HARMONIC MOTION
Background
• Equation of Motion for Mass-Spring System:
my 00(t) + µy 0(t) + ky(t) = F (t).
y is displacement, m is mass, µ is damping constant,
k is spring constant and F is external force.
• RLC Circuit Equation:
1
LI 00(t) + RI 0(t) + I(t) = E 0(t),
C
where I is current, L is inductance, R is resistance,
C is capacitance and E is voltage source.
• Harmonic Motion Equation:
x00 + 2cx0 + ω02x = f (t),
where damping constant c > 0, and
f (t) is forcing function.
HARMONIC MOTION CONTINUED
Homogeneous Equations :
x00 + 2cx0 + ω02x = 0.
• The Undamped Equation: for
simple harmonic motion
x00 + ω02x = 0.
General solution is
x(t) = a cos(ω0t) + b sin(ω0t).
This is oscillatory motion with
natural frequency ω0, and period T =
Solution can also be written as
x(t) = A cos(ω0t − φ);
√
with amplitude A = a2 + b2
and phase angle φ, with tan(φ) = b/a.
2
2π
ω0 .
HARMONIC MOTION CONTINUED
Example: spring with m = 1 kg, k = 4 kg/s2,
x(0) = −4 m, x0(0) = −6 m/s, x(t)?.
x’’+4x=0, x(0)=−4, x’(0)=−6, x(t) = 5cos(2t−3.785)
5
4
3
2
1
0
−1
−2
−3
−4
−5
0
1
2
3
4
5
t
6
7
8
9
Using Matlab commands
t = [0:.1:10];
plot(t,-4*cos(2*t)-3*sin(2*t),t,5*cos(2*t-3.785))
3
10
• The Damped Equation:
x00 + 2cx0 + ω02x = 0.
Characteristic equation roots are
q
r1,2 = −c ± c2 − ω02.
Solutions have the forms
– underdamped if c2 < ω02, and
x(t) = e−ct(C1cos(ωt) + C2 sin(ωt)),
p
where ω = ω02 − c2;
– overdamped if c2 > ω02, and
x(t) = C1er1t + C2er2t;
– critically damped if c2 = ω02, and
x(t) = (C1 + C2t)er1t.
Example: if 50g mass stretches a spring 20cm,
find damping constant µ for critical damping;
if x(0) = −15 cm, x0(0) = 0, find x(t).
Note: to find k for spring stretch s, use mg = ks.
4
HARMONIC MOTION CONTINUED
x’’+14x’+49x=0, x(0)=−.15, x’(0)=0, x(t) = −(.15+1.05t)exp(−7t)
0
−0.02
−0.04
−0.06
−0.08
−0.1
−0.12
−0.14
−0.16
0
0.5
1
1.5
2
2.5
t
3
3.5
Using Matlab commands:
t = [0:.1:5]; plot(t,(-.15-1.05*t).*exp(-7*t))
5
4
4.5
5
HARMONIC MOTION CONTINUED
Example: solve x00 +10x0 +9x = 0, with x(0) = −8, x0(0) = 0.
0
−1
−2
−3
−4
−5
−6
−7
−8
0
0.5
1
1.5
2
2.5
3
3.5
Using Matlab commands:
t = [0:.1:5]; plot( t, exp(-9*t)-9*exp(-t) )
6
4
4.5
5