MECHANICAL VIBRATIONS: - Vehicle Dynamics and Optimization

130 Pages·2012·1 MB·English

MECHANICAL VIBRATIONS:
LECTURE NOTES FOR COURSE EML
4220
ANIL V. RAO
University of Florida
Spring 2009
ii
Anil V. Rao earned his B.S. in mechanical engineering and A.B. in mathematics from
Cornell University, his M.S.E. in aerospace engineering from the University of Michi-
gan, and his M.A. and Ph.D. in mechanical and aerospace engineering from Princeton
University. After earning his Ph.D., Dr. Rao joined the Flight Mechanics Department
at The Aerospace Corporation in Los Angeles, where he was involved in mission sup-
port for U.S. Air Force launch vehicle programs and trajectory optimization software
development. Subsequently, Dr.RaojoinedTheCharlesStarkDraperLaboratory, Inc.,
in Cambridge, Massachusetts. As a Draper employee, Dr. Rao led numerous projects
related to trajectory optimization, guidance, and navigation of both space flight and
atmospheric flight vehicles. Concurrently, from 2001 to 2006 Dr. Rao was an Ad-
junct Professor of Aerospace and Mechanical Engineering at Boston University where
he taught the core undergraduate engineering dynamics course. While at Boston Uni-
versity, Dr. Rao was voted the 2002 and 2006 Mechanical and Aerospace Engineering
Faculty Member of the Year and was voted 2004 College of Engineering Professor of
theYearforoutstandingteaching.
cAnilVithalaRao2006
(cid:13)
VakratundaMahaakaayaSooryaKotiSamaprabha
NirvighnamKuruMeinDevaSarvaKaaryashuSarvadaa
vi
Contents
1 ResponseofSingleDegree-of-FreedomSystemstoInitialConditions 1
1.1 Mass-Spring-DamperSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 GeneralSolutionofaSecond-OrderLTIDifferentialEquation . . . . . . . . . . . . . . 3
1.3 GeneralSolutiontoSecond-OrderHomogeneousLTISystem . . . . . . . . . . . . . . 4
2 ForcedResponseofSingleDegree-of-FreedomSystems 9
2.1 ResponseofSingleDegree-of-FreedomSystemstoNonperiodicInputs . . . . . . . . 9
2.2 PhysicsofImpulsiveMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 ImpulseResponseofSecond-OrderLinearSystem . . . . . . . . . . . . . . . . . . . . . 10
2.4 StepResponseofSecond-OrderLinearSystem . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 ResponseofSingleDegree-of-FreedomSystemstoPeriodicInputs . . . . . . . . . . . 15
2.6 BaseMotionIsolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.7 FourierSeriesRepresentationofanArbitraryPeriodicFunction . . . . . . . . . . . . 48
2.8 ResponseofaSingleDegree-of-FreedomSystemtoanArbitraryPeriodicInput . . . 53
3 ResponseofMultipleDegree-of-FreedomSystemstoInitialConditions 55
3.1 UnforcedUndampedMultipleDegree-of-FreedomSystems . . . . . . . . . . . . . . . 55
3.2 UnforcedDampedMultipleDegree-of-FreedomSystems . . . . . . . . . . . . . . . . . 74
3.3 Non-SymmetricMassandStiffnessMatrices . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 ForcedResponseofMultipleDegree-of-FreedomSystems 93
4.1 GenericModelforForcedMultipleDegree-of-FreedomSystem . . . . . . . . . . . . . 93
4.2 ResponseofModallyDampedSystemstoNonperiodicInputs . . . . . . . . . . . . . . 93
4.3 ResponseofModallyDampedSystemstoPeriodicInputs . . . . . . . . . . . . . . . . 98
4.4 ResponseofSystemswithGeneralDampingtoPeriodicInputs . . . . . . . . . . . . .102
4.5 UndampedVibrationAbsorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103
A ReviewofLinearAlgebra 107
A.1 RowVectors,ColumnVectors,andMatrices . . . . . . . . . . . . . . . . . . . . . . . . .107
A.2 TypesofMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
A.3 SimpleAlgebraAssociatedwithMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . .110
A.4 NullSpaceandRangeSpaceofaRealMatrix . . . . . . . . . . . . . . . . . . . . . . . .112
A.5 EigenvaluesandEigenvectorsofaRealSquareMatrix . . . . . . . . . . . . . . . . . . .113
A.6 EigenvaluesandEigenvectorsofaRealSymmetricMatrix . . . . . . . . . . . . . . . .115
A.7 SymmetricWeightedEigenvalueProblem . . . . . . . . . . . . . . . . . . . . . . . . . . .117
A.8 DefinitenessofMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
Bibliography 121
viii Contents
Chapter 1
Response of Single Degree-of-Freedom
Systems to Initial Conditions
In this chapter we begin the study of vibrations of mechanical systems. Generally speaking a
vibrationisaperiodicoroscillatorymotionofanobjectorasetofobjects. Vibratingsystems
areubiquitousinengineeringandthusthestudyofvibrationsisextremelyimportant.
Themostbasicproblemofinterestisthestudyofthevibrationofaonedegree-of-freedom
(i.e.,asystemwhosemotioncanbedescribedusingasinglescalarsecond-orderordinarydif-
ferentialequation). Thegenericmodelforaonedegree-of-freedomsystemisamassconnected
toalinearspringandalinearviscousdamper(i.e.,amass-spring-dampersystem). Becauseof
itsmathematicalform,themass-spring-dampersystemwillbeusedasthebaselineforanalysis
of a one degree-of-freedom system. In particular, the differential equation of motion will be
derived for the mass-spring-damper system. It will then be shown that the time response of
thissystemisthesumofthezeroinputresponseandthezeroinitialconditionresponse. Inthis
chapterwewillfocusattentiononthezeroinputresponse,i.e.,theresponseofthesystemtoa
givensetofinitialconditions. Severalexamplesofsingledegree-of-freedomsystemswillthen
begiven. Ineachoftheseexamplesthedifferentialequationwillbederivedandwillbeshown
tohavethesamemathematicalformasthegenericmass-spring-dampersystem.
1.1 Mass-Spring-Damper System
The most basic system that is used as a model for vibrational analysis is a block of mass m
connectedtoalinearspring(withspringconstantK andunstretchedlengthℓ )andaviscous
0
damper(withdampingcoefficientc). Inaddition,anexternalforceP(t)isappliedtotheblock
andthedisplacementoftheblockismeasuredfromtheinertiallyfixedpointO,whereOisthe
pointwherethespringisunstretched. Finally,thespringanddamperarebothattachedatthe
inertiallyfixedpointQ. ThissystemisshowninFig.1–1Denotingunitvectorinthedirection
fromOtoQasE andtheinertialreferenceframeofthegroundby ,theinertialacceleration
x
F
oftheblockisgivenas
Fa x¨Ex (1–1)
=
Next,theforcesexertedbythespringanddamperaregiven,respectively,as
F K(ℓ ℓ )u (1–2)
s 0 s
= − −
F cv (1–3)
f rel
= −
First,becausethespringisattachedatpointQ,wehave
ℓ r r (1–4)
Q
=k − k
2 Chapter1. ResponseofSingleDegree-of-FreedomSystemstoInitialConditions
g
K
Q m P
c
O
x
ℓ
0
Figure1–1 Blockofmassmslidingwithoutfrictionalongahorizontalsurfacecon-
nectedtoalinearspringandalinearviscousdamper.
whererandr arethepositionsoftheblockandtheattachmentpointsofthespring,respec-
Q
tively. UsingacoordinatesystemwithitsoriginatpointOatE asthefirstprincipaldirection,
x
wehave
r xE (1–5)
x
=
r ℓ E (1–6)
Q 0 x
=
Therefore,
ℓ xE ℓ E (x ℓ )E x ℓ (1–7)
x 0 x 0 x 0
=k − k=k − k=| − |
Then,becausex<ℓ wehave
0
x ℓ ℓ x (1–8)
0 0
| − |= −
Finally,theunitvectorinthedirectionfromtheattachmentpointofthespringtotheposition
oftheblockis
us = rr−rrQ = (xℓ−ℓ0x)Ex =−Ex (1–9)
Q 0
k − k −
Theforceinthelinearspringisthengivenas
F K(ℓ x ℓ )( E ) KxE (1–10)
s 0 0 x x
=− − − − =−
Next, because the ground is already assumed to be inertial, the relative velocity between the
blockandthegroundissimplythevelocityoftheblock,i.e.,
vrel Fv x˙Ex (1–11)
= =
Therefore,theforceexertedbytheviscousdamperisobtainedas
F cx˙E (1–12)
f x
=−
Theresultantexternalforceactingontheparticleisthenobtainedas
F P F F PE KxE cx˙E (P Kx cx˙)E (1–13)
s f x x x x
= + + = − − = − −
ApplyingNewton’ssecondlawtotheparticle,weobtain
(P Kx cx˙)E mx¨E (1–14)
x x
− − =
DroppingE fromEq.(1–14)andrearranging,weobtainthedifferentialequationofmotionas
x
mx¨ cx˙ Kx P (1–15)
+ + =
1.2GeneralSolutionofaSecond-OrderLTIDifferentialEquation 3
Now historically it has been the case that the differential equation has been written in a form
thatisnormalizedbythemass,i.e.,wedivideEq.(1–15)bymtoobtain
c K P
x¨ x˙ x p(t) (1–16)
+ m + m = m =
where p(t) P(t)/m. Furthermore, it is common practice to define the quantities K/m and
=
c/masfollows:
K
ω2
n = m
c
2ζω
n = m
The quantities ω and ζ are called the natural frequency and damping ratio of the system,
n
respectively. Intermsofthenaturalfrequencyanddampingratio,thedifferentialequationof
motionforthemass-spring-dampersystemcanbewritteninthesocalledstandardformas
x¨ 2ζω x˙ ω2x p(t) (1–17)
+ n + n =
ItisseenthatEq.(1–17)isasecond-orderlinearconstantcoefficientordinarydifferentialequa-
tion. Often, the term “constant coefficient” is replaced with the term time-invariant, i.e., we
say that Eq. (1–17) is a called a second-order linear time-invariant (LTI) ordinary differential
equation. Theterminology“timeinvariant”stemsfromthefactthat,foragiveninputp(t)and
agivensetofinitialconditions(x(t ),x˙(t ) (x ,x˙ )attheinitialtimet t isthesameas
0 0 0 0 0
= =
the solution to the input p(t τ) for the initial conditions (x(t τ),x˙(t τ) (x ,x˙ ) at
0 0 0 0
+ + + =
the(shifted)initialtimet t τ. BecauseofthisfactassociatedwithanLTIsystem,without
0
= +
lossofgeneralitywecanassumethattheinitialtimeiszero,i.e.,t 0. Thus,whenstudying
0
=
the zero input response of an LTI system we can restrict our attention to initial conditions
(x(0),x˙(0) (x ,x˙ ).
0 0
=
1.2 General Solution of a Second-Order LTI Differential Equation
Eq.(1–17)canbewrittenas
d2x dx
2ζω ω2x p(t) (1–18)
dt2 + ndt + n =
whichcanbefurtherwrittenas
d2 d
2ζω ω2 x p(t) (1–19)
dt2 + ndt + n! =
Nowlet
d2 d
L 2ζω ω2 (1–20)
= dt2 + ndt + n
ThenwecanviewthesystemofEq.(1–17)asasystemoftheform
Lx f (1–21)
=
ItisseenthattheoperatorLdefinedinEq.(1–20)islinearbecause
L(αx βx ) αL(x ) βL(x ) (1–22)
1 2 1 2
+ = +
for all constants α and β. Then it is seen that Eq. (1–21) is a linear system whose general
solutionisofthenformEq.(1–17)isgivenas
x(t) x (t) x (t) (1–23)
h p
= +