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)
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