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Section 5.2. Damped Motion. DAMPED MOTION. The second-order differential equation where β is a positive damping constant, 2 λ = β / m , and ω 2 = k / m is the equation that describes free damped motion. AUXILIARY EQUATION. - PowerPoint PPT Presentation
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Section 5.2
Damped Motion
DAMPED MOTION
The second-order differential equation
where β is a positive damping constant, 2λ = β/m, and ω2
= k/m is the equation that describes free damped motion.
02or0 22
2
2
2
xdt
dx
dt
xdx
m
k
dt
dx
mdt
xd
AUXILIARY EQUATION
The auxiliary equation is m2 + 2λm + ω2 = 0, and the corresponding roots are
221
221 , mm
The solution to the differential equation is divided into three cases:
Case 1: λ2 − ω2 > 0Case 2: λ2 − ω2 = 0Case 3: λ2 − ω2 < 0
CASE 1 — OVERDAMPED MOTION
In this situation, the system is said to be overdamped since the damping coefficient β is large when compared to the spring constant k. The solution of the equation is
ttt ececetx2222
21)(
CASE 2 — CRITICALLY DAMPED MOTION
The system is said to be critically damped since any slight decrease in the damping force would result in oscillatory motion. The solution of the equation is
tccetx t21)(
In this case the system is said the be underdamped, since the damping coefficient is small compared to the spring constant k. The roots are now complex
and so the solution to the equation is
CASE 3 — UNDERDAMPED MOTION
imim 221
221 ,
tctcetx t 222
221 sincos)(
We can write any solution
in the alternative form
where , and the phase angle φ is determined from the equations
ALTERNATIVE FORM FOR THE SOLUTION OF UNDERDAMPED
MOTION
tctcetx t 222
221 sincos)(
tAetx t 22sin)(
2
121 tan,cos,sinc
c
A
c
A
c
22
21 ccA
DAMPED AMPLITUDE, QUASI PERIOD, QUASI FREQUENCY
The damped amplitude is:
The quasi period is:
The quasi frequency is:
2
2
22
22
tAe
HOMEWORK
1–23 odd
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