Forced linear equations
[1] Superposition II
[2] Harmonic sinusoidal response
[3] Exponential response formula
[4] Sinusoidal response using complex replacement
I drew the spring/mass/dashpot system and added a force to it:
the little sail comes back into play.
mx" + bx' + kx = F_ext (*)
Notice by the way that I can put the damper on the left in parallel
with the spring: it still opposes velocity. If x' < 0 , F_dash > 0
and so on: so you get exactly the same equation.
Also important will be the "associated homogeneous equation"
mx" + bx' + kx = 0
(*)_h
which we know all about after Lecture 12. Final comment on this:
We can "reduce" this by dividing by m:
x" + (b/m)x' + (k/m) = 0
If b = 0 we get solutions with circular frequency sqrt(k/m) .
If b > 0 , you get exponentially damped sinusoids, with smaller
circular frequency omega_d (or not oscillating at all, if b is big enough).
In general, even if b > 0 , we call sqrt(k/m) the "natural circular
frequency" of the system, and write omega_n for it. So in the underdamped
case, when there is an omega_d , omega_d < omega_n . So the reduced
homogeneous equation is
x" + (b/m) x' + omega_n^2 = 0 .
[1] The general strategy in finding solutions is based on "superposition."
[Slide:]
Superposition I: If x1 and x2 are solutions of a homogeneous
linear equation, then so is any linear combination c1 x1 + c2 x2 .
If the equation is of second order and neither of x1 , x2 is a multiple
of the other, then c1 x1 + c2 x2 is the general solution.
Now we have:
Superposition II: If xp is any solution to (*) and xh is a solution
to (*)_h , then xp + xh is again a solution to (*).
Proof: Plug x into (*):
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 Spring '09
 vogan
 Linear Equations, Equations, Quadratic equation, Elementary algebra, Quintic equation, input signal

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