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TRANSCRIPT
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Vehicle Dynamics Lecture-3
A course on Vehicle Dynamics
By
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Prof. Sarvesh Mahajan BITS, PILANI
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Learning !jec"ives #
Vibration Analysis Procedure
Natural Frequency
ystem Classifications
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Characteristics of Discrete ystem
Linear #rin$
Non-Linear #rin$
Lineari%ation of Non-Linear #rin$
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Vi!ra"ion Analysis Proce$ure #
te# by te# &e can analy%e Vibration beha'ior of system to ma(e it more accurate
and sim#le to sol'e) *he ste#s can be follo&ed as
Ma"hema"ical Mo$eling #
*he #ur#ose of mathematical modelin$ is to re#resent all the im#ortant features
of the system for the #ur#ose of deri'in$ the mathematical equations $o'ernin$
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t e systems e a' our) e mat emat ca mo e s ou nc u e enou$ eta s to
allo& describin$ the system in terms of equations &ithout ma(in$ it too com#le+)
Deriva"ion of %overning &'ua"ions.
,nce the mathematical model is a'ailable &e use the #rinci#les of dynamics and
deri'e the equations that describe the 'ibration of the system) *he equations of
motion can be deri'ed con'eniently by dra&in$ the free-body dia$rams of all themasses in'ol'ed) *he free-body dia$ram of a mass can be obtained by isolatin$ the
mass and indicatin$ all e+ternally a##lied forces the reacti'e forces and the
inertia forces)
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Vi!ra"ion Analysis Proce$ure #
Solu"ion of "he %overning &'ua"ions #
*he equations of motion must be sol'ed to find the res#onse of the 'ibratin$
system) De#endin$ on the nature of the #roblem &e can use one of the follo&in$techniques for findin$ the solution. standard methods of sol'in$ differential
equations La#lace transform methods matri+ methods1 and numerical methods)
/f the $o'ernin$ equations are nonlinear they can seldom be sol'ed in closed form)
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urt ermore t e so ut on o #art a erent a equat ons s ar more n'o 'e t an
that of ordinary differential equations)
In"er(re"a"ion of "he )esul"s #
*he solution of the $o'ernin$ equations $i'es the dis#lacements 'elocities and
accelerations of the 'arious masses of the system) *hese results must beinter#reted &ith a clear 'ie& of the #ur#ose of the analysis and the #ossible desi$n
im#lications of the results)
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Vi!ra"ion Analysis Proce$ure #
Ma"hema"ical Mo$eling can !e $one in s"ages "o un$ers"an$ i" more clear. As an
e*am(le +e are sho+ing here A mo"orcycle #
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Vi!ra"ion Analysis Proce$ure #
e can $o -irs" an$ secon$ Mo$el as follo+s #
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Vi!ra"ion Analysis Proce$ure #
-inal Mo$eling #
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Na"ural -re'uency #
Natural frequency is the frequency at &hich a system tends to oscillate in the absence of
any dri'in$ or dam#in$ force)
Free 'ibrations of any elastic body is called natural 'ibration and ha##ens at
a frequency called natural frequency)
Natural 'ibrations are different from forced 'ibration &hich ha en at fre uenc of a lied
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force 5forced frequency6)
/f forced frequency is equal to the natural frequency the am#litude of 'ibration increases
manifold) *his #henomenon is (no&n as resonance.
For a normal #rin$ Natural Frequency can be found out as
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Na"ural -re'uency #
The na"ural fre'uency is im(or"an" for many reasons
1) All thin$s in the uni'erse ha'e a natural frequency and many thin$s ha'e more than one)") /f you (no& an ob7ect8s natural frequency you (no& ho& it &ill 'ibrate)
3) /f you (no& ho& an ob7ect 'ibrates you (no& &hat (inds of &a'es it &ill create)
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)
) natural frequencies that match the &a'es you &ant)
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Sys"em /lassifica"ions 0
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Sys"em /lassifica"ions #
Lum(e$ Parame"er Sys"em # Can be Defined by ordinary Differential ;quation
Dis"ri!u"e$ Parame"er Sys"em 0 /nfinite Dimensional Problems
De"erminis"ic Sys"em # All the #arameters are (no&n e+actly
S"ochas"ic Sys"em 0 #arameters are (no&n #robabilistically
/on"inuous "ime Sys"em # Variables are defined for all 'alues of time
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Discre"e "ime Sys"em 0 Variables are defined only at discrete instances
Linear Sys"em # u#er#osition Princi#le is satisfied
Non0Linear Sys"em 0 u#er#osition Princi#le is not a##licable
Time0Invarian" Sys"em # All Para constant 5uch ystem can be defined byconstant coefficient differential equations6
Time0Varying Sys"em 0 uch ystem can be defined by time 'aryin$
coefficient differential equations6
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Sys"em /lassifica"ions #
Lum(e$ Parame"er Sys"em v1s Dis"ri!u"e$ Parame"er Sys"em
A lum#ed system is one in &hich the de#endent 'ariables of interest are a function of time
alone) /n $eneral this &ill mean sol'in$ a set of ordinary differential equations 5,D;s6
A distributed system is one in &hich all de#endent 'ariables are functions of time and one
or more s#atial 'ariables) /n this case &e &ill be sol'in$ #artial differential
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equations 5PD;s6
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Sys"em /lassifica"ions #
De"erminis"ic Sys"em v1s S"ochas"ic Sys"em
/n mathematical modelin$ deterministic simulations contain no random 'ariables and no
de$ree of randomness and consist mostly of equations
2uan"um mechanics, /haos Theory
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/n #robability theory a #urely stochastic system is one &hose state is randomly
determined ha'in$ a random #robability distribution or #attern that may be analy%ed
statistically but may not be #redicted #recisely)
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Sys"em /lassifica"ions #
/on"inuous "ime Sys"em v1s Discre"e "ime Sys"em
A system is continuous-time &hen its /
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Sys"em /lassifica"ions #
Linear Sys"em 3 Non0Linear Sys"em
Linear systems must 'erify t&o #ro#erties su#er#osition and homo$eneity)
*he #rinci#le of su#er#osition states that for t&o different in#uts + and y in the
domain of the function f
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f 5+ = y6 > f 5+6 = f 5 y6
*he #ro#erty of homo$eneity states that for a $i'en in#ut + in the domain of the
function f and for any real number (
f 5(+6 > (f 5+6
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Sys"em /lassifica"ions #
Time0Invarian" Sys"em 3 Time0Varying Sys"em
Linear *ime /n'ariant 5L*/6 systems are commonly described by the equation.
+ 5'ector6 > A+ = ?u
*ime Var in s stems are commonl described b the e uation.
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%eneral /onsi$era"ions #
?y no& &e ha'e learned D,F and ystem classifications)
/n $eneral a system defined by sin$le second order differential equation is (no&n as in$le-de$ree-of-freedom-system)
*he mathematical formulation associated &ith multi-D,F discrete and continuous system
can be reduced do&n to set of inde endent second-order-differential e uation hence
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thorou$h study of sin$le-D,F system throu$h second-order-differential equation is 'eryim#ortant for us) *his case is for Linear systems only)
*he res#onse of system to initial e+citation is (no&n as free res#onse)
*he res#onse to e+ternally a##lied force is (no&n as forced res#onse)
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/harac"eris"ics of Discre"e Sys"em /om(onen"s #
*he elements constitutin$ a discrete mechanical systems are mainly dis#lacement 'elocity
and accelerations)
*he most common e+am#le &ith &hich &e can understand this #rinci#al is a Free #rin$
ystem) #rin$s are $enerally assumed to be of ne$li$ible mass and dam#in$)
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/harac"eris"ics of Discre"e Sys"em /om(onen"s #
A s#rin$ is said to be linear if the elon$ation or reduction in len$th + is related to the
a##lied force F as
F > (+ 53)16
&here ( is a constant (no&n as the s#rin$ constant or s#rin$ stiffness or s#rin$ rate) *he
s rin constant ( is al&a s ositi'e and denotes the force ositi'e or ne ati'e re uired to
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cause a unit deflection 5elon$ation or reduction in len$th6 in the s#rin$)
*he &or( done 5@6 in deformin$ a s#rin$ is stored as strain or #otential ener$y
in the s#rin$ and it is $i'en by
@ > 1
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/harac"eris"ics of Discre"e Sys"em /om(onen"s #
Non0Linear S(rings
ost s#rin$s used in #ractical systems e+hibit a nonlinear force-deflection relation
#articularly &hen the deflections are lar$e) /f a nonlinear s#rin$ under$oes small deflections
it can be re#laced by a linear s#rin$ by usin$ the #rocedure discussed)
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/n 'ibration analysis nonlinear s#rin$s &hose force-deflection relations are $i'en by -
- 4 a* 5 !*67 a 8 9 53)36
*he s#rin$ is said to be hard if ! 8 9linear if !49
and soft if ! : 9
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/harac"eris"ics of Discre"e Sys"em /om(onen"s #
Non0Linear S(rings
*he force-deflection relations for 'arious 'alues of b are sho&n in Fi$) belo& -
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/harac"eris"ics of Discre"e Sys"em /om(onen"s #
Non0Linear S(rings
ome systems in'ol'in$ t&o or more s#rin$s
may e+hibit a nonlinear force-dis#lacement
relationshi# althou$h the
indi'idual s rin s are linear)
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/harac"eris"ics of Discre"e Sys"em /om(onen"s #
Lineari;a"ion of a Non0Linear S(rings
Actual s#rin$s are nonlinear and follo& ;q) 53)16 u# to ;lastic Limit as sho&n belo& and
beyond ;lastic Limit they start beha'in$ Non-Linear)
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/harac"eris"ics of Discre"e Sys"em /om(onen"s #
Lineari;a"ion of a Non0Linear S(rings
*o illustrate the lineari%ation #rocess let the static equilibrium load - actin$ on the s#rin$
cause a deflection of +B) /f an incremental force F is added to F the s#rin$ deflects by an
additional quantity +) *he ne& s#rin$ force can be e+#ressed usin$ *aylor s series e+#ansion
about the static equilibrium #osition as
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53)06
For small 'alues of + hi$her order deri'ati'es can be ne$lected
53)6
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/harac"eris"ics of Discre"e Sys"em /om(onen"s #
Lineari;a"ion of a Non0Linear S(rings
ince F > F5+B6 e can e+#ress F as follo&s
F > (+ 53)26
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here ( is Lineari%ed s#rin$ constant at +B and can be e+#ressed as
53)6
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/harac"eris"ics of Discre"e Sys"em /om(onen"s #
Lineari;a"ion of a Non0Linear S(rings
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