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Chapter 3: Problem 31

A vehicle of mass \(m\), with suspension of stiffness \(k\) and negligible damping, is traveling at constant speed \(v_{0}\) when it encounters the series of equally spaced semi-circular speed bumps shown. Assuming that no slip occurs between the wheel and the road, determine (a) the motion of the engine block while it is traveling over this part of the roadway and \((b)\) the force transmitted to the block.

Short Answer

Expert verified
Answer: The motion of the engine block can be described as a function of time using the harmonic motion equation: y(t) = A * cos(ωt), where A is the amplitude of the motion and ω is the angular frequency. The force transmitted to the engine block is determined using Newton's second law, F(t) = m * a(t), where a(t) is the acceleration found by taking the second derivative of the displacement y(t).

Step by step solution

01

Determine the angular frequency ω

When the vehicle encounters the speed bumps, it oscillates back and forth over the bumps, creating a sinusoidal pattern in its motion. The angular frequency ω is the rate at which the vehicle oscillates. As the vehicle moves forward at a constant speed v0, the bump frequency f can be determined by dividing the speed v0 by the distance between the bumps L, such that f = v0/L. We can find the angular frequency ω using the equation ω = 2πf.
02

Determine the amplitude A of the motion

The amplitude A of the motion is the maximum displacement of the vehicle from its equilibrium position. This can be found using the equation A = R, where R is the radius of the semi-circular bumps. Since the vehicle moves over semi-circular speed bumps, the radius R is equal to half the distance between the peaks of the bumps.
03

Calculate the motion of the engine block (a)

To determine the motion of the engine block, we will analyze its vertical displacement y(t) as a function of time. For this, we will use the following equation for harmonic motion: y(t) = A * cos(ωt). Since we have already found the amplitude A and angular frequency ω in previous steps 1 and 2, we can substitute them into the equation to determine the motion of the engine block as a function of time.
04

Calculate the force transmitted to the engine block (b)

To find the force transmitted to the engine block, we will first determine the acceleration a(t) of the vehicle, which can be found using the second derivative of the displacement y(t) that we calculated in step 3. The acceleration equation is a(t) = -ω^2 * A * cos(ωt). Next, we will use Newton's second law F = ma to find the force transmitted to the engine block. By substituting the mass m and the acceleration a(t) into the equation, we can determine the force F(t). Following these steps will help you find the motion of the engine block and the force transmitted to it as the vehicle travels over the series of semi-circular speed bumps.

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