Question

Seismometer Project Imagine that we have a delicate instrument carefully mounted on a low-friction surface, and held in place laterally by springs. Delicate Instrument Very low friction Mounting frame fixed to Earth Since the mounting frame is fixed to the Earth, seismic movements such as earthquakes are our input into the system! The delicate instrument of mass 'm', on the other hand, wants to stay fixed relative to inertial space. Let's assume that the inertial position of the delicate instrument is z as depicted in the figure. Our output which we can easily measure with lasers range-finding (or even the old-fashioned pen on trace paper) is ‘y', the distance from the instrument's center of mass to some point fixed on the frame. Since the mounting frame is fixed to the Earth, the mounting frame vibrates at some typical angular frequency of seismic movements say w=1 rads、The amplitude is obviously a function of the size of the semic movement. If you assume the vibration of the mounting frame is denoted by x(t), the output y is easily given as y yo+ (z-x) where yO is some equilibrium distance. The instrument is clearly a seismometer. Thus, we would like y-yO to describe the motion of the earth as closely as possible. The instrument itself needs to be carefully shielded from all seismic movements-in which case we would like the instrument to stay fixed with respect to inertial space so that y-y0 will describe the motion of the earth as closely as possible. Again, since yp)-z-x and x is the motion of the Earth we would like the magnitude of z to be as small as possible. This way if we measure y-y0 using range finders or the old fashioned pencil and paper we can assume that y-y0 is approximately-x There is always a small amount of friction such that Ffriction·-ce(zdot-xdot) where 'c' is the friction coefficient and is very small. Note that dry friction is more likely to be present however the problem becomes way more complex. Still though let's assume that c-t(m), let's say ㎡g.mu where mus some friction coefficient that is very small. Note that in this scenario if m is increased so does c. So, using only the topics covered so far in class-our knowledge of the differential equation of a 2 order system and how we can analyze it using the Laplace transform and Newtonian Physics- derive the transfer function describing y-y0 in response to the movement of the Earth. Examining the transfer function, what value of k and m will give us a value of y-y0 that is as close as possible to tracking the motion of the Earth x(t)? Does k need to be really big, really small or some relationship to m? Does m need to be really big? Really small? Try different values of k and m to determine what sort of trends you find when changing m and k.

Answer 1

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