POSSIBLE SOLUTION
POSSIBLE SOLUTION
Differential Equation
Input := ee = x''[t] + .2 x'[t] + 4.4 x[t] == Sin[w t]
Output = 4.4 x[t] + 0.2 x'[t] + x''[t] == Sin[t w]
Solution to Differential Equation
Input :=
x(t) = A(w) sin(w t) - B(w) cos(w t)
44000 - 10000 w2
where A(w) = -----------------------------------
193600 - 87600 w2 + 10000 w4
2000 w
where B(w) = -----------------------------------
193600 - 87600 w2 + 10000 w4
Input :=
X = (
(44000 - 10000 w^2) Sin[ w #]/
( 193600 - 87600 w^2 + 10000 w^4 ) -
2000 w Cos[ w #] /
( 193600 - 87600 w^2 + 10000 w^4 )
)&;
Verification of the Solution
Input := X''[t] + 2/10 X'[t] + 440/100 X[t] //Simplify
Output = Sin[t w]
Identification of the Coefficients A(w) and B(w)
Input := B = Coefficient[ X[t], Cos[w t] ]
Output =
-2000 w
----------------------------
2 4
193600 - 87600 w + 10000 w
Input := A = Coefficient[ X[t], Sin[ w t] ]
Output =
2
44000 - 10000 w
----------------------------
2 4
193600 - 87600 w + 10000 w
Step May be Given as a Hint!!!!!
If we start with
x(t) = A(w) sin(w t) - B(w) cos(w t)
we can re-write this as
A(w) B(w)
x(t) = Q(w) { ------- sin(w t) - ----------- cos(w t) }
Q(w) Q(w)
where we define Q(w) = [ A(w)2 + B(w)2]1/2
Next , let A(w) / Q(w) = cos(f) and B(w) / Q(w) = sin(f). So
f(w) = tan-1( B(w)/ A(w))
and we have
x(t) = Q(w) sin( w t - f(w))
so the function Q(w) is our amplitude.
Calculation of Amplitude as function of Driving Frequency
Input := Amp[w_] = Sqrt[ A^2 + B^2]
Output =
2
4000000 w
Sqrt[------------------------------- +
2 4 2
(193600 - 87600 w + 10000 w )
2 2
(44000 - 10000 w )
-------------------------------]
2 4 2
(193600 - 87600 w + 10000 w )
Plots of Amplitude Response
Find Resonant Frequency
Input :=
FindRoot[ Amp'[w] == 0, {w,2.1}]
Output =
{w -> 2.09284}