Damping Mass in Mountain Bike Suspension

Simulation and Visualization Damping Mass in Mountain Bike Suspension Mark Sleith

In this paper we discuss the problems posed by quantitative mathematical models of a physical system and their solution. The model in question is the design and control of the damping for the suspension of a mountain bike. The behaviour of such dynamic systems is best described using ordinary differential equations applying Laplace transform methods. We will discuss the springmass-damper system and observe its inputs and outputs in order to obtain relationships within its components and subsystems in the form of transfer functions. We will then demonstrate their behaviour using graphs and block diagrams for which we can graphically depict interconnections interconnections in a convenient way for designing and analysing control diagrams. We conclude by applying these methods to the real-life problem of the suspension of a mountain bike.

Damping Mass in a Mountain Bike Suspension Suspension

1. Contents 1.1 Table of Contents

1. Contents Conte nts ...................................................................... ........................................................................................................ ................................................................ ..............................i 1.1 Table of Contents Contents ........................................................................................... ..................................................................................................................... .......................... i 1.2 Table of Figures Figures ................................................................... .................................................................................................... .................................................. ................. iii 1.3 Table of Tables Tables .................................................. ..................................................................................... ..................................................................... .................................. iii 1.4 Table of Equation Equationss........................................................ .......................................................................................... .......................................................... ........................iii

2. A 2nd Order Solution to Damping Mass .............................................................................. 1 2.1 Introduc Introduction tion ................................................................... ...................................................................................................... .......................................................... ....................... 1 2.2 The Spring-Mass-Damper System............................................................................................ 2 2.3 The Laplace Laplace Transf Transform orm................................................. .................................................................................... ........................................................... ........................ 3 2.3.1

Definition Definition ..................................................................... ....................................................................................................... .............................................. ............ 3

2.3.2

Applying the Laplace Transfo Transforms rms to Solve Differential Equations Equations ............... ........ .............. ............... ........ 4

2.3.3

Solving Solving for Y(s) Y(s) .................................................................. .................................................................................................... ........................................ ...... 4

2.3.4

Using Using with a Specific Specific Case ............................................................................. ......................................................................................... ............ 5

2.3.5

Using Using the Look Look Up Table Table..................................................................... ............................................................................................ ....................... 5

2.3.6

Solution to Y(t) – Y(t) – Inverse Inverse Laplace Transform ............................................................. 6

2.4 The Damping Damping Ratio ................................................................................... ................................................................................................................. .............................. 7 2.4.1

Definition Definition ..................................................................... ....................................................................................................... .............................................. ............ 7

2.4.2

Derivation Derivation .................................................................... ....................................................................................................... .............................................. ........... 8

3. Block Diagram Diag ram Models Mode ls .................................................................... ...................................................................................................... ...................................... .... 10 3.1 Models and and Simulatio Simulation n ............................................................................... .......................................................................................................... ........................... 10 3.1.1

What are Models? Models? ........................................ .......................................................................... ............................................................. ........................... 10

3.1.2

What are Simulatio Simulations ns?? ......................................................................................... .............................................................................................. ..... 10

3.2 Modelling and Simulating with Simulink ............................................................................... 10 3.2.1

Creating Creating a Model with Simulin Simulink k .............................................................. ............................................................................... ................. 11

3.2.1.1 Express System System as First First Order Derivatives Derivatives ............... ....... ............... ............... .............. .............. ............... ............... ............. ..... 11 3.2.1.2 Add One Integrator Integrator per State, Label Label Inputs and and Outputs ............... ........ ............... ............... .............. .............. ....... 11

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3.2.1.3

Connect the Terms to Form the System ...................................................................... 12

3.2.1.4 Initial Conditions .............. ....... .............. ............... ............... .............. .............. ............... ............... .............. .............. ............... ............... .............. ......... 13

4. Applying it all to a Mountain Bike Simulation ................................................................. 14 4.1 Identifying Identifying the Problem Problem ................................................................... ..................................................................................................... ...................................... .... 14 4.2 Solving Solving the Problem Problem ............................................................................................. .............................................................................................................. ................. 15 4.3 Creating Creating a Model Simulatio Simulation n .............................................................. ................................................................................................. ................................... 16

5. Reference Refe rencess .................................................................. .................................................................................................... ............................................................. ........................... 21

ii


1.2 Table of Figures

Figure 2.1 – 2.1 – A A damped signal against its original ............................................................................... 1 Figure 2.2 – 2.2 – Sprin Spring-m g-mass ass-dam -damper per system system......................................................................... ........................................................................................... .................. 2 Figure 2.3 – 2.3 – Fre Free-bo e-body dy diagram diagram ............................................... ................................................................................. .......................................................... ........................ 2 Figure 2.4 – 2.4 – Grap Graph h of Equation Equation 2.21 ........................................ .......................................................................... .......................................................... ........................ 6 Figure 2.5 – 2.5 – Differ Different ent Damping Damping Values ...................................................................... ............................................................................................. ....................... 7 Figure 2.6 – 2.6 – Type Typess of Damping Damping ....................................................................................... ......................................................................................................... .................. 8 Figure 3.1 – 3.1 – Spring-mass-damper integrators ................................................................................... 12 Figure 3.2 – 3.2 – Second Second order using integrators ..................................................................................... 12 Figure 3.3 – 3.3 – A A Spring-mass-damping system model......................................................................... 12 Figure 3.4 – 3.4 – Trace Trace of a simple spring-mass-damper ......................................................................... 13 Figure 4.1 – 4.1 – Moun Mountain tain bike bike with parame parameters ters .................................................................... .................................................................................... ................ 14 Figure 4.2 – 4.2 – Sys System tem on road ...................................................................... ........................................................................................................ ...................................... .... 15 Figure 4.3 – 4.3 – Moun Mountain tain bike bike front front wheel wheel model model ................................................................. ................................................................................. ................ 16 Figure 4.4 – 4.4 – Outpu Outputt from from first first run .................................................................... .................................................................................................... ................................ 16 Figure 4.5 – 4.5 – Fron Frontt wheel wheel optimu optimum m solution solution ..................................... ...................................................................... ................................................. ................ 17 Figure 4.6 – 4.6 – Zoomed Zoomed in at peak of negative oscillation in optimum solution .............. ....... ............... ............... .............. ......... 18 Figure 4.7 – 4.7 – Com Complete plete Mountain Mountain Bike Model Model .................................................................. .................................................................................. ................ 19 1.3 Table of Tables

Table 2.1 - Important spring-mass-damping Laplace pairs ................................................................. 3 Table 3.1 - Simulink Simulink Blocks Blocks .............................................................. ................................................................................................... .............................................. ......... 11 1.4 Table of Equations

Equation Equation 2.1 ............................................................................................ .............................................................................................................................. .......................................... ........ 1 Equation Equation 2.2 ............................................................................................ .............................................................................................................................. .......................................... ........ 2 Equation Equation 2.3 ............................................................................................ .............................................................................................................................. .......................................... ........ 3 Equation Equation 2.4 ............................................................................................ .............................................................................................................................. .......................................... ........ 3 Equation Equation 2.5 ............................................................................................ .............................................................................................................................. .......................................... ........ 4 Equation Equation 2.6 ............................................................................................ .............................................................................................................................. .......................................... ........ 4 Equation Equation 2.7 ............................................................................................ .............................................................................................................................. .......................................... ........ 4 Equation Equation 2.8 ............................................................................................ .............................................................................................................................. .......................................... ........ 4 Equation Equation 2.9 ............................................................................................ .............................................................................................................................. .......................................... ........ 4 Equation Equation 2.10 ............................................................................................... .................................................................................................................................... ..................................... 4 Equation Equation 2.11 ............................................................................................... .................................................................................................................................... ..................................... 5 Equation Equation 2.12 ............................................................................................... .................................................................................................................................... ..................................... 5 Equation Equation 2.13 ............................................................................................... .................................................................................................................................... ..................................... 5 Equation Equation 2.14 ............................................................................................... .................................................................................................................................... ..................................... 5 Equation Equation 2.15 ............................................................................................... .................................................................................................................................... ..................................... 5 Equation Equation 2.16 ............................................................................................... .................................................................................................................................... ..................................... 5 Equation Equation 2.17 ............................................................................................... .................................................................................................................................... ..................................... 5 Equation Equation 2.18 ............................................................................................... .................................................................................................................................... ..................................... 5 Equation Equation 2.19 ............................................................................................... .................................................................................................................................... ..................................... 6 Equation Equation 2.20 ............................................................................................... .................................................................................................................................... ..................................... 6 Equation Equation 2.21 ............................................................................................... .................................................................................................................................... ..................................... 6

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Equation Equation 2.22 ............................................................................................... .................................................................................................................................... ..................................... 7 Equation Equation 2.23 ............................................................................................... .................................................................................................................................... ..................................... 7 Equation Equation 2.24 ............................................................................................... .................................................................................................................................... ..................................... 7 Equation Equation 2.25 ............................................................................................... .................................................................................................................................... ..................................... 8 Equation Equation 2.26 ............................................................................................... .................................................................................................................................... ..................................... 8 Equation Equation 2.27 ............................................................................................... .................................................................................................................................... ..................................... 8 Equation Equation 3.1 ............................................................................................ .............................................................................................................................. ........................................ ...... 11 Equation Equation 3.2 ............................................................................................ .............................................................................................................................. ........................................ ...... 11 Equation Equation 4.1 ............................................................................................ .............................................................................................................................. ........................................ ...... 15 Equation Equation 4.2 ............................................................................................ .............................................................................................................................. ........................................ ...... 15

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2. A 2nd Order Solution to Damping Mass 2.1 Introduction

In order to understand and control complex systems we must first reach quantitative mathematical models of these systems. It is therefore necessary for the relationships between the system variables to be analysed a nalysed and a mathematical mathematical model to t o be obtained. Due to the constantly changing nature of the t he system the equations to describe them are generally differential . If we are able to linearize a solution then then we can utilize the Laplace the Laplace transform to transform to simplify the method of solution. Due to the real life complexities of the systems that we will be investigating many assumptions should be made with regards to the system operation. For this reason we will consider the physical system and define the necessary necessary assumption in order to li nearize it. Then, we can obtain a set of l inear differential equations with the use of the physical physical laws describing the linear equivalent system. Finally, we will implement a Laplace transform which will give us a solution describing the operation of the physical system. We will apply this working method to get an understanding in the mechanisms of a real life system, a mountain bikes suspension. suspension. In summary, the approach for solving a dynamic systems problem is as follows: 1. 2. 3. 4. 5. 6.

Define the system system and its components Formulate the mathematical model and list the required assumptions Write the differential di fferential equations which which describe the model model Solve the equations for the desired d esired output variables Examine the solutions and the t he assumptions If necessary, reanalyse or redesign the system

Figure 2.1 – A A damped signal against its original

In physics In physics,, damping is the effect used to reduce the amplitude of oscillations in an oscillatory system as shown in Figure 2.1. The differential equations which describe the dynamic performance of a physical system system are obtained obtained by making use use of the physical physical laws of the process. process.



Equation 2.1

Newton’s Newton’s Second Second Law

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Mark Sleith

2.2 The Spring-Mass-Damper System

For our investigation we are interested int erested in the simple spring-mass-damper simple spring-mass-damper system system shown in Figure 2.2 as described by Newton’s second law of motion, motion , shown by Equation 2.1 on page 1 page 1.. This system will represent our shock absorbers within a mountain bikes suspension. A free body diagram of mass is shown in Figure 2.3. It should be noted however, that the t he knowledge one gains within the mechanical mechanical system, system, is i s equally applicable to electrical, electrical, fluid and thermodynamic thermodynamic systems. systems.



Figure 2.2 – Spring-mass-damper system

Figure 2.3 – Free-body Free-body diagram





In this spring-mass-damper example, the wall friction is modelled as a viscous damper ; meaning that the frictional force is linearly proportional to the velocity of the mass . In a more realistic example friction may behave more like dry friction. friction. Dry friction, also known as a coulomb damper, is a nonlinear function of the mass velocity and possesses a discontinuity around zero velocity. However, for our example a well-lubricated system, the viscous friction is appropriate. Summing the forces acting on differential equation:

  

Where

        

and making use of Newton’s second law yields the yields the second-order

is the mass applied to the spring,

ideal spring and is the t he friction constant. constant. nd

 

Equation 2.2



is Newton’s 2nd law, is the spring constant of the



Since is a 2 order differential equation with respect to position. It is clear t hat such a simple equation can be used for prediction i.e. to know . In general if we can write the equations of rate of change we often can solve the equation and make predictions.

2


2.3 The Laplace Transform 2.3.1

Definition

The ability to obtain linear approximations of physical systems allows the analyst to consider the use of the Laplace the Laplace transformation as mentioned in our introduction. introduction. The Laplace transform allows us to take a complex differential equation and turn it into easily solvable algebraic equations. Thus, allowing us to solve these complex systems with simple arithmetic. The time response solution is obtained as follows: 1. Obtain the differentia d ifferentiall equations 2. Obtain the Laplace t ransformation ransformation of t he differential equations 3. Solve the resulting algebraic transform of the variable of interest

    ∫   

Signals that are physically realizable will always have a Laplace transform. The Laplace transformation transformation for a function of time is: 

Equation 2.3

The inverse Laplace transform is written as:

     ∫  

Equation 2.4

The transformation integrals have been used to derive t ables of Laplace transforms transforms that are often used for the great majority of problems. A list of the Laplace transform pairs which relate to spring-massdamping systems systems can be found in Table 2.1. A more complete table goes beyond the scope of this paper but can be be found online. Time Domain

    ∫  

Laplace Domain

      

Table 2.1 - Important spring-mass-damping Laplace pairs

Notice how the dreadful maths maths becomes arithmetic. We can then use this with our spring-massdamper system described by Equation 2.2.

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Mark Sleith

2.3.2

Applying the Laplace Transforms to Solve Differential Equations

We will now demonstrate the usefulness of the Laplace transformation transformation and all of the s teps involved in the system analysis with respect to our spring-mass-damper system described by Equation 2.2 as shown on page 2. page 2.

                        ( (  ) ) 

We wish to obtain the response follows:

as a function of time. The Laplace transform of Equation Equation 2.2 is as

Equation 2.5

Laplace transform

Equation 2.6

when

   

,

and

  

and

 |   

,

we have

               .

Equation 2.7

Since we now have the Laplace transform of the differential equation where corresponds corresponds to t he 2 derivative, is a derivative and is the transform of the function that we are looking for.



2.3.3

nd

Solving for Y(s)

      

When we solve the solution for

we obtain

Equation 2.8

then

      We can then get a neater version by dividing the top and bottom by

            

Equation 2.9



This equation can now be used simply by plugging in the corresponding parameters.

4

Equation Equation 2.10 2 .10


2.3.4

Using with a Specific Case

Let us consider a specific case of the system when

  

  

, and

we make the assumption that

          i.e. at

then

,

Then Equation Then Equation 2.10 becom 2.10 becomes es


solving the quadratic in t he denominator denominator we get

     2.3.5


Using the Look Up Table

  

From a working knowledge of Laplace transforms we need to split into separate parts so that we can use the look up table (Table 2.1) on 2.1) on page 3, page 3, to to get the solution. We use partial use partial fractions so fractions so that we can get the following

        

by multiplying multiplying by

                    and


we get


by taking this a step step further. The fully fully expanded partial partial fraction of Equation 2.12, we 2.12, we obtain


by observation of the numerators numerators

let

 

             

let

 

  

  


  


i.e.

i.e.


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Mark Sleith

now

         2.3.6


Solution to Y(t) – Inverse Inverse Laplace Transform

The inverse Laplace transform of Equation Equation 2.15 is then

   {  }   {   } 




using the look up Table up Table 2.1 on page 3, page 3, we we find that

            


Thus we can conclude we have found a solution is a sum of exponentials with different decay constants. The choices of ratio to and to clearly indicate an underdamped system.

Figure 2.4 – Graph Graph of Equation Equation 2.21

6


2.4 The Damping Ratio 2.4.1

Definition

The damping ratio is a dimensionless measure describing how oscillations in a system decay after disturbance. Using our spring-mass-damper spring-mass-damper model, Figure model, Figure 2.2 on page 2, page 2, as as an example. If we were to pull down the t he mass and release it, i t, the spring would would cause the t he mass to bounce up and down as the system attempts to return to equilibrium. The damping ratio is a measure of describing how quickly the oscillations decay from one bounce to the next.

Figure 2.5 – Different Different Damping Values



The damping ratio is a parameter usually denoted by . This provides a mathematical means of expressing the level of damping in a system relative to critical damping as as demonstrated by by Figure 2.5. 2.5. For a damped system with mass , damping coefficient , and spring constant , it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient:





  




where the system differential equation is(notice the similarities with Equation with Equation 2.2)

        √ 


and the corres c orresponding ponding critical damping coefficient


being the ratio of two two coefficients of identical units, units, the damping damping ratio is dimensionless.

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Mark Sleith



The damping ratio is also related to the logarithmic decrement for underdamped vibrations via the relation

     

where


 

This relation is only meaningful for underdamped systems because the logarithmic decrement is defined as the natural log of the ratio of any two successive amplitudes, and only underdamped systems exhibit oscillation. 2.4.2

Derivation

Using the natural frequency of the simple harmonic harmonic oscillator:

   


and the definition of the damping ratio as given above, we can rewrite rewrite this as:

      


Figure 2.6 – Types Types of Damping

The value of the t he damping damping ratio determines determines the behaviour of the system. system. Figure 2.6 shows the different types of a damped harmonic oscillator which can be described as:

8










    

Overdamped (

): The system returns (exponentially decays) to equilibrium without oscillating. Larger values of the damping ratio return to equilibrium slower. ): The system returns to equilibrium as quickly as possible without Cr itical it ical ly dampe damped d ( oscillating. This is what we desire for a mountain bikes suspension. Underdamped ( ): The system oscillates (at reduced frequency compared to the undamped case) case) with the amplitude gradually decreasing to zero. Undamped (





): The system oscillates at its natural resonant frequency (

).

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Mark Sleith

3. Block Diagram Models 3.1 Models and Simulation

Understanding and testing the workings of a real life system would be tedious if we had to t o do the long complicated maths and plot our findings onto a chart for every time step of the running simulation. This would be made worse when we want to test and change different parameters to see what effects this has to variables within our system. Luckily for us we can use a block diagram model to help us when we are trying to represent a system in a simulation. 3.1.1

What are Models?

First we must ask ourselves ‘what is a model?’ A model is a representation of the construction and working of some system of interest. It is similar to but simpler than the system it represents. We will use our model for the purpose of allowing us to analyse and predict the effect of changes to the system. This means that our model will have to be dynamic and enable time-varying interactions among variables. 3.1.2

What are Simulations?

Now that we have a simple simple understanding of what what we mean by a model we now now have to ask ourselves ourselves ‘what do we mean by simulation?’ A simulation sim ulation of a system is the operation of a model of a system. The model can be reconfigured and experimented with. Usually this is impossible, too expensive or impractical to do so in the system it represents. The operation of t he model can be studied, and hence, properties concerning the behaviour behaviour of the actual system system or its subsystem can be inferred. inferred. In its broadest sense, simulation is a tool to evaluate performance performance of a system, system, existing or proposed, under different configurations of interest and over long periods of time. 3.2 Modelling and Simulating with Simulink

Now that we understand what is meant by models and simulation simulation we must decide on which software we will implement for our model simulation of the mass-spring-damper system. For this we have chosen Simulink. Simulink is an environment for multidomain simulation and Model-Based Design for dynamic and embedded systems. It provides an interactive graphical environment and a customizable customizable set of block libraries that let you design, simulate, simulate, implement, implement, and a nd test a variety of timevarying systems, including communications, communications, controls, signal processing, processing, video vid eo processing, processing, and image processing. processing. Table 3.1 shows the different blocks and their meaning from the Simulink library which we will be using to model our system. Block

Description

The Gain block multiplies the input by a constant value (gain). The input and the gain can each be a scalar, vector, or matrix. The Integrator block outputs the integral of its input at the current ti me step.

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The Sum block performs addition or subtraction on its inputs The Scope block displays signal inputs with respect to simulation simulation time. ti me. Table 3.1 - Simulink Blocks 3.2.1

Creating a Model with Simulink

To create the model we are going to follow three simple steps: 1. Rewrite the equation as a system of first order derivatives. 2. Add integrators to the model model labelling inputs and outputs. 3. Connect the terms of the equation to form the system. Since we are using t he mass-spring-dam mass-spring-damper per system let us again look at the t he equation,

The position of the mass is



̈  ̇   ̇ 

the velocity is

Equation 3.1

and the acceleration acceleration is

̈

3.2.1.1 E 3.2.1.1 E xpr ess ess Syste System m as F i r st Order Or der Der ivati ves ves

  ̇  ̇ ̈

  ̇  

To rewrite this as a system of first order derivatives, we want to substitute for and we can identify the two states as position and velocity . The equation becomes,

for . Then

Equation 3.2

and this is rewritten at two first derivatives,

̇  

, and

̇    

,

velocity and position are the states of our system. system. When thinki ng about ordinary differential equations in models, states are integrator blocks. 3.2.1.2 Add 3.2.1.2 Add One I ntegrator ntegrator per per State, tate, L abel abel I nputs and Outputs

It is be good practice to annotate our model with our equations so that we can refer to it as we add blocks to the canvas. canvas. Figure 3.1 shows the two integrator blocks for the mass-damping-system.

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Mark Sleith

Figure 3.1 – Spring-mass-damper Spring-mass-damper integrators

We draw signals from the ports and label inputs as the derivative variable.

̇    

and the output is the state

3.2.1.3 Con 3.2.1.3 Connec nectt the t he Terms to F or m the Syste System m

̇  

The first connection is simply so we connect the output of the velocity integrator to the input of the position int egrator. By aligning the integrators in t he model we can show that we have a second order system as shown in Figure in Figure 3.2.

Figure 3.2 – Second Second order using integrators

We can then implement the second equation by adding gains and sums to the model and linking up the terms as shown in Figure in Figure 3.3.

Figure 3.3 – A A Spring-mass-damping system model

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3.2.1.4 I 3.2.1.4 I niti al Conditions Conditions

 



In order to simulate, differential equations require initial conditions for each state. The initial states are set in the integrator blocks. These can be considered as initial values for and at time The simulation simulation computes the derivatives deri vatives at time zero using these initial conditions and then propagates propagates the system forward in time. We can see the initial conditions in the annotation box in Figure 3.3. Simulating the model for 50 seconds produces the results shown by Figure by Figure 3.4.

Figure 3.4 – Trace Trace of a simple spring-mass-damper

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Mark Sleith

4. Applying it all to a Mountain Bike Simulation

We now have a strong understanding of the spring-mass-damper system both mathematically and within model simulations. simulations. We can c an now apply what what we have learned l earned to solve a real life problem, in our case the spring mass damping system used for a mountain bikes suspension. 4.1 Identifying the Problem

A mountain bike requires suspension in order for the rider to have a more comfortable and safe ride on the rough terrain. The bikes suspension system would be made up of springs or pistons with compressed compressed air similar to Figure to Figure 2.2 on page 2. page 2. We We must investigate which values of the parameters in Equation 3.2, 3.2, when adjusted minimize the oscillations and the optimum values of and are selected for practical implementation.

 

Before we design our simulation we must first make assumptions about our bike model. We know that mountain bikes have two sets of suspension for both the front and back, we will assume that both the front and back suspension systems are the same with different values for the parameters. For the purpose of our simulation we will assume assume that t hat the mass mass is restricted to move only in the vertical direction and is connected to a fixed frame through a spring and a damper. We will assume that the spring is rigid and the spring and damper are massless. massless. We can also assume assume that weight distribution is 40:60 to the front and back respectively. The distance between wheel centres is 1m, tyre pressure and wheel sizes are both negligible. For our tests we will make the weight of the rider 80kg and have an average speed of 35km/h. See Figure See Figure 4.1.



Figure 4.1 – Mountain Mountain bike with parameters

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4.2 Solving the Problem

Figure 4.2 – System System on road

 

Displacement of and is brought about by the movement of the wheels on a bumpy road, see Figure 4.2. In 4.2. In order for the rider to travel safely and be in full control of the mountain bike the front wheel must return to equilibrium before the rear wheel reaches the point where the front wheel was disturbed.





Since we know the distance between between the front and rear wheels and we also know the average speed we can again use Newton’s 2 nd Law to predict how long it will take for the back wheel to get to the point in which the front wheel wheel was originally disturbed. This time however we we use,

  

Equation 4.1

  

Equation 4.2

by inserting the known known parameters we get,

  



Solving this we get (notice we have converted from km/h to m/s). We now know that our front mass damping system must return to equilibrium under 0.1s. With this in mind our system should be critically damped or or overdamped .

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Mark Sleith

4.3 Creating a Model Simulation

Developing on our model in Figure 3.3, 3.3, we will make a more complex system by allowing us to adjust the input and then store both the input and output to files to create more informative figures. We will use this to simulate the front wheel before moving on to the rear wheel, adjusting the parameters until unti l we find the most most optimum solution. Figure 4.3 shows our new model, the output boxes are are coloured green, these these store the variables variables to a file.

Figure 4.3 – Mountain Mountain bike front wheel model

Figure 4.4 – Output Output from first run

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Using the following initial parameters, parameters,

            ̇          

where is part of the damping ratio Equation ratio Equation 2.22 on page 7 page 7 and is the natural frequency. Plotting the inputs and outputs from file we get the output as shown shown Figure 4.4. We 4.4. We see that the simulation takes approximately 3seconds to return to equilibrium, not the solution that we are looking for, however it is better than the solution shown in Figure in Figure 3.4 which took almost 50seconds.

Figure 4.5 – Front Front wheel optimum solution

After much investigation we have found that the most most optimum opti mum parameters parameters are,



       ̇

Notice in Figure in Figure 4.5 that is now set to 1, it does not matter what the initial value is, it will still produce the same same solution in the same same amount amount of time. Furthermore Furthermore we have added onto the top graph which tells us the system’s speed. We say that this is the most optimum solution because it has returned to equilibrium at exactly 0.9s, because the system is in fact slightly underdamped we have a very slight oscillation where the system goes past equilibrium by 12.7% at its peak over a duration of 1.3seconds see Figure see Figure 4.6. In 4.6. In a real life scenario this would be acceptable on a mountain bike as the system rapidly returns to equilibrium with such a slight negative oscillation that it can be regarded as negligible by time the rear wheel reaches reaches the initial point of disturbance.

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Mark Sleith

Figure 4.6 – Zoomed Zoomed in at peak of negative oscillation in optimum solution

Now that we we have found a suitable solution for the front wheel we we must now add the rear wheel wheel to the model. Again we must make observations and assumptions about the system, since the wheels are coupled together by the bike frame, the observations and assumptions made for the front wheel apply equally to the rear. The only difference this time is that the mass is 60% on the rear suspension and it is not important for the system to return to equilibrium as quickly. Using the following values,

  We see the rear wheel returning to equilibrium a lot more smoothly and slightly later than the front wheel but because there is no other disturbance to the system the smoothness in which it returns is more important for the control and stability of the system. We see this as a full and complete result see Figure see Figure 4.7, we 4.7, we could use other values for the system which will allow for a smoother but slower return to its original state however we must assume that the road does not only have one bump.

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Figure 4.7 – Complete Complete Mountain Bike Model

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Mark Sleith

5. Conclusion

Within this paper we have shown how to solve complex systems using quantitative mathematical models. We have introduced the Laplace Laplace Transform Transform along with how it can be used to solve the Spring-Mass-Damper system of a mountain bike. We have shown that using complex math can be a tedious process and with modern software solutions can be found more easily by creating a model of the system within a simulation. Giving the example of the mountain bike problem, we have demonstrated how simulations can help inform and find a solution to the problem.

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6. References

http://mathworld.wolfram.com http://mathworld.wolfram.com/OverdampedSim /OverdampedSimpleHarmo pleHarmonicMotion.html nicMotion.html http://mathworld.wolfram.com http://mathworld.wolfram.com/LaplaceTrans /LaplaceTransform.html form.html http://www-math.mit.edu/daimp/ http://www-math.mit.edu/daimp/DampingRatio.html DampingRatio.html http://math.mit.edu/daimp/DampedVib.html Introduction to Modelling and Simulation, Simulation, Anu Maria http://www.machinehead-so http://www.machinehead-software.co.uk/bike/s ftware.co.uk/bike/speed_distance_ peed_distance_time_calc.h time_calc.html tml

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Damping Mass in Mountain Bike Suspension

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