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Free Vibrations - Dynamics and Vibrations - Lecture Slides, Slides of Dynamics

Punjab Engineering CollegeDynamics

The key points in the lecture slides of the Dynamics and Vibrations are:Free Vibrations, Simple Harmonic Motion, Natural Frequency, Vibration Mode, Linear and Nonlinear System, Damping Factor, Overdamped Motion, Free Vibration Response, Design-Type Problems, Combining Springs

Typology: Slides

2012/2013

Uploaded on 05/07/2013

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Free Vibrations – concept checklist
You should be able to:
1. Understand simple harmonic motion (amplitude, period, frequency,
phase)
2. Identify # DOF (and hence # vibration modes) for a system
3. Understand (qualitatively) meaning of ‘natural frequency’ and
‘Vibration mode’ of a system
4. Calculate natural frequency of a 1DOF system (linear and nonlinear)
5. Write the EOM for simple spring-mass systems by inspection
6. Understand natural frequency, damped natural frequency, and
‘Damping factor’ for a dissipative 1DOF vibrating system
7. Know formulas for nat freq, damped nat freq and ‘damping factor’ for
spring-mass system in terms of k,m,c
8. Understand underdamped, critically damped, and overdamped motion
of a dissipative 1DOF vibrating system
9. Be able to determine damping factor from a measured free vibration
response
10. Be able to predict motion of a freely vibrating 1DOF system given its
initial velocity and position, and apply this to design-type problems
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Download Free Vibrations - Dynamics and Vibrations - Lecture Slides and more Slides Dynamics in PDF only on Docsity!

Free Vibrations – concept checklist

You should be able to:

1. Understand simple harmonic motion (amplitude, period, frequency,

phase)

2. Identify # DOF (and hence # vibration modes) for a system

3. Understand (qualitatively) meaning of ‘natural frequency’ and

‘Vibration mode’ of a system

4. Calculate natural frequency of a 1DOF system (linear and nonlinear)

5. Write the EOM for simple spring-mass systems by inspection

6. Understand natural frequency, damped natural frequency, and

‘Damping factor’ for a dissipative 1DOF vibrating system

7. Know formulas for nat freq, damped nat freq and ‘damping factor’ for

spring-mass system in terms of k,m,c

8. Understand underdamped, critically damped, and overdamped motion

of a dissipative 1DOF vibrating system

9. Be able to determine damping factor from a measured free vibration

response

10. Be able to predict motion of a freely vibrating 1DOF system given its

initial velocity and position, and apply this to design-type problems

Number of DOF (and vibration modes)

If masses are particles:

Expected # vibration modes = # of masses x # of directions

masses can move independently

If masses are rigid bodies (can rotate, and have inertia)

Expected # vibration modes = # of masses x (# of directions

masses can move + # possible axes of rotation)

k

m m

k (^) k

x 1 x 2

Calculating nat freqs for 1DOF systems – the basics

EOM for small vibration of any 1DOF

undamped system has form

m

k,L y 0

2 2 2 n

d y y C dt

  • ω =
  1. Get EOM (F=ma or energy)
  2. Linearize (sometimes)
  3. Arrange in standard form
  4. Read off nat freq.

n ω (^) is the natural frequency

Useful shortcut for combining springs

k 1

k 2

Parallel: stiffness k^ =^ k 1 +k 2

k 1 k (^2)

Series: stiffness

k 1

k 2

m

k 1 +k 2

m

1 2

k k k

k 1

Are these in series on parallel?^ m

Linearizing EOM

2

2

( )

d y f y C dt

Sometimes EOM has form + =

We cant solve this in general…

Instead, assume y is small

2 2 0 2 2 0

(0) ...

1 (0)

y

y

d y df m f y C dt dy

d y df C f y dt m dy m

=

=

      • =

−

  • =

There are short-cuts to doing the Taylor expansion

Writing down EOM for spring-mass systems

s=L 0 +x

k, L (^0)

m

c

2

2

2 2 2

n n n

d x c dx k m x

dt m dt^ m

d x dx k c x

dt dt^ m^ km

F a

k 1

k 2

Commit this to memory! (or be able to derive it…)

x(t) is the ‘dynamic variable’ (deflection from static equilibrium )

Parallel: stiffness k^ =^ k 1 +k 2

c (^2)

c 1

Parallel: coefficient c^ =^ c 1 +c 2

k 1 k (^2)

Series: stiffness

1 2

k k k

c 1 c 2

Parallel: coefficient

1 2

c c c

Solution to EOM for damped vibrations

s=L 0 +x

k, L (^0)

m

c

2 2

2

n n n

d x dx k c x dt dt^ m^ km

Initial conditions : 0 0 0

dx x x v t dt

0 0 ( ) exp( ) 0 cos sin

n n d d d

v x x t t x t t

Underdamped: ς < 1

Critically damped: (^) ς = (^1) { [ ] } 0 0 0

( ) exp( )

n n x t = x + v + ω x t −ω t

0 (^ )^0 0 (^ ) 0 ( ) exp( ) exp( ) exp( ) 2 2

n d n d n d d d d

v x v x x t t t t

ςω ω ςω ω ςω ω ω ω ω

 + + + −  = − (^)  − −   

Overdamped: (^) ς > 1

Critically damped gives fastest return to equilibrium

Calculating natural frequency and damping

factor from a measured vibration response

Displacement

time

t 0 t 1 t 2 t 3

T

x(t 0 ) x(t 1 ) x(t 2 ) x(t 3 )

t (^4)

log ( (^) n)

x t

n x t

Measure log decrement:

2 2

2 2

n

T

δ π δ ς ω

π δ

Measure period : T

Then