Natural Vibration of Beam - PDE

In summary: How does the non homogeneity function just go away?2) How is it shown that the exponential is absorbed in the LHS of PDE?Thanks, that makes good sense. Now I can proceed :-)1) How does the non homogeneity function just go away?
  • #1
bugatti79
794
1
I am just wondering the author is doing in this calculation step.

Given ##\displaystyle \rho A \frac {\partial^2 w}{\partial x^2} - \rho I \frac{\partial^4 w}{\partial t^2 \partial x^2} +\frac {\partial^2 }{\partial x^2}EI \frac {\partial^2 w}{\partial x^2}=q(x,t)##

where ##w(x,t)=W(x)e^{-i \omega t}##

##\omega## is the frequency of natural transverse motion and ##W(x)## is the mode shape of the transverse motion.

He substitutes the above into the PDE to get the following

##\displaystyle \frac {d^2 }{d x^2}EI \frac {d^2 W}{d x^2} - \lambda (\rho A W -\rho I \frac {d^2 W }{d x^2} ) =0## where ##\lambda=\omega^2##

However, I calculate the second derivative ##w''(x)=e^{-i\omega t} W''(x)## and ##w''(t)=- \lambda e^{-i\omega t} W(x)##

What is incorrect on my part? Ie, where did the exponentials go?

thanks
 
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  • #2
If the non-homogeneity function q goes away, so that the PDE hat 0 in the RHS, you can show that the exponential gets factored in the LHS, so that the resulting ODE will no longer contain it.
 
  • #3
dextercioby said:
If the non-homogeneity function q goes away, so that the PDE hat 0 in the RHS, you can show that the exponential gets factored in the LHS, so that the resulting ODE will no longer contain it.
1) How does the non homogeneity function just go away?

2) How is it shown that the exponential is absorbed in the LHS of PDE?

Thanks
 
Last edited:
  • #4
bugatti79 said:
1) How does the non homogeneity function just go away?

2) How is it shown that the exponential is absorbed in the LHS of PDE?

Thanks

On second thoughts is it something along these lines...

If ##q(x,t)=0##then we can write the ODE in which each term on the LHS will have a ##e^{-i \omega t}## factor. Pull this out from each of the terms and thus we get

##e^{-i \omega t}[ODE]=0## but ##e^{-i \omega t}\ne 0## therefore

##[ODE]=0##...?
 
  • #5
bugatti79 said:
1) How does the non homogeneity function just go away?

If "natural transverse motion" means the same as "free vibration", then q(x,t) can't depend on t by definition, otherwise you would have forced vibration not free vibration.

So this is the same idea as a vertical mass-on-a-spring, where the spring is stretched by the weight of the mass. You can find the equilibrium position as the statics problem
$$\frac{d^2}{dz^2}EI\frac{d^2}{dz^2}w_0(x) = q(x)$$
Then you measure w(x) from the equilibrum position. That makes the right hand side = 0.

You answered your own question 2).
 
  • #6
AlephZero said:
If "natural transverse motion" means the same as "free vibration", then q(x,t) can't depend on t by definition, otherwise you would have forced vibration not free vibration.

So this is the same idea as a vertical mass-on-a-spring, where the spring is stretched by the weight of the mass. You can find the equilibrium position as the statics problem
$$\frac{d^2}{dz^2}EI\frac{d^2}{dz^2}w_0(x) = q(x)$$
Then you measure w(x) from the equilibrum position. That makes the right hand side = 0.

You answered your own question 2).

Thanks, that makes good sense. Now I can proceed :-)
 

1. What is the natural vibration of a beam?

The natural vibration of a beam refers to the inherent tendency of a beam to vibrate at certain frequencies without any external forces acting on it. These frequencies are determined by the properties of the beam, such as its length, material, and boundary conditions.

2. How is the natural vibration of a beam related to PDE?

The natural vibration of a beam is governed by the Partial Differential Equations (PDE) that describe the motion of the beam. These equations take into account the physical properties and boundary conditions of the beam to determine its natural frequencies and corresponding modes of vibration.

3. What factors affect the natural vibration of a beam?

The natural vibration of a beam is influenced by several factors, including its length, material, cross-sectional shape, boundary conditions, and any additional loads or forces acting on it. These factors can all impact the natural frequencies and modes of vibration of the beam.

4. How can the natural vibration of a beam be calculated?

The natural vibration of a beam can be calculated using analytical methods, such as solving the governing PDEs, or through numerical techniques, such as finite element analysis. These methods take into account the physical properties and boundary conditions of the beam to determine its natural frequencies and modes of vibration.

5. What are the practical applications of studying the natural vibration of a beam?

Understanding the natural vibration of a beam is important in various engineering fields, such as civil, mechanical, and aerospace engineering. It can help in the design and optimization of structures, such as bridges, buildings, and aircraft, to ensure they can withstand the effects of vibrations and avoid potential failures.

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