1) The suspension system of a 1800 kg automobile \"sags\" 9.6 cm when the chassi
ID: 1489639 • Letter: 1
Question
1)
The suspension system of a 1800 kg automobile "sags" 9.6 cm when the chassis is placed on it. Also, the oscillation amplitude decreases by 46% each cycle. Estimate the values of (a) the spring constant k and (b) the damping constant b for the spring and shock absorber system of one wheel, assuming each wheel supports 450 kg.
2)
The function
x = (1.5 m) cos[(5rad/s)t + /4 rad]
gives the simple harmonic motion of a body. At t = 4.6 s, what are the (a) displacement, (b) velocity, (c) acceleration, and (d) phase of the motion? Also, what are the (e) frequency and (f) period of the motion?
Explanation / Answer
1) solution
a)F = kx {spring resistance force}
F = M*g =450 *10 = 4500
x = spring deformation = 0.096 m
k = F/x = 4500/0.096 = 46875 N/m
b) damping ratio = 0.46
damping ratio = b/2mk {where b = damping coefficient
b = (0.46)(2)mk = 0.92(450)(46875) = 0.92 * 4592 = 4225 kg/s
2)
x = (1.5 m) cos[(5 rad/s)t + /4 rad].
(a) displacement at t = 4.6 sec
= x at 4.6 s
= (1.5) cos[5 * 4.6 + /4 rad] m
= 1.5 cos [ (23.25) rad] m
= 1.5 * (-0.732) = -1.098
Answer(a): The displacement is 1.098 m in negative x direction.
b) Comparing with the standard equation of S.H.M [ x = A cos(t + ) ],
we get: A(amplitude) = 1.5, = 5, (initial phase) = /4.
Velocity(v) = dx/dt = (d/dt)A cos(t + )
= A (-sin( t + )) Using chain rule, which I hope you know.
Putting values,
v = 1.5*5* (- sin( 5(4.6) + /6 )m/s
= 23.55* 0.680
= 16.014 m/s.
Answer(b): The velocity is 16.014 m/s in positive x direction.
c) Acceleration(a) = dv/dt = d²x/dt²
= (d/dt)( A (-sin( t + ))
= -A²cos(t + )
= x(-²).
Putting values,
a = 1.098 * (5)² m/s²
= 270.64 m/s².
Answer(c): The acceleration is 270.64 m/s² in positive x direction.
d) Phase of the motion is = t + = 5.45 rad = 73.005 rad (Putting values).
Answer(d): The phase of the motion is 73.005 radian.
e) Frequency(f) = / 2 [T ^ (-1)]
= 5/2 Hz
= 2.5 Hz.
Answer(e): The frequency of the motion is 2.5 Hz.
f) Time period(T) = f ^ (-1) = 0.4 second.
Answer(f): The time period of the motion is 0.4 second.
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