Question

Please help me with the attached homework assignment. In problem #5, I am asked to solve for the response with time. Could someone please explain what that means? Thank you

Answer 1

babyslapmyfro VIBRATION ANALYSIS
The pdf file looks very interesting.
Here's a guess.
If you leave something completely alone (were that possible), then it presumably would "age slowly".
When something is some sort of mechanical component, then it will probably be subjected to stresses and the associated strains over some sort of "duty cycle". Maybe a Formula 1 car, and its engine is a reasonable example. I believe that the engines for said cars are designed and built to, basically, last the course of a grand prix. Thereafter, they are discarded, or some such thing.
How, then would you put an F1 engine, or any other mechanical component through any sort of testing to see if it will "last the course", but in a laboratory/workshop not out on the track.
Perhaps one way is to subject it to the stresses and strains which it will experience in its duty cycle and "see how it behaves". Perhaps to mimic the demands of the duty cycle, and repeat the cycle, some form of "repeated stimulus" consisting of, say, belting it with a hammer - a pulsed input - and observing the amplitude (young's modulus type of thing) frequency (whippiness type of thing) of the subsequent vibrations and their time to die away (decay time constant) would be a start.
Then, if you repeated that test - once the vibrations from the first bang have decayed, belt it again, and, say, compare the graphs of the two decaying vibrations. (Much, in some ways, like "pinging" a piece of nice glass to hear the sound of "good quality glass").
Repeated belting, and recording of the frequency, amplitude and decay of the resulting vibrations over a large number of cycles - possibly to failure - would allow some sort of "fingerprint" or "signature" to be built up of the behaviour of the component under test.
So with the unused component, you'd get probably the "shelf life" out of it, with the component that has been tested, you would have probably put it through thousands of duty cycles, and so build up a picture of such things as the Mean Time Between Failures and other measures which are used.
The vague bits of what I've said (yup, probably most of it) become less vague if you analyse the component, and try to get an estimate of what you think should happen. Problem there could be that your analysis is an approximation and, well, you could be wrong. I suspect that there are standards published - in the UK there are "British Standards" also known as BS followed by a number.

I don't have the specialist knowledge to answer it in detail. Mind you, I wish I had.

Answer 2

They want you to get an equation in terms of \(\theta = \theta (t)\)

so it's a *complicated* pendulum, ie there is some damping bottom left

Answer 3

Personally, I would start at b) and get some kinda FBD

i would also google for a parallel damper/spring arrangement that could just be plugged into the usual vertical pendulum

not sure what ID is but that and the assumptions would come after I had solved the DE.

Answer 4

Yup, I can see the damping spring dashpot. But for me this is a "wood for the trees" question. Grinding through endless versions of dashpots and other dampers and oscillators etc etc etc may be good exam fodder, but the real point about the post to me is "what happens when the duty cycle tested limits are exceeded", especially, say, at 130mph, or when you're in orbit round the earth ? I also forgot to post the ideas of thermal cycling, and resonance as in singer breaks glass resonance, or even MILLENIUM BRIDGE resonance !!!!

Answer 5

yeah @osprey

i'm suggesting finding the standard bit for a parallel spring damper and just bolting it onto a pendulum thingy

and i'd stuff the DE into a computer too, because ...... well, why not?!?!?! why the hell not?!?!