(a) Without any mathematical detail, use your physical intuition to deduce the normal frequency for symmetric-stretch vibrations. (b) Show that the equations of motion of the three masses are: d?x, + wžx, – wix2 = 0, dt2 d?x2 wžx1 + 2wžx2 – wžx3 = 0 dt² d²x; and - wix2 + w}x3 = 0, where wi = and wž = k/M dt² (c) Assume x1 = A coswt, x2 = B cos wt, and x3 = C coswt. Show that the normal frequencies of the system are vk/m and vk(2m + M)/Mm
(a) Without any mathematical detail, use your physical intuition to deduce the normal frequency for symmetric-stretch vibrations. (b) Show that the equations of motion of the three masses are: d?x, + wžx, – wix2 = 0, dt2 d?x2 wžx1 + 2wžx2 – wžx3 = 0 dt² d²x; and - wix2 + w}x3 = 0, where wi = and wž = k/M dt² (c) Assume x1 = A coswt, x2 = B cos wt, and x3 = C coswt. Show that the normal frequencies of the system are vk/m and vk(2m + M)/Mm
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Transcribed Image Text:The below figure shows two identical masses of mass m connected to a third mass of mass M by two
identical springs of spring constant k. Consider vibrations of the masses along the line joining their
centres where x1, x2, and x3 are their respective displacements from equilibrium
M
m
m
(a) Without any mathematical detail, use your physical intuition to deduce the normal frequency for
symmetric-stretch vibrations.
(b) Show that the equations of motion of the three masses are:
d²x1
+ wix1 – wix2 = 0,
dt²
d²x2
wžx1 + 2wžx2 – wžx3 = 0
dt2
d²x3
and
wix2 + w}x3 = 0, where wi = and wž = k/M
dt2
m
(c) Assume x1 = A coswt, x2 = B cos wt, and x3 = C coswt. Show that the normal frequencies of the
system are vk/m and vk(2m + M)/Mm
(d) Determine the ratio of normal frequencies for m/M = 16/12.
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