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dynamic response spring mass system
3

dynamic response spring mass system

dynamic response spring mass system

(OP)
I have a design problem that I can do numerically but I'm looking for a more elegant method.

Here goes:

A spring, S  is interposed between 2 masses (M-S-m aligned vertically) and compressed by the gravity of the much larger mass, M which sits on top. The smaller mass, m which is much smaller than the spring mass is at the bottom and is initially restrained. Now the restraint is suddenly removed. The system is now moving vertically under gravity.
What is the response?

I am after the dynamic equations that include the spring mass.


Thanks for any ideas.

 

RE: dynamic response spring mass system

So, if I understand the problem correctly, after the restraint of the smaller mass is removed, there is no support for the system?  The whole system is falling in space due to gravity as the three masses respond internally to the lack of restraint to the system?

What is the real world application here?

Mike McCann
MMC Engineering
Motto:  KISS
Motivation:  Don't ask

RE: dynamic response spring mass system

Also, do you want to integrate the frictional damping effect due to wind resistance as the system falls?

Mike McCann
MMC Engineering
Motto:  KISS
Motivation:  Don't ask

RE: dynamic response spring mass system

Response to what?  It seems there is no Q to relate.  Are you asking for the mode of a spring with two different masses on each end?

Tobalcane
"If you avoid failure, you also avoid success."  

RE: dynamic response spring mass system

Zekeman...

Show us a picture that will make us believe this is a real situation and not an undergraduate student question.

 

RE: dynamic response spring mass system

I estimate that the first response is that the assembly begins to fall.

Regards,

SNORGY.

RE: dynamic response spring mass system

Quote:


I have a design problem that I can do numerically but I'm looking for a more elegant method.
What is a more elegant method?   Do you want a symbolic solution?   An equation as a function of time?  Will the exact solution be significantly better so it is worth the effort?

What are the equations you used for your numerical solution?
( This will let me know if you have worked out a numerical solution ).

There may not be an exact solution if the formula is non-linear or the solution may be very complex.

Have you tried wxMaxima?

A similar problem is stacked servo controlled hydraulic cylinders.   That is a real problem.
 

Peter Nachtwey
Delta Computer Systems
http://www.deltamotion.com

 

RE: dynamic response spring mass system

(OP)
"Zekeman...

Show us a picture that will make us believe this is a real situation and not an undergraduate student question."

It's been about 45 years since I was in academia. This is  a REAL  application.

RE: dynamic response spring mass system

(OP)
"Also, do you want to integrate the frictional damping effect due to wind resistance as the system falls? "

Ignore friction and windage.

In fact, the larger mass is >> than the spring mass and the end mass.
So the problem really reduces to  the usual spring-mass dynamic problem except that the mass<< spring mass.And the large mass, M is out of the equation.












, of the end mass of a compressed spring with the  end mass<< spring mass.

RE: dynamic response spring mass system

(OP)
"Wouldn't it be a supersposition of a moving reference frame and an oscillating mass-spring-mass system? "

Yes, but the greater and only concern is that

spring mass>> the end mass so the distributed mass of the spring comes into play.

Also, I have simplified the problem ignoring the free fall and am looking at a spring single mass system where

m<< spring mass.

 

RE: dynamic response spring mass system

(OP)
"What is a more elegant method?   Do you want a symbolic solution?   An equation as a function of time?  Will the exact solution be significantly better so it is worth the effort?"


I was thinking the wave equation with the end mass which has the distributed mass embedded.

My socalled numerical method is not really numerical but would be modeling the spring into a number of discrete masses separated by ideal springs which can be solved but the acuracy of this model is in question.
 

RE: dynamic response spring mass system

(OP)
I am repeating the simplified version of my original problem.

Ignore friction and windage.

In fact, the larger mass is >> than the spring mass and the end mass.
So the problem really reduces to  the usual spring-mass dynamic problem except that the mass<< spring mass.And the large mass, M is out of the equation.

 

RE: dynamic response spring mass system

(OP)
"A similar problem is stacked servo controlled hydraulic cylinders."

I DOUBT IT.
   

RE: dynamic response spring mass system

When the restraint is suddenly removed there is no longer an assurance that the contract between the small mass and the spring exists all the time it may separate. Since "m which is much smaller than the spring mass" to my opinion you may actually need to divide the springs into small segments and calculate the internal dynamics of the spring coils movement to find where the spring is no longer acting a force on the small mass.

RE: dynamic response spring mass system

Let's call:

k = spring constant
Ms =spring mass
m = mass << Ms
l = spring length

We can write the energy E of the system :

E = 1/2* m* (dl/dt)^2 + 1/2* k * l^2 + 1/2*(Ms/3)*l^2

Note that the term ½*(Ms/3)*l^2 represents the spring kinetic energy.

Rearranging:

E = 1/2* (m + Ms/3) * (dl/dt)^2 + 1/2*k * l^2

And this corresponds to a harmonic oscillator with frequency

f = 1/(2*PI) *SQRT[k/(m + Ms/3)]

Now being Ms>>m it will also be Ms/3 >> m and so

f = 1/(2*PI) *SQRT[k/(Ms/3)]
 

RE: dynamic response spring mass system

If you take the spring as a distributed mass and integrate over its length the equivalent spring mass/inertia for natural frequency calculations is 1/3 of the spring mass.  

RE: dynamic response spring mass system

First, apologies for my earlier "tongue in cheek" post.

The derivation of M/3 can be found here:

http://en.wikipedia.org/wiki/Effective_mass_(spring-mass_system)

A real world application might be a motocross bike with a broken shock absorber, or conversely, sizing a shock absorber for a motocross bike based on undamped dynamic response.
 

Regards,

SNORGY.

RE: dynamic response spring mass system

IRstuff,

Yes there is and israelkk is right. It has been reported by Lawrence Ruby in its "Equivalent mass of a coil spring" The Physics Teacher – March 2000 -- Volume 38, Issue 3, pp. 140-141 Issue Date: March 2000.

Mr. Ruby's work assumes the spring expands in a uniform manner. I'll try to report the way the kinetic energy of the spring has been calculated (this is trick).
Each element of the spring having a mass dMs moves with a velocity Vx proportional to the distance x from the spring edge:

Vx = Vm*(x/l)

Being:
l = spring length
Vm = velocity of the suspended mass m

Moreover:
dMs = (Ms/l)*dx

So the kinetic energy Ks of the spring, integrating between [0,l] is (INT stands for integral):

Ks = 1/2 * INT[(Vx)^2*dm]= 1/2 * INT[(Vm)^2* (x^2)/(l^2)*Ms/l*dx ]= 1/2 * Ms*(Vm)^2/(2*l^3)*iNT[ x^2 dx]= 1/2 * Ms*(Vm)^2/(2*l^3)*[l^3/3] = 1/2 * Ms/3*(Vm)^2


 

RE: dynamic response spring mass system

(OP)
"Mr. Ruby's work assumes the spring expands in a uniform manner."

I'm very skeptical about that assumption. This contradicts the wave equation which more accurately describes the motion.

I could accept it for a quasi static motion, but certainly not here-- at least without some proof.

As far as I can remember, and I have used it, the 1/3 assumption has been used where m was larger than the spring mass but I have never seen it for the problem at hand,

m<<spring mass

This looks more like a compressed spring suddenly released.

RE: dynamic response spring mass system

The 1/3 assumption does not rely on m being smaller than the mass of the spring.

If you have it, take a look at Harris' Shock and Vib Handbook 6th ed pages 242-248

=====================================
(2B)+(2B)'  ?

RE: dynamic response spring mass system

I should clarify for people with other editions of that book, it is chapter 7 of systems with distributed mass and elasticity.

=====================================
(2B)+(2B)'  ?

RE: dynamic response spring mass system

Well maybe there are some assumptions built into that 1/3 about relative masses.

At any rate table 7.10 gives solution (resonant frequency) for various cases similar to yours.

=====================================
(2B)+(2B)'  ?

RE: dynamic response spring mass system

Actually, not in the table. One would have to study the text to extract a solution for this specific case.  

=====================================
(2B)+(2B)'  ?

RE: dynamic response spring mass system

(OP)
"
The derivation of m/3 can be found here:

http://en.wikipedia.org/wiki/Effective_mass_(spring-mass_system)"
Good reference.fter the derivation there was a footnote caveat that said

'Real spring

However, the above calculations are only suitable for small values of \frac{M}{m}. Since Jun-ichi Ueda and Yoshiro Sadamoto have done the experiment[citation needed]. It is found that, as the ratio \frac{M}{m} increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value \frac{m}{3} and eventually reaches negative values. This unexpected behavior of the effective mass can be explainable in terms of the elastic'

m is the spring mass in this case and I assume the M is the fixed mass which is << m [ the reverse notation I have used]

Pretty much rules out the arbitrary 1/3 rule for general application. However, for the problem proposed, it looks like it might be valid. I will check further and post results if possible.

Thanks for your input as well as everybody else who responded. Your comments are most appreciated.

zeke

RE: dynamic response spring mass system

(OP)
Correction

I think the reverse is true. I must have gotten it backwards.i.e.

As the ratio of spring mass to fixed mass increases beyond 7 it looks like  the effective mass of the spring decreases below 1/3 and eventually goes negative.

I think I'll assume the 1/3 factor and test it . Analytic work is too  time consuming for this. In fact a solution to the free spring with similar boundaries I found* indicated that 0.4 is more appropriate

*
Mechanics Applied to Vibration and balancing, D laugharne Thornton, Chapmen & Hall, 1951

RE: dynamic response spring mass system

Way out of my expertise but here goes.  The mention of hydraulic cylinders above post hit neuron in my small brain.


Isn't there a very similar problem with mass dampening system, like earthquake mitigation and situations the infamous foot bridge bridge in England? Trying to recall a paper I saw about the swinging bridge in England.   

RE: dynamic response spring mass system

Is this school work or home work?  This sounds an awful lot like our senior lab project in ME Design.  We used analog computers to solve it and compose the equations.

rmw

RE: dynamic response spring mass system

(OP)
"Is this school work or home work?  This sounds an awful lot like our senior lab project in ME Design.  We used analog computers to solve it and compose the equations."

Yeah, mine too, in the 60's but not with analog computers. You must be talking the 50's.
 

RE: dynamic response spring mass system

Interesting discussion. When m is less than spring mass the problem is one of vibration of distributed or continuous systems. These are governed by a wave equation of the form

diff(diff(u,t),t) = E/Rho x diff(diff(u,x),x).

where u is the displacement of a point on the spring. In addition to this wave equation you need the appropriate boundary values and initial conditions (the two masses at the ends affect the boundary conditions). You can find more details in many Mechanical Vibration textbooks in the advanced chapters. I found a solution on page 239 of Vibration of continuous systems By Singiresu S. Rao (from Google books), which I'll summarize below:

The natural frequencies expression is

  wn_i = alpha_i x c / L

where alpha_i are the multiple solutions of

  alpha_i x tan(alpha_i) = beta

and
  beta = spring mass / end mass
and
  c = sqrt (E/Rho) of spring

In this problem we should be primarily interested in the first of these natural frequencies (i=1).

This equation actually results in the 1/3 factor for large values of end mass/spring mass (greater than 7). The other limit of end mass very small gives a factor of 4/Pi^2 which is approximate 0.4 as Zekeman mentioned.

It's important to gain that insight. However, in practice I would consider using finite elements or some other numerical tool to model this however.

I checked these equations and the final expression, but not thoroughly enough!

Nagi Elabbasi
Veryst Engineering

RE: dynamic response spring mass system

(OP)
Nagi,

Thank for your detailed post.
 
I believe Timoshenko did this problem first around 1910 and a detailed analysis can be found in his  book,"vibration Problems in Engineering". I have a copy of the second edition where it appears in pp 317-321.

He derives  the  1/3 factor based on assuming a small ratio of spring to lumped mass; however, this holds pretty good over a a larger range of ratios and tends toward  4/Pi^2 or about 0.4, my case.

Thanks again for the discussion and valuable comments.


zeke

 

RE: dynamic response spring mass system

The behaviour of a spring having a mass Ms and a mass m suspended, being  Ms>>m is similar to that of a slinky coil spring. So the assumption of a spring with uniform elongation that leads to an inertial equivalent mass Ms/3, is not valid anymore (my bad). In this case it is not possible to neglect the influence of waves and impulses which propagate over the spring.

E.E. Galloni and M. Kohen, in their "Influence of the mass of the spring on its static and dynamic effects," Am. J. Phys. 47, 1076–1078 (Dec. 1979), described the behaviour of a slinky coil spring and found a value for the inertial equivalent mass equal to [Ms* 4/(PI^2)]. This value has been already noticed above (Zekeman and Ellabasi).
 

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