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Natural Vibration of Beam  Algebraic Query 
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#1
Dec712, 04:16 PM

P: 652

Folks,
Just wondering what the author is doing here, given ##\displaystyle \left (\frac{2EI}{L^3} \begin {bmatrix} 6&3L\\3L&2L^2 \end {bmatrix} \omega^2 \frac{\rho A L}{420} \begin {bmatrix} 156&22L\\22L&4L^2 \end{bmatrix} \begin {bmatrix} W\\ \theta \end {bmatrix} \right )=\begin {bmatrix} 0\\ 0 \end {bmatrix}## He proceeds to set the determinant of the matrix above to 0 to obtain a quadratic characteristic equation in ##\omega^2## The equation being ##151201224 \lambda + \lambda^2=0##, where ##\displaystyle \lambda=\frac{\rho A L^4}{EI} \omega^2## Using wolfram http://www.wolframalpha.com/input/?i...%7D%7D%29+%3D0 (Z=EI) I dont see how he arrives at this quadratic expression or how he determined ##\lambda## ...thanks 


#2
Dec712, 04:27 PM

P: 652




#3
Dec812, 11:33 AM

P: 652

If we do a units balance on this to which I calculate ##\displaystyle\frac{\frac{kg}{m^4}m^2m^4 }{\frac{\frac{kgm}{s^2}}{m^2}m^4}\frac{rad^2}{s^2}=\frac{rad^2}{m}## where ##\rho## is the mass density per metre...thanks 


#4
Dec812, 12:38 PM

HW Helper
P: 1,391

Natural Vibration of Beam  Algebraic Query
Are you sure that ##\rho## is a linear mass density? It looks like it should be kg/m^{3}. Also, keep in mind that "radians" are dimensionless units (they have dimensions of length/length). 


#5
Dec812, 01:06 PM

P: 652

##u_h^e(x)=c_1+c_2x+c_3x^2## This is not dimensionless... 


#6
Dec812, 01:12 PM

P: 652

Its when an equation is set to 0 with different powers of ##\lambda## like you say, that they must be dimensionless, right..........? thanks 


#7
Dec812, 03:44 PM

Engineering
Sci Advisor
HW Helper
Thanks
P: 6,933




#8
Dec812, 04:05 PM

HW Helper
P: 1,391

In the quadratic for ##\lambda##, you assigned no numbers to the numerical coefficients, and since you were following a book that appears to have left all dimensionful quantities as variables, I assumed those coefficients were dimensionless. Are the coefficients supposed to have units? 


#9
Dec912, 02:05 PM

P: 652

I have never heard of a volumetric density... 


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