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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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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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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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e can $o -irs" an$ secon$ Mo$el as follo+s #


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-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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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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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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/on"inuous "ime Sys"em v1s Discre"e "ime Sys"em

A system is continuous-time &hen its /

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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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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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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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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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Non0Linear S(rings

*he force-deflection relations for 'arious 'alues of b are sho&n in Fi$) belo& -


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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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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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*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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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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