피어슨- 기계진동학(engineering vibration) 2판 솔루션입니다.

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피어슨-최신기계진동학 2판 솔루션입니다.
engineering vibration

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1.4 The amplitude of vibration of an undamped system is measured to be 1 mm. The phase
shift from t = 0 is measured to be 2 rad and the frequency is found to be 5 rad/s.
Calculate the initial conditions that caused this vibration to occur.
Solution:
Given: A = 1mm, φ = 2rad,ω = 5rad/s
For an undamped system:
x t A t t
v t x t A t t
n
n n
( ) = ( + ) = ( + )
( ) = ( ) = ( + ) = ( + )
sin sin
˙ cos cos
ω φ
ω ω φ
1 5 2
5 5 2
and
Setting t = 0 in these expressions yields:
x(0) = 1sin(2) = 0.9093 mm
v(0) = 5 cos(2) = - 2.081 mm/s
1.5 An undamped system vibrates with a frequency of 10 Hz and amplitude 1 mm. Calculate
the maximum amplitude of the system`s velocity and acceleration.
Solution:
Given: First convert Hertz to rad/s: ω π π π n n= 2 f = 2 (10) = 20 rad/s. We also have that
A= 1 mm.
For an undamped system:
x(t)= A (ω t +φ ) n sin
and differentiating yields the velocity: v t A n nt ( ) = ω cos(ω +φ ). Realizing that both the
sin and cos functions have maximum values of 1 yields:
= ω = 1(20π )= 62.8mm/s max n v A
Likewise for the acceleration: a(t)= −Aω (ω t +φ ) n n 2 sin
= = ( ) = 3948mm/s 2

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