ENGG1811 Assignment 2: The Millennium Bridge

Version: v1.00, 14 July, 2022

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Introduction

This assignment is inspired by the Millennium Bridge in London, England. The Millennium Bridge is a suspension bridge for pedestrians only. Many visitors walked across the bridge in the first day of its opening in 2000. However, these visitors were treated to something unusual. They noticed that, when there were many people on the bridge, the bridge started to sway a lot. You can watch the wobbling of the bridge in this YouTube video. Since the bridge did not sway much if there were few people on the bridge, the authority decided to limit the number of people on the bridge. Eventually, the bridge was closed a few days after its opening. The bridge was re-opened after additional dampers have been put on the bridge, see [1] for a report on how the bridge was stabilised.

Engineers and scientists wanted to understand why the Millennium Bridge started to sway when there were many people on the bridge. You probably have learnt about resonance in your Physics class and you are right to guess that the wobbling had something to do with resonance. However, the baffling part is how pedestrians could have caused resonance to occur. One theory, which was published in [2], is that the pedestrians would synchronise their walking with that the bridge's motion and their synchronised movement caused the bridge to sway more. There are also other theories, such as [3], which disputes the theory proposed in [2].

The aim of this assignment is to give you an opportunity to work on a small-scale engineering problem in Python. The engineering system that you will be working on is the Millennium Bridge motion model from [2]. (Note that we are not claiming that [2] is the correct model. We are merely using the model in [2] in a computing exercise.) You will first develop a Python function to simulate the motion of the bridge due to the pedestrians' movement. After that, you will use the simulation program to try out different modifications that can reduce the amount of swaying. On the whole, this assignment relates computer science to two important aspects of engineering, which are simulation and design.

The key objective of this assignment is to for you to learn and use Python and numpy to solve problems. At the same time, we would like to give the problem an engineering context so that you can get some ideas how computing is used in engineering. Note that we used the word inspired earlier because, for this assignment, we took the liberty to simplify the model in [2] and the engineering design problem.

Learning objectives

This assignment is designed to give you practice in

Intuition behind the bridge and pedestrian motion modelling

We will give you a basic mental picture that you can use to visualise the bridge and pedestrian motions. We will use Figure 1 below, which is taken from [2], for our description.

Figure 1. Taken from [2]. 

The left side of Figure 1 depicts the Millennium bridge. We are particularly interested in the lateral motion (i.e. side-to-side motion or sway) of the bridge.

For the modelling of the bridge, we can think about the bridge as a wheeled cart with mass M, see the top-right of Figure 1. This wheeled cart can only move in one dimension along the X-axis and this motion is representative of the sway of the bridge. The sway of the bridge is restricted by both stiffness and damping. A material is stiff if a force can only extend the material a little. Engineers like to think about stiffness as a spring where a stiff spring is hard to extend. If we go back to the picture of the cart in Figure 1, the quantity K represents the stiffness of the bridge. A damper is used by engineers to resist or to slow down motion. A common example of a damper in day-to-day life is a door closer which slows down a door so it will not slam. In the picture of the cart in Figure 1, the quantity B denotes damping where a larger B means more resistance.

With correctly chosen stiffness and damping, the sway of a bridge will eventually die down. If we simulate the lateral motion of bridge without pedestrians (which can be done with the simulation program which you will develop), we obtain the lateral displacement and lateral velocity in Figure 2 and you can see that their magnitudes become smaller over time. 


Figure 2. Displacement and velocity of the bridge without pedestrians

Since we will only consider lateral motion in this assignment, we will drop the word lateral for brevity from now on.

You can see from Figure 2 that the motion of the bridge is oscillatory. Engineers and scientists like to think about an oscillatory motion as a point going round in a circle. You can see how the motion of a circle maps to a sine curve here. We know that we can specify the position of a point on a circle by using an angle, so we can use an angle to describe the oscillatory motion of the bridge. This angle is commonly referred to as the phase angle or simply phase (in the same sense of the word "phase" in the expression "moon phase"). The bottom-centre picture in Figure 1 depicts an angle using the Greek alphabet Ψ (Psi). This angle is used to describe the phase of the oscillatory motion of the bridge.

The walking of a pedestrian can also described as a cycle since walking is a repetition of: lifting of the left foot, landing of the left foot, lifting of the right foot, landing of the right foot, and so on. For each pedestrian, we can use a phase to describe their motion, see the angle Θ i (Theta) in the bottom-right picture in Figure 1. Note that each pedestrian on the bridge has his/her own phase, the subscript i in Θ i is used to indicate the pedestrian whose index is i. 

The theory in [2] is that the pedestrians synchronised their motion with that of the bridge. If we simulate the motion of the bridge with the pedestrians (which can be done with the simulation program which you will develop), we obtain the displacement and velocity in Figure 3 and you can see that their magnitudes grow over time. If you simulate for longer, the magnitude of oscillation will grow larger until it reaches a steady value.


Figure 3. Displacement and velocity of the bridge with pedestrians

The bottom plot in Figure 3 shows the sine of the pedestrians' phase. There are 20 pedestrians and there are in fact 20 lines in different coloured lines in the plot. You can see the different coloured lines near time 0, but afterwards, these lines overlap and you can only see one coloured line. This means the pedestrians synchronised their walking.

Having seen that the displacement and velocity of the bridge will become large, a part of this assignment for you to see what modifications are needed to stabilise the bridge.

The above mental picture should give you the intuition you need for this assignment. In order to do simulation, we need a mathematical model which we will discuss next.

A mathematical model for the bridge and pedestrians

From the bridge and pedestrian motion that we have discussed above, you know that we are interested in a few quantities:

The mathematical model tells us how x(t), v(t), Ψ(t)  and Θi(t) evolve over time.

The model has six model parameters. You have seen M, B, K and N before. There are two more: G and C. The table below summarises all the model parameters and their meaning. The Python programs will use the same notation for these model parameters.

Constants Meaning and their unit
M Mass [kg]
B Damping [kg/s]
K Stiffness [kg/s/s]
N Number of pedestrians
G The maximum force that a pedestrian exerts on the bridge [N]
C Larger C value means the pedestrians take shorter time to synchronise with the bridge [/m/s]

We have placed the mathematical model for the bridge and pedestrians on a separate page. We believe it is best for you to get to know the different parts of the assignment before dwelling into the mathematical model. You should be able to get a big-picture understanding on what you need to do for this assignment without going into the mathematical model at this stage. (The model is here and you can read it later.) The mathematical model is modified from [2]. If you would like to read [2], click on this which will take you to the reference and there are links to download the paper.

Overview of tasks 

We have divided the assignment into four tasks where each task corresponds to the writing of a Python function.

Task 1: Simulation  

The aim of this task is to write a Python function sim_bridge() to simulate the bridge and pedestrian motion. In the yellow box below, you can see some code for sim_bridge() which you can use to start your work.

import numpy as np

def sim_bridge(t_array,M,B,K,G,N,C,Omega_array,dis0,vel0,ped0):

    # BEGIN: Supplied code   ************************************
    # Time increment
    dt = t_array[1] - t_array[0]
   
    # Initialise dis_array, vel_array, ped_array
    dis_array = np.zeros_like(t_array)
    vel_array = np.zeros_like(t_array)
    ped_array = np.zeros((N,len(t_array)))
   
    # Initialise for index 0
    dis_array[0] = dis0
    vel_array[0] = vel0
    ped_array[:,0] = ped0
    # Compute Y_0 [from eq 4]
    Y_0 = np.sqrt(K/M)
    # END: Supplied code   ************************************
    # Your code to compute the entries in dis_array, vel_array, ped_array
    #
    # Hint: Should use arctan2() from the numpy library to calculate Psi(t)



    # BEGIN: Supplied code  ************************************
    # Return the array   
    return dis_array, vel_array, ped_array
    # END: Supplied code  ************************************

The function sim_bridge() takes on a number of inputs. We will provide you with the parameter values to use so the important thing for you is to understand what they are referring to. The meaning of M, B, K, N, G and C have already been explained earlier at here. The meaning of the other inputs are explained in the table below.

Python variable name Meaning
t_array A numpy array of regularly spaced points. They are the time instances in simulation.
Omega_array This is a numpy array with N entries. The entry Omega_array[i] is the value of Ωi in the mathematical model.
The quantity Ωi is related to the walking speed of the the pedestrian with index i.
We will explain how you can use Omega_array on the page where we describe the mathematical model. It is here and it is best that you read that later.
dis0 Initial displacement of the bridge. This is a scalar.
vel0 Initial velocity of the bridge. This is a scalar.
ped0 This is a numpy array with N entries. The entry ped0[i] is the initial phase of the pedestrian with index i.

The function sim_bridge() returns three outputs, see the second last line of the code above. All the three outputs are numpy arrays and their meanings are explained in the table below.

Python variable name Shape Purpose
dis_array
 The same as t_array To store the displacement of the bridge
dis_array[j] is the displacement at the time given by t_array[j]
vel_array The same as t_array To store the velocity of the bridge
vel_array[j] is the displacement at the time given by t_array[j]
ped_array A two-dimensional numpy array
The shape is N by the number of entries in t_array
 To store the phase of the pedestrians
ped_array[i,j] is the phase of pedestrian i at the time given by t_array[j]

Note that the template file has included code to create these three arrays with the specific shape mentioned above. Please do not change these lines of code. In addition, the template also includes lines of code which initialise these arrays for time 0. Again, please do not change these lines.

In order for you to complete sim_bridge(), what you need to do is to add the for-loop for simulation.

In the simulation, you will need to use arctangent (or inverse tangent) to compute the phase Ψ(t) of the bridge. We ask you to use the numpy function arctan2() for this purpose. More explanation is on this page.

(Testing and incremental development) You can test this function by using the test file test_sim_bridge.py. There are four test cases in this test file. We have developed these test cases so that you can develop your sim_bridge() incrementally. We will be explaining these test cases on a separate page because it requires some understanding of the mathematical model. Our suggestion is that you keep going with this document first to get an understanding of the whole assignment. After that, you can read the mathematical model and when you are ready to think about how to implement sim_bridge(), you can read how you can incrementally develop it on this page, which is also where the mathematical model is.

Task 2: A function to calculate the design objective

The function sim_bridge() allows you to compute the displacement and velocity from the bridge parameters. In the next task (Task 3), you will explore different designs by varying the values of stiffness and damping. In order for us to choose a design later on, we need a way to measure how good a design is. This measure is known as the design objective. The aim of this task (Task 2) is to develop the function comp_obj() whose aim is to compute the design objective.

The def line of comp_obj() is:

def comp_obj(dis_array,vel_array,M,K):

The names of the inputs have been chosen to reflect their roles. The function is expected to return one number as the design objective. We will use an example to explain how you compute the design objective.

(Example) Note that both dis_array and vel_array are expected to be 1-dimensional numpy arrays of the same shape. For this example, we assume:

We compute the following three numbers

square root of ( (d02 + (M / K) v02 ),      square root of ( d12 + (M / K) v12 ),        square root of ( d22 + (M / K) v22 )

where d02 denotes the square of d0 etc. The design objective is the maximum of these three numbers. In general, if there are H entries in each of dis_array and vel_array, you will be computing H numbers and finding the maximum of them.

As a numerical example, if

then you first compute

    square root of ( (-1.1)2 + (M / K) (-4.1)2 ),      square root of ( (2.1)2 + (M / K)  (-2.1)2 ) ,        square root of ( (3.1)2 + (M / K) (1.3)2 )    

Their numerical values are approximately 8.09, 4.61 and  4.01. The maximum is 8.09 which is the design objective. Your comp_obj() will need to return this number.

You can test this function by using test_comp_obj.py.

Note that it is possible to do all of the computation of this function with merely one line of code in numpy. The lectures in Week 8 will give you some inspiration.

Task 3: Calculating the design objective for many pairs of B and K

Assuming that you are able to modify both the damping B and stiffness K of the bridge, you want to compute the design objective for many pairs of B and K values. The aim of this task is to develop the function  run_different_designs()  whose template is:

def run_different_designs( t_array, M, G, N, C, Omega_array, dis0, vel0, ped0, B_min, B_max, K_min, K_max, B_num, K_num):


     return B_array, K_array, sway_array

There are many inputs to this function. The first group of inputs, which is typeset in blue, has been discussed before. The second group of inputs, which is typeset in maroon, is new. 

This function returns three numpy arrays, see the return line in the template code above.

In this task, you will use many different pairs of B and K values for simulation. For each pair of B and K, you will use sim_bridge() to obtain the displacement and velocity. (Note that you assume that the other parameters, M, G, N, ... remain the same.) Once the displacement and velocity have been obtained, you use them together with M and K to compute the design objective using comp_obj().

Note that the template file for run_different_designs() does not include default arguments. The last two parameters B_num and K_num can have default arguments. The default argument for B_num is 5, and that of K_num is 10. This is a piece of work that you need to complete for this task.

The steps inside the function run_different_designs() are:
  1. Create a numpy array B_array of equally spaced values. The first and last values of this array are given, respectively, by B_min and B_max. The number of entries is determined by B_num. For example, if, at the time when the function is called, the specified values of B_min,  B_max and  B_num  are, respectively, 11000.0, 27500.0 and 6, then B_array is expected to be array([11000., 14300., 17600., 20900., 24200., 27500.]). Note that the numbers in this array are equally spaced because the difference between any two consecutive entries is 3300. Note that if the value of B_num is not specified when the function is called, the default value of B_num, which is 5, should be used; in this case, we expect B_array to be array([11000., 15125., 19250., 23375., 27500.]).
  2. Create a numpy array K_array of equally spaced values. The first and last values of this array are given, respectively, by K_min and K_max. The number of entries is determined by K_num. Note that K_num has a default argument.
  3. Perform simulations for all possible pairs of B and K combinations that can be obtained from B_array and K_array. The number of possible B-K pairs is B_num times K_num. Let B_new and K_new be a particular pair of combination where B_new is an element in B_array and K_new is an element of K_array. For each pair of B_new and K_new, you need to do the following:

You can use the file test_run_different_designs_0.py, test_run_different_designs_1.py and test_run_different_designs_2.py to test your function. Note that for both test_run_different_designs_0.py, test_run_different_designs_1.py, the value of B_num by K_num are specified. For test_run_different_designs_2.py, both B_num by K_num are not specified so the default arguments are expected to be used.

Note that for all these three test files, a number of simulations will be run so the computation may take a bit of time. The files test_run_different_designs_0,  test_run_different_designs_1 and test_run_different_designs_2 run, respectively, 12, 24 and 50 different designs. If you need computing power, you can use a vlab computer. 

Hint: You will need to use nested-for for this function. An example of using nested-for can be found in the file nested_for.py in Week 5B. 

Hint: please read the examples in the file numpy_2d_examples.py


Remark: We want to remark that the design problem that we have posed above is not entirely realistic. It is hard to modify the stiffness of a bridge, see [1]. If stiffness and damping are modified, the other bridge parameters, such as the mass M, may need to be adjusted at the same time. We have taken the liberty to keep the problem simple.

Task 4: Choose the best design 

The aim of this task is to choose the best design from all those designs that you have simulated in Task 3. In this assignment, we define the best design as the pair of B and K that minimises the design objective. In this task you will develop the function find_best_design() whose template is:

def find_best_design(sway_array,B_array,K_array):
   

    return B_chosen, K_chosen

The inputs sway_array, B_array and K_array are the outputs of  run_different_designs(). The function is expected to return two outputs which are the chosen pair of B and K.

Let us use an example to illustrate the expected behaviour of find_best_design(). For this example, we assume that sway_array, B_array and K_array are given by:

sway_array = np.array( [ [3.1, 2.1, 1.1],
                                                 [4.1, 1.6, 2.4],
                                                 [2.2, 3.2, 3.6],
                                                 [1.5, 2.5, 3.5] ] )

 B_array = np.array([3.7, 4.7, 5.7, 6.7])

K_array = np.array([1.5, 1.8, 2.1])

Note that B_array has 4 entries, K_array has 3 entries, and from  run_different_designs() we expect that sway_array should have a shape of (4,3). Recall also that sway_array[i,j] is computed from B_array[i] and K_array[j], e.g. sway_array[2,1] , which has the value of 3.2 (highlighted in blue), is the design objective computed by using B_array[2] and K_array[1].

Since our goal is to find the design that minimises the design objective, we look for the smallest element in sway_array, which is sway_array[0,2] and has a value of 1.1 (highlighted in magenta). The best design therefore comes from a value of B equals to B_array[0] and a value of K equals to K_array[2]. You should assign B_array[0] to B_chosen  and  K_array[2] to K_chosen.

You can test find_best_design() by using the test file  test_find_best_design().

Note that you can use numpy to write this function without using any loops.

Remark: We have used a rather brute force method to determine the design parameters. You will learn better optimization methods in later years.

Putting all functions into action  

We have provided you with a Python program called run_all.py. In this file, you will use run_different_designs() to simulate a number of different designs and use find_best_design()  to choose the best design. After that, the file will use sim_bridge() to simulate the bridge, first with the original parameters and then with new parameters obtained from the design. If your programs work correctly, you should be able to see that the new design has stabilised the bridge.

This is not part of the assessment but we think it is good for you to see how you can put everything together.

Getting Started

  1. Download the zip file assign2_prelim.zip, and unzip it. This will create the directory (folder) named 'assign2_prelim'.
  2. Rename/move the directory (folder) you just created named 'assign2_prelim' to 'assign2'. The name is different to avoid possibly overwriting your work if you were to download the 'assign2_prelim.zip' file again later.
  3. First browse through all the files provided including the test files.
  4. (Incremental development) Do not try to implement too much at once, just one function at a time and test that it is working before moving on.
  5. Start implementing the first function, properly test it using the given testing file, and once you are happy, move on to the the second function, and so on.
  6. Please do not use 'print' or 'input' statements. We won't be able to assess your program properly if you do. Remember, all the required values are part of the parameters, and your function needs to return the required answer. Do not 'print' your answers.

The zip file assign2_prelim.zip contains altogether 23 files. The assumption is that all these 23 files must be in same directory. Here is an overview of what the files are:

Testing

Test your functions thoroughly before submission.

You can use the provided Python programs (files like test_sim_bridge.py etc.) to test your functions. Please note that each file covers a limited number of test cases. We have purposely not included all the cases because we want you to think about how you should be testing your code. You are welcome to use the forum to discuss additional tests that you should use to test your code.

The test files will calculate the difference between the expected results and your results. The test file will inform you the maximum absolute difference. The difference should be less than 1e-6.

We will test each of your files independently. Let us give you an example. Let us assume we are testing three files: prog_a.py, prog_b.py and prog_c.py. These files contain one function each and they are: prog_a(), prog_b() and prog_c(). Let us say prog_b() calls prog_a(); and prog_c() calls both prog_b() and prog_a(). We will test your files as follows:

Submission

You need to submit the following four files. Do not submit any other files. For example, you do not need to submit your modified test files.

Instructions on how to submit the above files will be posted here later.

Assessment Criteria

We will test your program thoroughly and objectively. This assignment will be marked out of 23 where 19 marks are for correctness and 4 marks are for style.

Correctness

The 19 marks for correctness are awarded according to these criteria.

Criteria Nominal marks
Function sim_bridge() (Case 0: G = 0, C = 0, non-zero initial conditions. Need to get dis_array and vel_array correct) 2
Function sim_bridge() (Case 1: nonzero G. C = 0. Need to get dis_array, vel_array and ped_array correct) 3
Function sim_bridge() (Case 2: nonzero G and C. Need to get dis_array, vel_array and ped_array correct) 4
Function comp_obj() 2
Function run_different_designs() (Case 0: All arguments are specified. No default arguments will be used.)
4
Function run_different_designs() (Case 1: Testing default arguments.) 1
Function find_best_design() 3

(Assessment and incremental development) We told you earlier that the function sim_bridge() can be developed incrementally, which are Step 0, Step 1 and Step 2 mentioned here. Each step corresponds to a more complicated simulation model with Step 0 being the simplest and Step 2 being the complete model. We want to let you know that there is a one-to-one correspondence between the incremental development of sim_bridge() and the assessment of sim_bridge(). The assessment of sim_bridge() is divided into 3 cases, see the table above. Case 0 in the assessment corresponds to Step 0 of the incremental development, and so on. You need to know.

The important message is this: Just in case you find doing the simulation model difficult, you can get a partially completed simulation model going first and then move on to work on the other functions first. Do not think that you need to complete the entire simulation model before working on the other functions.

Style

  Four (4) marks are awarded by your tutor for style and complexity of your solution. The style assessment includes the following, in no particular order:

Assignment Originality

You are reminded that work submitted for assessment must be your own. It's OK to discuss approaches to solutions with other students, and to get help from tutors, but you must write the Python code yourself. Sophisticated software is used to identify submissions that are unreasonably similar, and marks will be reduced or removed in such cases.

Tutor Assessment

You need to properly explain your submission to a tutor. Your ability to properly explain your answers/solutions will determine your final grade for this assignment.

Assignment Conditions

Violation of any of the above conditions may result in an academic integrity investigation, with possible penalties up to and including a mark of 0 in ENGG1811, and exclusion from future studies at UNSW. For more information, read the UNSW Student Code, or contact the course account.

Further Information

Reference

[1] T. Fitzpatrick and R. R. Smith. Stabilising the London Millennium Bridge. Ingenia online pdf at (https://www.ingenia.org.uk/getattachment/Ingenia/Issue-9/Stabilising-the-London-Millennium-Bridge/Fitzpatrick.pdf)

[2] Strogatz, S. H., Abrams, D. M., McRobie, A., Eckhardt, B. & Ott, E. 2005 Crowd synchrony on the Millennium Bridge. Nature 438. (doi:10.1038/43843a). Links to the paper and its supplementary information.

[3] J. H. G. MacDonald. Lateral excitation of bridges by balancing pedestrians. Proc. of Royal Society A. (2009)