- #1
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Homework Statement
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http://www.dartmouth.edu/~sullivan/22files/New Laplace Transform Table.pdf
(see item 26a)
homogenous solution to underdamped in amplitude phase form: (see attached image)
2. Relevant info
- non zero initial conditions: x(t=0) = xo AND dx/dt(t=0) = vo
- unforced motion: F(t) = 0
- second order dynamic system (1DOF & in one dimension x)
- ODE and final solution is written in terms of damping ratios (z), and natural frequencies (wn)
- damping frequency: wd = wn*sqrt(1-z^2)
-Underdamped system: |z| < 1
The Attempt at a Solution
-Given
d^2x/dt^2 + 2*z*wn*dx/dt + wn^2*x = 0
-Now I take laplace
[X(s)*s^2 - s*xo - vo] + 2*z*wn*[X(s)*s - xo] + wn^2*X(s) = 0
-Algebra
X(s) (s^2+2*z*wn*s+wn^2) = s*xo + vo + 2*z*wn*xo
-Solving for X(s)
X(s) = [s*xo + vo + 2*z*wn*xo] / [s^2 + 2*z*wn*s + wn^2]
More algebra so that I may use item 26a in the extended Laplace tables
X(s) = xo*[ s + (vo+2*z*wn*xo)/xo ] / [s^2+2*z*wn*s+wn^2]
I will define alpha (as listed in the Extended Laplace Table item 26a):
alpha = (vo+2*z*wn*xo)/xo
When I directly apply this formula from the Extended Laplace Tables I get an inconsistent answer (see attached)
Just looking at the Magnitude (magnitude is off)
sqrt{ [( alpha/wn - z*wn )^2] / (1-z^2) + 1 }
Replacing alpha with (vo+2*z*wn*xo)/xo
= sqrt{ [ [(vo+2*z*wn*xo)/xo)/wn - z*wn]^2/(1-z^2) + 1 }
Making common denominator, and making 1-z^2 = (wd/wn)^2
= sqrt{ [vo+2*z*wn*xo]/(xo*wn) - z*wn^2*xo/(xo*wn)]^2/ (wd/wn)^2 + 1}
Bringing up wd/wn into the square & notice wn and 1/wn cancel
Factor out 1/xo^2
= sqrt{ [vo+2*z*wn*xo] - z*wn^2*xo]^2/ (wd*xo)^2 + }
Making 1 have common denominator with other stuff
= sqrt{ [vo+2*z*wn*xo] - z*wn^2*xo]^2/ (xo*wd)^2 + 1*(xo*wd)^2/(xo*wd)^2 }
Factor out 1/(xo*wd)
= 1/(xo*wd)* sqrt{ [vo+2*z*wn*xo] - z*wn^2*xo]^2 + (xo*wd)^2}
This magnitude (just above) does not agree with the correct magnitude (next couple of lines) (can also see attached source):
= 1/(xo*wd)* sqrt{ [vo+2*z*wn*xo - z*wn*xo]^2 + (xo*wd)^2}
= 1/(xo*wd)* sqrt{ [vo+z*wn*xo ]^2 + (xo*wd)^2}
I believe that item 26a may not be general enough. As the z*wn term within the square thats within the square root should just be z
QUESTION:
What's going on on here? Thank you.