cm lecture 4

8/10/2019 CM Lecture 4

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Lecture 1: Single Degree of

Freedom Systems (III)

Prof. Kuo-Shen Chen

Department of Mechanical EngineeringNational Cheng-Kung University

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Outline

Rotordynamics, an introduction

Impulse Responses

Arbitrary excitation

Transfer function and Laplace Domain

Shock Isolation

Simple problems

Youtube Demos

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Part I: Rotordynamics: an Introduction

Rotor phenomenon

Unbalance

Campbell diagrams

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Rotordynamics

A specialized branch concerned with the behavior

and diagnosis of rotating structures.

commonly used to analyze the behavior of structures

ranging fromjet engines and steam turbines to auto

engines and computer disk storage.

Vibration, noise, bearing damages

Key issues to be introduced

Critical speed

Whirling

Campbell diagram

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Jeffcott Rotors

A single disk mounted on a flexible and

massless shaft with rigid bearings

Serves as the fundamental model for

studying rotor phenomena

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Whirling

Rotation of bent shaft

: rotating speed, : whirling speed

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Critical Speeds

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Whirling: Mathematics

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Whirling Behavior

Synchronous whirling:

As rotating speed passes the critical speed, the imbalance

location would actually move toward the center

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Campbell Diagram

known as "Whirl Speed

Map" or a "Frequency

Interference Diagram

Basic concept

Natural frequencies

depends on rotating

speed

Resonance occurs as

natural frequencies hit

rotating speed

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Turbine Example

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Dynamic Balancing

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Responses Under a General Force (I)

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Responses Under a General Force (II)

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Response Under a Periodic Force with

Irregular Form

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Part III: Vibration Subject to Arbitrary

Excitations

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Introduction

It is important to evaluate the response to a

general, non-periodic input

No, exact analytical solutions available

However, the task can be performed by either

Convolution approach

Fourier transform or Laplace transform approach

Transfer function

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Impulse Response

The response of a vibration system subjected

to a unit impulse input

Impulse function

0

0

0

0 t t

( ) t=tt t

0( ) 1

t t dt

2

F

- +

is a small positive number

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Effect on spring-mass-damper ?

Ft mv

impulse=momentum change

m[v( t0) v(t0

)]

?F mv0 v0?F

m

Ft

m

Just after

impulse

Just before

impulse

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For an underdamped system

x(t)?Fent

md

sindt (response with zero I.C.)

x(t) ?F h(t), where h(t)e

nt

md

sindt

impulse response function

0 10 20 30 40-1

-0.5

0

0.5

1

Time

h(t)

m

k

x(t)

c

F

h(t )1

mn

sinn(t)

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h(t )

0 te

n ( t)

md

sind(t ) t

for the case that the impulse occurs at

note that the effects of non - zero initial conditions

and other forcing terms must be super imposed on

this solution

0 10 20 30 40-1

0

1

h1

0 10 20 30 40-1

0

1

h2

0 10 20 30 40-1

0

1

Time

h1+h2

=0

=10

For example: If two

pulses occur at two

different times then

their impulseresponses will

superimpose

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Impulse Response of a

Vibration System (I)

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Impulse Response

of a Vibration

System (II)

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Convolution Integrals: Introduction

Decompose a general input to the

combination of a series of impulse train

Superimpose the impulse response to form

the final response The method itself can be treated as an

analytical relation

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Convolution Integral (I)

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Convolution Integral (II)

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Linear Superposition Method

A complicate input can be decomposed into a

few simple input with

scaling, multiplexing, and time shift operations

The response are then becomes the linear

superposition of the simple output after these

linear operations

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Example: Pulse Input

0

2

1( ) 1 cos( )

1

nt

d

Fx t e t

k

0( )00

2

1( ) 1 cos( ( ) )

1

n t t

d

Fx t e t t

k

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Time Shift Operation

E.g., for the pulse input

The total response = x1 + x2

Where

X1

x2

0

2

1( ) 1 cos( )

1

nt

d

Fx t e t

k

0( )00

2

1( ) 1 cos( ( ) )

1

n t t

d

Fx t e t t

k

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Part IV: Transfer Function and

Laplace Transform

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Transfer Functions

A black box approach to correlate the input-

output relation

Based on linear superposition principle

Usually performed by Laplace Transform

approach

Or Fourier Transform

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Laplace Transform

Changes ODE into algebraic equation

Solve algebraic equation then compute the

inverse transform

Rule and table based in many cases

Is used extensively in control analysis to

examine the response

Related to the frequency response function

dtetxtxsX st

0

)())(()( L

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22

02

0

s

sFsXkcsms

tFkxxcxm

)()(

cos

Take the transform of the equation of motion:

Now solve algebraic equation in s for X(s)

))((

)(222

0

skcsms

sFsX

To get the time response this must be inverse

transformed

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Laplace Transform: Fundamentals (I)

Consider a SDOF vibration system

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Laplace Transform: Fundamentals (II)

Transfer Functions: Z(s) Impedance; Y(s): Admittance

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Typical

Laplace

Transforms

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Propertiesof Laplace

Transform

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Example: Rao Ex. 4.16

Find the response

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where

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Part V: Response Spectrum

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Introduction

Shock

A sudden application of a force input to a SDOF

system to result a transient response

The maximum value of the response can be used

to measure the shcok sensitivity

Response spectrum is a plot of the maximum

peak response of the SDOF oscillator as a

function of natural frequency

Different shock inputs result in different response

spectra

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Typical Response Spectra

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General Response Spectra (1)

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Part VI: Simple Problems

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Problem 1. Periodic Vibration of a

Hydraulic Valve (Rao 4.1)

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Problem 2. Response of a Structure under

Impact (Rao 4.4, 4.5)

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Problem 3. Step Force of a

Compacting Machine (Rao 4.6)

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Problem 4. Blast Load on a Building

Frame (Rao 4.10)

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Problem 5: Water Tank Subjected to

Base Acceleration (Rao 4.12)

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Problem 6. Response of a Building

Frame to an Earthquake (Rao. 4.13)

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Part VII: Youtube Demonstrations

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