advanced engineering mathematic differential equations flexible ope with a lengt

Question

advanced engineering mathematic differential equations

flexible ope with a length Lthat haags them the ceilings undergravity.g Suppose that 2) Consider a the lower end of the ope is pulled to the side and then let zo. The rope will then surt to oscillate plicated Pattem which amounts uperposition of different mode iltapes. The equation of motion is govemed by where is the angular frequency. general solution the above equation terms of Bessel functions a) Derive b) If L 50 cm and the lower end of the rope is pulled to the side by cm while the upper end is fixed, find and plot the first three modes shapes of the oscillations. ms (g-9

Explanation / Answer

Let   f(t)f(t)   be a given function which is defined for   t0t0. If there exists a function   F(s)F(s) so that

F(s)=0estf(t)dtF(s)=0estf(t)dt,

then   F(s)F(s)   is called the Laplace Transform of   f(t)f(t), and will be denoted by   L{f(t)}L{f(t)}. Notice the integrator   estdtestdt   where   ss   is a parameter which may be real or complex.

Thus,

L{f(t)}=F(s)L{f(t)}=F(s)

The symbol   LL   which transform   f(t)f(t)   into   F(s)F(s)   is called the Laplace transform operator.

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