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

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