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cantilever beam model
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Nikyu
science forum beginner


Joined: 17 Oct 2005
Posts: 4

PostPosted: Tue Apr 11, 2006 8:49 pm    Post subject: cantilever beam model Reply with quote

I'm trying to figure out a simple way of determining the first natural
frequency of a cantilever beam. I figured it would be just like a
mass-spring system where the frequency is equal to the square root of
spring constant over mass...but how do I determine the equivalent mass
and spring constant of a cantilever beam?

Thanks,
Nikyu
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Olin Perry Norton
science forum addict


Joined: 13 May 2005
Posts: 58

PostPosted: Tue Apr 11, 2006 10:14 pm    Post subject: Re: cantilever beam model Reply with quote

eryk33@netscape.net wrote:

Quote:
I'm trying to figure out a simple way of determining the first natural
frequency of a cantilever beam. I figured it would be just like a
mass-spring system where the frequency is equal to the square root of
spring constant over mass...but how do I determine the equivalent mass
and spring constant of a cantilever beam?

Thanks,
Nikyu



You know, I'm sure, that the most accurate way to solve this

problem is to work with the PDE that describes the beam deflection
Y as a function of distance X and time T.

You say you want a simple method, based on a mass-spring
type of model. Some people call this a "lumped-parameter"
model -- if you Google that term together with cantilever, beam,
and vibrate I bet you'll find your answer.

To find the spring constant, use the equation for (static) bending
of a beam with a force applied at the end, and your spring constant
is just the force divided by the deflection.

To get the mass, you lump a certain fraction (I forgot how much,
sorry) of the beam mass at the free end.
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BobK207
science forum addict


Joined: 12 May 2005
Posts: 52

PostPosted: Wed Apr 12, 2006 12:21 am    Post subject: Re: cantilever beam model Reply with quote

Olin Perry Norton wrote:
Quote:
eryk33@netscape.net wrote:

I'm trying to figure out a simple way of determining the first natural
frequency of a cantilever beam. I figured it would be just like a
mass-spring system where the frequency is equal to the square root of
spring constant over mass...but how do I determine the equivalent mass
and spring constant of a cantilever beam?

Thanks,
Nikyu



You know, I'm sure, that the most accurate way to solve this
problem is to work with the PDE that describes the beam deflection
Y as a function of distance X and time T.

You say you want a simple method, based on a mass-spring
type of model. Some people call this a "lumped-parameter"
model -- if you Google that term together with cantilever, beam,
and vibrate I bet you'll find your answer.

To find the spring constant, use the equation for (static) bending
of a beam with a force applied at the end, and your spring constant
is just the force divided by the deflection.

To get the mass, you lump a certain fraction (I forgot how much,
sorry) of the beam mass at the free end.

get a copy of Mark's Mechaincal Engineering Handbook

check out

http://en.wikipedia.org/wiki/Rayleigh-Ritz_method

my recollection is that Meffective = 1/3 mass total but my memory
could be faulty

http://wwwex.physik.uni-ulm.de/lehre/physikalischeelektronik/phys_elektr/node298.html


has the answer embeded in the equations but I'm too lazy to back
calculate it.

cheers
Bob
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Greg Locock
science forum Guru Wannabe


Joined: 28 May 2005
Posts: 111

PostPosted: Wed Apr 12, 2006 7:43 am    Post subject: Re: cantilever beam model Reply with quote

"BobK207" <rkazanjy@gmail.com> wrote in
news:1144801283.748453.132650@j33g2000cwa.googlegroups.com:

Quote:

Olin Perry Norton wrote:
eryk33@netscape.net wrote:

I'm trying to figure out a simple way of determining the first
natural frequency of a cantilever beam. I figured it would be just
like a mass-spring system where the frequency is equal to the square
root of spring constant over mass...but how do I determine the
equivalent mass and spring constant of a cantilever beam?

Thanks,
Nikyu



You know, I'm sure, that the most accurate way to solve this
problem is to work with the PDE that describes the beam deflection
Y as a function of distance X and time T.

You say you want a simple method, based on a mass-spring
type of model. Some people call this a "lumped-parameter"
model -- if you Google that term together with cantilever, beam,
and vibrate I bet you'll find your answer.

To find the spring constant, use the equation for (static) bending
of a beam with a force applied at the end, and your spring constant
is just the force divided by the deflection.

To get the mass, you lump a certain fraction (I forgot how much,
sorry) of the beam mass at the free end.

get a copy of Mark's Mechaincal Engineering Handbook

check out

http://en.wikipedia.org/wiki/Rayleigh-Ritz_method

my recollection is that Meffective = 1/3 mass total but my memory
could be faulty

http://wwwex.physik.uni-ulm.de/lehre/physikalischeelektronik/phys_elekt
r/node298.html


has the answer embeded in the equations but I'm too lazy to back
calculate it.


And for that you use the tip stiffness?

That's a fine wiki article, a model of accuracy, brevity and wit, worth every
penny. It would be nice if someone could check it, actually.

Another approach is by considering the upper and lower bounds.

An upper bound solution would be to take all the mass and the stiffness half way
along the beam.

A lower bound solution would be to use all the mass, and the tip stiffness.

So you know the frequency must fall into that band.

sqrt(8*k/m)>F/2/pi>sqrt(k/m)

Bob's suggestion of sqrt(3*k/m) would fall nicely in the middle of that band.

A better approximation might be to say (after thinking about the likely mode
shape) that the stiffness 3/4 of the way along, and 1/2 the mass, is probably
pretty close.

F/2/pi ~ sqrt(k*(4/3)^3/(m*.5)) = sqrt((128/27)*k/m) ~ sqrt(4.3*k/m)

and so on and so forth

Another way of calculating it would be to build an FEA model.

SO the OP has at least 3 good options (PDE, Rayleigh Ritz, FEA, guessing).

In practice R-R is terrific for real conditions, such as strangely shaped beams,
and springy foundations.

Cheers

Greg Locock
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Gordon153
science forum beginner


Joined: 19 Oct 2005
Posts: 28

PostPosted: Wed Apr 12, 2006 12:30 pm    Post subject: Re: cantilever beam model Reply with quote

eryk33@netscape.net wrote:
Quote:
I'm trying to figure out a simple way of determining the first natural
frequency of a cantilever beam. I figured it would be just like a
mass-spring system where the frequency is equal to the square root of
spring constant over mass...but how do I determine the equivalent mass
and spring constant of a cantilever beam?

The simplest way is to look up the correct formula. For beams with a
variety of B.C., the natural frequency is
omega=2 pi f=a sqrt(EI/(mL^4)), where m=mass per unit length. For a
cantilever, a=3.52.
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BobK207
science forum addict


Joined: 12 May 2005
Posts: 52

PostPosted: Wed Apr 12, 2006 7:43 pm    Post subject: Re: cantilever beam model Reply with quote

Been a LONG time since I checked this but I'm pretty sure that

using 1/3 of the total mass AND the tip stiffness (3EI) / (L^3)

gives a very good approximation, so good that FEA is hardly worth the
effort

cheers
Bob


Sorry, I only took a 5 sec look at the WIki article...............it
was just something to get him thinking about energy methods
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