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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online