Concept explainers
The suspension of an automobile can be approximated by the simplified spring-and-dashpot system shown. (a) Write the differential equation defining the vertical displacement of the mass m when the system moves at a speed v over a road with a sinusoidal cross-section of amplitude δm and wavelength L. (b) Derive an expression for the amplitude of the vertical displacement of the mass m.
Fig. P19.151
(a)
Write the differential equation defining the vertical displacement of the mass m when the system moves at a speed v over a road with a sinusoidal cross section of amplitude
Answer to Problem 19.151P
The differential equation defining the vertical displacement of the mass m when the system moves at a speed v over a road with a sinusoidal cross section of amplitude
Explanation of Solution
Calculation:
Show the free body diagram of the system of automobile, spring and dashpot as in Figure (1).
The expression for the weight of the automobile (W) as follows:
Here,
The expression for the acceleration of the automobile (a) as follows:
Refer Figure (1),
The expression for the force by considering the vertical equilibrium condition as follows;
Substitute
Substitute
The expression for the time interval needed to travel
The expression for the forced circular frequency
Substitute
The expression for the motion of the wheel which is sine curve
Differentiate the above equation with respect to time ‘t’.
Substitute
Substitute
Therefore, the differential equation defining the vertical displacement of the mass m when the system moves at a speed v over a road with a sinusoidal cross section of amplitude
(b)
Derive an expression for the amplitude of the vertical displacement of the mass m.
Answer to Problem 19.151P
The expression for the amplitude of the vertical displacement of the mass m is
Explanation of Solution
Calculation:
The expression for the general solution from the identity as follows:
Here,
The expression for the force transmitted (F) to the automobile as follows:
Substitute
The expression for the differential equation of the motion for the damped forced vibration as follows:
Compare the equation (3) and (4).
The expression for the steady state of motion of the system as follows:
The expression for the steady state of motion of the system as follows:
Substitute
The expression for the phase angle
The expression for the Eulerian angle
Therefore, the expression for the amplitude of the vertical displacement of the mass m is
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Chapter 19 Solutions
<LCPO> VECTOR MECH,STAT+DYNAMICS
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