Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3...

15
MDPN471 Mechanical Vibrations CHAPTER 3 Single degree-of-freedom Forced Vibrations

Transcript of Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3...

Page 1: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

MDPN471

Mechanical Vibrations CHAPTER

3 Single degree-of-freedom Forced Vibrations

Page 2: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations 3.6 Response Under the Harmonic Motion of Base

)64.3(0)()( =−+−+ yxkyxcxm

tYty ωsin)( =

)65.3()sin(cossin

αωωωω

−=+=+=++

tAtYctkYyckykxxcxm

−=+= −

kcckYA ωαω 122 tan and )(

If ,

From the figure, the equation of motion is

where

Page 3: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations REMEMBER

0( ) ( ) ( ) cosmx t cx t kx t F tω+ + =

The system with equation of motion:

the particular solution is :

( ) cos( ) px t X tω φ= −

( )0

2 2 2 22 2 2 (1 ) (2 )stFX

r rk m c

δζω ω

= =− +− +

1 12 2

2tan tan1

c rk m r

ω ζϕω

− − = = − −

Page 4: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations

[ ] )66.3()sin()()(

)()( 12/1222

22

αφωωω

ω−−

+−

+= t

cmkckY

txp

−= −

21

1 tanω

ωφmk

c

)67.3()sin()( φω −= tXtxp

)68.3()2()1(

)2(1)()(

)(2/1

222

22/1

22

22

+−

+=

+−

+=

rrr

cmkck

YX

ζζ

ωωω

The steady-state response of the system

can be expressed as

where

or

where

and )69.3()14(1

2tan)()(

tan 22

31

22

31

−+

=

+−

= −−

rr

cmkkmc

ζζ

ωωωφ

3.6 Response Under the Harmonic Motion of Base

sin( )mx cx kx A tω α+ + = −

Page 5: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations

The variations of displacement transmissibility is shown in the figure below.

3.6 Response Under the Harmonic Motion of Base

222

2

)2()1()2(1

rrr

YXTd ζ

ζ+−

+==

−+

= −22

31

)14(12tan

rr

ζζφ

Page 6: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations

1. The value of Td is unity at r = 0 and close to unity for small values of r.

2. For undamped system (ζ = 0), Td →∞ at resonance (r = 1).

3. The value of Td is less than unity (Td < 1) for values of r >√2 (for any amount of damping ζ ).

4. The value of Td = 1 for all values of ζ at r =√2.

5. For r <√2, smaller damping ratios lead to larger values of Td. On the other hand, for r >√2, smaller values of damping ratio lead to smaller values of Td.

6. The displacement transmissibility, Td, attains a maximum for 0 < ζ < 1 at the frequency ratio r = rm < 1 given by:

The following aspects of Td can be noted from the figure: 3.6 Response Under the Harmonic Motion of Base

18121 2 −+= ζζmr

Page 7: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations 3.6 Response Under the Harmonic Motion of Base

2

( ) ( ) (3.72)Xsin( ) sin( )T

F k x y c x y mxm t F tω ω ϕ ω ϕ

= − + − = −

= − = −

•Force transmitted: (force transmitted to base from spring and damper)

The force transmissibility is given by:

Disturbing force

22

2 2 2

1 (2 )(1 ) (2 )

rrr r

ζζ

+=

− +T

fo

FTF

= =

kY=

2

2 2 2

1 (2 )(1 ) (2 )

X rY r r

ζζ

+=

− +

Page 8: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations

2 sin (3.75)mz cz kz my m Y tω ω+ + = − =

The equation of motion can be written as

•Relative Motion:

3.6 Response Under the Harmonic Motion of Base

( ) ( ) 0

For :

mx c x y k x y

z x y

+ − + − =

= −

Page 9: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations

1( ) sin( ) (3.76)z t Z tω ϕ= −

)75.3(sin2 tYmymkzzczm ωω=−=++

)77.3()2()1()()( 222

2

222

2

rrrY

cmkYmZ

ζωωω

+−=

+−=

=

−= −−

21

21

1 12tantan

rr

mkc ζω

ωφ

For

The steady-state solution is given by:

where, the amplitude Z:

•Relative Motion:

And phase angle

3.6 Response Under the Harmonic Motion of Base

Page 10: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations 3.6 Response Under the Harmonic Motion of Base

222

2

)2()1( rrrYZ

ζ+−=

Page 11: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations Example 3.3 Vehicle Moving on a Rough Road

The figure shows a simple model of a motor vehicle that can vibrate in the vertical direction while traveling over a rough road. The vehicle has a mass of 1200kg. The suspension system has a spring constant of 400 kN/m and a damping ratio of ζ = 0.5. If the vehicle speed is 20 km/hr, determine the displacement amplitude of the vehicle. The road surface varies sinusoidally with an amplitude of Y = 0.05m and a wavelength of 6m.

Page 12: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations

rad/s 290889.061

3600100022 vvf =

×

== ππω

rad/s 2574.181200

104002/13

=

×==

mk

318653.02574.18

81778.5===

n

rωω

The frequency can be found by

For v = 20 km/hr, ω = 5.81778 rad/s. The natural frequency is given by,

Hence, the frequency ratio is

Example 3.3 Solution

Page 13: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations

469237.1)318653.05.02()318653.01(

)318653.05.02(1)2()1(

)2(12/1

22

22/1

222

2

=

××+−××+

=

+−+

=rr

rYX

ζζ

m 073462.0)05.0(469237.1469237.1 === YX

Thus, the displacement amplitude of the vehicle is given by

The amplitude ratio can be found from Eq.(3.68):

This indicates that a 5cm bump in the road is transmitted as a 7.3cm bump to the chassis and the passengers of the car.

Example 3.3 Solution

Page 14: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations Quiz

Page 15: Single degree-of-freedom Forced Vibrations 2021. 1. 18. · SDOF Forced Vibrations . Example 3.3 Vehicle Moving on a Rough Road . The figure shows a simple model of a motor vehicle

SDOF Forced Vibrations Quiz