A natural frequency of child's swing is determined by its height (if  neglect inertia etc). So a swing with constant height will have the only natural frequency. Change of forced frequency around the resonant one will mean removal from the resonance conditions.

Dear dr. Lazimov,

first let me thank you very much for your answer. Free oscillations of both, the electrical RLC series circuit and the swing, are at their natural frequency. However, forced electrical oscillations can also be induced with reduced amplitude at lower as well as at higher frequencies. They are phase shifted according to phase diagrams. My question is if such out of resonance behavior can be induced in a swing, too. If it can, it will be out of phase, of course.

My question arose from reading on pag. 4 of the following interesting article:

http://staff.kfupm.edu.sa/PHYS/tahmed/How%20to%20pump%20a%20swing.pdf

A swing is an example of a parametric resonance. A child must bend her knees twice per period of oscillations. Read about it deeper.

A pendulum is better analogy to the electric oscillations.  Small oscillations of the pendulum are isochronic, which means their frequency does not depend on amplitude. For the arbitrary oscillations the frequency does depend on amplitude. There is an analytic solution of for the pendulum which describes anything in it including the mode of rotation. 

The  top is a problem of  axial solid body rotation in the gravitational field with one point fixed. It has the analytic solution. I don'k know if anybody bothered to consider what happens if that one fixed point is lost and it turns upside down. 

Dear Sara,

In a children's swing, the excitation is caused by the kicking off, e.g. by the mother of the child. The time point of the kick is important. If this is not done exactly at the moment of the reversal, a phase shift occurs first. In this case, a resonance appears to me to be difficult because the exciter (the mother) does not initially possess a natural frequency.

Otherwise the law of the thread pendulum can be used.

Regards

Michael Lersow

Dear dr. Lersow,

thank you for your kind answer. The reason for my question was to see if it is possible to reduce the motion due to heart beat on MRI images. Nowadays the correction for motion is via synchronization of acquisitions with the heart beat, for example.

We put a metronome on a swing. With the swinging metronome the case is that of a child swinging without the help of its mother. We found that, although the pull due to the metronome is quite weak, it was enough for the swing to start oscillation. The resonance peak of the swing is quite sharp, indeed. Thus, it is correct that a child hardly could come up with the idea to swing out-of-resonance. Yet, with the metronome, it was possible to deviate a couple of Hz from resonance on either side, and to observe onset of oscillation in counter-phase (which is less stable). In my opinion, this behavior is analogous to that of electrical oscillators, and the counter-phase oscillations would in principle counterbalance heart motion.

Dear Sara,

Now, first of all, my answer was that there was an energy input on the child pendulum from the outside. The case you describe is that the child gives a self-impulse within the system (bend and stretching the legs). For this case, there is a resonance (child`s swing for larger children). I am not an expert but from the sport I know, that the child then experiences both a heart rate increase and an increase in the blood pressure. To simulate this, two types of people have to are considered:

A) which are essentially limited by the heart rate and

B) which are essentially limited by the blood pressure.

If I understood you correctly, then you wanted to prove with your attempt, that a displacement of inner momentum of the child pendulum by the oscillator is effected, and that the child pendulum behave analogous to an electric oscillator.

I think I have saw already something like that. If I find it, I'll send it to you.

Regards

Michael Lersow

I would say that the child's swing movement, rhythmically bending and stretching his legs but without someone else assistance, can be very roughly approached by that of a simple pendulum with varying length. That because the child alternately changes his center of mass up and down, what imparts parametric excitation to the mentioned modelling system. The governing equation for this system can be found at: Ali Hasan Nayfeh, Dean T. Mook, "Nonlinear Oscillations", John Wiley & Sons, 1979, p. 351.