Ch. 3: Forced Vibration of 1-dof system



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for this normalized transfer function).

3.2 Frequency Response Function

2

1 2



n

ω ω


ζ

=



0

1 / 2


ζ

< <

1 / 2


ζ



Ch. 3: Forced Vibration of 1-DOF System

3.2 Frequency Response Function

Ex. 3  Consider the pivoted mechanism with k=4x10

3

N/m,



l

1

=0.05 m, l



2

=0.07 m, l=0.10 m, and m=40 kg.

The mass of the beam is 40kg which is pivoted

at point O and assumed to be rigid.  Calculate c

so that the damping ratio of the system is 0.2.

Also determine the amplitude of vibration of the

steady-state response if a 10 N force is applied

to the mass at a frequency of 10 rad/s.



Ch. 3: Forced Vibration of 1-DOF System

( )


( )

(

)



1

2

2



1

1

2



2

1

2



1

  

2



                           

12

2



0.1 10 cos10

0.5


0.0049

59.05


0.0049

15.37,  0.2

,  627.3 

Ns/m


2 0.5

O

O

n

n

l

l

M

I

Fl

mgl

Mg

c l

l

k l

l

l

l

l

l

ml

M

M

t

c

c

c

θ

θ



θ

θ

θ



θ

θ

θ



θ

ω

ζ



ω

ω





=









+





=

+

+









×

=



+

+

=



=

=

=



×

×

=



( )


(

)

(



)

10,  0.6506

1

0.02677


24.268

59.05 0.5767

0.26

0.02677 cos 10



0.424

ss

r

H i

i

t

ω

θ



=

=

=



°

+



=

3.2 Frequency Response Function



Ch. 3: Forced Vibration of 1-DOF System

Ex. 4  A foot pedal for a musical instrument is modeled

as in the figure.  With k=2000 kg/s

2

, c=25 kg/s,



m=25 kg, and F(t)=50cos2πt N, compute the

steady-state response assuming the system starts

from rest.  Use the small angle approximation.

3.2 Frequency Response Function



Ch. 3: Forced Vibration of 1-DOF System

(

)



(

)

( )



(

)

2



2

 0.15


0.05

0.05


0.05

0.1


0.15

5

100



3.75

50 cos 2


,  positive CW

6

3



Find the parameters

2.98,  0.0373,  2 ,  2.108

1

0.0087


177.4

1

2



since 

0, the transie



O

O

n

M

I

F

k

c

m

t

r

H i

k

r

i

r

θ

θ



θ

θ

θ



θ

θ

π



ω

ζ

ω



π

ω

ζ



ζ



=

×



×

×



= ×



+

+

=



=

=

=



=

=

=



°

− +



( )



(

)

(



)

0

nt response will die out



cos

0.435 cos 2

3.096

ss

F H i

t

t

θ

ω



ω θ

π

=



+

=



3.2 Frequency Response Function

Ch. 3: Forced Vibration of 1-DOF System

3.3 


Applications

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications



Ch. 3: Forced Vibration of 1-DOF System

3.3 Applications

(

)

(



)

(

)



(

)

2



2

2

2



2

2

2



2

2

2



measured acc.

10

1



actual acc.

9.81


1

2

1



2

0.962


From the problem statement, 

628 rad/s, 

1

628 rad/s



1

1

758 rad/s



,  

0.56


2

1

5745.6 N/m,  



8.49 Ns/

n

d

n

d

d

n

n

z

y

r

r

r

r

r

k

c

m

m

k

c

ω

ζ



ζ

ω

ω



ω

ζ

ω



ω

ζ

ω



ω

ζ

ω



ζ

=

=



=

+



+

=



=

=



=

=

=



=

=



=

=

=



=

=



m

Ch. 3: Forced Vibration of 1-DOF System

3.4 


Periodic Excitation

A periodic function is any function that repeats itself in 

time, called period T.

It is more general than the harmonic function.  Here, we

will find the response to the input that is a periodic 

function.  The idea is to decompose that periodic input

into the sum of many harmonics.  The response, by the

superposition principle of linear system, is then the sum 

of the responses of individual harmonic.  The response 

of a harmonic function was studied in section 3.1

3.4 Periodic Excitation

( )


(

)

f t



f t T

=

+



Ch. 3: Forced Vibration of 1-DOF System

Fourier found the way to decompose the periodic

function into sum of harmonic functions (sine & cosine)

whose frequencies are multiples of the fundamental 

frequency.  The fundamental frequency is the frequency 

of the periodic function.

3.4 Periodic Excitation


Ch. 3: Forced Vibration of 1-DOF System

Fourier series

3.4 Periodic Excitation

( )


(

)

( )



( )

( )


0

0

0



0

0

1



0

0

0



0

Fourier series in real form:

2

cos


sin

,   


2

Fourier coefficients:

2

cos


 ,  

  0,1, 2,

2

sin


 ,  

  1, 2, 3,

Fourier series in complex form:

n

n

n

T

n

T

n

in

t

n

n

a

f t

a

n

t

b

n

t

T

a

f t

n

t dt

n

T

b

f t

n

t dt

n

T

f t

C e

ω

π



ω

ω

ω



ω

ω



=

=−∞



=

+

+



=

=

=



=

=

=





( )



0

0

0



2

,    


Fourier cofficients (complex):

1

  ,   



, 2, 1, 0,1, 2,

T

in

t

n

T

C

f t e

dt

n

T

ω

π



ω

=



=

=

− −







Ch. 3: Forced Vibration of 1-DOF System

Some properties of Fourier series

3.4 Periodic Excitation

)

( )



)

( )


)

( )


)

( )


( )

0

0



0

1

1  If 



 is an even function, 

0.

2  If 



 is an odd function, 

0.

3  



 is the average value of 

 over one period.

2

4  If 


 is real, 

  

  



2 Re

n

n

in

t

k

k

n

n

f t

b

f t

a

a

f t

f t

C

C

f t

C

C e

ω



=

=



=



=

=



+





Ch. 3: Forced Vibration of 1-DOF System

Frequency spectrum

tells how much each harmonic

contributes to the periodic function        .

Plot of the amplitude of each harmonic vs. its frequency

is the (discrete) frequency spectrum.

3.4 Periodic Excitation

( )


f t

( )


2

2

0



0

In real form, the harmonic at 

 has the amplitude 

In complex form, the harmonic at 

 has the amplitude 2 Re

 

n



n

n

n

a

b

n

C

ω

ω



+

Ch. 3: Forced Vibration of 1-DOF System

3.4 Periodic Excitation



Ch. 3: Forced Vibration of 1-DOF System

Superposition principle of linear system

3.4 Periodic Excitation


Ch. 3: Forced Vibration of 1-DOF System

Response to harmonic excitation

3.4 Periodic Excitation


Ch. 3: Forced Vibration of 1-DOF System

3.4 Periodic Excitation

( )

(

)



(

)

( )



(

)

0



0

0

0



0

0

2



0

0

0



1

0

0



1

From section 3.1,  

1

where  


1

2

2 Re



by superposition, 

2 Re


in

t

n

in

t

ss

n

n

n

in

t

n

n

in

t

ss

n

n

mx cx

kx

F t

C e

x

C H in

e

H in

n

n

k

i

mx cx

kx

F t

C

C e

C

x

C H in

e

k

ω

ω



ω

ω

ω



ω

ω

ω



ζ

ω

ω



ω

=



=

+



+

=

=



=

=







+











+



+

=

=



+





=

+







Ch. 3: Forced Vibration of 1-DOF System

3.4 Periodic Excitation

response frequency spectrum

excitation frequency spectrum

system frequency response


Ch. 3: Forced Vibration of 1-DOF System

Ex.  


Calculate the response of a damped system to

the periodic excitation f(t) depicted in the figure

by means of the exponential form of the Fourier

series.  The system damping ratio is 0.1 and the

driving frequency is ¼ of the system natural freq.

3.4 Periodic Excitation



Ch. 3: Forced Vibration of 1-DOF System

3.4 Periodic Excitation

( )

( )


( )

( )


( )

0

0



0

0

0



/ 2

0

0



0

/ 2


odd

Expand 


 as sum of harmonic series

1

1



2

   


 

,  


0,  even

1

1



2

,  odd


2

2

in



t

n

n

T

T

T

in

t

in

t

in

t

n

T

n

n

i n

in

t

n

f t

f t

C e

C

f t e

dt

Ae

dt

Ae

dt

T

T

T

n

iA

C

i A

n

n

n

i A

A

f t

e

e

n

n

ω

ω



ω

ω

ω



ω

π

ω



π

π

π



π

=−∞





=

=



=

=



+

=





=





=

− −


= ⎨



=

⎪⎩



=



×

=

×





( )



( )

(

)



(

)

(



)

(

)



( )

(

)



0

2

0



odd

1,3,


0

2

2



1

2

2



2

2

0



1,3,

4

1



sin

1

,   0.1,  



1

2

4



1

1

/ 4



0.05

1

0.05



,    

tan


1

0.25


1

0.25


0.05

4

1



sin

t

n

n

n

n

n

n

n

n

ss

n

n

n

A

n

t

n

n

n

G i

r

r

i

r

G i

n

i

n

n

G

G

n

n

n

A

x

t

G

n

t

G

n

π

ω



π

ω

ω



ω

ζ

ζ



ω

ω

ω



ω

π







=

=



=



=

=

=



=

=

=



− +

=



+

=



=



+





=

+







Ch. 3: Forced Vibration of 1-DOF System

3.4 Periodic Excitation



Ch. 3: Forced Vibration of 1-DOF System

Ex.  


The cam and follower impart a displacement y(t)

in the form of a periodic sawtooth function to the

lower end of the system.  Derive an expression


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