Dartmouth Extended Laplace Tables -- Not general enough? item26.a

[ltkach2015]

Homework Statement

[/B]
http://www.dartmouth.edu/~sullivan/22files/New%20Laplace%20Transform%20Table.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 that's within the square root should just be zQUESTION:

What's going on on here? Thank you.

 

[collinsmark]

I'm actually a little suspicious that the entry 26a in the table might not even be correct at all.

Unless I'm making a large mistake myself, here is my logic:

I suspect that the constant $\alpha$ should have dimensionality of $\frac{1}{[\mathrm{time}]}$. (Again, that is if I'm not mistaken). If so, then one of the terms in 26a doesn't make sense to me: in particular, the part of term that is $\frac{\alpha}{\omega_n} - \zeta \omega_n$.

$\omega_n$ also has units of $\frac{1}{[\mathrm{time}]}$. That makes $\frac{\alpha}{\omega_n}$ dimensionless (if I'm correct about $\alpha$), and $\zeta$ is dimensionless. But that gives $\zeta \omega_n$ dimensionality of $\frac{1}{[\mathrm{time}]}$. So that operation is subtracting a $\frac{1}{[\mathrm{time}]}$ value from a dimensionless number.

With that I conclude (or at least suspect) that something doesn't look right with 26a to me.

 

[ltkach2015]

Yes I agree the dimensions do look off