prescribed by Jacobi all-the non diagonal terms may be the ffnal diagonal matrk gives all the Eigen values along the diagonal.
i. Whlt are the steps to be followed to the dynamic onrtysis of structure? The dlmanric analysis of any structure basically consists of the following steps.
1.
2.
Idealize tlrc structure for the purpose of analysis, as an assemblage of discree! elements which are intercornected at tlre nodal points. Evaluate the s.tiffness, iaertia and damping property matfices of the elemeats chosen supporting the element prop€rty matrices appropriately formulate the corresponding marrices representing the stiffiiess, inertia and damping of the whole structure,
3. By
. Write a short note on Inertia force - Mass matrix [M] On the same analogy, the inertia forces can be represented in terms of rfluence co efficient, the matrix r.epresertation of which is given by {fi} : [M] {Y} Mij a typical elemed
mass
matrix M is defined as tbe force corresponding to co -ordipate due as the force corresponding to coordinate i due to uit acceleration applied to the co rdinate
of
j.
. ll'ha-t
are the efiects of Damping? ; The presence of damping in the system afects the nah:ral frequencies only to a rargindl sxtenl. h ij conventional therefore to ignore darnpiirg in tlte computation. s for natural :equencies and mode shapes
. 'Write
;
da4ping force - Damping force matrir. If damping is assuming to be of the viscous t)pg, th" damping forces may kewile be represgnted by mqan$ of a general damping inlluence co efficient, C1. In matrix a short note on
lfln rhis san
be repres€rfed
d trD):tc1{Y}
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10. What are the steps to be followed to the dynamic anatysis of structure? The
I' 2.
Idealize the skucture for the purpose sf analysis, as an assemblage of discreet elements which are intercontrected at the nodal points. Evaluate the stiffness, inertia and damping property matrices of the elements chosen
3. By
supporting the element propedy matrices appropriatelg formulate the cofr.eqponding inatrices represenling tlic stiftess, iirertia and damping of the whole structure.
11. lVhat are lorraal modes ofvibration? If in the principal mode of vibration, the amplitude of one of the masses is qnity, it is known as nor-mal modes of vibration 12. Define Shear
building.
Shear buildiqg is defuied as a g.tructure in which no rotation of a horizontal merrber at the
14.
15.
" {dlltklt@};
:
0, this condition is called orthogonality principles.
16. Explain Damped system. o The response to the damped MDOF system subjected to free vibration is govemed by
lM){il} + tcl{ri} + t rl{u} =
o
o In which [c] is damping matrix and {ri} is velocity vector.
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fs*allyr srnall srnolpi 6f drmping is always present in rear structure and it does not have nri inEnce on rhe detemination of natural frequencies and mode rhup".;;;; sy$cm-
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TIrc ffiuxBlly frcquencies and mode shapes for the damped systein are carcurated : -' by using the same procerfirre adopterl for undamped system.
17. What is meant by first and second mode of vibration?
The lowest frequency of the vibration is cafled fimdamental frequency and the :onesponding displacement shape of the vibration is ialled first mode or firndamental mode of ribration. The displacement shape corrasponding to second higher natural t"qr*cy is called lecond mode of vibration. [8. write the equation of motion for an undamped two degree of freedom system.
Iml{r{+[k]{a}=0 This is ca ed equation ubjected to free vibration.
of motion for an
lIkJ- aztmll.=
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of
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.-'--..._ l: Define 'Nqrmal mode of vibr.afion?? In a trvq degree freedom system ther.e are two natural frequeircies 'stem--at its-.lowest. or first *etJle] frequency its first and next higher :quericy is called its second mode.
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s.AZllAOAP,sAl Y At.Tech, /IIISTE ASST. PROFESSOR Deportmenf of Civil En99.
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23. What is a two degree system? In a two degree freedom system, any point in the system may execute hamonic of the two natural frequencies and these are knoun of vibration Let us assume. the motion of two masses is simple barmonic and is represented as
s, -_X1&ot .ta =X2$nart
Where Xl and X2are the amplitddes principal co-ordinal.
of two
masses respectively aod are rcferred as
24. What are Vhrious mbthods available for vibratiori control? Various methods ofvibration contol are
l.
.
vibration Absorbers (centrifugal pendulum absorber, Lanchester damper, Houdaille
damper). 2. Vibration Isolation materials like rubber, cork,
fel! pad etc.
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Write any two poinfs of asstumption are made in the shi:ar s[ building' Nov/Dec 20rr r To dercrmin" the eff"ct ort*,l::,_,:-'*'n aequencv of rhe .nderparab",r; Momcnr
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ffit'":ffi}"{,tkiT#*,Tiifi;::*'#i:{:::,ryffi re sonle gxamples of vibrations
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Enumerate dynamic equilibrium? May/June 2009 A dynamic eqtrilibrium exisb once a reversible reaction ceases to change its ratio oi r.-eactantvproducts, but substances move between the chemicaLs at an eqr_lal *t", me"o;r,g there is no nbt change. It is a particurar example ofa system in a steady state.
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Deline transitional gro .nd motion? May/June 2009 A general term including afl aspects of grorm_ d modon, namely particle accereration, velocity, or displacement, fiom an earthquaku o, oth", .oogy *,*". '
7.
rrow fiequency is affpcted irr the free vibration of a shear frame? Nov/Dec 2007 o Modal analysis is the study of the dynamic properties of structures under vibrational excitation.
the dynamic response -of
o
o Iqnkire frrce is a force which acts on an objbct for a very short interval during c
pTtoc;m- E is equals to tbe
e Are hsfun in a string friction and other non.corservative
collision
forces impllsive forces for all ron-conservative forces being impulsive Impulsive force is the erornous amormt of forbe acting for a very small time on a body. fcr e.g. when batsman hits the ball, the period of contact is .,r"ry i"., but force is hug", the friction carurot be impulsive force becarise friction acts for longer +ime to reshict the motion of the body, neither tension in the sfing
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l. What is meant by coupled and uncoupled equations of motion? May/June 2013 'fhe equations of urotiorr arc rucoupled and koowu as the modal e$Bdons
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2. \Yhat.is meant by mode shape? May/June 2013 y' The response cif the structwe is different at each.,of the different natural frequencies. These defonlation patterns are called mgde shapes. A norinal mode of an oscillating system is a pattern oimotion in which all parts of the system move sinusoidally with the same,frequency.
/
iiffiJ5i:'JJ"1,1-iili#;,"**abodypossessesarethosenecessarytecompretery Lefine its position and
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LJ WORI(ED EXAMPLE ILo.3
A spring of stiflhess 20 kI.I/m supports a mass of 4 kg. The mass is pulled dovm 8 mm
and relcased to produce linear oscillations. Calculate the frequency and periodic time. Skctch the graphs of displacemen! velocity and acseleration. Calculate the displacemeng Velocity acceleration 0.05 s after beiog released-
SOLUTION
,=,H=,'P
=to.7t.dtl
r=9=t.zsnu 2tt T
,.
=:f = 0.0899s
:'
:
The oscillatidn starts at the bottom ofthe cycle so xe -8 min The resulting graph of time will be a negative cosine cuws with an amplifude of 8 mm. The equations describing the motion are as follows.
x = xocosot cos(7071 x 0.05) Wheh t = 0.05 seconds x :7 .iB7 rom. (Note angles:ab in radian) = This is confirnred by the graph"
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we differentiate.once we get the equaaion forvelocity. v = -oxosin . v= -oxosin 10.71 (-8)sitr(Tifzf x o.Os;
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