m yright dan segalman, 1998 …brannon/public/djsegal/vibclass/...yright dan segalman, 1998...

31
Advanced Vibrations /home/djsegal/UNM/VibCourse/slides/Lecture8.frm 11/30/98 Copyright Dan Segalman, 1998 1 Slides of Lecture 8 Today’s Class: Review Of Homework From Lecture 7 Hamilton’s Principle More Examples Of Generalized Coordinates Calculating Generalized Forces Via Virtual Work

Upload: leminh

Post on 24-May-2019

214 views

Category:

Documents


0 download

TRANSCRIPT

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

1

Slid

es o

f Lec

ture

8

Toda

y’s

Cla

ss:

Rev

iew

Of H

ome

wor

k F

rom

Lec

ture

7H

amilt

on’s

Prin

cipl

eM

ore

Exa

mpl

es O

f Gen

eral

ized

Coo

rdin

ates

Cal

cula

ting

Gen

eral

ized

For

ces

Via

Vir

tual

Wor

k

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

2

Hom

ew

ork

from

Lec

ture

7

In th

e m

ost r

ecen

t cla

ss, w

e de

rived

the

gove

rnin

geq

uatio

ns f

or th

e co

mpo

und

pend

ulum

. The

hom

ew

ork

assi

gnm

ent

was

tove

rify

the

deriv

atio

nan

dto

linea

rize

the

resu

lting

equ

atio

ns.

We

foun

d th

e go

vern

ing

equa

tions

to b

e

and

R1

R2

m1

m2

θ 2

θ 1

m1

m2

+(

)R12θ 1

m2R

1R

2θ 2

θ 2θ 1

–(

)co

s+

m–2R

1R

2θ 22

θ 2θ 1

–(

)si

ng

m1

m2

+(

)R1

θ 1si

n+

0=

m2R

22θ 2

m2R

1R

2θ 1

θ 2θ 1

–(

)co

s+ m2R

1R

2θ 12

θ 2θ 1

–(

)si

ng

m2R

2θ 2

sin

+0

=+

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

3

Hom

ew

ork

from

Lec

ture

7

1.E

stab

lish

the

loca

tion

of s

tab

le e

quili

briu

m a

bout

whi

ch

to

linea

rize.

By

obse

rva

tion,

that

loca

tion

is.

2.S

ubst

itute

and

into

the

gove

rnin

g eq

uatio

ns a

nd e

xpan

d w

ith T

aylo

r se

ries.

and

θ 1θ 2

0=

=

θ 1θ 1s

∆θ 1

+=

0

θ 2θ 2s

∆θ 2

+=

0

m1

m2

+(

)R12θ 1

m2R

1R

2θ 2

θ 2θ 1

–(

)co

s+

m–2R

1R

2θ 22

θ 2θ 1

–(

)si

ng

m1

m2

+(

)R1

θ 1si

n+

0=

1

θ 1θ 2

-θ1

m2R

22θ 2

m2R

1R

2θ 1

θ 2θ 1

–(

)co

s+

m2R

1R

2θ 12

θ 2θ 1

–(

)si

ng

m2R

2θ 2

sin

+0

=+

1

θ 2θ 2

-θ1

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

4

Hom

ew

ork

from

Lec

ture

7

3.D

elet

e al

l ter

ms

invo

lvin

g po

wer

s an

d pr

oduc

ts o

f,

,, a

nd

and

θ 1θ 2

θ 1θ 2 m1

m2

+(

)R12θ 1

m2R

1R

2θ 2

+

m–2R

1R

2θ 22

θ 2θ 1

–(

)g

m1

m2

+(

)R1θ 1

+0

=0

m2R

22θ 2

m2R

1R

2θ 1

+

m2R

1R

2θ 12

θ 2θ 1

–(

)g

m2R

2θ 2

+0

=+

0

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

5

Hom

ew

ork

from

Lec

ture

7

4.G

roup

term

s to

for

m th

e m

ass

and

stiff

ness

mat

rix.

Not

e th

at b

oth

the

mas

s an

d st

iffne

ss m

atric

es a

re s

ymm

etric

and

posi

tive-

defin

ite.

The

mas

s eq

uatio

n w

ill b

e po

sitiv

e so

long

as

we

avo

id m

assl

ess

degr

ees

of fr

eedo

m, t

hat i

s w

e al

way

s w

ant t

o c

hoos

e ou

r de

gree

s of

free

dom

so

that

.

The

stiff

ness

mat

rixw

illal

way

sbe

non-

nega

tive

defin

ite.I

fth

ere

are

norig

id-b

ody

mod

es,

itw

illbe

posi

tive

defin

ite.W

esh

all

disc

uss

this

mor

ela

ter.

m1

m2

+(

)R12

m2R

1R

2

m2R

1R

2m

2R

22

θ 1 θ 2

gm

1m

2+

()R

10

0g

m2R

2

θ 1 θ 2+

0 0=

q r∂∂T

0≠

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

6

Ham

ilton

’s P

rinci

ple

We

cons

ider

a m

echa

nica

l sys

tem

who

se c

onfig

urat

ion

at a

ny

time

is

char

acte

rized

by

the

N g

ener

aliz

ed c

oord

inat

es. T

he s

yste

m is

subj

ect t

o po

tent

ial e

ner

gy a

nd a

ddi

tiona

l for

ces

and

evo

lves

over

the

inte

rva

l a

ccor

ding

to th

e La

gran

ge

equa

tions

for

eac

h.

We

can

ima

gine

the

evo

lutio

n of

the

syst

emco

nfigu

ratio

nov

erth

atin

terv

alby

pict

urin

gth

em

otio

nof

a po

int w

hose

coo

rdi

nate

s ar

e

in a

n N

-dim

ensi

onal

Car

tesi

an s

yste

m.

q{}

VF

rA{

}t 1

t 2,(

)

tdd

q r∂∂T

q r

∂∂T–

q r∂∂V

+F

rA=

r

q(t 2

)

q(t 1

)

q(t)

q1

q2

q3

qt()

{}

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

7

Ham

ilton

’s P

rinci

ple

We

cont

ract

thes

esc

alar

equa

tions

with

test

func

tions

whi

ch

are

as

yetu

ndet

erm

ined

exce

ptfo

rth

eco

nditi

ons

for

each

r, a

nd th

en s

um th

em.

η rη r

t 1()

η rt 2(

)0

==

tdd

q r∂∂T

q r

∂∂T–

q r∂∂V

+F

rA–

η rt()

td

t 1t 2 ∫r

1=N ∑

0=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

8

Ham

ilton

’s P

rinci

ple

Inte

grat

ion

by

part

s yi

elds

Rec

allin

g th

at, t

he a

bove

inte

gral

sim

plifi

es

q r∂∂T

–q r

∂∂V+

FrA

–η r

t()

td

t 1t 2 ∫r

1=N ∑

tdd

q r∂∂T

ηr

q r∂∂T

η r–

td

t 1t 2 ∫r

1=N ∑

+0

=

η rt 1(

)η r

t 2()

0=

=

q r∂∂T

–q r

∂∂V+

FrA

–η r

t()

q r∂∂T –

η rî

td

t 1t 2 ∫r

1=N ∑

0=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

9

Ham

ilton

’s P

rinci

ple

The

abo

ve is

true

for

all

test

func

tions

.

Let

whe

re is

ano

ther

pat

h

from

to“n

ear”

the

path

take

n b

y.

Our

inte

gral

s ca

n no

w b

e w

ritte

n:

Obs

erve

that

the

term

s in

volv

ing

pote

ntia

l ene

rgy

are

a c

ompl

ete

diff

eren

tial.

η r

q(t 2

)

q(t 1

)

{q(t

)}~ {q(t

)}

q1

q2

q3

η rt()

q rt()

q rt()

–=

δqr

t()

=q

t()

{}

qt 1(

){

}q

t 2()

{}

qt()

{}

q r∂∂T

δqr

q r∂∂T

δqr

+

q r∂∂V

FrA

–δq

r+

î

td

t 1t 2 ∫r

1=N ∑

0=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

10

Ham

ilton

’s P

rinci

ple

Rea

rran

ging

the

abo

ve:

Whe

re.

Thi

s fo

rm o

f Ham

ilton

’s

prin

cipl

e as

ser

ts th

at th

e ac

tual

pat

h is

one

abou

t whi

ch

q r∂∂T

δqr

q r∂∂T

δqr

+

q r

∂∂V–

FrA

+δq

r+

î

r1

=N ∑

td

t 1t 2 ∫

δTδV

–δW

+(

)td

t 1t 2 ∫0

==

δWt()

FrA

t()

q rt()

q rt()

–(

)= δT

δV–

δW+

()

td

t 1t 2 ∫0

=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

11

Ham

ilton

’s P

rinci

ple:

Spe

cial

Cas

e

For

the

spec

ial c

ase

whe

re th

e g

ener

aliz

ed f

orce

s a

re p

resc

ribed

load

s, w

e ca

n de

fine

the

“Pot

entia

l Ene

rgy

of L

oadi

ng”

and

the

“Tot

al P

oten

tial E

ner

gy”

is.

In th

is c

ase

, Ham

ilton

’s P

rinci

ple

beco

mes

: The

true

pat

h in

confi

gura

tion

spac

e of

the

syst

em m

akes

the

quan

tity

sta

tiona

ry.

FrA

AF

rAq r

r1

=N ∑–

=

VT

VA

+=

JT

VT

–(

)td

t 1t 2 ∫=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

12

Ham

ilton

’s P

rinci

ple:

Exa

mpl

eB

eam

Ben

ding

Equ

atio

n

The

str

ain

ener

gy in

an

Eul

er-

Ber

noul

li be

am is

. The

pote

ntia

l ene

rgy

of l

oadi

ng is

And

the

Kin

etic

Ene

rgy

is w

here

is th

e m

ass

per

unit

leng

th o

f the

bea

m.

M0 Q

0

ML

QL

p(x)

L

V1 2---

EI

x2

2 ∂∂y

2

xd

0L ∫=

AM

0x∂∂y

0M

Lx∂∂y

L–

Q0y

0()

–Q

Ly

L()

px()y

x()

xd

0L ∫–+

=

T1 2---

mt∂∂y

2

xd

0L ∫=

m

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

13

Ham

ilton

’s P

rinci

ple:

Exa

mpl

eB

eam

Ben

ding

Equ

atio

n

Lets

eva

luat

e th

e vi

rtu

al q

uant

ities

, beg

inni

ng w

ithK

inet

ic E

ner

gy:

and

δT1 2---

mt∂∂

yδy

+(

)2

x1 2---

mt∂∂y

2

xd

0L ∫–

d

oL ∫=

mt∂∂y

t∂∂δy

xd

0L ∫≅

δTtd

t 1t 2 ∫m

t∂∂y

t∂∂δy

xd

0L ∫

td

t 2t 2 ∫=

mt∂∂

t∂∂yδy

yδy

–td

t 1t 2 ∫

xd

0L ∫=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

14

Ham

ilton

’s P

rinci

ple:

Exa

mpl

eB

eam

Ben

ding

Equ

atio

n

Str

ain

Ene

rgy

:

δTtd

t 1t 2 ∫m

yδy

[]

t 1

t 2x

mt22 ∂∂y

δyxd

td

0L ∫t 1t 2 ∫

–d

0L ∫=

0

δV1 2---

EI

x22

∂∂y

δy+

()

2

x1 2---

EI

x2

2 ∂∂y

2

xd

0L ∫–

d

0L ∫=

EIy

''x22

∂∂δy

xd

0L ∫≅

EI

y''

x∂∂δy

0LE

Iy'

''δy

()

0L–

EIy

IVδy

xd

0L ∫+

=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

15

Ham

ilton

’s P

rinci

ple:

Exa

mpl

eB

eam

Ben

ding

Equ

atio

n

The

str

ain

ener

gy te

rm b

ecom

es:

δVtd

t 1t 2 ∫= E

Iy'

'x∂∂δy

0LE

Iy'

''δy

()

0L–

EIy

IVδy

xd

0L ∫+

td

t 1t 2 ∫

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

16

Ham

ilton

’s P

rinci

ple:

Exa

mpl

eB

eam

Ben

ding

Equ

atio

n

In a

sim

ilar

man

ner

, we

find

the

cont

ribut

ion

from

Pot

entia

l Ene

rgy

of

Load

ing

δAtd

t 1t 2 ∫

M0

x∂∂δy

0M

Lx∂∂δy

L–

Q0δy

0()

–Q

Lδy

L()

+td

t 1t 2 ∫=

px()δ

yxd

0L ∫td

t 1t 2 ∫–

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

17

Ham

ilton

’s P

rinci

ple:

Exa

mpl

eB

eam

Ben

ding

Equ

atio

n

We

can

grou

p te

rms.

We

star

t with

the

term

s in

volv

ing

,,

, and

δy0(

)δy

L()

x∂∂δy

0x∂∂δy

L

M0

x∂∂δy

0M

Lx∂∂δy

L–

Q0δy

0()

–Q

Lδy

L()

+td

t 1t 2 ∫

EI

y''

x∂∂δy

0LE

Iy'

''δy

()

0L–

td

t 1t 2 ∫+

0=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

18

Ham

ilton

’s P

rinci

ple:

Exa

mpl

eB

eam

Ben

ding

Equ

atio

n

Fro

m w

hic

h w

e de

duce

.

A g

eom

etric

bou

ndar

y co

nditi

on s

peci

fyin

g im

plie

s th

at

. If d

ispl

acem

ent i

s no

t spe

cifie

d th

ere

, the

n

. Thi

s is

a “

natu

ral”

boun

dar

y co

nditi

on.

Sim

ilar

inte

rpre

tatio

ns a

re m

ade

of,

and

,

Q0

E–Iy

'''0(

)[

]δy

0()

0=

y0(

)δy

0()

0=

Q0

EIy

'''0(

)=

QL

E–Iy

'''L(

)[

]δy

L()

0=

M0

E–Iy

''0(

)[

]x∂∂δy

00

=M

LE–

Iy''

L()

[]

x∂∂δy

L0

=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

19

Ham

ilton

’s P

rinci

ple:

Exa

mpl

eB

eam

Ben

ding

Equ

atio

n

Mat

chi

ng te

rms

in th

e sp

acia

l int

egra

l we

have

from

whi

ch

we

conc

lude

that

my

EIy

IVp

–+

[]δ

yx

t,(

)xd

0L ∫td

t 1t 2 ∫0

=

my

EIy

IV+

px

t,(

)=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

20

Ham

ilton

’s P

rinci

ple

Ham

ilton

’s p

rinci

ple

is g

ener

al a

nd a

lway

s w

orks

, tho

ugh

som

etim

es it

is h

ard

to e

valu

ate

.

In p

artic

ular

, not

e ho

w H

amilt

on’

s P

rinci

ple

is u

sed

to d

eriv

e th

e pa

rtia

ldi

ffer

entia

l go

vern

ing

equa

tions

.

Als

o, w

e sa

w h

ow

to d

efine

the

pote

ntia

l ene

rgy

of l

oadi

ng a

nd to

use

that

with

Ham

ilton

’spr

inci

ple

.We

will

see

that

we

can

also

use

itin

with

Lagr

ang

e’s

equa

tions

.

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

21

Cha

nge

of S

ubje

ct.

The

fol

low

ing

is a

n in

trod

uctio

n to

the

met

hod

ofA

SS

UM

ED

MO

DE

S.

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

22

Mor

e O

n G

ener

aliz

ed D

egre

es o

f Fre

edom

Dis

trib

uted

Dis

plac

emen

t

Lets

cons

ider

anE

uler

Ber

noul

libe

am s

impl

y su

ppor

ted

at e

ach

end.

Initi

ally

, we

assu

me

that

all

forc

es a

re c

onse

rva

tive

. We

post

ulat

e a

disp

lace

men

tdi

strib

utio

n of

the

sor

t

We

shal

l der

ive

Lagr

ang

e eq

uatio

ns f

or th

e e

volu

tion

of a

nd

. The

se a

re o

ur g

ener

aliz

ed d

egre

es o

f fre

edom

.L

yx

t,(

)A

1t()x

Lx

–(

)L

2----

--------

--------

A2

t()x2

Lx

–(

)L

3----

--------

--------

--+

=

A1

t()f

1x()

A2

t()f

2x()

+=

A1

t()

A2

t()

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

23

Mor

e O

n G

ener

aliz

ed D

egre

es o

f Fre

edom

Dis

trib

uted

Dis

plac

emen

t

Kin

etic

Ene

rgy

:

whe

re,

and

T1 2---

my2

xd

0L ∫1 2---

mA

1t()f

1x()

A2

t()f

2x()

+[

]2xd

0L ∫=

= 1 2---A

1(

)2I 1

2A

1A

2I 2

A2

()2

I 3+

+[

]=

I 1m

f 1x()2

xd

0L ∫m

L30--------

==

I 2m

f 1x()f

2x

xd

0L ∫m

L60--------

==

I 3m

f 2x()2

xd

0L ∫m

L10

5----

-----=

=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

24

Mor

e O

n G

ener

aliz

ed D

egre

es o

f Fre

edom

Dis

trib

uted

Dis

plac

emen

t

Str

ain

Ene

rgy

:

whe

re

, a

nd

V1 2---

EI

y''

()2

xd

0L ∫1 2---

EI

A1

t()f

1''

x()

A2

t()f

2''

x()

+[

]2xd

0L ∫=

= 1 2---A

1(

)2I 4

2A

1A

2I 5

A2

()2

I 6+

+[

]=

I 4E

If 1

''x()2

xd

0L ∫E

I4 L3

--------

-=

=I 6

EI

f 2''

x()2

xd

0L ∫E

I4 L3

--------

-=

=

I 5E

If 1

''x()f

2''

x()

xd

0L ∫E

I2 L3

--------

-=

=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

25

Mor

e O

n G

ener

aliz

ed D

egre

es o

f Fre

edom

Dis

trib

uted

Dis

plac

emen

t

Lagr

ang

e E

quat

ions

:

and

In m

atrix

for

m:

.

Not

e th

at b

oth

mat

rices

are

sym

met

ric, p

ositi

ve d

efini

te.

tdd

A1

∂∂T

A1

∂∂T–

A1

∂∂V+

A1I 1

A2I 2

A1I 4

A2I 5

++

+0

==

tdd

A2

∂∂T

A2

∂∂T–

A2

∂∂V+

A1I 2

A2I 3

A1I 5

A2I 6

++

+0

==

mL

15--------

1 2---1 4---

1 4---1 7---

A1

A2

EI

L3

------

42

24

A1

A2

+0 0

=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

26

Gen

eral

ized

For

ces,

Cal

cula

ted

by

Met

hod

of V

irtu

al W

ork

Rec

all t

hat t

he g

ener

aliz

ed f

orce

ass

ocia

ted

with

the

gen

eral

ized

coor

dina

te is

We

exam

ine

the

incr

emen

tal w

ork

asso

ciat

ed w

ith in

crem

ents

of

:

The

gen

eral

ized

for

ce a

ssoc

iate

d w

ith th

e g

ener

aliz

ed c

oord

inat

e is

q rF

rF

nq r

∂∂xn

⋅n∑

=

q r

δWF

rδq r

Fn

q r∂∂x

nδq

r⋅

n∑F

nδx

n⋅

n∑=

==

q r

Fr

δW δqr

--------

=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

27

Gen

eral

ized

For

ces,

Cal

cula

ted

by

Met

hod

of V

irtu

al W

ork

Lets

cons

ider

anE

uler

Ber

noul

libe

am s

impl

y su

ppor

ted

at e

ach

end.

We

cons

ider

mom

ents

M0

and

ML a

pplie

d at

the

ends

and

adi

strib

uted

trac

tion

appl

ied

alon

g th

e le

ngth

of t

he b

eam

.

We

post

ulat

e a

disp

lace

men

t dis

trib

utio

n of

the

sor

t

Lets

cal

cula

te th

e g

ener

aliz

ed f

orce

s as

soci

ated

with

the

gen

eral

ized

coor

dina

tes

and

.

M0

ML

p(x)

L

yx

t,(

)A

1t()x

Lx

–(

)L

2----

--------

--------

A2

t()x2

Lx

–(

)L

3----

--------

--------

--+

=

A1

t()f

1x()

A2

t()f

2x()

+=

A1

A2

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

28

Gen

eral

ized

For

ces,

Cal

cula

ted

by

Met

hod

of V

irtu

al W

ork

The

wor

k do

ne to

the

stru

ctur

e b

y th

e fo

rces

act

ing

thro

ugh

a vi

rtu

al d

ispl

acem

ent

, is

so

can

be

calc

ulat

ed s

imila

rly.

δyδA

1(

)f1

x()

=

δWδA

1M

0f 1

'0(

)–

ML

f 1'

L()

px()f

1x()

xd

0L ∫+

+

= FA

1M

0f 1

'0(

)–

ML

f 1'

L()

px()f

1x()

xd

0L ∫+

+

=

FA

2

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

29

Hom

ew

ork

for

Lect

ure

8A

num

eric

al e

xper

imen

t with

line

ariz

atio

n

Man

y tim

es w

e ha

ve d

eriv

ed th

e eq

uatio

ns f

or a

spr

ing

rein

forc

ed p

endu

lum

:.

The

line

ariz

ed f

orm

is w

here

.

We

use

the

linea

rized

freq

uenc

y to

non

-dim

ensi

onal

ize

the

time

para

met

er. D

efine

, defi

ne, a

nd d

efinem

R

θ

κθ

κm

R2

--------

--θ

g R---θ

sin

+

+

0=

θω

L2θ

+0

=

ωL2

κm

R2

--------

--g R---

+=

τω

Lt

τ()

θτ

ωL

⁄(

)=

αg

R⁄ ωL2

--------

--=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

30

Hom

ew

ork

for

Lect

ure

8co

ntin

ued

The

n a

nd

1.S

olve

num

eric

all

y th

e di

men

sion

less

go

vern

ing

equa

tion

for

the

initi

al c

ondi

tions

: a

nd o

ver

the

perio

d

for

the

thre

e ca

ses:

,, a

nd.

2.D

oth

esa

me

asab

ove

but

for

the

initi

alco

nditi

ons

and

3.C

ompa

re a

nd d

iscu

ss th

e y

our

resu

lts f

or p

arts

1 a

nd 2

.

τ22

ddφ

t22

ddθ

1 ωL2

------

=τ22

ddφ

–(

)φα

φsi

n+

[]

+0

=

φ0(

ddφ0

0=

06

π,

()

α0

12⁄

1=

φ0(

)π 6---

=

τddφ

00

=

Ad

van

ced

Vib

rati

on

s

/hom

e/dj

sega

l/UN

M/V

ibC

ours

e/sl

ides

/Lec

ture

8.fr

m11

/30/

98C

op

yrig

ht

Dan

Seg

alm

an, 1

998

31

Nex

t Tim

e

Exa

mpl

epr

oble

ms

stud

ents

choi

cean

ddi

scus

sion

ofpa

st m

ater

ial.

Dis

cuss

ion

of m

id-t

erm

exa

m.