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ENGG1811 Computing for Engineers
Questions:
Introduction
This assignment gives you an opportunity to work on a small-scale engineering design problem in python. The engineering system that you will be working on is a passive suspension, which is used in vehicles to reduce the amount of vertical vibration. A passive suspension is made of multiple components and the parameter of each component must be chosen correctly so that the passengers get a comfortable ride. The engineering design problem is to choose the right parameters and you will use python programming to solve this problem. The first step is to write a program to simulate the motion of a vehicle with suspension. You will then use the simulation result to evaluate the level of comfort for different choices of suspension parameters.
Learning Objectives
• Applying programming to solve a simple engineering design problem
• Writing a python program to simulate an engineering system
• Applying a number of python features, which include vectorisation, built-in functions and others
• Applying good software engineering practices, proper documentation, program style
Assignment Overview
This assignment is divided in 3 main tasks
• Task-1 and Task-2 (Simulation): Simulation of a vehicle with a passive suspension.
• Task-3 (Design) Evaluating the comfort level of different suspension designs, and choose the designs that give the most and least comfortable rides.
Note: The normal practice is to look at the best few designs and don’t care about the worst. As an exercise, we look at the best and the worst so that you can see the contrast.
In the following, we will first give an introduction to passive suspension and vehicle modelling. The introduction is meant to give you some intuition on the design problem. After that we will tell you what you need to do for each task.
Passive Suspensions
Passive suspensions are used to reduce the vibration experienced by passengers, but they can also be used to improve the tyre grip and other performance measures. For this assignment, we will only be concerned about vibration reduction or comfort. There are many designs for passive suspension. For this assignment, we consider a passive suspension consisting of a spring, a damper and an inerter in parallel, as depicted in Figure 1.
The classical method to reduce vibration is to use a spring and a damper where the damper is used to slow down the motion of the car body. A new method is to use a new mechanical device called an inerter [1]. An inerter can store kinetic energy temporarily. Intuitively, a shaking car has excessive kinetic energy in the vertical axis, so the inerter can absorb and store some of this energy temporarily. The energy in the inerter can be released later in an orderly manner without the passengers feeling sudden movements. The first ever deployment of inerter was in the 2005 Spanish Grand Prix by the MaLaren Formula 1 racing team, which also happened to have won that race. There are a few interesting tidbits surrounding the use of inerters, including McLaren invented the decoy name J-damper so that its rivals would not know that it was actually an inerter and a spy scandal, see [1] if you are interested.
The passive suspension has three parameters:
• Spring stiffness k. (A large k means a stiff spring which is hard to stretched.) (python variable k)
• Damping coefficient c. (A larger c means a larger resistance to movement.) (python variable c)
• Inertance b. (A large b means the interter can store more kinetic energy.) (python variable b)
The design problem is to choose values of k, c and b so that the ride is comfortable. A passive suspension can only be designed together with a vehicle, so we need to look at the vehicle model now.
The vehicle model
For this assignment, we will use a quarter car model for the vehicle. The quarter car model consists of one wheel/tyre and a quarter of the car body. It is commonly used to evaluate suspension designs at the initial stage. Of course, a full car model will be used for the final design but it is too complicated for this assignment.
Engineers very often have to make simplified models in order to derive mathematical models for real-life engineering systems. You will learn how to do this in later years. Figure 2 shows a simplified quarter car model.
The quarter car model consists of two lumps of masses. The mass ms (python variable name ms), which is on top, is the mass of 1/4 of the car body. (Note: The technical name is the sprung mass, hence the subscript s.) The lower mass mu (variable name mu) is the mass of the tyre and wheel. (Note: Technically this is the unsprung mass, hence subscript u.) The tyre is a bit elastic and is represented by a spring with stiffness kt (variable name kt). The passive suspension sits between the car body and the wheel/tyre.
There are three vertical displacements (see Figure 2) that we are interested in:
• yr is the height of the road surface from a reference level (variable name y_road)
• yu is the vertical displacement of the center of wheel/tyre from a reference level (variable name yu)
• ys is the vertical displacement of the center of the quarter car body from a reference level (variable name ys)
As the quarter car moves, yr, yu and ys change. That means these displacements are functions of time and we should write them as yr(t), yu(t) and ys(t). The corresponding python variables y_road, yu and ys are therefore arrays.
We also need two velocities for the quarter car model:
•Vu is the vertical velocity of the wheel/tyre (variable name vu)
•Vs is the vertical velocity of the quarter car body (variable name vs)
These velocities should be functions of time and we write them as vu(t) and vs(t). The corresponding python variables vu and vs are arrays.
The aim of the mathematical model for the quarter car is to determine ys(t), yu(t), vs(t) and vu(t). The mathematical model assumes the following are given:
• The height of the road yr(t) over time. We call this the road profile.
• The parameters: k, c and b (from the passive suspension) and ms, mu and kt (from the quarter car)
We have placed the mathematical model for the quarter car on a separate page. We believe it is best for you to understand what you need to do for this assignment first before dwelling into the mathematical model. You should be able to understand what you need to do for the assignment without going into the mathematical model at this stage. (The model is here and you can read it later.) We will now describe what you need to do for the three tasks.
Overview of Tasks
We have divided the work into a number of tasks.
Hint: You are strongly encouraged to read (or re-read!) the lecture notes, the code examples and the labs on numpy. In particular, look for numpy functions relevant to the following tasks. See the code examples on numpy from week-05 (click), week-07 (click) and week-08 (click) .
Task 1: Simulation of the Quarter Car
The aim of this task is to write a python function simulate_qc (which should be in a file with name simulate_qc.py) to simulate the quarter car. You can find a template for this function in simulate_qc_template.py (in assign2.zip). You should rename it as simulate_qc.py before you start. The declaration of the function simulate_qc is:
Outputs:
Ys the verticle displacement of the center of the car body from a reference level (array of floats)
Yu the verticle displacement of the center of the wheel/tyre from a reference level (array of floats)
Vs the verticle velocity of the quarter car body (array of floats)
Vuthe verticle velocity of the wheel/tyre (array of floats)
Task 2: A Function to Calculate Discomfort
The aim of Task 2 is to determine the discomfort level for a given set of suspension parameters. Intuitively, a comfortable ride means the passengers are not experiencing much vibration. We can express this quantitatively by calculating how much acceleration the car body experiences. The higher or longer the acceleration is, the more uncomfortable the ride is. (Note: An important part of using computers to perform engineering design is to express the design objective quantitatively. You will learn that in later years but this assignment will show you how to do that.)
Since the function simulate_qc gives us the velocity of the car body, we can use it to determine the acceleration and subsequently the discomfort level. For this task, you are asked to write a python function def calc_discomfort(vs , dt) The above function should be in a file calc_discomfort.py .
Task 3: Calculating Discomfort Level for Many Pairs
The function calcdiscomfort allows you to determine the discomfort level for each set of suspension parameters: spring stiffness k, damping coefficient c and inertance b. For simplicity, we will not change the value of k. We will calculate the discomfort level for many different pairs of (inertance,damping coefficient) or (b,c) values. def explore_qc(time_array, y_road, ms, mu, kt, k, inerter_array, damping_coefficient_array): The above function should be in a file explore_qc.py .
ENGG1811 Computing for Engineers
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