Mechanical Vibration by Palm

CHAPTER 4: HARMONIC RESPONSE WITH A SINGLE DEGREE OF

FREEDOM

Instructor: Dr Simin Nasseri, SPSU

© Copyright, 2010 1Based on a lecture from Brown university (Division of engineering)

External Forcing Base Excitation

Types of Forcing:

Rotor Excitation

All of these situations are of practical interest. Some subtle but important distinctions to consider, so we will look at each.

But the strategy is simple: derive Equation of Motion and put into the “Standard Form”

3

Base Excitation (Seismic motion)

4

Base Excitation (Seismic motion)

Base Excitation – the Earthquake Problem

Here, base supporting object is subjected to motion.

How does the object respond?

Forces in the spring, dashpot are proportional to the motion RELATIVE to the base

Draw F.B.D. and get equation of motion….

)sin()( tYty

)(tx

2

2( ) ( )vertical

dx dy d xF k x y c m

dt dt dt

2

2

d x dx dym c kx ky c

dt dt dt

2

2sin( ) cos( )

d x dxm c kx kY t c Y t

dt dt

22 2

2( ) ( ) sin( )

d x c dx k kY c Yx t

dt m dt m m m

22 2 2 2

22 ( ) (2 ) sin( )n n n n

d x dxx Y Y t

dt dt

22 2 2

22 1 (2 / ) sin( )n n n n

d x dxx Y t

dt dt

Now in the “standard form” but with a new “driving force” 22 )/2(1 nno YmF

2222

21

1

nn

n

o

m

FX

222

2

21

)/2(1

nn

n

Y

X

(Displacement Amplitude of body)/(Displacement Amplitude of Base)

22 2

2( ) ( ) sin( )

d x c dx k kY c Yx t

dt m dt m m m

Harmonic Base Excitation

Displacement transmission ratio: 222

2

21

)2(1

rr

r

Y

X

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

Y

X

rn /

1.0

2.0

3.0

0.11.0

Base Isolation Concept:

“soft” spring

Given an expected frequency of a driving force,

Design spring/dashpot coupling to minimize response

Clearly want to get in the regime 1/ n n

“soft” springs (small k)

Non-isolated Isolated

04_03_02

9

Design of Car Suspension for Wavy Roads:

Car weighing 3000 lbs drives over a road with sinusoidal profile shown

k c

m

s

16”

33 ft

Design the suspension so that:

1. The vibration amplitude of the car is < 14” at all speedsand2. The vibration amplitude of the car is < 4” at 55 mph

Select springs (k) and shocks (c) to satisfy requirements of maximum car vibration amplitude when driving on a wavy roadx(t)

What is the “base excitation” here?

Equation of road profile?16”

33 ft

As car drives along at constant speed, it is as if the road is vibrating up and down underneath the car

OK, but how do we represent that? ftLftY 33 ; 667.0"8

)2

sin( L

VtYy

L

V 2Driving frequency of base is

Design requirements are now:

(Vibration amplitude < 14” at all speeds)

1. at all frequencies75.18

14

Y

X

5.08

4

Y

X2. at frequency s

rad

ft

sftx36.15

33

)3600/528055(2

t

VLYt

TYtYy

/

2sin

2sin)sin(

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

Let’s do this graphically, using our magnification plot:

Y

X

50.0,.....15.0,10.0,05.0

1.75 35.0~

0.5

35.0

n /2/55 nMPh

2/55 nMPh will satisfy criterion #2 for =0.35

35.0 will satisfy criterion #1

2/55 nMPh

35.0

nMPh 2/55

sradn /2

36.15 sradn /68.7

Spring k required: ftlbsft

lbsmk n /495,5)

/2.32

3000()/68.7(

222

Damping c required: ftslbssft

lbmc n /500)/68.7)(

/2.32

3000)(35.0(22

2

14

Summary:

Harmonic Base Excitation – Motion Relative to Base

Sometimes the motion relative to the base is of interest 2

2( ) ( )

dx dy d xk x y c m

dt dt dt

Introducing the relative displacement z = x – y, the equation of motion becomes:2 2

2 2( )

dz d z d ykz c m

dt dt dt

2 2

2 2

d z dz d ym c kz m

dt dt dt Or:

22 2

22 sin( )n n

d z dzz Y t

dt dt

mFo /

2222

21

1

nn

n

o

m

FX

222

2

21nn

n

Y

Z

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8

9

10

MY

Zn

nn

n 2

222

2

21

Y

Z

nLow frequencies: body

moves with base – no relative motion

High frequencies: base is moving but body is not, so relative motion = 1

17

Goal: Detect “Low Frequency Earthquake” tremors in the 1-5 Hz frequency range along the San Andreas fault.

Constraints: Background vibrations over a wide range of higher frequencies occur with typical amplitudes of 0.1mm, so tremor amplitudes comparable to or smaller than this cannot be detected.

Design a mass/spring/dashpot system (choose m, k, c) to:(i)reliably detect tremors at a frequency of 3 Hz and having earth motion amplitudes of 0.01mm or larger,(ii) ensure that the maximum amplitude will not exceed 30 mm for earth motion amplitudes of 1.0 mm.

Design of a Seismograph:

Note: device measures motions relative to its base

FYI

Example: The motion of the outer cart is varying sinusoidally

as shown.

For what range of is the amplitude of the motion of the mass m, relative to the cart less than 2b?

2

22 2

2 1

n

n n

Z

Y

2

2

1

n

n

Z

Y

when no damping

2

22

1

n

n

Z

Y

Two solutions:

(When )n And: (When )n

2

2

1

n

n

Z

Y

We want 2

2Z b

Y b

2

2 2

1

n

n

2

3n

2

2 2

1

n

n

2n

n for

n for 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

(0.817)

(1.414)