I suggest reading E.P. Popov Mechanics of Materials, he covers beams and helical springs. Looking at the springs section it would appear that it is more of a beam torsion problem then a beam bending problem. You will learn more effectively by reading, deriving the equations,and figuring it our for yourself then from any abbreviated explanation on a forum. There is no need for an ANSYS FEM model, calculations in MathCad or Excel (I hate Excel, IMO it is an accountant's tool not an engineering's tool) should allow you to explore and identify the important parameters. That would be the path I would take if I wanted a deeper understanding.
For defining the natural frequencies of a beam rigidly coupled at both ends, we are given the equation:
F(X)=(Kn/2pi)*sqrt(EI/Rho*A*(L^4))
where:
Kn=mode number
E=elastic modulus
I=area moment of inertia (ie second moment of inertia, I believe)
Rho=mass density
A=cross sectional area
L=length
However, what second moment of inertia equation do we want? I believe we can calculate it with respect to both X and Y, where (assuming a rectangular cross-section) I(x)=b(h^3)/12 and I
=h(b^3)/12 and h=height (ie y) and b=width (ie x).
F(X)=(Kn/2pi)*sqrt(EI/Rho*A*(L^4))
where:
Kn=mode number
E=elastic modulus
I=area moment of inertia (ie second moment of inertia, I believe)
Rho=mass density
A=cross sectional area
L=length
However, what second moment of inertia equation do we want? I believe we can calculate it with respect to both X and Y, where (assuming a rectangular cross-section) I(x)=b(h^3)/12 and I
=h(b^3)/12 and h=height (ie y) and b=width (ie x).A factor with helical springs is that above a certain ratio of width to length they are subject to buckling, and the stress on them is a combination of a torsional and shear stress.
Being made from an elastic solid they can support both compressional and shear waves, but the stress is not compressional but torsional.
The question then arises is there in a coil spring anything analogious to the standing wave patterns we would expect on a cantelever spring, in this way the cantelever is similar to an organ pipe with one end stopped.
rcw.
Being made from an elastic solid they can support both compressional and shear waves, but the stress is not compressional but torsional.
The question then arises is there in a coil spring anything analogious to the standing wave patterns we would expect on a cantelever spring, in this way the cantelever is similar to an organ pipe with one end stopped.
rcw.
A cantilever and a organ pipe are completely different things. One is a compressional or longitudinal wave and the other is a transverse bending wave. They follow entirely different sets of equations.
The coil spring has both transverse bending and tortion by the nature of its construction. This makes its detailed analysis quite difficult requiring very sophistcated FEA. Under small displacements and considered as a bulk stiffness, it acts linearly, but this is only an approximation.
The coil spring has both transverse bending and tortion by the nature of its construction. This makes its detailed analysis quite difficult requiring very sophistcated FEA. Under small displacements and considered as a bulk stiffness, it acts linearly, but this is only an approximation.
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Interesting.
For theory to apply accurately for a large duration of cycles, I believe we must consider the fatigue limit rather than the proportionality limit.
I believe steel approaches a value and below this it can approach operating for an infinite number of cycles. However, materials such as Aluminum do not possess this trait.
How are we then able to determine an acceptable level of stress for a beam/spring not possessing that particular attribute of steel? Would we simply quantify the expected number of cycles? Is there something I'm overlooking?
For a given deflection, a beam will experience a finite stress/strain (and thus failure). I believe this may be independent of the thickness of the beam. A thicker beam will simply require a larger force to overcome the lower compliance and achieve the finite deflection.
For determining the variables that contribute to failure, would we not consider those which directly relate to stress/strain for a finite deflection. Assuming this is true, what variables should we consider?
Thanks,
Thadman
"I(x)=b(h^3)/12 and I
That's correct
I'm supposed to use both equations?
"For theory to apply accurately for a large duration of cycles, I believe we must consider the fatigue limit rather than the proportionality limit."
What theory? Fatigue is a totally different animal? We've been talking only about frequencies, which depend on mass/stiffness/geometry.
Fatigue prediction is much more empirical than theoretical; it's based on S-N curves derived from exhaustive testing and must be done for each material/alloy.
"I'm supposed to use both equations?"
Yes, one will give the vertical stiffness and the other, lateral.
It sounds like you need to take some courses or read Wikipedia.
What theory? Fatigue is a totally different animal? We've been talking only about frequencies, which depend on mass/stiffness/geometry.
Fatigue prediction is much more empirical than theoretical; it's based on S-N curves derived from exhaustive testing and must be done for each material/alloy.
"I'm supposed to use both equations?"
Yes, one will give the vertical stiffness and the other, lateral.
It sounds like you need to take some courses or read Wikipedia.
The properties of the beam may change once cracks (however infinitesimal they may appear to us) begin to form. The theory we used to define its behavior thus may not apply under those circumstances.
Exactly. Something like this?
http://en.wikipedia.org/wiki/File:S-N_curves.PNG
We would then attempt to quantify the expected number of cycles and place it below this curve?
"The theory we used to define its behavior thus may not apply under those circumstances."
So? You can hypothesize any number of situations that change the behavior for any kind of system you care to discuss.
I don't see this discussion going anywhere, at least anywhere I want to go, so I'm bowing out.
So? You can hypothesize any number of situations that change the behavior for any kind of system you care to discuss.
I don't see this discussion going anywhere, at least anywhere I want to go, so I'm bowing out.
If we are interested in achieving the most complete understanding of a system, should we not consider all possible deviations of the applicability of our theory? Should we not question all of our assumptions?
Theory, as I see it, is based upon a finite number of assumptions. If we diverge from these assumptions, our theory may no longer apply.
For our analysis, an assumption is made that fatigue is negligible. How are we able to make this assumption?
Under what conditions would fatigue be negligible? I believe this is non-trivial. Assuming it is quantified, we could reach the conclusion that it is either applicable or non-applicable to our particular model. However, it has not been quantified, therefore I believe it should be considered for an accurate understanding of the behavior of a system.
For S-N curves, we are establishing stress with respect to the number of cycles, correct?
Also, is not mass/stiffness/geometry related to stress? Assuming this is true, would not fatigue be an appropriate and relevant tangent to consider?
By suggesting the fatigue limit as the first appropriate limit to consider, I was simply suggesting that we restrict our model to conditions where our assumptions are valid and true.
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Excellent
For theory, the least required assumptions is desirable.
However, how are we able to reduce our system to the least number of assumptions? Surely we are not expected to dismiss or ignore effects.
We should and can eliminate superfluous assumptions, however without superfluous being defined, how can we differentiate with respect to the triviality of our assumptions?
I believe we should thus consider all possible circumstances and from this we can reduce our system. This will establish the required assumptions.
As Noah and Martin have stated, true curiosity of this nature is best directed at a library rather than a forum.
Perhaps the first question that should be asked is how to integrate a spring into a loudspeaker suspension. I can think of a few simple ways, all of which open a can of worms and don't provide centering given a very compliant spring in the low mm/N range. One can mount a few (3+) radially to approximate a spider, but then aren't you really just adding mass and gaining secondary vibration issues?
Since we are having fun with thought experiments where others do the work, why not go nuts and think about electromagnetic suspension?
Perhaps the first question that should be asked is how to integrate a spring into a loudspeaker suspension. I can think of a few simple ways, all of which open a can of worms and don't provide centering given a very compliant spring in the low mm/N range. One can mount a few (3+) radially to approximate a spider, but then aren't you really just adding mass and gaining secondary vibration issues?
Since we are having fun with thought experiments where others do the work, why not go nuts and think about electromagnetic suspension?
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