dynamics and vibration, wahab,2008-chapter 8

8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8

http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 1/46



Introduction

Two co-ordinates and two

euations o! motion

Two independent motions

k "

m"

m#

k #

x#

x"

Trailer

Suspension

$%le

Tyre

k r

y f yr

k f

yC

CG

C θ

k r

y f yr

k f

yC

CG

C θ

Chapter 8: Systems With Two Degrees Of Freedom (page 3!"

© John Wiley & Sons, Lt



#$uations of motion (page 3%"

k "

x"

m" m#k #

x#

f " f #

on!iguration

'sing (ewton)s #nd law

x=m F ..

x """∑ ###

..

x x=m F ∑

""""##""

..

x=m )+f x(x+k x-k −

###"##

..

x=m )+f x(x-k −

*ree body diagrams

+x

"" xk + "## x xk −

f "

+ "## x xk −

f #




#$uations of motion

""""##""

..

x=m )+f x(x+k x-k −###"##

..

x=m )+f x(x-k −e-arranging

"##"#""" =f x-k )x+k +(k xm..

###"### =f x+k x-k xm..

$nd in matri% !orm

#

"

#

"

##

##"

#

"

#

"

=

+

f

f

x

x

k -k

-k +k k

x

x

m

m..

..




#$uations of motion (page 3&"

a+ on!iguration

cθ

C y

k "

L"

k #

$

L#

yC f

C M θ

b+ *ree body diagram

$

L"

L#

mg

+" A A yk δ + +# B B yk δ +

yC f

C M θ

+x

+y

θ +

C

..

yC B B A A y=mmg+f y-k yk −++ ++- #" δ δ

&

..

##"" ++ θ δ δ θ C C B B A A =I +M L y-k L yk ++


C

..

y=m F ∑ y &

..

θ C C =I M ∑




#$uations of motion

+x

+y

θ +

k "

L"

k #

$

L#

y A

y B

yC C θ

d+ 0e!ormed bar

" tan L

y y AC

C

−

=θ

tan" C C A -L=y y θ C C B +L=y y θ tan#

*or small 1ibration C C θ θ ≈tan

" C C A -L=y y θ C C B +L=y y θ #

Substituting in the euation o! motions

$nd in matri% !orm


2##""#" yC C C c

..

f ) Lk L(k )yk +(k ym θ −−+

2

#

##

#

""##""&

..

C C C C M ) Lk L+(k )y Lk L(k I θ θ θ +−−

23

&

#

##

#

""""##

""###"

&

..c

..

+−

−+

C

yC

C C M

f y

Lk Lk Lk Lk

Lk Lk k k y

I

m

θ θ θ



#$uations of motion (page 3&'"

$ spring-suspended mass system )+y(x=-k P iiii θ θ sincos

x=m +f θ P ..

x

i=

ii∑4

"

cos y=m +f θ P ..

y

i=

ii∑4

"

sin


) =f θ θ +yθ (xk + xm xii

i=

ii

..

sincoscos4

"

#∑

) =f θ +yθ θ (xk + ym yi

i=

iii

..#

4

"

sincossin∑


k "

mk 3

k #

#θ

4θ

"θ f

x

f y

x

y

P "#θ

4θ

"θ f

x

f y

P #

P 4

x

y



#$uations of motion

=

+

∑= f

f

y

x

θ θ θ

θ θ θ k

y

x

m

m

y

x

iii

iii

i

i..

..

#

#4

" sincossin

cossincos

$nd in matri% !orm

The general matri% !orm +S D=F D M

..

F

F

D

D

S S

S S

D

D

M

M ..

..

=

+

#

"

#

"

###"

"#""

#

"

##

""


P "#θ

4θ

"θ f

x

f y

P #

P 4

x

y



#$uations of motion

+S D=F D M ..

5n case o! !ree 1ibration

+S D= D M ..

D

D

S S

S S

D

D

M

M ..

..

=

+

#

"

###"

"#""

#

"

##

""

The displacement time response solutions +sin"" ψ ω += t D D m +sin## ψ ω += t D D m

Where Dm" and Dm# are the ma%imum 1alues or amplitudes+

0i!!erentiation twice

n)damped free *i+ration (page 3&,"


+sin"

#..

" ψ ω ω +−= t D D m+sin#

#..

# ψ ω ω +−= t D D m



#$uations of motion

=

−

−

D

D

M S S

S M S

m

m

#

"

##

#

###"

"#""

#

""

ω

ω Substituting in the euation o! motions

*or non-tri1ial solutions

##

#

###"

"#""

#

"" =−

− M S S

S M S

ω ω *rom which

+- #

"###""

#

""####""

6

##"" =−++ S S S S M S M M M ω ω

Sol1ing using the uadratic !ormula




#$uations of motion

!!c"-"

#62

##

#," −±ω Where

!=M M # "=-(M S M S )# c=S S S"" ## "" ## ## "" "" ## "#

#+ −

#

"###""

#

"#####""

# 66 S M M )S M S !c=(M -" +−The term is always positi1e

=

−

−

D

D

M S S

S M S

m

m

#

"

##

#

###"

"#""

#

""

ω

ω

The amplitude ratios

S-SS-

#"

###"##

""

#

"""

"#

#

""

M S M D

D=r m

m ω ω

−=−

= S-S

S- 2#"

######

""

#

#""

"#

#

"#

M S M D

D=r m

m ω ω

−=−




#-amp.e 8/,: Free *i+ration I (page 3&0"

"7 8(/m

x"

# 8g " 8g

x#

#7 8(/mTwo masses are eual to #

8g and " 8g

The two springs) constants are eual

to "7 8(/m and #7 8(/m alculate the natural !reuencies and the mode

shape amplitude ratios

S$%&ti$'

=

=

#

"

##

""

m

m

M

M M M ""2m"2# 8g, M ##2m#2" 8g




#-amp.e 8/,: Free *i+ration I

"7 8(/m

x"

# 8g " 8g

x#

#7 8(/m

=

##

##"

###"

#"""

k -k

-k +k k

S S

S S S=

S ""

2k "

3k #

26 (/m, S "#

2S #"

2-k #

2-#7 (/m,

S ##2k #2#7 (/m

9#

"###""

7

""####""##"" ":7.4";# ×−×−=+= =S S # c=S )S M S # "=-(M M !=M

rad/s<4.7#672rad/s,#".777<2 ,##

":7.4#6+";";2

#"

9#77#

#,"

ω ω ω

×

×××−×±×

π ω

#" π

ω #

# f "2 24.64 =>, f

#2 2"."" =>




#-amp.e 8/,: Free *i+ration I

The amplitude ratios are?

phase+-anti .<"6-2

#+7#67.<46

#7

S

S- 2

phase+-in .9"62#+777<.#"6

#7- 2

#

""

#

#""

"##

#

""

#

"""

"#"

×−

=

−

×−=

−

M

r

M S

S r

ω

ω

x" x#

.9"6 "

x" x#

-.<"6"




#-amp.e 8/0: Free *i+ration II (page 3&%"

$ racing car has a mass o! #4 8g

The sti!!ness o! the !ront and rear

wheel/suspension are both eual to

":< (/m The mass moment o! inertia o! the

car is 67 8g.m#

0etermine the !reuencies and mode shape amplitude

ratios o! the car


*igure @9.#-"

.# m

A

".: m # m .4 m



#-amp.e 8/0: Free *i+ration II

S$%&ti$'

=

=

C I

m

M

M M

##

""

M ""

2m"

2#4 8g, M ##

2 I

267 8g.m#

S ""2k "3k #247# (/m, S "#2S #"2k # L#-k " L"27#9 (, S ##2k " L"

#3k # L##2"#"#<6 (.m,


+−

−+=

#

##

#

""""##

""###"

###"

#"""

Lk Lk Lk Lk

Lk Lk k k

S S

S S S=

*igure @9.#-"

.# m

A

".: m # m .4 m



#-amp.e 8/0: Free *i+ration II

"#6.6

"#"6.6;4"7

;#

"###""

:

""####""##""

×−

×−=+=

=S S c=S

# )S M S # "=-(M M !=M

rad/s":.479;2rad/s,"#.#;2

,;4"7#

"#6.6;4"76+"#"6.6"#"6.62

#"

;#::#

#,"

ω ω

ω ×

×××−×±×

#.:<=>2#

".;<=>,2#

##

""

π

ω

π

ω == f f


-#24<

#"m/rad-"".69"92#4+#;."#47#

7#9

S

S- 2 4#

""

#

"""

"#"

π

ω ×××−

−=− M r mm/degree anti-phas

#.:24<

#"m/rad."7692

#4+479;.":47#

7#9

S

S- 2 4

#

""

#

#""

"##

π

ω ××

×−

−=

− M r mm/degree in-phase+

-# mm

"o

cθ

C y

Bibration mode "

+x

+y

θ +

#.: mm

"o

cθ

C y

Bibration mode #



#-amp.e 8/3: Free *i+ration III (page 3&&"

x"

m"

x#

k m#

$ semide!inite or unrestrained system

two masses m" and m# are attached to

each other with a spring o! sti!!ness k

0etermine the two natural !reuencies o! the system

S$%&ti$'

M ""2m", M ##2m#

=

k -k

-k k

S S

S S S=

###"

#""" S ""2k , S "#2S #"2-k , S ##2 k




Torsiona. 1i+ration

k " #G" #I P "

"", θ

"

##,( θ

k # #G

# #I

P #

L" L#

#

*igure 9.:? Torsional two degrees o! !reedom system

""θ k

"

+ "## θ θ −k

31e

#(

#(

Where the torsional sti!!ness

"

"""

L

I Gk

*=

#

###

L

I Gk

*=

#

"

#

"

##

##"

#

"

#

"

=

+

k -k

-k +k k

..

..

θ

θ

θ

θ

5n matri% !orm

$nd !or !ree 1ibration

#

"

##

##"

#

"

#

"

=

+

k -k

-k +k k

..

..

θ

θ

θ

θ




#-amp.e 8/2: Free torsiona. *i+ration (page 38"

.7 m.7 m

# 8g

" 8g

Two dis8s o! masses " 8g and

# 8g, ha1ing radii o! 7 cm

and " cm

Counted on a steel solid sha!t o!diameter "7 mm and length " m

0etermine the two torsional natural !reuencies o!

the system and the corresponding amplitudes ratios




#-amp.e 8/2: Free torsiona. *i+ration

.7 m.7 m

# 8g

" 8g

S$%&ti$'

The polar moment o! inertia o!

the sha!t

6;

66

m"-;:.64#

-"7.-

4#

−×=

×==

π π D

I *

The dis8s) mass moments o! inertia

4##

""" "#7."

#

7."

#

−×=×

==r m

8g.m#

".#

".#

#

####

# =×

==r m

8g.m#




#-amp.e 8/2: Free torsiona. *i+ration

.7 m

.7 m

# 8g

" 8g

The torsional sti!!ness o! the

sha!t is constant

49.:<77.-

"-;:.6"-::;;

#" =×××

===−

L

GI k k *

(.m

=

=

#

"

##

""

M M M

M ""

2 "2 4

"-#7." −× 8g.m#, M ##

2 #2." 8g.m#

=

##

##"

###"

#"""

k -k

-k +k k

S S

S S

S=

S ""

2k "3k

#2"74.:< (.m, S

"#2S

#"2-k

#2-:<7.49 (.m, S

##2k

#2:<7.49 (/m




n)damped forced *i+rations

k "

x"

m" m#

k #

x#

f " f #

on!iguration

The steady state solutions

t A D ω sin"" = sin## t A D ω =

0i!!erentiating twice with

respect to time

t A D ω ω sin"

#"

..

−= sin#

##

..

t A D ω ω −=

Substituting in the euation o! motions

=

−

−

F

F

A

A

M S S

S M S

$

$

#

"

#

"

##

#

###"

"#""

#

""

ω

ω

Sol1ing !or D AE SS-

S-S"

#

"

""

#

""#"

"###

#

##

#

"

−

−=

F

F

M

M

C A

A

$

$

ω

ω




#-amp.e 8/,': Forced *i+ration I (page 32'"

".7 m .7 m

$

*igure F9."-"

The bar has a length o! # m and

a mass o! " 8g

k A2< (/m and k B2; (/m

a+ alculate the two natural !reuencies o! the bar.b+ 0etermine the amplitude ratios in mm/degree+.

c+ 5! a harmonic !orce o! magnitude P 2# ( and !orcing

!reuency o! " rad/s is applied at A, determine the

absolute 1alues o! the two amplitudes.




#-amp.e 8/,': Forced *i+ration I

".7 m .7 m

$

*igure F9."-"

S$%&ti$'

=

=

C I

m

M

M M

-

-

-

-

##

""

+−

−+=

#

##

#

""##""

##""#"

###"

#"""

Lk Lk Lk Lk

Lk Lk k k

S S

S S S=

M ""

2" 8g, M ##

2 I 2mL#/"#24.44 8g.m#

× ×"- ; .72"7 (/m,S ""

2<3;2"7 (/m # S "#

2S #"

2<

× "#3; .7#29#7 (/m.S ##

2 < ×




".7 m .7 m

$

(+t "-sin#-


The amplitude ratios

phase+-in /mm7#.#4<

#"m/rad."66#2

"+;4:."7"7

"7

S

S- 2

phase+-anti /mm#7.64<

#"m/rad-#.4<2

"+;:;."""7

"7

S

S- 2

4

#

""

#

#""

"##

4

#

""

#

"""

"#"

°=×××−

−=

−

°−=×××−

−=

−

π

ω

π

ω

M r

M r

'sing F o"2# (, F o#2

mm66m66.44.##4444

444.;944

"#"7+""4#7+"444.44

#+444.4+"9#7

S-+

-#6

#

"##

##""

#

""####""

6

##""

#"#"##

#

##

"

===

+×−×

××−

=++−== S S S M S M M M

F -S )F M (S

A$$

C ω ω

ω





°−===

+×−×

×−=

++−==

::.rad"464.##.##4444

:

"#"7+""4#7+"444.44

#"7

S-+

#6"##

##""

#

""####""

6

##""

"#"#""

#

""#

S S S M S M M M

F -S )F M -(S A $$

C ω ω

ω θ

*igure F9."-#

".7 m .7 m

$

(+t "-sin#-




#-amp.e 8/,,: Forced *i+ration II (page 320"

"7 8(/m

x"

# 8g " 8g

x#

*igure @9.""

#7 8(/m

" (

Two-degree o! !reedom system

=armonic loading o! magnitudes

F o"2" ( and F $#2

0raw the steady-state amplitudes 1ersus

ω cur1e response spectrum+

S$%&ti$'

9#

"###""

7

""####""##"" ":7.4";# ×−×−=+= =S S # c=S )S M S # "=-(M M !=M

'sing F o"2# (, F o#2




#-amp.e 8/,,: Forced *i+ration II

"7 8(/m

x"

# 8g " 8g

x#

*igure @9.""

#7 8(/m

" (

'sing F o"2# (, F o#2

9#76

<

9#76#

9#76

#<

9#76

#

"

":7.4";-#

"#.7-

":7.4";-#

"#7+-2

":7.4";-#

"-"#.7

":7.4";-#

"+"-#72

×+×

×=

×+×

×

×+×

×=

×+×

×

ω ω ω ω

ω ω

ω

ω ω

ω

A

A

*igure @9.""-4

rad/s+ω

" A # A

ω rad/s




1i+ration +sor+ers (page 32!"

k "

x"

m" m#

k #

x#

f " f #

on!iguration

The two amplitudes

#"###

###""

#""

#"#"###

##"

-

)-S M - )(S M -(S

F -S )F M (S = A

$$

ω ω

ω

2$#"######""#""

"#"#""#

""

# )-S M - )(S M -(S

F -S )F M -(S $$

ω ω

ω

$$ F F =" -# =$ F 'sing the !orce amplitudes as

""" m M = ### m M =#"# k S −=### k S =The mass and stiffness elements

###

##"

##"

###"

-

)-k m- )(k m-k (k

)F m(k = A $

ω ω

ω

+ 2$

###

##"

##"

##

)-k m- )(k m-k (k

F k $

ω ω +


t F f $ ω sin=



1i+ration +sor+ers

Bibration absorption is achie1ed when

the amplitude o! the mass m" is >ero

-

-#

###

###

#"#

#"

##

#" =⇒=

+ )m(k

)-k m- )(k m-k (k

)F m(k = A $ ω

ω ω

ω

*rom which ##

#

m

k

=ω

x"

m"

x#

k "/# k "/#

x"

m"

m#

x#

k "/#

t F f $ ω sin=

k "/#

k #

when A"2

#

#

k

F A $=


t F f $ ω sin=



1i+ration +sor+ers

x"

m"

x#

k "/# k "/#

x"

m"

m#

x#

k "/#

t F f $ ω sin=

k "/#

k #

rad/s+

Griginal system without

1ibration absorber+

Codi!ied system with1ibration absorber+

" A

ω




#-amp.e 8/,0: 1i+ration a+sor+ers I

Cotor

x"

k #

x#

m#

S$%&ti$'

'sing S0G*

<:9.<9<

#<7" =×=

π ω rad/s

"<:9.<9 "

"

""

k

m

k =⇒=ω

*rom which

<" "<44.6 ×=k (/m




#-amp.e 8/,0: 1i+ration a+sor+ers I

94.#"7"------

-<:9.<9 #

#

#

#

## =⇒=⇒= mmm

k ω 8g

<"<44.7 × M ""

2m"2" 8g, M

##2m

#2#"7.94 8g, S

""2k

"3k

#2 (/m,

<" <"N/m, S ##2k #2 (/mS "#2S #"2-k #2-

The two natural !reuencies o! the modi!ied system

"##

"###""

;

""####""

7

##"" "<44.6"#"7:.#""79.# ×−×−=+×= =S S # c=S )S M S # "=-(M M !=M

rad/s97.<;;:2rad/s,76.<62 ,""79.##

"<44.6""79.#6+"#"7:.#"#"7:.#2 #"7

"#7#;;

#

#," ω ω ω ××

××××−×±×

π ω

#" π

ω #

#

f "2 29.<=>, f

#2 2"4.<6 =>




#-amp.e 8/,!: 1i+ration a+sor+ers III (page !''"

x"

"< tons

m#

x#

k #

k /#

$ bridge is modelled as a singledegree o! !reedom

=as a mass o! "<, tons and

sti!!ness o! k 2#", 8(/m

$ harmonic !orce o! ; (

0esign an undamped 1ibration

absorber so that its amplitude does

not e%ceed #.7 cm




#-amp.e 8/,!: 1i+ration a+sor+ers III

x"

"< tons

m#

x#

k #

k /#

'sing S0G*

rad/s67<6.""

""<

"#"<

9

=

×

×==

m

k ω

The amplitude A#

##

#

;----#7.-

k k

F A $ =⇒=

*rom which4<----# =k N/m=360 kN/m

S$%&ti$'




#-amp.e 8/,!: 1i+ration a+sor+ers III

x"

"< tons

m#

x#

k #

k /#

*rom which

The suppressed e%citation !reuency

#

#

#

## 4<+#9#4."

mm

k =×⇒= π ω

#:74# =m kg


dynamics and vibration, wahab,2008-chapter 8

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