Research Article Stability Analysis of Pneumatic Cabin

Research ArticleStability Analysis of Pneumatic Cabin Pressure RegulatingSystem with Complex Nonlinear Characteristics

Xinhua Zheng Lili Xie and Lizhuo Liu

School of Automation Northwestern Polytechnical University Xirsquoan 710129 China

Correspondence should be addressed to Xinhua Zheng parisz163com

Received 26 October 2014 Accepted 11 February 2015

Academic Editor Petko Petkov

Copyright copy 2015 Xinhua Zheng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Stability of pneumatic cabin pressure regulating system with complex nonlinear characteristics is considered The mathematicalmodel of each component is obtained and given in detailThe governing equations of the considered system consist of 8 differentialequations In the circumstance commonly used methods of nonlinear system analysis are not applicable Therefore a new methodis proposed to construct phase plane trajectories numerically The calculation steps are given in detail And convergence region ofnumerical calculation and limits on step size is definedThemethod is applied constructing phase plane trajectories for consideredcabin pressure regulating system Phase plane analysis shows that there exists a limit cycle which is responsible for pressurepulsating in aircraft cabin After parameters adjustment excellent stability characteristics are acquired And the validity of thismethod is confirmed by the simulation

1 Introduction

It is known that cabin pressure regulating system is usedto provide a safe and comfortable surrounding for crewand passengers at high altitude [1] Now cabin pressureregulating system is necessary for commercial airplanes andmilitary aircrafts cruising at high altitude For commercialairplanes digital pressurization systems had become a newdeveloping trend And in some newly designed airplanesit is integrated with other systems [2] The implement ofthis new technology advantages improvement in intellec-tualization and automation of control At the same timeit improves comfort of the passenger and reduces laborcost in maintenance work But these numerical systems areelectrical energy dependent and sensitive to electromagneticinterference Therefore in military aircrafts pneumatic cabinpressure regulating systems are still preferable

The cabin pressure regulating system considered inpresent paper is an all pneumatic system It regulates cabinpressure automatically and no human interference is neededIt works pneumatically and needs no electrical energy Thismakes it a system of high reliability even if electromagneticinterference of high intensive cannot have any influence on its

performance And it adapts automatically for use on high alti-tude airports It also has additional desirable characteristicsof light weight small volume and low maintenance Thesecharacteristics satisfy exactly the needs of advanced militaryairplanes Now it is used widely in newly developed militaryairplanes in China

But at certain flying conditions pulsating of cabin pres-sure arises The pulsating of cabin pressure is undesirable forit may lead to common symptoms of airplane ear includinghearing loss discomfort in the ear or mild to moderate painContinuous existence of such symptoms distracts attentionof pilots and reduces their work efficiency or even leads topermanent hearing impairment [3] Therefore the stabilityof cabin pressure must be ensured The main object of thepresent paper is to define the cause of the cabin pressurepulsating and eliminate it

2 Cabin Pressure Regulating System and ItsDynamic Equation

21 Diagram of Cabin Pressure Regulating System As Figure 1displays the considered cabin pressure regulating system

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2015 Article ID 871621 8 pageshttpdxdoiorg1011552015871621

2 Journal of Control Science and Engineering

Schedulepressuregenerator

middot middot middot

Flyingparameters

Ratelimiter

Pneumaticamplifier

Outflowvalve

Cabinl4 p4p3

p2p1

Figure 1 The diagram of cabin pressure regulating system

comprises five main parts schedule pressure generator ratelimiter pneumatic amplifier outflow valve and cabin Com-prised of two independent components schedule pressuregenerator is the core of the system Its main function is toproduce cabin pressure schedule signal to control outflowvalve The rate limiter limits maximum rising rate andmaximum dropping rate of scheduled pressure The outputsignal of rate limiter is too weak to drive pneumatic outflowvalve therefore pneumatic amplifier is used to amplify itsdriven capability And as a pneumatically sourced elementit is capable of driving outflow valve directly Outflow valveis the actuator of the system Its function is to open valve inaccordance with pressure in its control chamber which is apneumatic load of amplifier At certain altitude the flow rateof cabin air to ambient circumstance is directly proportionalto the opening of outflow valve And by this means cabinpressure is regulated

22 Mathematical Model of Components The differentialequations of the cabin pressure regulating system are built onbasis of the following assumptions [4]

(1) The cabin temperature 119879119901 keeps invariant and 119879119901 =20∘C

(2) The cabin volume keeps constant(3) The air filling the cabin is ideal gas(4) The leakage area of the cabin is constant

Because the cabin temperature keeps constant the flowquantity can be measured in volume in the mathematicalmodel built for each component This greatly simplifiescomputation

In the following mathematical models 1199011 1199012 1199013 and1199014 are the output pressures of schedule pressure generatorrate limiter pneumatic amplifier and cabin 1198971 is the openingof valve 1 in the active component of schedule pressuregenerator 11989731 11989732 are the opening of valve 31 and valve 32which connect pneumatic amplifier to higher and lowerpressure source respectively 1198974 is the opening of outflowvalve 1199041 11990431 11990432 and 1199044 are the corresponding effective flowarea of valve 1 valve 31 valve 32 and outflow valve and 119875119886 isthe air pressure around the aircraft

221 Mathematical Model for Schedule Pressure GeneratorSchedule pressure generator is composed of two components

pht

p1c

mc1

mc2

p4

p1

Diaphragm

Diaphragm

p3

Valve 1

p3-C1

p4 Linkage

Feedbackchamber

Inputchamber

Modulatingchamber

Figure 2 Functional structure of differential pressure generator

proportional pressure generator and differential pressuregenerator The two components take part in schedule pres-sure modulating at different altitude of flight Proportionalpressure generator works at lower altitude of flight anddifferential pressure generator takes over its job a height about5100 meter The two components have similar structure andsimilar governing equations Therefore in the present paperonly differential pressure generator is analyzed Correspond-ingly only stability at altitude above 5100 meter is consideredThe stability at lower altitude can be analyzed similarly

The functional structure of differential pressure generatoris shown in Figure 2 Having a double nozzle-flapper struc-ture disposed in modulating chamber differential pressuregenerator has a similar structure as proportional pressuregenerator [5] The differential equation of its valversquos opening1198971 is

1198971 =

(119901119886 minus 1199011119888) 1198601 + 11986510 + 11989611198971

1198911

0 lt 1198971 lt 1198711198721

0 1198971 le 0 or 1198971 ge 1198711198721(1)

where 1198601 is the effective area of pressure applied ondiaphragms 11986510 is the preload spring force on diaphragms1198961 is the equivalent elasticity coefficient of the springs and 1198911is the kinetic friction coefficient of valve 1 movement

In the situation of dynamic balance volumes flowing inand out of feedback chamber are equal According to flow rate

Journal of Control Science and Engineering 3

calculation formula equation (3) in [6] and equation (1) in [7]1199011119888 is dynamic solution of the following equation

11990211988811199011119888radic1 minus (

11990131199011119888 minus 1198871

1 minus 1198871

)

2

= 11990211988821199014radic1 minus (

11990111198881199014 minus 1198872

1 minus 1198872

)

2

(2)

where 1199021198881 is the saturation volume flow of capillary pipe mc11199021198882 is the saturation volume flow of capillary pipe mc2 1198871 isthe critical pressure ratio of capillarymc1 and 1198872 is the criticalpressure ratio of capillary mc2 Therefore 1199011119888 is a variablewhich lies between 1199013 and 1199014

In accordance with (15) in [8] the change rate of 1199011 is

1 =

240119904111199014radic1199011 (1199014 minus 1199011)

1198811

minus

240 (11987810 minus 11990411) 1199011radic(1199013 minus 1198621) (1199011 minus 1199013 + 1198621)

1198811

(3)

where 11990411 = 119888 sdot 1198971 is the effective flow area of valve 1 and 11987810 isthe maximum effective flow area of valve 1 1198621 is a constantwhich stands for the pressure drop of 1198753 1198811 is the volume ofmodulating chamber of differential pressure generator

222 Mathematical Model for Rate Limiter The rate limiteris composed of a capillary pipe mc3 connecting to a gascontainer By limits volume flowing through capillary pipethe change rate of output pressure in gas container is limitedDynamic equation of 1199012 is as follows

2 =

11990211988831199011

1198812

radic1 minus (11990121199011 minus 1198873

1 minus 1198873

)

2

1199012 lt 1199011

minus11990211988831199012

1198812


1 minus 1198873

)

2

1199012 gt 1199011

(4)

where 1199021198883 is the saturation volume flow of capillary pipe mc31198873 is the critical pressure ratio of capillary pipe mc3 and1198812 isthe volume of gas container

223 Mathematical Model for Pneumatic Amplifier Thefunctional structure of pneumatic amplifier is shown inFigure 3 It also has a similar structure as proportionalpressure generator In pneumatic amplifier there are twoindependent valves valve 31 and valve 32 Suppose the

p2

Diaphragm

Diaphragm

Valve 31

Valve 32

p3

p4

pa

Figure 3 Functional structure of pneumatic amplifier

opening of valve 31 and valve 32 are 11989731 and 11989732 respectivelythe relevant differential equations are as follows

11989731 =

(1199012 minus 1199013) 1198603 minus 119865310 minus 1198963111989731

11989131

0 lt 11989731 lt 11987111987231

0 11989731 le0 or 11989731 ge 11987111987231

11989732 =


11989132

0 lt 11989732 lt 11987111987232

0 11989732 le0 or 11989732 ge 11987111987232

3 =

240119904311199014radic1199013 (1199014 minus 1199013)

1198813

minus120119904321199013

1198813

119901119886

1199013

le 05

240119904311199014radic1199013 (1199014 minus 1199013)

1198813

minus

240119904321199013radic119901119886 (1199013 minus 119901119886)

1198813

119901119886

1199013

gt 05

(5)

where 11990431 = 119888 sdot 11989731 is the dynamic charging area of valve 3111990432 = 119888 sdot 11989732 is the dynamic discharging area of valve 32 and1198813 is the total volume of modulating chamber and pneumaticload of pneumatic amplifier

224 Mathematical Model for Outflow Valve and CabinPressure Cabin pressure and its change rate is the controlobject of cabin pressure regulating systemThe velocity of theoutflow valve opening is

1198974 =


1198914

0 lt 1198974 lt 1198711198724

0 1198974 le 0 or 1198974 ge 1198711198724(6)

where 1198604 is the effective area of air pressure applied ondiaphragm of outflow valve 11986540 is the force preload onoutflow valve by reset spring in control chamber of outflowvalve 1198964 is the elasticity coefficient of the reset spring and 1198914is the kinetic friction coefficient of outflow valve movement


The volume flow rate of outflow valve 119866Out is

119866Out =

12011990441199014119901119886

1199014

le 05

24011990441199014radic119901119886

1199014

(1 minus119901119886

1199014

)119901119886

1199014

gt 05

(7)

According to (13) in [9] and (2) in [10] the change rate ofcabin pressure is

4 =

119866119878 (ℎ ℎ) minus 119866119871 minus 119866Out

119881119862

sdot 1199014(8)

where119881119862 is the volume of aircraft cabin119866119871 is the volume flowrate caused by cabin leakage and 119866119878(ℎ ℎ) is the volume flowinto the cabin caused by air supply system

3 Scheme of Stability Analysis andIts Realization

31 Expression of the System in General Form Set 1199101 = 11990111199102 = 1199012 1199103 = 1199013 1199104 = 1199014 1199105 = 1198971 1199106 = 11989731 1199107 = 11989732and 1199108 = 1198974 and according to (1)sim(8) write their differentialexpressions in a general form [11] as shown in

1198891199101

119889119905= 1198911 (119905 1199101 1199102 1199108)

1198891199102

119889119905= 1198912 (119905 1199101 1199102 1199108)

1198891199108

119889119905= 1198918 (119905 1199101 1199102 1199108)

(9)

and if the initial condition is given in

1199101 (0) = 1205721 1199102 (0) = 1205722 1199108 (0) = 1205728 (10)

then stability analysis of the system described by differentialequation set (9) with initial condition (10) is the object of thepresent paper

32 Equilibrium Point of the System An equilibrium pointof system (9) with initial condition (10) is a solution ylowast =(1199101lowast 1199102lowast 1199108

lowast) of the following algebraic equation

1198911 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

1198912 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

1198918 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

(11)

At certain height (ht) the considered cabin pressureregulating systemhas only one equilibriumpoint Suppose ht

is higher than switching height at which differential pressurecomponent takes over the modulating of schedule pressureThe equilibrium point of the system (9) at height ht is

1198971lowast=1

2119871Max 1198972

lowast= 1198973lowast= 0

1199011lowast= 1199012lowast= 1199013lowast= 119901 ht + Δ 119904119889 = 119901119886 + Δ 119904119889

1199014lowast=20 (11986510 + 119901 ht + 0511989611198711198721)

119904minus 191199013

lowast

=20 (11986510 + 119901119886 + 0511989611198711198721)

119904minus 191199013

lowast

1198974lowast=(1199014lowastminus 1199013lowast) 1198604 minus 11986540

1198964

(12)

where 119901ht is pressure of atmosphere at height ht 119901 ht = 119901119886

33 Definition of Stability The stability in present paper is akind of global uniform asymptotic stability and its definition[12] is as follows

System is said to be asymptotically stable about its equilib-rium if it is stable in the sense of Lyapunov and furthermorethere exists a constant 120575 = 120575(1199050) gt 0 such that

1003817100381710038171003817x (1199050)1003817100381710038171003817 lt 120575 997904rArr x (119905) 997888rarr 0 as 119905 997888rarr infin (13)

The asymptotic stability is said to be uniform if the existingconstant is independent of 1199050 over [0infin] and is said to beglobal if the convergence x(119905) rarr 0 is independent of theinitial state 119909 over the entire domain on which the system isdefined

On the basis of the above definition the cabin pressureregulating system has its special requirement on stability thatis the transient must complete within 25 seconds

34 Scheme of Stability Analysis and Its Realization As statedabove the considered cabin pressure control system is anonlinear system consisting of 8 differential equations andform of each equation is very complicatedTherefore stabilityanalysis of the system is difficult

Describing function method phase plane method andLyapunov method are the three methods commonly used toanalyze stability of systems But to a complex system con-taining multiple nonlinearities describing function methodis very complicated and unique for special problem [13]Therefore it is not a universal method Lyapunov method isan important technique that is used for not only analyzingbut also designing stability of the system And it can beapplied conveniently to systems containing new controltechniques and systems with special characteristics [14 15]But the difficulty of using Lyapunov function lies in foundingand screening Lyapunov function In this circumstanceLyapunov functionmethods cannot be used as it is impossibleto find a 119881 function And classic meaning of phase planemethod designed for second order systems cannot solve theproblem too Therefore in the present paper a numericalmethod is proposed and applied constructing phase plane


trajectories by approximating time-domain solution of thesystem Then on this basis system stability can be analyzedin accordance with phase plane method

For System (9) with initial condition (10) within cer-tain time interval 0 lt 119905 lt 119879 there are 8 functions1199101(119905) 1199102(119905) 1199108(119905) that satisfy each of the differential equa-tions together with all the initial conditions But it is difficultto find such functions analytically Therefore we should firstapproximate 1199101(119905) 1199102(119905) 1199108(119905) numerically at specifiedtime intervals The existence and uniqueness of the timedomain solution are assured by Definition 516 andTheorem517 in [11]

To approximate time-domain solution of a complexnonlinear system subject to initial condition the uniquenessof time-domain solution needs to be ensured firstly Then asuitable numerical algorithm should be selected and adaptedfor special needs of considered problem And selected stepsize should fulfill requirements of numerical stability of thecalculation

Therefore the secondary object of present paper is topropose a new method to construct phase plane numericallywhich extends the scope of phase plane analysis from two-dimension to multidimension

If the system of first-order differential equations (9)subject to the initial conditions (10) has a unique solution1199101(119905) 1199102(119905) 119910119898(119905) then numerical methods could beused approximating its time-domain solution within certainperiod of time

Taking into consideration computational complexity andaccuracy the improved Euler method is selected and itscomputational formulas are as follows

119910119894(119899+1) = 119910119894119899 +ℎ

2(1198961198941 + 1198961198942)

1198961198941 = 119891 (119905119899 119910119894119899)

1198961198942 = 119891 (119905119899 + ℎ 119910119894119899 + ℎ1198961198941)

for each 119899 = 0 1 119873 119894 = 1 2 8

(14)

where 119873 is number of steps needed to completed computa-tion at time interval [0 119879]

In order to construct phase plane trajectories for con-sidered cabin pressure regulating system approximated timedomain solution as well as its derivative needs to be acquiredTherefore the computation formula is adapted as shown in

119910119894(119899+1) = 119910119894119899 +ℎ

2(1198961198941 + 1198961198942)

1198961198941 = 119891 (119905119899 119910119894119899)

1198961198942 = 119891 (119905119899 + ℎ 119910119894119899 + ℎ1198961198941)

119910119894(119899+1) = 119891 (119905119899+1 119910119894(119899+1))

for each 119899 = 0 1 119873 119894 = 1 2 8

(15)

In present paper an m-file is programmed on MATLABplatform to acquire data needed according to equation set(15) Use the notation119872119910119894119895 and 119863119910119894119895 for each 119895 = 0 1 119873

and 119894 = 1 2 8 to denote an approximation to 119910119894(119905119895) andits derivative 1199101015840

119894(119905119895) respectively Set

11987211991011 = 1205721 11987211991021 = 1205722 11987211991081 = 1205728 (16)

The process of computation is described as follows

(1) Set number of cycle119873 in accordance with step size ℎand time limit 119879119873 = 119879ℎ

(2) Set initial value as

11987211991011 = 1205721 11987211991021 = 1205722 11987211991081 = 1205728

11986311991011 = 119891 (011987211991011) 11986311991021 = 119891 (011987211991021)

11986311991081 = 119891 (011987211991081)

(17)

(3) While 119904 increase from 1 to 119873 carry out computationfrom Step 4 to Step 7

(4)

[[[[[[

[

11989611

11989621

11989681

]]]]]]

]

=

[[[[[[

[

1198631199101119904

1198631199102119904

1198631199108119904

]]]]]]

]

(18)

(5)

[[[[[[

[

11989612

11989622

11989682

]]]]]]

]

=

[[[[[[

[

1198911 (119905119904 + ℎ119872119910119904 + ℎ11989611)

1198912 (119905119904 + ℎ119872119910119904 + ℎ11989621)

1198918 (119905119904 + ℎ119872119910119904 + ℎ11989681)

]]]]]]

]

(19)

(6)

[[[[[[

[

1198721199101(119904+1)

1198721199102(119904+1)

1198721199108(119904+1)

]]]]]]

]

=

[[[[[[

[

1198721199101119904

1198721199102119904

1198721199108119904

]]]]]]

]

+ℎ

2

[[[[[[

[

11989611 + 11989612

11989621 + 11989622

11989681 + 11989682

]]]]]]

]

(20)

(7)

[[[[[[

[

1198631199101(119904+1)

1198631199102(119904+1)

1198631199108(119904+1)

]]]]]]

]

=

[[[[[[

[

1198911 (119905119904+11198721199101(119904+1))

1198912 (119905119904+11198721199102(119904+1))

1198918 (119905119904+11198721199108(119904+1))

]]]]]]

]

(21)

(8) Plot interested 119894th pair of variable (119872119910119894 119863119910119894) ofacquired data (here 119894 may be any number between 1and 8) Phase plane trajectories are now constructedand stability of system with initial-value may beanalyzed in accordance with phase plane methods


35 Convergence of Calculation Define model equation ofabsolute stability as linear differential equation

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205971198911

1205971199101

0 sdot sdot sdot 0

01205971198912

1205971199102

sdot sdot sdot 0

d

0 0 1205971198918

1205971199108

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205821 0 sdot sdot sdot 0


d


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

(22)

The region of absolute stability for the improved Eulermethod is |1 + ℎ120582 + (ℎ120582)22| lt 1 and the interval of absolutestability is minus2 lt ℎ120582 lt 0 for each of 1205821 1205822 1205828

Set 120582119894119883 = max0le119905le119879|120597119891119894120597119910119894| 120582119883 = max(120582119894119883) (119894 =

1 2 8) then to ensure the results of calculation approxi-mate to the time-domain solution119910(119905) the interval of ℎ valueis limited by 0 lt ℎ lt 2|120582119883|

4 Results

The method stated above is applied to construct phase planetrajectories of cabin pressure regulating system with 2 initialconditions

At height ht = 7000m the first initial condition is givenas

11990110 = 80 11990120 = 80 11990130 = 80 11990140 = 805

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(23)

And at height ht = 7000m the second initial conditionis

11990110 = 78 11990120 = 78 11990130 = 78 11990140 = 795

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(24)

According to Section 33 the calculated |120582119883| lt 3084then the interval of ℎ value is 0 lt ℎ lt 00064 Then stepsize ℎ is chosen as ℎ = 0001 s Take into consideration theactual requirement of the system set 119879 = 30 s and then119873 =

119879ℎ = 30000 And data is acquired by an m-file programmedin accordance with computation process listed in Section 32

In the cabin pressure regulating system pressure stabilityof 1199012 and 1199014 is variables interested The phase plane trajecto-ries of 1199012 and 1199014 are shown in Figures 4 and 5 respectively

Figure 4 shows that under the 2 given initial conditionsa limit cycle exists in pressure 1199012 That means pressure 1199012will not be stable The fluctuation in 1199012 will pass on and

775 78 785 79 795 80

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

p2 (kPa)

p998400 2

(kPa

s)

p20 = 78

p20 = 80

Figure 4 Phase plane trajectories of 1199012

785 79 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 795


be reproduced in cabin pressure may be in reduced orcomparable amplitude Figure 5 shows an amplitude reducedlimit cycle exists in 1199014 phase plane trajectories This is notallowed for the fluctuation may be aggravated in the complexenvironment of the flying aircrafts

In order to ensure stability of cabin pressure 1199014 theparameters of the system should be adjusted According todesign experience enhancing energy storage capability in thepressure regulating process helps to realize stability of systemAnd reducing dead zone threshold of pneumatic amplifierhelps in eliminating limit cycle Therefore by substitutingnew type of polymermaterial for old one and using new typeof spring with bigger elastic coefficient in differential pressuregenerator some system parameters are changed And preload


785 7978 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 785

Figure 6 Phase plane trajectories of cabin pressure 1199014 afterparameters modification

forces on amplifierrsquos valve are reduced slightly New set ofparameters is as follows

1198961 = 4Ncm 11989631 = 02Ncm 11989632 = 02Ncm

1198911 = 01 11989131 = 11989132 = 01 1198914 = 05

1198651 = 94N 119865310 = 015N

119865320 = 015N 1198654 = 15N(25)

And the corresponding replaced old set of parameters isas follows

1198961 = 35Ncm 11989631 = 017Ncm

11989632 = 017Ncm

1198911 = 006 11989131 = 11989132 = 005 1198914 = 05

1198651 = 94N 119865310 = 02N

119865320 = 02N 1198654 = 15N

(26)

After parameters adjustment the new phase plane tra-jectories of 1199014 are shown in Figure 6 Figure 6 shows thatno limit cycle exists in the system and the phase planetrajectories goes to the same equilibrium point even thoughthe initial conditions are different The simulating resultshown in Figure 7 confirms this conclusion Figure 7 showsthat the stability of cabin pressure is achieved quickly andlittle fluctuation will happen On this base we come tothe conclusion that the system is stable after parametermodification

It needs to point out that comparing with two-dimensional system in a multidimensional system somedifference exists in their phase plane trajectories that is thetrajectoriesmay come across limit cycle before it finally entersinto it as shown by Figure 4

0 2 4 6 8 10 12 14 16 18 20

78

785

79

795

80

805

Time (s)

p4

(kPa

)

p40 = 805

p40 = 785

Figure 7 Simulation result of cabin pressure 1199014

5 Conclusion

The considered cabin pressure regulating system in presentpaper is a multidimensional complex nonlinear system Toanalyze stability of cabin pressure regulating system whichis used in advanced military airplane a numerical methodof construct phase plane trajectories is proposed in thepresent paper This method extends the scope of phaseplane analysis from two-dimension to multidimension Themethod is applied to stability analysis of cabin pressureregulating system and the validity of analysis is verified bysimulation result The method is applicable to complex non-linear system of any dimension and may provide an effectiveway for stability analysis of complex nonlinear systems It isinstructive for researchers facing similar problems

Nomenclature

1198601 Effective area of pressure applied ondiaphragms in differential pressuregenerator

1198603 Effective area of pressure applied ondiaphragms of pneumatic amplifier

1198604 Effective area of pressure applied ondiaphragm of outflow valve

1198911 Kinetic friction coefficient of valve 1movement



1198914 Kinetic friction coefficient of outflow valve11986510 Preload force on diaphragm of differential

pressure generator119865310 Preload force on valve 31119865320 Preload force on valve 3211986540 Preload force on outflow valve


119866119878 Volume flow rate caused by air supplysystem

119866119871 Volume flow rate caused by cabin Leakage119866Out Volume flow rate caused by outflow valveht Height of the aircraft1198961 Equivalent elasticity coefficient of springs

in differential pressure generator11989631 Equivalent elasticity coefficient of valve 31

movement11989632 Equivalent elasticity coefficient of valve 32

movement1198964 Equivalent elasticity coefficient of the reset

spring in outflow valve1198971 Opening of valve 1 in differential pressure

generator11989731 Opening of valve 3111989732 Opening of valve 321198974 Opening of outflow valve1198711198721 Maximum opening of valve 11199011 Output pressure of differential pressure

generator1199012 Output pressure of rate limiter1199013 Output pressure of pneumatic amplifier1199014 Pressure in cabin of airplane1199011119888 Pressure in control chamber of differential

pressure generator119901119886 Ambient pressure around airplane1199021198881 Saturation volume flow of capillary mc11199021198882 Saturation volume flow of capillary mc21199021198883 Saturation volume flow of capillary mc311990411 Dynamic flow area of valve 111990431 Dynamic flow area of valve 3111990432 Dynamic flow area of valve 321199044 Dynamic flow area of outflow valve11987810 Maximum effective flow area of valve 11198871 Critical pressure ratio of capillary mc11198872 Critical pressure ratio of capillary mc21198873 Critical pressure ratio of capillary mc31198811 Volume of modulating chamber in

differential pressure generator1198811119888 Volume of feedback chamber in

differential pressure generator1198812 Volume of gas container in rate limiter1198813 Total volume of modulating chamber and

pneumatic load of pneumatic amplifier1198814 Volume of cabin

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] SAE Aerospace ldquoAircraft cabin pressurization criteriardquo TechRep ARP1270B 2010

[2] K Linnet and R Crabtree ldquoWhat is next in commercial aircraftenvironmental control systemrdquo SAE Technical Paper Series932057 1993

[3] B Alagha S Ahmadbeigy S A J Moosavi and S M JalalildquoHypoxia symptoms during altitude training in professionaliranian fighter pilotsrdquo Air Medical Journal vol 31 no 1 pp 28ndash32 2012

[4] Z Lei F Yongling and Z Jingquan ldquoResearch on the controllerof the digital cabin pressure regulating system based on FIMFrdquoin Proceedings of the IEEE International Conference onAdvancedComputer Control (ICACC rsquo10) vol 5 pp 454ndash458 March 2010

[5] Z Xinhua X Lili L Lizhuo and Y Liangpu ldquoStatic character-stics of proponional pressure regulating components for aircratcabin pressure regulatorrdquo Machine Design and Research vol 2pp 94ndash96 104 2013

[6] K Kawashima Y Ishii T Funaki and T Kagawa ldquoDetermi-nation of flow rate characteristics of pneumatic solenoid valvesusing an isothermal chamberrdquoTransactions of theASME Journalof Fluids Engineering vol 126 no 2 pp 273ndash279 2004

[7] K Oneyama T Takahashi Y Terashima K Kuroshita and TKagawa ldquoStudy and suggestion on flow-rate characteristics ofpneumatic componentsrdquo in Proceedings of the 7th internationalSymposium on Fluid Control Measurement and Visualization2003

[8] C Maolin ldquoModern gasdynamics theory techniques andpractice lecture 1 flow-rate characteristics of pneumatic com-ponentsrdquo Hydraulics Pneumatics amp Seals vol 2007 no 2 pp44ndash48 2007

[9] K Li W Liu J Wang and Y Huang ldquoAn intelligent controlmethod for a large multi-parameter environmental simulationcabinrdquo Chinese Journal of Aeronautics vol 26 no 6 pp 1360ndash1369 2013

[10] F D Ramos C R de Andrade and E L Zaparoli ldquoComputa-tional simulation of an aircraft cabin pressure control systemrdquoin Proceedings of the 18th International Congress of MechanicalEngineering Ouro Preto Brazil November 2005

[11] R L BurdenNumerical Analysis BrooksCole Cengage Learn-ing Australia 2010

[12] C Guanrong ldquoStability of nonlinear systemrdquo in Encyclopedia ofRF and Microwave Engineering pp 4881ndash4896 John Wiley ampSons New York NY USA 2004

[13] R Gran and M Rimer ldquoStability analysis of systems with mul-tiple nonlinearitiesrdquo IEEE Transactions on Automatic Controlvol 10 no 1 pp 94ndash97 1965

[14] Y Chen W X Zheng and A Xue ldquoA new result on stabilityanalysis for stochastic neutral systemsrdquoAutomatica vol 46 no12 pp 2100ndash2104 2010

[15] G Garcia and S Keshmiri ldquoAdaptive and resilient flight controlsystem for a small unmanned aerial systemrdquo InternationalJournal of Aerospace Engineering vol 2013 Article ID 28935725 pages 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Schedulepressuregenerator

middot middot middot

Flyingparameters

Ratelimiter

Pneumaticamplifier

Outflowvalve

Cabinl4 p4p3

p2p1

Figure 1 The diagram of cabin pressure regulating system

comprises five main parts schedule pressure generator ratelimiter pneumatic amplifier outflow valve and cabin Com-prised of two independent components schedule pressuregenerator is the core of the system Its main function is toproduce cabin pressure schedule signal to control outflowvalve The rate limiter limits maximum rising rate andmaximum dropping rate of scheduled pressure The outputsignal of rate limiter is too weak to drive pneumatic outflowvalve therefore pneumatic amplifier is used to amplify itsdriven capability And as a pneumatically sourced elementit is capable of driving outflow valve directly Outflow valveis the actuator of the system Its function is to open valve inaccordance with pressure in its control chamber which is apneumatic load of amplifier At certain altitude the flow rateof cabin air to ambient circumstance is directly proportionalto the opening of outflow valve And by this means cabinpressure is regulated

22 Mathematical Model of Components The differentialequations of the cabin pressure regulating system are built onbasis of the following assumptions [4]

(1) The cabin temperature 119879119901 keeps invariant and 119879119901 =20∘C

(2) The cabin volume keeps constant(3) The air filling the cabin is ideal gas(4) The leakage area of the cabin is constant

Because the cabin temperature keeps constant the flowquantity can be measured in volume in the mathematicalmodel built for each component This greatly simplifiescomputation

In the following mathematical models 1199011 1199012 1199013 and1199014 are the output pressures of schedule pressure generatorrate limiter pneumatic amplifier and cabin 1198971 is the openingof valve 1 in the active component of schedule pressuregenerator 11989731 11989732 are the opening of valve 31 and valve 32which connect pneumatic amplifier to higher and lowerpressure source respectively 1198974 is the opening of outflowvalve 1199041 11990431 11990432 and 1199044 are the corresponding effective flowarea of valve 1 valve 31 valve 32 and outflow valve and 119875119886 isthe air pressure around the aircraft

221 Mathematical Model for Schedule Pressure GeneratorSchedule pressure generator is composed of two components

pht

p1c

mc1

mc2

p4

p1

Diaphragm

Diaphragm

p3

Valve 1

p3-C1

p4 Linkage

Feedbackchamber

Inputchamber

Modulatingchamber

Figure 2 Functional structure of differential pressure generator

proportional pressure generator and differential pressuregenerator The two components take part in schedule pres-sure modulating at different altitude of flight Proportionalpressure generator works at lower altitude of flight anddifferential pressure generator takes over its job a height about5100 meter The two components have similar structure andsimilar governing equations Therefore in the present paperonly differential pressure generator is analyzed Correspond-ingly only stability at altitude above 5100 meter is consideredThe stability at lower altitude can be analyzed similarly

The functional structure of differential pressure generatoris shown in Figure 2 Having a double nozzle-flapper struc-ture disposed in modulating chamber differential pressuregenerator has a similar structure as proportional pressuregenerator [5] The differential equation of its valversquos opening1198971 is

1198971 =

(119901119886 minus 1199011119888) 1198601 + 11986510 + 11989611198971

1198911

0 lt 1198971 lt 1198711198721

0 1198971 le 0 or 1198971 ge 1198711198721(1)

where 1198601 is the effective area of pressure applied ondiaphragms 11986510 is the preload spring force on diaphragms1198961 is the equivalent elasticity coefficient of the springs and 1198911is the kinetic friction coefficient of valve 1 movement

In the situation of dynamic balance volumes flowing inand out of feedback chamber are equal According to flow rate



11990211988811199011119888radic1 minus (

11990131199011119888 minus 1198871

1 minus 1198871

)

2

= 11990211988821199014radic1 minus (

11990111198881199014 minus 1198872

1 minus 1198872

)

2

(2)



1 =

240119904111199014radic1199011 (1199014 minus 1199011)

1198811

minus


1198811

(3)



2 =

11990211988831199011

1198812


1 minus 1198873

)

2

1199012 lt 1199011

minus11990211988831199012

1198812


1 minus 1198873

)

2

1199012 gt 1199011

(4)



p2

Diaphragm

Diaphragm

Valve 31

Valve 32

p3

p4

pa



11989731 =


11989131

0 lt 11989731 lt 11987111987231

0 11989731 le0 or 11989731 ge 11987111987231

11989732 =


11989132

0 lt 11989732 lt 11987111987232

0 11989732 le0 or 11989732 ge 11987111987232

3 =

240119904311199014radic1199013 (1199014 minus 1199013)

1198813

minus120119904321199013

1198813

119901119886

1199013

le 05

240119904311199014radic1199013 (1199014 minus 1199013)

1198813

minus

240119904321199013radic119901119886 (1199013 minus 119901119886)

1198813

119901119886

1199013

gt 05

(5)



1198974 =


1198914

0 lt 1198974 lt 1198711198724

0 1198974 le 0 or 1198974 ge 1198711198724(6)




119866Out =

12011990441199014119901119886

1199014

le 05

24011990441199014radic119901119886

1199014

(1 minus119901119886

1199014

)119901119886

1199014

gt 05

(7)


4 =


119881119862

sdot 1199014(8)




1198891199101

119889119905= 1198911 (119905 1199101 1199102 1199108)

1198891199102

119889119905= 1198912 (119905 1199101 1199102 1199108)

1198891199108

119889119905= 1198918 (119905 1199101 1199102 1199108)

(9)


1199101 (0) = 1205721 1199102 (0) = 1205722 1199108 (0) = 1205728 (10)




1198911 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

1198912 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

1198918 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

(11)



1198971lowast=1

2119871Max 1198972



1199014lowast=20 (11986510 + 119901 ht + 0511989611198711198721)

119904minus 191199013

lowast

=20 (11986510 + 119901119886 + 0511989611198711198721)

119904minus 191199013

lowast


1198964

(12)
















119910119894(119899+1) = 119910119894119899 +ℎ

2(1198961198941 + 1198961198942)

1198961198941 = 119891 (119905119899 119910119894119899)

1198961198942 = 119891 (119905119899 + ℎ 119910119894119899 + ℎ1198961198941)

for each 119899 = 0 1 119873 119894 = 1 2 8

(14)



119910119894(119899+1) = 119910119894119899 +ℎ

2(1198961198941 + 1198961198942)

1198961198941 = 119891 (119905119899 119910119894119899)

1198961198942 = 119891 (119905119899 + ℎ 119910119894119899 + ℎ1198961198941)

119910119894(119899+1) = 119891 (119905119899+1 119910119894(119899+1))

for each 119899 = 0 1 119873 119894 = 1 2 8

(15)




11987211991011 = 1205721 11987211991021 = 1205722 11987211991081 = 1205728 (16)




11987211991011 = 1205721 11987211991021 = 1205722 11987211991081 = 1205728

11986311991011 = 119891 (011987211991011) 11986311991021 = 119891 (011987211991021)

11986311991081 = 119891 (011987211991081)

(17)


(4)

[[[[[[

[

11989611

11989621

11989681

]]]]]]

]

=

[[[[[[

[

1198631199101119904

1198631199102119904

1198631199108119904

]]]]]]

]

(18)

(5)

[[[[[[

[

11989612

11989622

11989682

]]]]]]

]

=

[[[[[[

[

1198911 (119905119904 + ℎ119872119910119904 + ℎ11989611)

1198912 (119905119904 + ℎ119872119910119904 + ℎ11989621)

1198918 (119905119904 + ℎ119872119910119904 + ℎ11989681)

]]]]]]

]

(19)

(6)

[[[[[[

[

1198721199101(119904+1)

1198721199102(119904+1)

1198721199108(119904+1)

]]]]]]

]

=

[[[[[[

[

1198721199101119904

1198721199102119904

1198721199108119904

]]]]]]

]

+ℎ

2

[[[[[[

[

11989611 + 11989612

11989621 + 11989622

11989681 + 11989682

]]]]]]

]

(20)

(7)

[[[[[[

[

1198631199101(119904+1)

1198631199102(119904+1)

1198631199108(119904+1)

]]]]]]

]

=

[[[[[[

[

1198911 (119905119904+11198721199101(119904+1))

1198912 (119905119904+11198721199102(119904+1))

1198918 (119905119904+11198721199108(119904+1))

]]]]]]

]

(21)




[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205971198911

1205971199101

0 sdot sdot sdot 0

01205971198912

1205971199102

sdot sdot sdot 0

d

0 0 1205971198918

1205971199108

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



d


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

(22)


Set 120582119894119883 = max0le119905le119879|120597119891119894120597119910119894| 120582119883 = max(120582119894119883) (119894 =


4 Results



11990110 = 80 11990120 = 80 11990130 = 80 11990140 = 805

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(23)


11990110 = 78 11990120 = 78 11990130 = 78 11990140 = 795

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(24)





775 78 785 79 795 80

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

p2 (kPa)

p998400 2

(kPa

s)

p20 = 78

p20 = 80


785 79 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 795





785 7978 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 785



1198961 = 4Ncm 11989631 = 02Ncm 11989632 = 02Ncm

1198911 = 01 11989131 = 11989132 = 01 1198914 = 05

1198651 = 94N 119865310 = 015N

119865320 = 015N 1198654 = 15N(25)


1198961 = 35Ncm 11989631 = 017Ncm

11989632 = 017Ncm

1198911 = 006 11989131 = 11989132 = 005 1198914 = 05

1198651 = 94N 119865310 = 02N

119865320 = 02N 1198654 = 15N

(26)



0 2 4 6 8 10 12 14 16 18 20

78

785

79

795

80

805

Time (s)

p4

(kPa

)

p40 = 805

p40 = 785


5 Conclusion


Nomenclature
























References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











11990211988811199011119888radic1 minus (

11990131199011119888 minus 1198871

1 minus 1198871

)

2

= 11990211988821199014radic1 minus (

11990111198881199014 minus 1198872

1 minus 1198872

)

2

(2)



1 =

240119904111199014radic1199011 (1199014 minus 1199011)

1198811

minus


1198811

(3)



2 =

11990211988831199011

1198812


1 minus 1198873

)

2

1199012 lt 1199011

minus11990211988831199012

1198812


1 minus 1198873

)

2

1199012 gt 1199011

(4)



p2

Diaphragm

Diaphragm

Valve 31

Valve 32

p3

p4

pa



11989731 =


11989131

0 lt 11989731 lt 11987111987231

0 11989731 le0 or 11989731 ge 11987111987231

11989732 =


11989132

0 lt 11989732 lt 11987111987232

0 11989732 le0 or 11989732 ge 11987111987232

3 =

240119904311199014radic1199013 (1199014 minus 1199013)

1198813

minus120119904321199013

1198813

119901119886

1199013

le 05

240119904311199014radic1199013 (1199014 minus 1199013)

1198813

minus

240119904321199013radic119901119886 (1199013 minus 119901119886)

1198813

119901119886

1199013

gt 05

(5)



1198974 =


1198914

0 lt 1198974 lt 1198711198724

0 1198974 le 0 or 1198974 ge 1198711198724(6)




119866Out =

12011990441199014119901119886

1199014

le 05

24011990441199014radic119901119886

1199014

(1 minus119901119886

1199014

)119901119886

1199014

gt 05

(7)


4 =


119881119862

sdot 1199014(8)




1198891199101

119889119905= 1198911 (119905 1199101 1199102 1199108)

1198891199102

119889119905= 1198912 (119905 1199101 1199102 1199108)

1198891199108

119889119905= 1198918 (119905 1199101 1199102 1199108)

(9)


1199101 (0) = 1205721 1199102 (0) = 1205722 1199108 (0) = 1205728 (10)




1198911 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

1198912 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

1198918 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

(11)



1198971lowast=1

2119871Max 1198972



1199014lowast=20 (11986510 + 119901 ht + 0511989611198711198721)

119904minus 191199013

lowast

=20 (11986510 + 119901119886 + 0511989611198711198721)

119904minus 191199013

lowast


1198964

(12)
















119910119894(119899+1) = 119910119894119899 +ℎ

2(1198961198941 + 1198961198942)

1198961198941 = 119891 (119905119899 119910119894119899)

1198961198942 = 119891 (119905119899 + ℎ 119910119894119899 + ℎ1198961198941)

for each 119899 = 0 1 119873 119894 = 1 2 8

(14)



119910119894(119899+1) = 119910119894119899 +ℎ

2(1198961198941 + 1198961198942)

1198961198941 = 119891 (119905119899 119910119894119899)

1198961198942 = 119891 (119905119899 + ℎ 119910119894119899 + ℎ1198961198941)

119910119894(119899+1) = 119891 (119905119899+1 119910119894(119899+1))

for each 119899 = 0 1 119873 119894 = 1 2 8

(15)




11987211991011 = 1205721 11987211991021 = 1205722 11987211991081 = 1205728 (16)




11987211991011 = 1205721 11987211991021 = 1205722 11987211991081 = 1205728

11986311991011 = 119891 (011987211991011) 11986311991021 = 119891 (011987211991021)

11986311991081 = 119891 (011987211991081)

(17)


(4)

[[[[[[

[

11989611

11989621

11989681

]]]]]]

]

=

[[[[[[

[

1198631199101119904

1198631199102119904

1198631199108119904

]]]]]]

]

(18)

(5)

[[[[[[

[

11989612

11989622

11989682

]]]]]]

]

=

[[[[[[

[

1198911 (119905119904 + ℎ119872119910119904 + ℎ11989611)

1198912 (119905119904 + ℎ119872119910119904 + ℎ11989621)

1198918 (119905119904 + ℎ119872119910119904 + ℎ11989681)

]]]]]]

]

(19)

(6)

[[[[[[

[

1198721199101(119904+1)

1198721199102(119904+1)

1198721199108(119904+1)

]]]]]]

]

=

[[[[[[

[

1198721199101119904

1198721199102119904

1198721199108119904

]]]]]]

]

+ℎ

2

[[[[[[

[

11989611 + 11989612

11989621 + 11989622

11989681 + 11989682

]]]]]]

]

(20)

(7)

[[[[[[

[

1198631199101(119904+1)

1198631199102(119904+1)

1198631199108(119904+1)

]]]]]]

]

=

[[[[[[

[

1198911 (119905119904+11198721199101(119904+1))

1198912 (119905119904+11198721199102(119904+1))

1198918 (119905119904+11198721199108(119904+1))

]]]]]]

]

(21)




[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205971198911

1205971199101

0 sdot sdot sdot 0

01205971198912

1205971199102

sdot sdot sdot 0

d

0 0 1205971198918

1205971199108

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



d


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

(22)


Set 120582119894119883 = max0le119905le119879|120597119891119894120597119910119894| 120582119883 = max(120582119894119883) (119894 =


4 Results



11990110 = 80 11990120 = 80 11990130 = 80 11990140 = 805

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(23)


11990110 = 78 11990120 = 78 11990130 = 78 11990140 = 795

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(24)





775 78 785 79 795 80

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

p2 (kPa)

p998400 2

(kPa

s)

p20 = 78

p20 = 80


785 79 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 795





785 7978 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 785



1198961 = 4Ncm 11989631 = 02Ncm 11989632 = 02Ncm

1198911 = 01 11989131 = 11989132 = 01 1198914 = 05

1198651 = 94N 119865310 = 015N

119865320 = 015N 1198654 = 15N(25)


1198961 = 35Ncm 11989631 = 017Ncm

11989632 = 017Ncm

1198911 = 006 11989131 = 11989132 = 005 1198914 = 05

1198651 = 94N 119865310 = 02N

119865320 = 02N 1198654 = 15N

(26)



0 2 4 6 8 10 12 14 16 18 20

78

785

79

795

80

805

Time (s)

p4

(kPa

)

p40 = 805

p40 = 785


5 Conclusion


Nomenclature
























References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119866Out =

12011990441199014119901119886

1199014

le 05

24011990441199014radic119901119886

1199014

(1 minus119901119886

1199014

)119901119886

1199014

gt 05

(7)


4 =


119881119862

sdot 1199014(8)




1198891199101

119889119905= 1198911 (119905 1199101 1199102 1199108)

1198891199102

119889119905= 1198912 (119905 1199101 1199102 1199108)

1198891199108

119889119905= 1198918 (119905 1199101 1199102 1199108)

(9)


1199101 (0) = 1205721 1199102 (0) = 1205722 1199108 (0) = 1205728 (10)




1198911 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

1198912 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

1198918 (119905 1199101lowast 1199102lowast 1199108

lowast) = 0

(11)



1198971lowast=1

2119871Max 1198972



1199014lowast=20 (11986510 + 119901 ht + 0511989611198711198721)

119904minus 191199013

lowast

=20 (11986510 + 119901119886 + 0511989611198711198721)

119904minus 191199013

lowast


1198964

(12)
















119910119894(119899+1) = 119910119894119899 +ℎ

2(1198961198941 + 1198961198942)

1198961198941 = 119891 (119905119899 119910119894119899)

1198961198942 = 119891 (119905119899 + ℎ 119910119894119899 + ℎ1198961198941)

for each 119899 = 0 1 119873 119894 = 1 2 8

(14)



119910119894(119899+1) = 119910119894119899 +ℎ

2(1198961198941 + 1198961198942)

1198961198941 = 119891 (119905119899 119910119894119899)

1198961198942 = 119891 (119905119899 + ℎ 119910119894119899 + ℎ1198961198941)

119910119894(119899+1) = 119891 (119905119899+1 119910119894(119899+1))

for each 119899 = 0 1 119873 119894 = 1 2 8

(15)




11987211991011 = 1205721 11987211991021 = 1205722 11987211991081 = 1205728 (16)




11987211991011 = 1205721 11987211991021 = 1205722 11987211991081 = 1205728

11986311991011 = 119891 (011987211991011) 11986311991021 = 119891 (011987211991021)

11986311991081 = 119891 (011987211991081)

(17)


(4)

[[[[[[

[

11989611

11989621

11989681

]]]]]]

]

=

[[[[[[

[

1198631199101119904

1198631199102119904

1198631199108119904

]]]]]]

]

(18)

(5)

[[[[[[

[

11989612

11989622

11989682

]]]]]]

]

=

[[[[[[

[

1198911 (119905119904 + ℎ119872119910119904 + ℎ11989611)

1198912 (119905119904 + ℎ119872119910119904 + ℎ11989621)

1198918 (119905119904 + ℎ119872119910119904 + ℎ11989681)

]]]]]]

]

(19)

(6)

[[[[[[

[

1198721199101(119904+1)

1198721199102(119904+1)

1198721199108(119904+1)

]]]]]]

]

=

[[[[[[

[

1198721199101119904

1198721199102119904

1198721199108119904

]]]]]]

]

+ℎ

2

[[[[[[

[

11989611 + 11989612

11989621 + 11989622

11989681 + 11989682

]]]]]]

]

(20)

(7)

[[[[[[

[

1198631199101(119904+1)

1198631199102(119904+1)

1198631199108(119904+1)

]]]]]]

]

=

[[[[[[

[

1198911 (119905119904+11198721199101(119904+1))

1198912 (119905119904+11198721199102(119904+1))

1198918 (119905119904+11198721199108(119904+1))

]]]]]]

]

(21)




[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205971198911

1205971199101

0 sdot sdot sdot 0

01205971198912

1205971199102

sdot sdot sdot 0

d

0 0 1205971198918

1205971199108

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



d


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

(22)


Set 120582119894119883 = max0le119905le119879|120597119891119894120597119910119894| 120582119883 = max(120582119894119883) (119894 =


4 Results



11990110 = 80 11990120 = 80 11990130 = 80 11990140 = 805

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(23)


11990110 = 78 11990120 = 78 11990130 = 78 11990140 = 795

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(24)





775 78 785 79 795 80

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

p2 (kPa)

p998400 2

(kPa

s)

p20 = 78

p20 = 80


785 79 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 795





785 7978 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 785



1198961 = 4Ncm 11989631 = 02Ncm 11989632 = 02Ncm

1198911 = 01 11989131 = 11989132 = 01 1198914 = 05

1198651 = 94N 119865310 = 015N

119865320 = 015N 1198654 = 15N(25)


1198961 = 35Ncm 11989631 = 017Ncm

11989632 = 017Ncm

1198911 = 006 11989131 = 11989132 = 005 1198914 = 05

1198651 = 94N 119865310 = 02N

119865320 = 02N 1198654 = 15N

(26)



0 2 4 6 8 10 12 14 16 18 20

78

785

79

795

80

805

Time (s)

p4

(kPa

)

p40 = 805

p40 = 785


5 Conclusion


Nomenclature
























References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119910119894(119899+1) = 119910119894119899 +ℎ

2(1198961198941 + 1198961198942)

1198961198941 = 119891 (119905119899 119910119894119899)

1198961198942 = 119891 (119905119899 + ℎ 119910119894119899 + ℎ1198961198941)

for each 119899 = 0 1 119873 119894 = 1 2 8

(14)



119910119894(119899+1) = 119910119894119899 +ℎ

2(1198961198941 + 1198961198942)

1198961198941 = 119891 (119905119899 119910119894119899)

1198961198942 = 119891 (119905119899 + ℎ 119910119894119899 + ℎ1198961198941)

119910119894(119899+1) = 119891 (119905119899+1 119910119894(119899+1))

for each 119899 = 0 1 119873 119894 = 1 2 8

(15)




11987211991011 = 1205721 11987211991021 = 1205722 11987211991081 = 1205728 (16)




11987211991011 = 1205721 11987211991021 = 1205722 11987211991081 = 1205728

11986311991011 = 119891 (011987211991011) 11986311991021 = 119891 (011987211991021)

11986311991081 = 119891 (011987211991081)

(17)


(4)

[[[[[[

[

11989611

11989621

11989681

]]]]]]

]

=

[[[[[[

[

1198631199101119904

1198631199102119904

1198631199108119904

]]]]]]

]

(18)

(5)

[[[[[[

[

11989612

11989622

11989682

]]]]]]

]

=

[[[[[[

[

1198911 (119905119904 + ℎ119872119910119904 + ℎ11989611)

1198912 (119905119904 + ℎ119872119910119904 + ℎ11989621)

1198918 (119905119904 + ℎ119872119910119904 + ℎ11989681)

]]]]]]

]

(19)

(6)

[[[[[[

[

1198721199101(119904+1)

1198721199102(119904+1)

1198721199108(119904+1)

]]]]]]

]

=

[[[[[[

[

1198721199101119904

1198721199102119904

1198721199108119904

]]]]]]

]

+ℎ

2

[[[[[[

[

11989611 + 11989612

11989621 + 11989622

11989681 + 11989682

]]]]]]

]

(20)

(7)

[[[[[[

[

1198631199101(119904+1)

1198631199102(119904+1)

1198631199108(119904+1)

]]]]]]

]

=

[[[[[[

[

1198911 (119905119904+11198721199101(119904+1))

1198912 (119905119904+11198721199102(119904+1))

1198918 (119905119904+11198721199108(119904+1))

]]]]]]

]

(21)




[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205971198911

1205971199101

0 sdot sdot sdot 0

01205971198912

1205971199102

sdot sdot sdot 0

d

0 0 1205971198918

1205971199108

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



d


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

(22)


Set 120582119894119883 = max0le119905le119879|120597119891119894120597119910119894| 120582119883 = max(120582119894119883) (119894 =


4 Results



11990110 = 80 11990120 = 80 11990130 = 80 11990140 = 805

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(23)


11990110 = 78 11990120 = 78 11990130 = 78 11990140 = 795

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(24)





775 78 785 79 795 80

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

p2 (kPa)

p998400 2

(kPa

s)

p20 = 78

p20 = 80


785 79 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 795





785 7978 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 785



1198961 = 4Ncm 11989631 = 02Ncm 11989632 = 02Ncm

1198911 = 01 11989131 = 11989132 = 01 1198914 = 05

1198651 = 94N 119865310 = 015N

119865320 = 015N 1198654 = 15N(25)


1198961 = 35Ncm 11989631 = 017Ncm

11989632 = 017Ncm

1198911 = 006 11989131 = 11989132 = 005 1198914 = 05

1198651 = 94N 119865310 = 02N

119865320 = 02N 1198654 = 15N

(26)



0 2 4 6 8 10 12 14 16 18 20

78

785

79

795

80

805

Time (s)

p4

(kPa

)

p40 = 805

p40 = 785


5 Conclusion


Nomenclature
























References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205971198911

1205971199101

0 sdot sdot sdot 0

01205971198912

1205971199102

sdot sdot sdot 0

d

0 0 1205971198918

1205971199108

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



d


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[[

[

1199101

1199102

1199108

]]]]]]

]

(22)


Set 120582119894119883 = max0le119905le119879|120597119891119894120597119910119894| 120582119883 = max(120582119894119883) (119894 =


4 Results



11990110 = 80 11990120 = 80 11990130 = 80 11990140 = 805

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(23)


11990110 = 78 11990120 = 78 11990130 = 78 11990140 = 795

11989710 = 0 119897310 = 0 119897320 = 0 11989740 = 0

(24)





775 78 785 79 795 80

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

p2 (kPa)

p998400 2

(kPa

s)

p20 = 78

p20 = 80


785 79 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 795





785 7978 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 785



1198961 = 4Ncm 11989631 = 02Ncm 11989632 = 02Ncm

1198911 = 01 11989131 = 11989132 = 01 1198914 = 05

1198651 = 94N 119865310 = 015N

119865320 = 015N 1198654 = 15N(25)


1198961 = 35Ncm 11989631 = 017Ncm

11989632 = 017Ncm

1198911 = 006 11989131 = 11989132 = 005 1198914 = 05

1198651 = 94N 119865310 = 02N

119865320 = 02N 1198654 = 15N

(26)



0 2 4 6 8 10 12 14 16 18 20

78

785

79

795

80

805

Time (s)

p4

(kPa

)

p40 = 805

p40 = 785


5 Conclusion


Nomenclature
























References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










785 7978 795 80 805

minus5

minus4

minus3

minus2

minus1

0

1

p4 (kPa)

p998400 4

(kPa

s)

p40 = 805

p40 = 785



1198961 = 4Ncm 11989631 = 02Ncm 11989632 = 02Ncm

1198911 = 01 11989131 = 11989132 = 01 1198914 = 05

1198651 = 94N 119865310 = 015N

119865320 = 015N 1198654 = 15N(25)


1198961 = 35Ncm 11989631 = 017Ncm

11989632 = 017Ncm

1198911 = 006 11989131 = 11989132 = 005 1198914 = 05

1198651 = 94N 119865310 = 02N

119865320 = 02N 1198654 = 15N

(26)



0 2 4 6 8 10 12 14 16 18 20

78

785

79

795

80

805

Time (s)

p4

(kPa

)

p40 = 805

p40 = 785


5 Conclusion


Nomenclature
























References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Stability Analysis of Pneumatic Cabin