Research Paper

Atar DERJ and Wiam SAMIR 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 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.

Friday, December, 13th 2013Supervised by: Dr: Lahcen Laayouni[Type text] [Type text] [Type text]

Damping Mass in Mountain Bike Suspension

1. Int rod uc tio n

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.

Dumping Mass in Mountain Bike Suspension 2

2. Mathematical perspective

2.1 A 2nd Order Solution to Damping Mass

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

1. Define the system and its components2. Formulate the mathematical model and list the required assumptions3. Write the differential equations which describe the model4. Solve the equations for the desired output variables5. Examine the solutions and the assumptions6. 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 aphysical system are obtained by making use of the physical laws of the process.

F=maNewton’s Second Law

Equation 2.1

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2.2 The Spring-Mass-Dampe System

For our investigation we are interested in the simple spring-mass-damper system shown in Figure 2.2 as described by Newton’s 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 b is linearly proportional to the velocity of the mass M . 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 veloc ity. However, for our example a well-lubricated system, the viscous friction is appropriate.

Summing the forces acting on M and making use of Newton’s second law yields the second-order differential equation:

M d ² y (t)dt ²

+b dy ( t )dt

+ky (t )=r (t ) Equation 2.2

Where M is the mass applied to the spring, dy (t )

dt is Newton’s 2nd law, k is the spring constant

of the ideal spring and b is the friction constant. Since F=ma 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 x(t). In general if we can write the equations of rate of change we often can solve the equation and make predictions


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 equations2. Obtain the Laplace transformation of the differential equations3. 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 f(t) is:

F ( s)=∫0

∞

f ( t)e−st dt=L{f ( t )} Equation 2.3

The inverse Laplace transform is written as:

f ( t )= 12 π j ∫

σ− j ∞

σ+ j ∞

F ( s) e+st ds 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.

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.


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

Equation 2.5

Laplace transform

Equation 2.6When

We have Equation 2.7

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


2.3.5 Using the Look Up TableFrom a working knowledge of Laplace transforms we need to split Y(s) into separate parts

so that we can use the look up table (Table 2.1), to get the solution. We use partial fractions

so that we

can get the following

by multiplying by (s+1) and (s+2) we get

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

by observation of the numerators

now


2.3.6 Solution to Y(t) – Inverse Laplace Transform

Figure 2.4 – Graph of Equation 2.21


3 . A pp ly i n g i t a l l t o a M o u nt a i n Bi k e S i m u la t i o n

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.

3.1 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.

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 m 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 is40:60 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 3.1.

Figure 3.1 – Mountain bike with parameters


3.2 Solving the Problem

Figure 3.2 – System on road

Displacement of Y1 and Y2 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 d between the front and rear wheels and we also know

the average speed v we can again use Newton’s 2nd Law to predict how long it will take for the back wheel to getto the point in which the front wheel was originally disturbed. This time however we use,


4 . C o n cl u s i o n

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.


6 . R e f e r e n ce s

http://mathworld.wolfram.com/OverdampedSimpleHarmonicMotion.html http://www.math.mit.edu/daimp/DampingRatio.html http://math.mit.edu/daimp/DampedVib.html http://www.machinehead-software.co.uk/bike/speed_distance_time_calc.html http://mathworld.wolfram.com/LaplaceTransform.html


http://www.math.mit.edu/daimp/DampingRatio.htm

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