Periodic force applied to pendulum-like motion

kwas

When a force is applied to a pendulum, the pendulum sways back and forth until it eventually stops. In this problem however, a force is applied at uneven time intervals while the pendulum is still in motion. The force is always applied in the same direction.

A data set is given containing values for the position of the pendulum sampled every n milliseconds.

Image link:

http://www.tiikoni.com/tis/view/?id=f9efe29

The problem consists in recovering the exact times when the force is applied to the pendulum.

Simon Bridge

Welcome to PF;
This sounds like the "random kicked rotor" problem ... the applied force is an impulse (or of short duration compared with the natural period of the pendulum) right?
http://en.wikipedia.org/wiki/Kicked_rotator
... a couple of guys in the office next to mine were working on realizing this experimentally while I was doing a thesis on something less interesting.

So what do you think will characterize the point in time when the force strikes?

(What is the context of the problem? i.e. is it part of coursework?)

kwas

The force is applied for a fraction of a second. The interval is approximately every second. This is for a ballistocardiograph.

Simon Bridge

There you have it then.

kwas

There I have what? This system is not chaotic, it is a pendulum not a rotator, and I still need to find out the times in which force is applied.

Simon Bridge

Ideal pendulum - short duration force at random intervals ... sounded like a random kicked rotator to me.

So if it is not a rotator, then a particular kick (or combination) could make the string go slack... it will have vibration and spin components to it's equation of motion (due to non-rigid "string" and non-point mass.)

But since you are sure - then you should be able to figure out the regular motion without the force and look for the differences ... which leads you back to that "So what do you think will characterize the point in time when the force strikes?" from post #2.

And what is the context?
note: the machine does not give me the context - what do you need to know for? The answer tells me what sort of answer to give you ... i.e. you are probably not doing an exercize in chaos theory so what are you doing?

Seems you know a bit about the statistics of the force: roughly every second?
What else do you know about it?
Is it always the same?

kwas

There's a guy sleeping on a mattress. Under the mattress there is an air chamber connected to a pressure sensor whose signal is amplified. When the subject's heart beats the force creates an oscillatory motion due to the air compressibility and mattress elasticity. The next heart beat will interrupt this oscillatory motion. The beats aren't perfectly constant. They are also not necessarily identical in strength. But very similar.

Consider the possibility that the sinusoidal waveform created by one single oscillation does not have a fixed period, but slows down slightly as energy is dissipated (I do not know that this is so but it is a possibility). Also consider the possibility that the heart's force is actually two bursts in short succession; first a positive one then a negative one in the opposite direction and weaker.

I need to recover the time distance between heart beats.

kwas

too bizarre?

Simon Bridge

The heart beat is the driving force - you want to know the period of the driving force knowing the physics of the rest of the system?

OK - in that case you want the general harmonic motion equation ... f(x) will be something that approximates a heart-beat with the frequency parameter left variable.

kwas

Nope... that's not the solution... the motion does not approximate a heart beat because of elasticity of the layers between the person and the sensor, and the force can be applied either in phase or out of phase.

Simon Bridge

How do you know?That's what the damping part is for. Do you know the impulse response ?Doesn't matter - this is already discussed in the stuff about the randomly kicked rotor.

Note: you cannot solve the problem if you do not have any idea about the shape of the driving force.

Simon Bridge

I've checked with a couple of collegues just in case I'm missing something and none have come up with another approach. Sorry - it is either this or complete guesswork. Good luck.