285lect21 - 1 1.1 Lecture 21 Free undamped motion mx00 kx =...

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1 Lecture 21 1.1 Free undamped motion mx 00 + kx = 0 : Set k m = ! 2 0 . Then we have x 00 + ! 2 0 x = 0 : Characteristic equation: r 2 + ! 2 0 = 0 ) r = & ! 0 i . General solution: x ( t ) = A cos ! 0 t + B sin ! 0 t: We can rewrite the foregoing formula as follows: x ( t ) = A cos ! 0 t + B sin ! 0 t = p A 2 + B 2 & A p A 2 + B 2 cos ! 0 t + B p A 2 + B 2 sin ! 0 t ± : Because & A p A 2 + B 2 ± 2 + & B p A 2 + B 2 ± 2 = 1 ; there is & such that cos & = A p A 2 + B 2 and sin & = B p A 2 + B 2 : So, we have x ( t ) = p A 2 + B 2 (cos & cos ! 0 t + sin & sin ! 0 t ) = p A 2 + B 2 cos ( ! 0 t ± & ) : Set C = p A 2 + B 2 : x ( t ) = C cos ( ! 0 t ± & ) = C cos ! 0 & t ± & ! 0 ± 1
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is called simple C ! 0 & 2 ^ / g 0 J / g 0 x 00 + ! 2 0 x = 0 : x ( t ) = C cos ! 0 & t & & ! 0 ± Let T denote the time (in seconds) needed to perform one cycle. Complete cycle corresponds to the angle 2 ± radians. So, T = 2 ± ! 0 is the period of vibration in seconds. 1 T = ! 0 2 ± & frequency of vibration in Hertz=cycle/second. Example 1 Determine the period and frequency of the simple harmonic mo- tion of a body of mass 0.75 kg on the end of a spring with spring constant k = 48 n=m . Neglect friction.
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285lect21 - 1 1.1 Lecture 21 Free undamped motion mx00 kx =...

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