Natural Frequency

[EddyC]

Hi Folks. I have a system that I need to find out the natural frequency of. It consists of a mass sliding horizontally back & forth on a frictionless surface. At each end of the mass is a spring. But there is a gap between each spring and the mass. So the mass makes intermittant contact with each spring. Perhaps someone has a solution?

 

[Twoballcane]

That will all depend on the weight of the mass and the accelration of the mass going into the wall and the stifness of the springs and the distance it travels. So it looks like your standard Fn equations should do it.

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

 

[40818]

You have a system whereby your pretty close to a normal "one mass, 2 spring system".
Using usual notation:
The first natural frequency is "root of 2k/m"
and the restoring force on the mass is -2kx

 

[ivymike]

that ain't right... the mass is only ever in contact with one of the springs, based on the post

I believe that the system as described has no single natural frequency, although for infinitesimally small gaps it may behave as though it does.

The reason I say that is that the energy of the mass is invariant as it traverses the gap between contacting one spring and contacting the other, and the time it takes to traverse the gap will depend on the size of the gap and the net energy within the system. If you had low energy and a large gap, it might take a month for the mass to get from side to side. Once at either side, it would deflect the spring only slightly, and begin its return at a snail's pace. With high energy, it would traverse the gap rapidly, deflect the spring a great deal, and return rapidly. Thus, the "natural frequency" depends on net energy, which is not what I typically think of as a natural frequency.

For very very small gaps, the natural frequency would be the same regardless of energy, and would be calculated using sqrt(k/m), using k from only one spring.

 

[hacksaw]

hey it oscillates just fine, but it is a nonlinear problem. you can use simple laws of mechanics to sort out the period as a class room exercise

 

[EddyC]

Hey Folks,

Thanks for your assistance. I did a little more digging on my own and found the exact solution to this particular system. It was in the book: Mechancial Vibrations, Den Hartog, McGraw-Hill. It is indeed a non-linear system with the natural frequency dependent on the ratio between the gap and the spring deflection.

 

[Denial]

If the two springs are of equal stiffness, I think the solution is readily obtainable by extending the basic SDOF formulae.

If the gap is exactly zero, the natural frequency is sqrt(K/M) where K is the stiffness of a single spring. The motion's period is thus
2*pi*sqrt(M/K)
and the only "trick" is the use of one spring rather than two, as pointed out by ivymike above.

Now envisage a gap (G). Let E be the total energy of the system. The parts of the motion during which the mass is in contact with either spring are unchanged. All that happens is that we have a time between contacts during which the mass traverses the gap at a speed of
sqrt(2*E/M)
taking time G/sqrt(2*E/M) to do so.

Each "cycle" includes two such traverses, so the period will be
2*pi*sqrt(M/K) + 2*G*sqrt(M/(2E))

 

[electricpete]

Interesting solution.

Also as Mike pointed out, the presence of E in the solution indicates the same system can be made to oscillate at many different fundamental frequencies depending on how much energy you put in to get it moving.

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[Rob45]

This is certainly an oscillatory system, but non-sinusoidal.

And if you need to "find out" the natural frequency, why, that's what accelerometers and such are for.