Lecture 2

Normal Modes

♦ Coupled pendula (recap):

• 2 linear ODEs in x(t), y(t) −→ 2 normal frequencies ω1, ω2

at which system can oscillate as a whole.

⇒ x+ y and x− y oscillate independently at frequencies ω1 and ω2

(normal modes)

• any generic motion of the system is linear superposition of normalmodes : GS = c1 NM1 + c2 NM2

Coupled pendula

m!!x = !mgx

l+ k(y ! x)

m!!y = !mgy

l! k(y ! x)

x + y = 2A1cos !

1t + "

1( )

x ! y = 2A2cos "

2t + #

2( )

Normal modes

centre of mass motion

m(!!x + !!y) = !mg

l(x + y)

!!q1= !"

1

2q1, q

1= x + y( ), "

1

2=g

l

m(!!x ! !!y) = (!mg

l! 2k)(x ! y)

!!q2= !"

2

2q2, q

2= x ! y( ), "

2

2=g

l+ 2

k

mrelative motion

normal coordinates

Solution II - Decoupling method : (simpler in this case)

(Matrix method)

q1= A

1cos !

1t + "

1( )

q2= A

2cos !

2t + "

2( )

x

y

!"#

$%&=1

1

!"#$%&A1cos '

1t + (

1( ) +1

)1!"#

$%&A2cos '

2t + (

2( )

m(!!x + !!y) = !mg

l(x + y)

centre of mass motion !!q1= !"

1

2q1, q

1= x + y( ), "

1

2=g

l

m(!!x ! !!y) = (!mg

l! 2k)(x ! y)

!!q2= !"

2

2q2, q

2= x ! y( ), "

2

2=g

l+ 2

k

mrelative motion

q1= A

1cos !

1t + "

1( )

q2= A

2cos !

2t + "

2( )

General solution

x

y

!"#

$%&=1

1

!"#$%&A1cos '

1t + (

1( ) +1

)1!"#

$%&A2cos '

2t + (

2( )

Constants determined by initial condtions

Ex 1 x(0) = y(0) = a !x(0) = !y(0) = 0

A1cos(!

1) + A

2cos(!

2) = a

A1cos(!

1) " A

2cos(!

2) = a

A1#1sin(!

1) + A

2#2sin(!

2) = 0

A1#1sin(!

1) " A

2#2sin(!

2) = 0

! A2= 0, A

1= a, "

1= 0

x

y

!"#

$%&= acos '

1t( )1

1

!"#$%&, t > 0 1st Normal mode excitation

x

y

General solution

x

y

!"#

$%&=1

1

!"#$%&A1cos '

1t + (

1( ) +1

)1!"#

$%&A2cos '

2t + (

2( )

Constants determined by initial condtions

Ex2 x(0) = y(0) = 0 !x(0) = !v, !y(0) = v

A1cos(!

1) + A

2cos(!

2) = 0

A1cos(!

1) " A

2cos(!

2) = 0

A1#1sin(!

1) + A

2#2sin(!

2) = -v

A1#1sin(!

1) " A

2#2sin(!

2) = v

! A1= 0, "

2= # / 2, A

2= $

v

%2

x

y

!"#

$%&=v

'2

sin '1t( )

1

(1!"#

$%&, t > 0 2nd Normal mode excitation

General solution

x

y

!"#

$%&=1

1

!"#$%&A1cos '

1t + (

1( ) +1

)1!"#

$%&A2cos '

2t + (

2( )

Constants determined by initial conditions

Ex3 x(0) = a y(0) = 0 !x(0) = !y(0) = 0

A1cos(!

1) + A

2cos(!

2) = a

A1cos(!

1) " A

2cos(!

2) = 0

A1#1sin(!

1) + A

2#2sin(!

2) = 0

A1#1sin(!

1) " A

2#2sin(!

2) = 0

! "1= "

2= 0, A

1= A

2=a

2

x

y

!"#

$%&=a

2cos '

1t( )1

1

!"#$%&+a

2cos '

2t( )

1

(1!"#

$%&

both modes excited

both modes excited

x = acos!1+!

2

2t

"#$

%&'cos

!1(!

2

2t

"#$

%&'

y = asin!1+!

2

2t

"#$

%&'sin

!1(!

2

2t

"#$

%&'

= acos(! t)cos("! t / 2)

= asin(! t)sin("! t / 2)

where ! =!1+!

2

2and "!=!

1#!

2

x

y

!"#

$%&=a

2cos '

1t( )1

1

!"#$%&+a

2cos '

2t( )

1

(1!"#

$%&

!" <<"

x = acos!1+!

2

2t

"#$

%&'cos

!1(!

2

2t

"#$

%&'

y = asin!1+!

2

2t

"#$

%&'sin

!1(!

2

2t

"#$

%&'

= acos(! t)cos("! t / 2)

= asin(! t)sin("! t / 2)

Note transfer of energy between the two pendula “Beats”

Energy of motion P.E. V + K.E. K

m!!x = !mgx

l+ k(y ! x) Conservative force = !

"V

"x

(Potential energy+Kinetic energy)

Energy of motion

m!!x = !mgx

l+ k(y ! x) = !

"V

"x

m!!y = !mgy

l! k(y ! x) = !

"V

"y

V (x, y) = mgx2

2l+1

2kx

2! kxy + f (y)

= kx !df

dy! f (y) = mg

y2

2l+1

2ky

2+ C

V (x, y) =1

2m

g

l+k

m

!"#

$%&(x

2+ y

2) ' kxy + C

P.E. V + K.E. K

K =1

2m !x

2+ !y

2( )

In normal coordinates:

q1=1

2(x + y), q

2=1

2(x ! y)

x =1

2(q1+ q

2), y =

1

2(q1! q

2)

...with 1

2 q1,q2 rotated axis w.r.t. x,y

Energy of motion

m!!x = !mgx

l+ k(y ! x) = !

"V

"x

m!!y = !mgy

l! k(y ! x) = !

"V

"y

V (x, y) = mgx2

2l+1

2kx

2! kxy + f (y)

= kx !df

dy! f (y) = mg

y2

2l+1

2ky

2+ C

V (x, y) =1

2m

g

l+k

m

!"#

$%&(x

2+ y

2) ' kxy + C

P.E. V + K.E. K

K =1

2m !x

2+ !y

2( )

V =1

2mg

lq1

2+1

2m

g

l+2k

m

!"#

$%&q2

2

=1

2m'

1

2q1

2+1

2m'

2

2q2

2

K =1

2m !q

1

2+ !q

2

2( )

In normal coordinates:

q1=1

2(x + y), q

2=1

2(x ! y)

x =1

2(q1+ q

2), y =

1

2(q1! q

2)

V =1

2mg

lq1

2+1

2m

g

l+2k

m

!"#

$%&q2

2

=1

2m'

1

2q1

2+1

2m'

2

2q2

2

K =1

2m !q

1

2+ !q

2

2( )

!!q1= !"

1

2q1, q

1=1

2x + y( ), "

1

2=g

l

!!q2= !"

2

2q2, q

2=1

2x ! y( ), "

2

2=g

l+ 2

k

m

E =1

2m!

1

2q1

2+1

2m !q

1

2"#$

%&'+1

2m!

2

2q2

2+1

2m !q

2

2"#$

%&'

= E1+ E

2

Total energy = sum of energies of the normal modes

Parseval’s theorem