Question

An automobile is traveling at constant speed \(v_{0}\) along a flat road when it encounters a speed bump described by \(y(\xi)=-h_{0}[1-\cos (2 \pi \xi / \lambda)]\) where \(\lambda\) is the wavelength of the bump and \(h_{0}\) is the height. Determine the motion of the car as it traverses the bump. Assume that the frame of the car may be modeled as a uniform rod of mass \(m\) and length \(L\) and the combined stiffiness of the tires and suspension in both the front and the back is \(k\).

Short answer

Question: Describe in brief the steps involved in determining the motion of a car as it moves over a speed bump represented by the equation \(y(\xi)=-h_{0}[1-\cos(2 \pi \xi/\lambda)]\), where the car's frame is modeled as a uniform rod of mass \(m\) and length \(L\), and the combined stiffness of the tires and suspension is \(k\). Answer: To determine the motion of the car, we follow these steps: 1. Understand the bump function and its effect on the car, where the bump introduces a time-varying force onto the car's frame. 2. Set up the forces acting on the car, considering it as a point mass suspended on two springs with equal stiffness, and derive the spring forces from the bump function. 3. Write the equation of motion for the car using Newton's second law and the derived spring forces, and solve the equation to find the vertical motion of the car's center of mass. 4. Analyze the motion of the car to understand the response of the car to the bump, and gain insights into the car's suspension system and ride comfort.

Step-by-step solution

Understand the bump function and its effect on the car

The bump is described by the function \(y(\xi) = -h_{0}[1-\cos(2 \pi \xi/\lambda)]\), with \(\xi\) being the position along the road. This equation represents the shape of the bump, and the car moves over it with a constant speed \(v_0\). The bump causes the car's frame to oscillate vertically as it moves over the bump. This means that the bump introduces a time-varying force onto the car's frame.

Set up the forces acting on the car

To simplify the problem, we consider the car to be a point mass \(m\) suspended on two springs with equal stiffness \(k\), connected front and back. Let \(x_1\) and \(x_2\) be the vertical displacement of the front and back tires, respectively. The forces that act on the car due to the bump are spring forces, and can be described as: \(F_1 = k(x_1 - y(\xi_1))\) (front tire, connected at \(\xi = L/2\)) \(F_2 = k(x_2 - y(\xi_2))\) (back tire, connected at \(\xi = L/2 - L\)) Note that \(\xi_1 = v_0 t + L/2\), and \(\xi_2 = v_0 t - L/2\).

Write the equation of motion for the car and solve it

We apply Newton's second law to the car's vertical motion: \(m \ddot{x}_{c} = F_1 + F_2\), where \(x_c\) is the center of mass of the car. Using the forces derived in Step 2, and substituting in the values of the bump function, we get: \(m \ddot{x}_c = k(x_1 - (- h_0 [1 - \cos(\frac{2\pi}{\lambda}(v_0 t + \frac{L}{2})])) + k(x_2 - (- h_0 [1 - \cos(\frac{2\pi}{\lambda}(v_0 t - \frac{L}{2})]))\). Now, we need to solve this equation to find the motion of the car. This may require simplification and applying techniques such as Fourier analysis or numerical methods to find \(x_c(t)\).

Analyze the motion of the car

With the vertical motion of the car's center of mass, \(x_c(t)\), determined in Step 3, we can further analyze the response of the car to the bump, such as how oscillations induced by the bump propagate through the car, and how quickly the car settles back to its steady-state motion after traversing the speed bump. Insights from this analysis can help us better understand and optimize the performance of the car's suspension system and ride comfort.

Key concepts

Engineering Vibrations

When an automobile encounters a bump, it experiences engineering vibrations, which are oscillations induced by external forces specific to mechanical systems. These vibrations, like the ones initiated by the speed bump in the given exercise, provide a practical context where the principles of mechanical vibrations can be examined. In the case of our automobile, the tires and suspension system act as springs that absorb and dissipate energy, leading to vibrational motion.

Understanding and controlling these vibrations are crucial for the design and comfort of the vehicle’s ride. Engineers study these oscillatory phenomena to ensure safety and functionality, aiming to minimize undue vibrations while maintaining adequate responsiveness of the car’s suspension to road irregularities.

Equation of Motion

The equation of motion for our problem is derived from applying Newton's second law, which states that the force on an object is equal to the mass of the object multiplied by its acceleration. In our scenario, we simplify the model of the car to focus on the vertical forces and displacements: the spring forces from the suspension system versus the force introduced by the speed bump's profile.

Creating the equation of motion gives us a mathematical representation of how the car will behave as it traverses the bump. It encapsulates the dynamic interplay between the force exerted by the bump and the car’s inertial resistance, mediated by the car's mass and suspension. Solving this equation offers insights into the amplitude and frequency of vibrations and is essential for predicting the vehicle's response to road imperfections.

Harmonic Oscillator

The presented situation, where a car traverses a bump, can be modeled using the concept of a harmonic oscillator, which is a system that, when displaced from its rest position, experiences a restoring force proportional to the displacement. Our car's suspension system can be simplified to two springs with stiffness k, and the car itself acts like a mass m.

In a perfect scenario, this could result in simple harmonic motion — predictable, sinusoidal oscillations. However, real-world conditions may complicate the motion, necessitating a more thorough investigation into damping, which limits the oscillation over time, and factors like the car’s speed and bump profile that influence the motion’s characteristics.

Fourier Analysis

To solve the complete equation of motion derived from the problem, Fourier analysis may be employed. This mathematical technique decomposes complex periodic functions, like the force exerted by the speed bump, into a series of simple waves. It's a powerful tool because it allows us to understand the vibration characteristics in terms of individual frequency components.

In our automobile scenario, since the force has a sinusoidal dependency on position, Fourier analysis would reveal the fundamental frequencies contributing to the car's vertical oscillations as it crosses the bump. This information is instrumental for engineers to design suspension systems capable of attenuating unwanted vibrations, contributing to the car’s performance in terms of stability and comfort.