Question

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

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

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.

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.

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.

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.

Key concepts

Angular Frequency

In vehicle dynamics, understanding angular frequency is essential when analyzing oscillations, such as the ones caused by traversing speed bumps. Angular frequency, represented as \(\omega\), describes how quickly an object oscillates in harmonic motion. It is related to the concept of frequency \(f\), which is the number of oscillations per second.

• To find the angular frequency, use the formula \(\omega = 2\pi f\).
• In the context of a vehicle, \(f\) can be determined by dividing the vehicle's speed \(v_{0}\) by the distance \(L\) between bumps.
• The result is \(\omega = 2\pi \left(\frac{v_{0}}{L}\right)\), showing how vehicle speed and bump spacing influence oscillation rate.

Angular frequency is crucial for predicting how quickly a vehicle's suspension system will react to bumps, thereby affecting ride comfort and vehicle control.

Harmonic Motion

A vehicle's motion over speed bumps can be modeled as harmonic motion, characterized by oscillations similar to those of a pendulum or a spring. In this type of motion, the vehicle's position changes over time, often following a sinusoidal pattern. The key elements include

• Amplitude \(A\): This is the maximum displacement from the equilibrium position, determined by the bump radius, \(R\). Thus, \(A = R\).
• Displacement Equation: The vertical displacement \(y(t)\) of the vehicle can be described by \(y(t) = A \cos(\omega t)\).

This equation indicates how the vehicle moves up and down as it travels over each bump. Understanding harmonic motion helps quantify the impact forces and motion responses of the suspension system, contributing to safer and more comfortable rides.

Newton's Second Law

Newton's Second Law connects the concepts of force, mass, and acceleration in dynamic systems. For a vehicle moving over bumps, the equation \(F = ma\) becomes particularly useful to calculate the force transmitted to a part of the vehicle, such as the engine block.

• First, find acceleration \(a(t)\) by taking the second derivative of the displacement equation: \(a(t) = -\omega^2 A \cos(\omega t)\).
• Then apply Newton's Second Law using the known mass \(m\) of the vehicle's engine block to calculate the force: \(F(t) = m \cdot a(t)\).

This relationship between force and motion is essential for designing effective vehicle suspension systems that minimize the transmission of bump-induced forces to sensitive vehicle components.

Suspension Stiffness

Suspension stiffness, denoted by \(k\), is a fundamental aspect of vehicle dynamics, indicating how resistant a suspension system is to compression. The stiffness largely determines how a vehicle responds to uneven road surfaces, like speed bumps.

• Stiffer suspensions can better support heavy loads and reduce body roll during maneuvers.
• However, they tend to transmit more road shock to the vehicle occupants, leading to harsher ride quality over bumps.
• On the other hand, softer suspensions provide a smoother ride but may compromise vehicle handling and stability.

Understanding suspension stiffness is key to achieving a balance between ride comfort and vehicle performance, making it a pivotal consideration in vehicle design.