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 spring-mass-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 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

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1. Contents

1.1 Table of Contents

1. Contents ...................................................................................................................................... i

1.1 Table of Contents ..................................................................................................................... i

1.2 Table of Figures ..................................................................................................................... iii

1.3 Table of Tables ....................................................................................................................... iii

1.4 Table of Equations .................................................................................................................. iii

2. A 2nd Order Solution to Damping Mass .............................................................................. 1

2.1 Introduction ............................................................................................................................. 1

2.2 The Spring-Mass-Damper System............................................................................................ 2

2.3 The Laplace Transform ............................................................................................................ 3

2.3.1 Definition ................................................................................................................... 3

2.3.2 Applying the Laplace Transforms to Solve Differential Equations .............................. 4

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

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

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

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

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

2.4.1 Definition ................................................................................................................... 7

2.4.2 Derivation .................................................................................................................. 8

3. Block Diagram Models .......................................................................................................... 10

3.1 Models and Simulation .......................................................................................................... 10

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

3.1.2 What are Simulations? .............................................................................................. 10

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

3.2.1 Creating a Model with Simulink ............................................................................... 11

3.2.1.1 Express System as First Order Derivatives ................................................................ 11

3.2.1.2 Add One Integrator per State, Label Inputs 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 the Problem ......................................................................................................... 14

4.2 Solving the Problem .............................................................................................................. 15

4.3 Creating a Model Simulation ................................................................................................. 16

5. References ............................................................................................................................... 21


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1.2 Table of Figures

Figure 2.1 A damped signal against its original ............................................................................... 1

Figure 2.2 Spring-mass-damper system ........................................................................................... 2

Figure 2.3 Free-body diagram ......................................................................................................... 2

Figure 2.4 Graph of Equation 2.21 .................................................................................................. 6

Figure 2.5 Different Damping Values ............................................................................................. 7

Figure 2.6 Types of Damping ......................................................................................................... 8

Figure 3.1 Spring-mass-damper integrators ................................................................................... 12

Figure 3.2 Second order using integrators ..................................................................................... 12

Figure 3.3 A Spring-mass-damping system model......................................................................... 12

Figure 3.4 Trace of a simple spring-mass-damper ......................................................................... 13

Figure 4.1 Mountain bike with parameters .................................................................................... 14

Figure 4.2 System on road ............................................................................................................ 15

Figure 4.3 Mountain bike front wheel model ................................................................................. 16

Figure 4.4 Output from first run .................................................................................................... 16

Figure 4.5 Front wheel optimum solution ...................................................................................... 17

Figure 4.6 Zoomed in at peak of negative oscillation in optimum solution ..................................... 18

Figure 4.7 Complete Mountain Bike Model .................................................................................. 19

1.3 Table of Tables

Table 2.1 - Important spring-mass-damping Laplace pairs ................................................................. 3

Table 3.1 - Simulink Blocks ............................................................................................................ 11

1.4 Table of Equations

Equation 2.1 ...................................................................................................................................... 1

Equation 2.2 ...................................................................................................................................... 2

Equation 2.3 ...................................................................................................................................... 3

Equation 2.4 ...................................................................................................................................... 3

Equation 2.5 ...................................................................................................................................... 4

Equation 2.6 ...................................................................................................................................... 4

Equation 2.7 ...................................................................................................................................... 4

Equation 2.8 ...................................................................................................................................... 4

Equation 2.9 ...................................................................................................................................... 4

Equation 2.10 .................................................................................................................................... 4

Equation 2.11 .................................................................................................................................... 5

Equation 2.12 .................................................................................................................................... 5

Equation 2.13 .................................................................................................................................... 5

Equation 2.14 .................................................................................................................................... 5

Equation 2.15 .................................................................................................................................... 5

Equation 2.16 .................................................................................................................................... 5

Equation 2.17 .................................................................................................................................... 5

Equation 2.18 .................................................................................................................................... 5

Equation 2.19 .................................................................................................................................... 6

Equation 2.20 .................................................................................................................................... 6

Equation 2.21 .................................................................................................................................... 6

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Equation 2.22 .................................................................................................................................... 7

Equation 2.23 .................................................................................................................................... 7

Equation 2.24 .................................................................................................................................... 7

Equation 2.25 .................................................................................................................................... 8

Equation 2.26 .................................................................................................................................... 8

Equation 2.27 .................................................................................................................................... 8

Equation 3.1 .................................................................................................................................... 11

Equation 3.2 .................................................................................................................................... 11

Equation 4.1 .................................................................................................................................... 15

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 and a mathematical model to be obtained. Due to the constantly changing nature of the

system the equations to describe them are generally differential. If we are able to linearize a solution

then then we can utilize the Laplace 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 assumption in order to linearize it. Then, we can obtain a set of linear differential equations

with the use of the 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.

In summary, the approach for solving a dynamic systems problem is as follows:

1. Define the system and its components

2. Formulate the mathematical model and list the required assumptions

3. Write the differential equations which describe the model

4. Solve the equations for the desired output variables

5. Examine the solutions and the assumptions

6. If necessary, reanalyse or redesign the system

Figure 2.1 A damped signal against its original

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 are obtained by making use of the physical laws of the process.

Equation 2.1 Newtons Second Law

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2.2 The Spring-Mass-Damper System

For our investigation we are interested in the simple spring-mass-damper system shown in Figure 2.2

as described by Newtons second law of motion, shown by Equation 2.1 on 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 knowledge one gains within the mechanical

system, is equally applicable to electrical, fluid and thermodynamic systems.

Figure 2.2 Spring-mass-damper system Figure 2.3 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. 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 and making use of Newtons second law yields the second-order

differential equation:

( )

( )

( ) ( ) Equation 2.2

Where is the mass applied to the spring, ( )

is Newtons 2nd law, is the spring constant of the

ideal spring and is the friction constant. Since is a 2nd order differential equation with respect to position. It is clear that 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.


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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 transformation as mentioned in our 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 differential equations 2. Obtain the Laplace transformation of the 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 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 tables of Laplace transforms that are often used

for the great majority of problems. A list of the Laplace transform pairs which relate to spring-mass-damping systems can be found in Table 2.1. A more complete table goes beyond the scope of this paper but can be found online.

Time Domain Laplace Domain

( ) ( )

( )

( )

( )

( )

Table 2.1 - Important spring-mass-damping Laplace pairs

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

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2.3.2 Applying the Laplace Transforms to Solve Differential Equations

We will now demonstrate the usefulness of the Laplace transformation and all of the steps involved in

the system analysis with respect to our spring-mass-damper system described by Equation 2.2 as

shown on page 2.

We wish to obtain the response as a function of time. The Laplace transform of Equation 2.2 is as

follows:

( )

( )

( ) ( ) 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 to the 2nd

derivative, is a derivative and ( ) is the transform of the function that we are looking for.

2.3.3 Solving for Y(s)

When we solve the solution for ( ) we obtain

( )( ) ( ) Equation 2.8

then

( ) ( )

( ) Equation 2.9

We can then get a neater version by dividing the top and bottom by

( ) (

)

Equation 2.10

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


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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 2.10 becomes

( )

Equation 2.11

solving the quadratic in the denominator we get

( )

( )( ) Equation 2.12

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 page 3, to get the solution. We use partial fractions so that we

can get the following

( )

Equation 2.13

by multiplying by ( ) and ( ) we get

( ) ( )

( )( )

( )


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

( )


by observation of the numerators

( ) ( ) ( ) Equation 2.16

let

( ) i.e. Equation 2.17

let

i.e. Equation 2.18

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now

( )

Equation 2.19

2.3.6 Solution to Y(t) Inverse Laplace Transform

The inverse Laplace transform of Equation 2.15 is then

( ) {

} {

} Equation 2.20

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

( ) Equation 2.21

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 of Equation 2.21


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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 model, Figure 2.2 on page 2, as an example. If we were to

pull down the mass and release it, the spring would cause the 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 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 demonstrated by Figure 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:

Equation 2.22

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

Equation 2.23

and the corresponding critical damping coefficient

Equation 2.24 being the ratio of two coefficients of identical units, the damping ratio is dimensionless.

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

( ) Equation 2.25

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 oscillator:

Equation 2.26

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

Equation 2.27

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


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Overdamped ( ): The system returns (exponentially decays) to equilibrium without

oscillating. Larger values of the damping ratio return to equilibrium slower.

Critically damped ( ): The system returns to equilibrium as quickly as possible without

oscillating. This is what we desire for a mountain bikes suspension.

Underdamped ( ): The system oscillates (at reduced frequency compared to the

undamped case) with the amplitude gradually decreasing to zero.

Undamped ( ): The system oscillates at its natural resonant frequency ( ).

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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 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 understanding of what we mean by a model we now have to ask ourselves

what do we mean by simulation? A simulation 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 the model can be studied, and hence,

properties concerning the behaviour of the actual system or its subsystem can be inferred. In its

broadest sense, simulation is a tool to evaluate performance of a 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 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 set of block libraries that let you design, simulate, implement, and test a variety of time-

varying systems, including communications, controls, signal processing, video processing, and image

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 time 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 time.

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 labelling inputs and outputs.

3. Connect the terms of the equation to form the system.

Since we are using the mass-spring-damper system let us again look at the equation,

Equation 3.1

The position of the mass is the velocity is (

) and the acceleration is (

)

3.2.1.1 Express System as First Order Derivatives

To rewrite this as a system of first order derivatives, we want to substitute for and for . Then

we can identify the two states as position ( ) and velocity ( ). The equation becomes,

Equation 3.2

and this is rewritten at two first derivatives,

, and

( ),

velocity and position are the states of our system. When thinking about ordinary differential equations

in models, states are integrator blocks.

3.2.1.2 Add One Integrator per State, Label Inputs 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. Figure 3.1 shows the two integrator blocks for the mass-damping-system.

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Figure 3.1 Spring-mass-damper integrators

We draw signals from the ports and label inputs as the derivative ( ) and the output is the state

variable.

3.2.1.3 Connect the Terms to Form the System

The first connection is simply so we connect the output of the velocity integrator to the input

of the position integrator. By aligning the integrators in the model we can show that we have a second

order system as shown in Figure 3.2.

Figure 3.2 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 3.3.

Figure 3.3 A Spring-mass-damping system model


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3.2.1.4 Initial 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 computes the derivatives at time zero using these initial conditions and then 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 3.4.

Figure 3.4 Trace of a simple spring-mass-damper

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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. We can now apply what we have learned 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 air similar to Figure 2.2 on page 2. We must investigate which values of the parameters in

Equation 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 that the 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. We can also 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 4.1.

Figure 4.1 Mountain bike with parameters


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4.2 Solving the Problem

Figure 4.2 System on road

Displacement of and is brought about by the movement of the wheels on a bumpy road, see

Figure 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 the front and rear wheels and we also know the average

speed we can again use Newtons 2nd Law to predict how long it will take for the back wheel to get

to the point in which the front wheel was originally disturbed. This time however we use,

Equation 4.1

by inserting the known parameters we get,

Equation 4.2

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 overdamped.

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4.3 Creating a Model Simulation

Developing on our model in Figure 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 we find the most optimum solution. Figure 4.3 shows our new model, the output

boxes are coloured green, these store the variables to a file.

Figure 4.3 Mountain bike front wheel model

Figure 4.4 Output from first run


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Using the following initial parameters,

where is part of the damping ratio Equation 2.22 on page 7 and is the natural frequency. Plotting

the inputs and outputs from file we get the output as shown Figure 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 3.4 which took almost 50seconds.

Figure 4.5 Front wheel optimum solution

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

Notice 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 solution in the same amount of time. Furthermore we have added onto the top

graph which tells us the systems 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 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 the initial point of disturbance.

Mark Sleith

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

Now that we have found a suitable solution for the front wheel we must now add the rear 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 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 Mountain Bike Model

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5. Conclusion

Within this paper we have shown how to solve complex systems using quantitative mathematical

models. We have introduced the Laplace 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/OverdampedSimpleHarmonicMotion.html

http://mathworld.wolfram.com/LaplaceTransform.html

http://www-math.mit.edu/daimp/DampingRatio.html

http://math.mit.edu/daimp/DampedVib.html

Introduction to Modelling and Simulation, Anu Maria

http://www.machinehead-software.co.uk/bike/speed_distance_time_calc.html

damping mass in mountain bike suspension

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