A2 Free Vibration

8/3/2019 A2 Free Vibration

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A2- Free Vibration

Single Degree of Freedom

A2- Free VibrationSingle Degree of Freedom


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Chapter Outline

1. Single Degree of Freedom

2. EOM and Natural Frequency3. Energy Method

4. Rayleigh Method

5. Viscously Damped Vibration6. Underdamped: Oscillatory Motion

7. Overdamped and Critically Damped

8. Logarithmic Decrement9. Coloumb Damping


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1. Single Degree of Freedom (SDOF)

Lumped Mass

Stiffness displacement Damping velocity

Linear time variant

2nd order differential equation

( )mx cx kx F t + + =


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2. EOM and Natural Frequency Undamped spring-mass system

2 k

n m =

2

0nx x+ =

Circular frequency (n)

Homogeneous second-order linear differential equation has a

general solution:

x =A sin n t + B cos n t


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+

+


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3. Energy Method

In a conservative system, the total energy is constant.

The kinetic energy Tis stored in the mass by virtue of itsvelocity, whereas the potential energy Uis stored in the form

of strain energy in elastic deformation or by a spring or work

done in a force field such as gravity. The total energy being

constant, its rate of change is zero.

T+ U= constant Tmax = Umax

d/dt( T+ U) = 0


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4. Rayleigh Method

Effective Mass

In systems in which masses are joined by rigid links, levers, orgears, the motion of various masses can be expressed in termsof the motionx of some specific point and the system is simplyone of a single DOF, because only one coordinate is necessary.The kinetic energy, Tcan then be written as

where meffis the effective mass or an equivalent lumped massat the specified point. If the stiffness at that point is also

known, the natural frequency can be calculated from theequation

n =

2

2

1xm eff

effm

k


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KE= 0.5 m (v)2

2 2 2 2

2 2 21 3

22 2 2

0.5 0.5 0.5 0.5

0.5 ( ) ( ) }{

p p v v r r r r

r

v p r v

KE m x m x J m x

J l lKE m m m x

l l l

= + + +

= + + +

4. Rayleigh Method


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5. Viscously Damped Vibration

( )mx cx kx F t + + =

2 2c nc m km= =

Define critical damping

c

c

c =

Damping ratio2 12 ( )

n n x x x F t

m + + =

Homogeneous solution

(3 cases)

Underdamped 1

2( 1)n n

Roots are:


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6. Underdamped: Oscillatory motion

( ) ( sin sin )nt

d dx t e A t B t

= +

0 0.5 1 1.5 2

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time, s

Displacement


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0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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Time, s

Displacement

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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Time, s

Displacement

7. Overdamped and critically damped


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8. Logarthmic Decrement

0 0.5 1 1.5 2-1

-0.5

0

0.5

1

Force

Time (s)0 0.5 1 1.5 2

-1

-0.5

0

0.5

1

Position

2

2

1

=

For small damping, =2


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9. Coulomb dampingSliding or dry friction

Constant force opposite to velocity

Decay in amplitude per cycle is 4mg/k

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (s)

Displacement(m)


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A2 Free Vibration

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