30

CHAPTER 3

QUARTER AIRCRAFT MODELING

3.1 GENERAL

In this chapter, the quarter aircraft model is developed and the

dynamic equations are derived. The quarter aircraft model is two degrees of

freedom model to deal with a single landing gear. The most commonly and

widely used model for representing any vehicle suspension system is a quarter

model. A quarter model is a simple model with two translational degree of

freedom, which can depict the basic principle involved in a ride problem. This

model essentially consists of a proper representation of controlling the

aircraft-body and landing gear variations during landing impact and taxing on

the surfaces with undulations. The advantage of using this model is that it

allows in controlling or modifying the landing gear parameters in a simple

manner, since it does not take into account any complex dynamics. However,

this model contains no representation of the geometric effects of an aircraft

and hence the effects of longitudinal and lateral interconnections cannot be

studied. Initially in this chapter, two system models representing the passive

and active landing gear system were modeled with the runway bump input for

parametric analysis of landing gear and for numerical simulations to compare

the dynamic response of passive and active landing gear.

Passive control of vibration is relatively simple, reliable, robust and

economical but it has its limitations. The control force generated in the

passive device, depends entirely on the natural dynamics. Once the device is

designed, after choosing the values of mass, stiffness coefficient, damping

coefficient, location, it is not possible to adjust the control forces that are

31

naturally generated in real time. Further, there is no supply of power from an

external source. Hence even the magnitude of the control forces cannot be

changed from their natural values. Then PID controller was designed and

implemented in the system model individually and, series of the simulation

runs were carried out for the system for different runway excitations. The

results which were obtained were then analyzed and a comparative study was

done to compare the system responses of the passive system with the active

system using PID controller. During a series of simulations, the effectiveness

of the controllers were validated and the controllers optimum tuning values

are obtained.

3.2 MODEL FORMULATION OF PASSIVE LANDING GEAR

Passive landing gear system consists of upper and lower chambers.

These two chambers are connected by an orifice. The upper chamber is filled

with pressurized nitrogen or air and the remaining upper and lower volume is

filled with hydraulic oil. The oil flow is regulated in the orifice area by

metering pin. This absorber produces spring and damping characteristics.

During the aircraft landing, the shock strut experiences compression and

extension. This motion forces the oil to pass through the orifice which

dissipates the large amount of energy created by landing impact. The oil flows

from the lower to upper chamber, compressing the nitrogen that stores the

remaining impact energy. This stored energy extends the shock strut and the

oil flows from the upper chamber thus dissipating the residual impact energy.

This compression and extension oscillation continues until all landing impact

energy dissipates. The schematic diagram of passive landing gear is as shown

in Figure 3.1.

33

Figure 3.2 Mathematical model of passive landing gear

3.2.1 Linear Model Assumptions

1. All motions are in the vertical direction.

2. Vehicle seat cushioning is neglected.

3. Small motions.

4. Gravity ignored (measurement from equilibrium position).

5. Ground contact is maintained.

6. Rigid suspension linkages and vehicle body.

7. Damping in the tire.

8. Linear tire behavior.

9. No friction in suspension members.

10. Linear springs and dampers.

34

3.2.2 Dynamic Equation of Motion

By Newton’s second law of force equilibrium method the dynamic

equations of motion are derived by using the free body diagram concept of

sprung mass and unsprung mass .The second order differential equation

describing dynamics of the passive landing gear is written by using free body

diagram.

Figure 3.3 Free body diagram of sprung mass

From the Figure 3.3, the equation of motion of sprung mass

representing the aircraft body is written as Equation (3.1)

+ ( ) + ( ) = 0 (3.1)

Figure 3.4 Freebody diagram of unsprungmass

35

From the Figure 3.4, the equation of motion of un sprung

representing the wheel components is written as Equation (3.2)

+ ( ) + + ( ) + = 0 (3.2)

The equations of motion for the two degree of freedom system can

be written in matrix form as Equation (3.3)

00 + +

y +

+ +0

= 0 (3.3)

3.2.3 Quarter Aircraft Model Parameters

The model parameters taken from the Fokker aircraft for numerical

simulation are given in Table 3.1 to analyze the vertical vibration levels such

as acceleration, displacement and shock strut travel of the passive landing

gear.

Table 3.1 Quarter aircraft model parameters

Description Value

sprung mass ( ) 8800 kg

unsprung mass ( ) 260 kg

sprung mass stiffness rate ( ) 4.08 × 10 N/m

sprung mass damper rate ( ) 41944 N.s/m

unsprung mass tire stiffness rate ( ) 1. 59 × 10 N/m

unsprung mass tire damper rate ( ) 37411 N.s/m

36

Assuming sprung mass damping ratio = 0.35,

The landing gear damping coefficient is

= 2 = 41944 /

The unsprung mass damping ratio and the tire damping coefficient as

=c

2 (k + k )m

c2 22 k2m=37411Ns

m

These parameters will yield the frequency for the body mode and

the wheel mode.

12

= 1.083

=1

2+

= 13.951

3.2.4 Bump Model

A widely used method to construct fortified runways is the casting of

large plates using liquid concrete. These plates are separated from each other by

gaps filled with rubber. Aging of concrete runways causes the plates to settle

unevenly, leading to long wavelength bumps, ramps and steps at the gaps. Figure

3.5 illustrates an assumed half sine type runway bump of height 0.06 m and wave

length 44 m (0.8*55 m/s) over which the airplane travels. Figure 3.5 shows the

runway profile, generated as a function of time for simulations based on the

relation Time=Distance/velocity. The ride dynamic behavior of the aircraft

37

due to a sinusoidal excitation is investigated. The excitation frequency based

on the vehicle speed and the wavelength is computed as approximately 1.25

Hz (7.85 rad/sec) (frequency=velocity/wave length). The generated bump

input is used for the analysis of model parameters.

Figure 3.5 Bump input for simulation

3.3 SIMULINK MODELING OF PASSIVE LANDING GEAR

As MATLAB is a high–level technical computing language and

interactive environment for algorithm development, data visualization, data

analysis, numerical computation and control system. Tool box software

provides tools for systematically analyzing, designing, and tuning linear

control systems. Simulink software is closely integrated with the MATLAB

environment. It requires MATLAB to run, to define and evaluate model and

block parameters. Simulink can also utilize many MATLAB features.

There are six steps to modeling any system in the Simulink:

1. Defining the System 2. Identifying System Components

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

time(s)

Runway bump input

38

3. Modeling the System with Equations

4. Building the Simulink Block Diagram

5. Running the Simulation

6. Validating the Simulation Results

The first three steps of this process is performed outside of the Simulink software before building the model. Appendix 1.5 represents the Simulink model in the block diagram form of the sprung mass equations of motion given in Equation (3.1) and unsprung mass equations of motion given in Equation (3.2). The model assumes that the sprung mass is free to move through vertically and the unsprung masses have contact with the runway surface. Thus, the vertical acceleration, velocity and displacement of the aircraft center of gravity are functions of the vertical displacement of the quarter aircraft model. The vertical acceleration, velocity, and vertical displacement from the sprung mass dynamics and un sprung masses dynamics are calculated by simulations.The developed simulink model of the passive landing gear is validated with the active suspension using PID controller (Mouleeswaran Senthil kumar 2008).The simulation of simlink model for parametric analysis of passive landing gear system is done through the matlab programme in Appendix 1.1. The simulink block diagram of the passive landing gear system for the simulation is shown in Appendix 1.5.

3.4 PARAMETRIC ANALYSIS OF PASSIVE LANDING GEAR

3.4.1 Unsprung to Sprung Mass Ratio Effect

The analysis has been performed by maintaining the aircraft parameters constant, as specified in the Table 3.1 and changing the unsprung mass. The quarter model un sprung mass changes from 50% of its nominal value and through its nominal value and to a final 200% of the nominal mass. These changes correspond to an unsprung mass to sprung mass change of 400%.The changing of un sprung mass is given in the Table 3.2.

39

Table 3.2 Unsprung mass range for analysis

Mass Value(kg) Un sprung to sprung mass ratio

50% of unsprung mass 130 0.0147Base line 260 0.0295200% of sprung mass 520 0.0591

The response to an excitation of a bump input with amplitude of

0.06 m is used to simulate the sprung mass and unsprung mass system.

Figures 3.6 and 3.7 show that the body response is marginally changed by the

variation in the unsprung to sprung mass ratio and the results are tabulated in

Table 3.3.

Figure 3.6 Acceleration response for different unsprung mass

0 1 2 3 4 5 6-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time(s)

Fuselage acceleration for different unsprung mass

50% of unsprungmaasBase line200% of unsprung mass

40

Figure 3.7 Displacement response for different unsprung mass

Table 3.3 Dynamic response for change in unsprung mass

Characteristic 50% of unsprung mass Base line 200 % of

unsprung mass outcome Body response is slightly affected

Body mode frequency(rad/s)

6.0723 6.0705 6.0667

Body settling time(s) 5.8 5.8 5.9Body peak

displacement(m)0.0772 0.0773 0.0776

Body peak acceleration(m/s²)

1.555 1.555 1.565

0 1 2 3 4 5 6-0.04

-0.02

0

0.02

0.04

0.06

0.08

time(s)

Fuselage displacement for different unsprung mass

50% of unsprung massBase line200% of unsprung mass

41

3.4.2 Landing Gear Spring Stiffness Effect

The analysis was done by varying the spring stiffness while

maintaining all other parameters constant, as shown in Table 3.1. The quarter

model spring stiffness changes from 50% of its value through the original

value and to a final 200% of the stiffness value. These changes correspond to

a stiffness change range of 400%.The sprung mass response to a bump input

shows that as the stiffness increases, the body mode is less damped and the

natural frequency value increases. Figure 3.8 shows the higher amplitude of

vibration when the stiffness value increases and Figure 3.9 shows the

increases in displacement. The results are tabulated in the Table 3.4.

Figure 3.8 Acceleration response for different landing gear stiffness

0 1 2 3 4 5 6 7 8 9 10-4

-3

-2

-1

0

1

2

3

4

time(s)

Fuselage acceleration of different landing gear stiffness

50% of stiffnessBase line200% of stiffness

42

Figure 3.9 Displacement response for different landing gear stiffness

Table 3.4 Dynamic response for change in landing gear stiffness

Characteristic 50% of spring stiffness Base line 200 % of spring

stiffness Outcome Body response is less damped .Higher natural

frequency and higher acceleration Body mode

frequency(rad/s)4.5310 6.0705 7.814

Body settling time(s) 4.6 5.8 6.4Body peak

displacement(m)0.067 0.0773 0.0855

Body peak acceleration(m/s²)

0.939 1.555 3.072

0 1 2 3 4 5 6 7 8 9 10-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

time(s)

Fuselage displacement for different landing gear stiffness

50% of stiffnessBase line200% of stiffness

43

3.4.3 Landing Gear Damping Coefficient Effect

The analysis was performed by maintaining all the aircraft

parameters constant, as specified in Table 3.1, and changing damping

coefficient. The quarter model damping coefficient was changed from 50% of

its nominal value through the nominal value to a final 200% of the nominal

damping coefficient. The sprung mass response to a bump input shows that as

the landing gear damping increases, the body mode is more damped, with no

change in natural frequency, and lower amplitude. From the Figure 3.10 and

Figure 3.11 base line is suitable, there is too much overshoot for lower values

of damping coefficient and the system gets too fast for higher values of

damping coefficient and tabulated in Table 3.6.

Figure 3.10 Acceleration response for different landing gear damping coefficient

0 1 2 3 4 5 6 7 8 9 10-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

time(s)

Fuselage acceleration for different landing gear damping coefficient

50% of damping coefficientBase line200% of damping coefficient

44

Figure 3.11 Displacement response for a change in landing gear damping coefficient

Table 3.5 Dynamic response for change in damping coefficient

Characteristic 50% of sprung mass damping

coefficient Base line

200 % of sprung mass damping

coefficient Outcome Less damping time with the damping increases, no

change in body frequency Body mode frequency

(rad/s)6.0705 6.0705 6.0705

Body settling time (s) 8.6 5.8 3.6Body peak

displacement(m)0.0834 0.0773 0.0732

Body peak acceleration(m/s²)

2.061 1.555 1.545

0 1 2 3 4 5 6 7 8 9 10-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

time(s)

Fuselage displacement for different landing gear damping coefficient

50% of damping coefficientBase line200% of damping coefficient

45

3.5 MODELING OF ACTIVE LANDING GEAR SYSTEM

Figure 3.12 shows the active landing gear system consisting of low

pressure reservoir, hydraulic pump, high pressure accumulator, servo actuator

and electronic controller (Howell et al 1991). The passive system does not

include servo actuator, transducers and electronic controllers. When an

aircraft lands, the shock absorber stroke is influenced by the aircraft’s payload

and varies depending on runway excitations. The generation of active control

energy is to attenuate the vibrations to improve the ride comfort.

Figure 3.12 Schematic diagram of active landing gear system

Active landing gear is mathematically modeled (Irwin Ross &

Edson 1983, Horta et al 1999) and the active force is controlled by electronic

controller which is activated by the sensors fitted in the landing gear. Energy

is supplied through the hydraulic fluid to the landing gear system and also

withdrawn from the system depends on load requirements by the servo

system. In the active landing gear system, the stroke is measured by the

46

transducers and their signal input into the PID controller. This controller

directs the servo valve to regulate the oil flow into or out of the shock

absorber, hence producing the active control force to reduce the vibration

level and also the force transferred to the airplane (Freymann & Johnson

1985, Freymann 1987, 1991). The mathematical modeling of the active

landing gear system is as shown in Figure 3.13. By Newton’s second law, the

dynamic equation of motion is derived, is the active control force. The

equations of motion are written by using free body diagrams.

Figure 3.13 Mathematical model of active landing gear system

Figure 3.14 Free body diagram of sprung mass

47

From the Figure 3.14, the equation of motion for sprung mass is

written as Equation (3.4)

+ ( ) + ( ) = 0 (3.4)

Figure 3.15 Free body diagram of un sprung mass

From the Figure 3.15, the equation of motion for unsprung mass is

written as Equation (3.5)

+ ( ) + + ( ) + + = 0

(3.5)

Dynamic equations can be written in matrix form as Equation (3.6)

00 + +

y + +

+0

+ 11 = 0 (3.6)

The general equation is written as Equation (3.7)

[ ]{ } + [ ]{ } + [ ]{ } = { } (3.7)

48

where,

[ ] is the mass matrix given by

[ ] = 00

[ ] is the damping matrix given by

[ ] = +

[ ] is the stiffness matrix given by

[ ] = +

{ } is the displacement vector given by

{ }=

{ } is the velocity vector given by

{ }=y

{ } is the acceleration vector given by

{ }=

and

{ } is the force vector given by

49

{ } = +

The governing equation can be simplified as Equation (3.8)

{ } = [ ] { } [ ] [ ]{ } [ ] [ ]{ } (3.8)

3.6 SHOCK STRUT FORCES

During operation of oleo pneumatic shock strut, damping effect is

created by compressing the oil through metering orifice whose area is varied

by the metering piston on various loading conditions. The air/nitrogen in the

pneumatic chamber area is compressed by the hydraulic oil which provides

air cushion spring effect throughout its operation. Sliding movement of parts

in the system induces frictional forces adding to the shock strut forces. The

gear forces are obtained as follows

Air spring force is the force simulating the pressure of nitrogen gas

in the upper chamber of the cylinder (Jayarami Reddy et al 1984). It is

assumed that the pressure and volume of the gas satisfies the state of

polytrophic equation of gas (3.9).

= (3.9)

where

= pressure in the cylinder

= area of the piston

= Initial volume of the cylinder

= stroke of the piston

n = polytrophic constant

50

Damping force is provided by oil flow forced through an orifice by

vertical strut position. The hydraulic oil flow is controlled by means of

metering pin, The equation is written as Equation (3.10)

= | | (3.10)

where

=Density of hydraulic fluid

= area of the piston

= velocity of the piston stroke

= orifice coefficient

=area of the orifice

Friction force: The friction is proportional to the velocity and the air spring

force developed in the system. This friction model is accurate in dynamic

loading circumstances. The equation is written as Equation (3.11).

y

where

= co-efficient of friction

=air spring force

= velocity of the piston stroke

The total axial force in the shock absorber = + + (3.11)

51

3.7 CONTROLLER DESIGN

3.7.1 PID Controller

Proportional-Integral-Derivative controller (PID) is a generic

control loop feedback mechanism widely used in industrial control system. It

is commonly used feedback controller. The error value is calculated as the

difference between a measured variable and reference point. The controller

has got three control parameters called the proportional, the integral and

derivative values. It is denoted as P, I and D.P depends on the present error, I

is the accumulation of past errors and D is a prediction of future errors based

on the current rate of change. The weighted sum of these three actions is used

to adjust the active control force by controlling the servo valve.

3.7.2 PID Controller Theory

The PID control is named by three terms viz, the proportional (P),

integral (I) and derivative (D) (Shinners 1964) are summed to calculate the

output of the PID controller. It is written as Equation (3.12)

= ( ) + ( ) + ( ) (3.12)

3.7.2.1 Proportional term

The proportional term is used to change the output. The output

is proportional to the current error value. The proportional value is calculated

by multiplying the error by a constant .This is called proportional gain.

The proportional term is given by the Equation (3.13)

= ( ) (3.13)

52

If the proportional gain is too high, the controller system will

become unstable. If the proportional gain is too low, the control will be very

small and will not respond to the system disturbances.

3.7.2.2 Integral term

The integral term is proportional to the magnitude of the error and

the duration of the error. The integral is the sum of instantaneous error over

time .which is known as accumulated error. The integral term is calculated by

multiplying the accumulated error and the integral gain( ).It is given by the

Equation (3.14)

= ( ) (3.14)

The integral term accelerates the movement of the process towards

reference point and eliminates the residual steady state error that occurs with a

pure proportional controller.

3.7.2.3 Derivative term

It is calculated by determining the slope of the error over time and

multiplying this rate of change by the derivative gain .The derivative term

is given by the Equation (3.15).

= ( ) (3.15)

The derivative term slows the rate of change of the controller

output. Derivative control is used to reduce the magnitude of the overshoot

produced by the integral component and improve the controller stability.

53

3.7.3 Tuning Method

Tuning a control loop is the adjustment of its control parameters to

the optimum values for the desired control response. PID tuning is a difficult

problem, because it must satisfy complex criteria within the limitations of PID

control. Generally stability of response is required and the process must not

oscillate for any combination of process conditions and set points, though

sometimes marginal stability is accepted or desired. There are several

methods (Datta et al 2000) of tuning of PID controller. The most effective

methods generally involve the development of some form of process model,

with appropriate P, I and D based on the dynamic model parameters. The

different methods are manual method, Ziegler- Nichols method, Cohen –coon

method and software tools as given in the Table 3.6. Manual tuning methods

can be relatively inefficient, particularly if the loops have response times on

the order of minutes or longer. In this work, Ziegler Nichols method is simple

and often used.

Table 3.6 Various tuning methods

Method Advantages Disadvantages

Manual tuning No mathematical knowledge required. Online method

Requires experienced personnel

Ziegler-Nichols Proven method. Online method Process upset some trial

and error. Very aggressive tuning

Software tools

Consistent tuning, online or offline method. May include valve and

sensor analysis. Allow simulation before downloading. Can support

non steady state tuning.

Some cost and training involved.

Cohen-coon Good process models Some math, offline

method. Only good for first order process.

54

3.7.3.1 Ziegler-Nichols method

This method is introduced by John Ziegler & Nathaniel Nichols

(1940). First the and gains are set to zero. The gain is increased until it

reaches the ultimate gain at which the output of the loop starts to oscillate.

and the oscillation period are used to tune the gains as shown in

Table 3.7.

Table 3.7 Ziegler – Nichols method

Control type

P 0.5

PI 0.45 1.25 /

PID 0.60 2 / /8

The PID controller design Haitao Wang et al (2008) is defined by

Equation (3.16)

= ( ) + ( ) + ( ) (3.16)

is the current input from the controller. is the proportional

gain, and is the integral and derivative gain of the PID controller.

( ) represents a reference signal and is the feedback signal

measured from the sensors fitted in the landing gear. The simulink modeling

of PID controller is shown in Figure 3.16. The error function is written as

Equation (3.17)

( ) = ( ) ( ) (3.17)

55

Figure 3.16 Simulink model of PID controller

The output signal of the controller gives the displacement of the

servo valve as Equation (3.18)

( ) = { ( ) [ ( ) ( )]} + { ( ) ( ) ( )}

+ { ( ) [ ( ) ( )]} (3.18)

The feedback coefficients viz , , are adjusted by Ziegler-

Nichols tuning rules to obtain the best control over the servo valve.

3.8 HYDRAULIC POWER SUPPLY SYSTEM

The following subsections comprise a brief description of the

Principal Hydraulic Elements that make up a typical position controlled

system. The block diagram of the hydraulic system is shown in Figure 3.17.

3.8.1 Low Pressure Reservoir

A hydraulic reservoir is a tank or container designed to store

sufficient hydraulic fluid for all conditions of operation. Reservoirs have

additional storage place to have a reserve of fluid for the emergency operation

of the landing gears, flaps, etc. The reservoir is pressurized to provide a

56

continuous supply of fluid to the pumps. The reservoir may be pressurized by

spring pressure, air pressure or hydraulic pressure. The desired pressure to be

maintained ranges from 10 psi to 90 psi approximately.

3.8.2 Hydraulic Gear Pump

Gear pump is commonly used in the hydraulic system. It is a

positive displacement pump. The gears of the pump are driven by the power

source, which could be an engine drive or electric motor drive. The fluid

trapped in the clearance between the gears and casing is forced through the

out port.

3.8.3 High Pressure Accumulator

An accumulator is basically a chamber for storing hydraulic fluid

under pressure. It can serve one or more purposes. It dampens pressure surges

caused by the operation of an actuator. It can aid or supplement the system

pump when several units are operating at the same time and demand is

beyond the pump capacity. An accumulator can also store power for limited

operation of a component if the pump is not operating. Finally it can supply

fluid under pressure for small system leaks that would cause the system to

cycle continuously between high and low pressure. The accumulators are of

the diaphragm, bladder, and piston types. The pressure in the accumulator is

approximately 3000 psi.

3.8.4 Servo Actuator

Servo actuator is designed to provide hydraulic power and it

includes an actuating cylinder, a multiport flow control valve, check valves

and relief valves together with connecting linkages. The movement of the

57

piston in the servo actuator depends on the control signal from the electronic

controller.

Figure 3.17 Block diagram of hydraulic servo system

3.8.5 Active Control Force

Active control force is a function of the flow output of the servo

valve. The servo valve displacement ( ) is controlled by the PID controller.

The controller actuates the servo valve by the velocity signal , measured

by the transducers. There is no exact relationship between the active control

force and the flow quantity from the servo valve (Sharp 1988) .It is

often determined through experiments or by empirical formula. It is assumed

that the active control force (Haitao Wang et al 2008) is described by

Equation (3.19)

= (3.19)

The flow quantity is calculated by Equation (3.20)

58

= | | (3.20)

when the displacement of the servo valve ( ) > 0, the hydraulic oil would

have positive flow from the accumulator in to the landing gear system and a

positive control force > 0. When ( ) < 0, oil is drawn from the landing

gear in to LP reservoir so that < 0 , where ( ) is the displacement

determined from the controller as in Equation (3.18).

3.9 SIMULINK MODELING OF THE ACTIVE LANDING

GEAR SYSTEM

Simulink model of the single active landing gear is the block

diagram form of the equations of motion given in Equation (3.8).The model

assumes that the sprung mass is free to move in the vertically and the un

sprung masses have contact with the runway surface. Thus, the vertical

acceleration, velocity and displacement of the aircraft center of gravity are

functions of the vertical displacement of the quarter aircraft model. The

simulation of this simulink model is done through the mat lab program in

Appendix 1.2. The Simulink block diagram of the active landing gear system

for the simulation is shown in Appendix 1.6.

3.10 BUMP MODEL

An assumed half sine type runway bump of height 0.10 m and wave

length 44 m (0.8*55 m/s) were generated over which the airplane travels. The

runway profile is generated as a function of time for simulations based on the

relation Time=Distance/velocity. 'The ride dynamic behavior of the aircraft

due to a sinusoidal excitation is investigated (Catt et al 1992). The excitation

frequency based on the vehicle speed and the wavelength is computed as

59

approximately 1.25 Hz (7.85 rad/sec) (frequency=velocity/wave length).The

equation is written as Equation (3.21)

= 100(1 cos ) 1.0 1.8

0 otherwise (3.21)

The half sine wave bump model with a height of 0.1m is designed

in Matlab/Simulink. The model is generated based on the above equation. The

product of step block and sine wave block is used in the bump model

generator. The profile generator of bump input for simulation is shown in

Figure 3.18.

Figure 3.18 Simulink model of bump input

1Output

Step2

Step1

Sine Wave

Product0.05

Constant