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[(5πrad/s)t + π/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?

Answer 1

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/2√mk {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[(5π rad/s)t + π/4 rad].

(a) displacement at t = 4.6 sec

= x at 4.6 s

= (1.5) cos[5π * 4.6 + π/4 rad] m

= 1.5 cos [ (23.25)π 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(ωt + φ) ],

we get: A(amplitude) = 1.5, ω = 5π, φ(initial phase) = π/4.

Velocity(v) = dx/dt = (d/dt)A cos(ωt + φ)

= Aω (-sin( ωt + φ )) Using chain rule, which I hope you know.

Putting values,

v = 1.5*5π* (- sin( 5π(4.6) + π/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)( Aω (-sin( ωt + φ ))

= -Aω²cos(ωt + φ)

= x(-ω²).

Putting values,

a = 1.098 * (5π)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 = ωt + φ = 5.45π rad = 73.005 rad (Putting values).

Answer(d): The phase of the motion is 73.005 radian.

e) Frequency(f) = ω / 2π [T ^ (-1)]

= 5π/2π 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.