Research Article Increase in Equilibrium Price by Fast ...downloads.hindawi.com/archive/2014/593254.pdfResearch Article Increase in Equilibrium Price by Fast Oscillations ... Using

Research ArticleIncrease in Equilibrium Price by Fast Oscillations

Babar Ahmad1 and Khalid Iqbal Mahr2

1 COMSATS Institute of Information Technology Islamabad 44000 Pakistan2Muhammad Ali Jinnah University Islamabad 44000 Pakistan

Correspondence should be addressed to Babar Ahmad babarsmsgmailcom

Received 15 January 2014 Revised 25 April 2014 Accepted 25 April 2014 Published 20 May 2014

Academic Editor Ivo Petras

Copyright copy 2014 B Ahmad and K I Mahr This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The dynamics of a market can be described by a differential equation Using the concept of fast oscillation the system (typicalmarket) can also oscillate around a new equilibrium price with an increase Previously that increase was established by applyingharmonic force In present work harmonic force is replaced by an arbitrary periodic force with zero mean Hence the increase inequilibrium price can be controlled by varying the external arbitrary periodic force

1 Introduction

The statistical physics and nonlinear dynamics can beemployed as tools in economics and social studies [1] to buildup econophysics [2] and statistical finance [3] Examples aretrading and price formation [4] excess and clustering ofstochastic volatility [5 6] investigation of scaling [7] of thecompetitive equilibrium [8 9] and role of noise to increasestability [10 11] in many physical systems

Using Kapitza method [12] Landau and Lifshitz dis-cussed the stability of the inverted pendulumunder fast oscil-lation He showed that when the suspension of a pendulumhas vertical modulation with harmonic force the position120601 = 0 is always stable and 120601 = 120587 is conditionally stable [13]

Using this approach in the market Holyst and Woj-ciechowski have shown that due to fast oscillation a newequilibrium price can occur Using external harmonic forcethis new equilibrium price is proportional to the difference11986310158401015840(119901) minus 11987810158401015840(119901) Hence due to fast oscillation the equilibriumprice of the market will increase [14]

In 2009 Ahmad and Borisenok extended the idea of sta-bility for arbitrary periodic force and stabilized the invertedpendulum with relatively low frequency They used periodickicking pulses in place of harmonic force Then the condi-tional stable point is controlled by varying external periodicforce [15]

In this paper the dynamics of the market is studied alongwith external arbitrary periodic force with zero mean Thenanother equilibriumprice can be establishedwith an increaseThis increase can be controlled by applying a particularperiodic force

2 Kapitza Method for ArbitraryPeriodic Force

A particle of mass 119898 is moving under a force due to time-independent potential 119880(119909)

1198911(119909) = minus

119889119880

119889119909(1)

and a periodic fast oscillating force with zero mean This fastoscillating force in Fourier expansion is

1198912(119909 119905)

=infin

sum119896=1

[119886119896(119909) cos (119896120596119905) + 119887

119896(119909) sin (119896120596119905)]

(2)

Here 120596 equiv 2120587119879 ≫ 21205871198790equiv 1205960and 120596

0is the frequency of

motion due to 1198911 Themean value of a function is denoted by

bar and is defined as

119891 =1

119879int119879

0

119891 (119909 119905) 119889119905 (3)

Hindawi Publishing CorporationJournal of Nonlinear DynamicsVolume 2014 Article ID 593254 8 pageshttpdxdoiorg1011552014593254

2 Journal of Nonlinear Dynamics

O

f(x t)

l

g

120601 = 0

120601

Figure 1 Vertical modulation

Also the Fourier coefficient 1198860is

1198860(119909) =

2

119879int119879

0

1198912(119909 119905) 119889119905 (4)

Since we are choosing a force with zero mean then from (3)and (4) it follows that

119891 cong 1198860= 0 (5)

In (2) 119886119896and 119887119896are the Fourier coefficients given by

119886119896(119909) =

2

119879int119879

0

1198912(119909 119905) cos 119896120596119905 119889119905

119887119896(119909) =

2

119879int119879

0

1198912(119909 119905) sin 119896120596119905 119889119905

(6)

Due to (1) and (2) the equation of motion is

119898 = 1198911(119909) + 119891

2(119909 119905) (7)

Equation (7) represents that the system has two motions at atime one along a smooth path and the other small but fastoscillations So the path can be written as

119909 (119905) = 119883 (119905) + 120585 (119905) (8)

Here 119883(119905) represents smooth path and 120585(119905) represents fastoscillations By averaging procedure the effective potentialenergy function is [15]

119880eff = 119880 +1

41198981205962

infin

sum119896=1

(1198862119896+ 1198872119896)

1198962 (9)

21 Stability Kapitza pendulum modulated vertically byarbitrary periodic force 119891(119909 119905) is illustrated in Figure 1Where 119892 is acceleration due to gravity 119897 is the length of themassless string The system is stabilized by minimizing (9)The position 120601 = 0 is always stable (see Figure 2(a)) whilethe position 120601 = 120587 is stable if 1205962 gt 05119892119897suminfin

119896=1(1198962(1198862

119896+ 1198872119896))

(see Figure 2(b)) The sum suminfin

119896=1((1198862119896+ 1198872119896)1198962) is associated

with external force It follows that due to fast oscillation theinverted position may also be stable [15]

The concept of fast oscillation is used in the marketThen another equilibrium price with an increase can beselected This increase can be controlled by varying theexternal periodic force

3 Effect of Fast Oscillation onEquilibrium Price

Almost every commodity hasmore price than its actual priceas some external factors are involved in that increase Inthis paper a mathematical approach is presented how anauthority can increasedecrease the price of the commodity

Let 119901(119905) be the commodity price at any time 119905 The timerate of change of 119901(119905) can be assumed proportional to thedifference between the demand 119863(119901) and the supply 119878(119901)functions that is

= 120573 [119863 (119901) minus 119878 (119901)] (10)

where 120573 is proportionally constant Let 119901 = 119901lowast be theequilibrium price then

119863(119901lowast) = 119878 (119901

lowast) (11)

Next a fast oscillating periodic forcewith zeromean is appliedto the dynamics of the market then (10) takes the form

= 120573 [119863 (119901) minus 119878 (119901)]

+infin

sum119896=1

(119886119896cos 119896120596119905 + 119887

119896sin 119896120596119905)

(12)

Here 120596 = 2120587119879 ≫ 1205960= 2120587119879

0is the frequency of market

perturbation with 1205960which is the approaching speed of (10)

towards equilibrium price In (12)

119886119896= 119886119896(119901)

119887119896= 119887119896(119901)

(13)

are price dependent amplitudesLike Kapitza method the price 119901(119905) can be split into its

slow 120601(119905) and fast 120585(119905) components as

119901 (119905) = 120601 (119905) + 120585 (119905) (14)

and its time derivative is

= 120601 + 120585 (15)

We assume that 119886119896= 119886119896(120601) and 119887

119896= 119887119896(120601) Using above

transformations defined in (14) and (15) (12) becomes

120601 + 120585 = 120573 [119863 (120601 + 120585) minus 119878 (120601 + 120585)]

+infin

sum119896=1

(119886119896cos 119896120596119905 + 119887

119896sin 119896120596119905)

(16)

Next using Taylorrsquos series expansion up to 2nd order term(16) can be rewritten as

120601 + 120585 = 119865 (120601) + 1205851198651015840(120601) +

1

2120585211986510158401015840(120601)

+infin

sum119896=1

(119886119896cos 119896120596119905 + 119887

119896sin 119896120596119905)

(17)

where 119865(120601) = 120573[119863(119901) minus 119878(119901)]

Journal of Nonlinear Dynamics 3

minus120587 120587

120601

00

Uef

fmgl

(a)

0minus120587 120587

120601

0

Uef

fmgl

(b)

Figure 2 Minimization of dimensionless effective potential energy function

Now the slow and fast part must be separately equal forfast part we simply put

120585 =infin

sum119896=1

(119886119896cos 119896120596119905 + 119887

119896sin 119896120596119905) (18)

Integrating (18) with initial condition 1205850= 0 the fast

component is

120585 =1

120596

infin

sum119896=1

1

119896(119886119896sin 119896120596119905 minus 119887

119896cos 119896120596119905) (19)

Using (3) the mean values of 120585 and 120585 are zero while 120601 and 120601are unaltered Then the mean values of (14) and (15) can begiven as

119901 = 120601

= 120601(20)

Hence 120601 describes the slow price and 120601 describes the rate ofslow price averaged over rapid oscillations

Next the mean values of 1205852 over the time interval [0 119879]are

1205852 =1

21205962

infin

sum119896=1

(1198862119896+ 1198872119896

1198962) (21)

Also 119865(120601) and its derivatives remain unaltered during thistime averaging Next time averaging of (17) will give afunction of 120601 only That is

120601 = 119865 (120601) +1

41205962

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)11986510158401015840(120601) (22)

or

= 120573([119863 minus 119878] (120601) +1

41205962

infin

sum119896=1

(1198862119896+ 1198872119896

1198962) [11986310158401015840minus 11987810158401015840] (120601))

(23)

It shows that after averaging the influence of fast periodicoscillations on the slow price component is also dependingon curvatures of demand and supply functions and thiseffect vanishes when both functions are linear Due to thisfast oscillation the averaged equilibrium price can be easilycalculated Expanding first term on right hand side of (23)into the power series around the value 119901lowast and consideringlinear part only we have

= 120573( [1198631015840(120601) minus 119878

1015840(120601)] (120601 minus 120601

lowast)

+1

41205962[11986310158401015840(120601) minus 119878

10158401015840(120601)]infin

sum119896=1

(1198862119896+ 1198872119896

1198962))

(24)

Following (11) at equilibrium price we must have = 0

0 = 120573( [1198631015840(120601) minus 119878

1015840(120601)] (120601 minus 120601

lowast)

+1

41205962[11986310158401015840(120601) minus 119878

10158401015840(120601)]infin

sum119896=1

(1198862119896+ 1198872119896

1198962))

(25)

consequently

120601 minus 120601lowast=1

4120596211986310158401015840 (120601) minus 11987810158401015840 (120601)

1198781015840 (120601) minus 1198631015840 (120601)

infin

sum119896=1

(1198862119896+ 1198872119896

1198962) (26)

The right hand side of (26) is a function of 120601 only Sinceafter averaging 120601 changes slightly that is 120601 asymp 120601 Also at


equilibrium 120601 is very near to 120601lowast while 119901lowast can be located farfrom 119901lowast Hence for good approximation we can write

119901lowast minus 119901lowast= Δ119901lowast=1

4120596211986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)

(27)

Equation (27) gives the shift of equilibrium price of themarket due to fast oscillations

This shift can increase the equilibrium price of a typicalmarket if

(a) the infinite sum increases(b) the frequency of small oscillation is small(c)

11986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)gt 0 (28)

In Kapitza method of averaging for arbitrary periodic forcethe increase in infinite sum will decrease the frequency ofoscillation at 120601 = 120587 To follow (28) we must suppose that1198631015840(119901) lt 0 11986310158401015840(119901) gt 0 1198781015840(119901) gt 0 and 11987810158401015840(119901) lt 0 Thenit follows that as price 119901(119905) increases the demand 119863(119901)decreases and 119878(119901) increases in a slow manner As a resultsaturation in demand and supply is established Hence byaveraging procedure the equilibrium price has shifted upChoose

11986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)= 1198622gt 0 (29)

Then (27) can be rewritten as

Δ119901lowast=

1

412059621198622

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)

= 025119860119878119896

(30)

where 119860 = 11986221205962

4 Shift of Market Equilibrium by ExternalArbitrary Periodic Force

In this section we will select some arbitrary periodic forcesand study the increase in equilibrium price by using (30)

41 Harmonic Force First we choose the harmonic force

119891 (119905) = sin120596119905 (31)

as external force (see Figure 3) Using (4) the Fouriercoefficient 119886

0= 0 indicates that the mean value of 119891(119905) about

its period 119879 = 2120587120596 is zero Next using (6) the other Fouriercoefficients for (31) are

119886119896= 0

119887119896=

0 119896 = 1

1 119896 = 1

(32)

1

05

0

minus05

minus1

2120587120587

t

1205874 1205872 31205874 51205874 31205872 71205874

f(t)

Figure 3 Sine type external force

Quantity

Pric

e

0

Equilibrium price due to external periodic forceEquilibrium price without external periodic force

D

Q

P1

P2

S

Figure 4 Change in equilibrium due to external force

Using these coefficients in (30) the old equilibrium price isshifted to new equilibrium price by [14]

Δ119901lowast= 025119860 (33)

This shift is illustrated in Figure 4

42 Triangular Force Next we apply periodical triangulartype force 119877

119904(119905) = 119877

119904(119905 + 119879)

119877119878(119905) =

4119905

119879if 0 le 119905 lt 119879

4

4

119879(119879

2minus 119905) if 119879

4le 119905 lt

3119879

4

4 (119905 minus 119879)

119879if 31198794le 119905 lt 119879

(34)


with the same property 119877119878= 0 (see Figure 5) Next using (6)

the other Fourier coefficients for (34) are

119886119896= 0

119887119896=

4

11989621205872[1 minus (minus1)

119896]

(35)

or

119887119896=

0 119896 is even8

11989621205872119896 is odd

(36)

Using these coefficients in (30) the shift in equilibrium priceis

Δ119901lowast= 025119860

64

1205874

infin

sum119896=1

1

(2119896 minus 1)6

=16

12058741205876

960119860

= 0164119860

(37)

Here (37) gives the shift of new equilibrium price from theold one due to fast oscillation with triangular external forceThis increase is lower than the increase with harmonic force

43 Rectangular Force The next force is rectangular typeforce 119877

119897(119905) = 119877

119897(119905 + 119879) (see Figure 6) given by

119877119897(119905) =

1 0 le 119905 le119879

2

minus1119879

2le 119905 le 119879

(38)

with the same property 119877119897= 0

Next using (6) the other Fourier coefficients for (38) are

119886119896= 0

119887119896=

0 119896 is even4

119896120587119896 is odd

(39)

or

1198872119896minus1

=4

(2119896 minus 1) 120587 (40)


Δ119901lowast= 025119860

16

1205872

infin

sum119896=1

1

(2119896 minus 1)4

= 025119860 (1645)

= 0411119860

(41)

Here (41) gives the shift of new equilibrium price for rect-angular force It has raised up the equilibrium price thanthe previous forces Hence by applying a different force anincreasedecrease in equilibrium price is possible

0

1

TT4 3T4

minus1

Figure 5 Triangular type force

0

1

TT2

minus1

Figure 6 Rectangular type force

5 Conclusions

A method similar to Kapitza method of averaging for anarbitrary periodic force is used in the dynamics of themarketThen another equilibrium price with an increase may bepossible Previously harmonic force was used to raise theequilibrium price of the market In this work an arbitraryperiodic force with zero mean is applied to raise it Now thisincrease can be controlled by varying the external arbitraryperiodic force On the same pattern when the governmentneeds money it announces an increase in the rates of existingtaxes or adds new taxes and when it wants to give relief topublic it decreases the rates of existing taxes or deletes sometaxes in the price of commodity

In September 2012 compressed natural gas (CNG) wasbeing sold at Rs 9253 per kg in region I Details are in Table 1[16]

In Table 1 if 119860 was gas price then some external factorswere involved and 119864 was the next price again price wasdecided by adding some factors (federal price compressioncost profit and taxes) and the consumer price was 119871 Here 119864and 119871 were new equilibrium prices with an increase

Next Consumer Rights Commission of Pakistan did notconsider it a justified price so they pursued the SupremeCourt of Pakistan who on October 26 2012 passed an orderdeclaring to implement the July 1 2012 prices That pricedetail is in Table 2 [17]

Then the consumer was paying Rs 61 instead of Rs9253 per kg in region I


Table 1 CNG consumer price detailed break up wef October 22 2012

S number Components Region I Region IIRsKg RsMMBtu RsKg RsMMBtu

119860Average well headpricecost of gas paid toexploration companies

1804 35887 92 1648 35887 92

119861Operating cost of gascompanies 114 2269 6 104 2269 6

119862Return on investment togas companies Otherincomes (net of prior year

1804 35887 92 1648 35887 92

119863 adjustment if any etc) minus053 minus1063 minus3 minus049 minus1063 minus3

119864 = 119860 to119863 Average prescribed price ofnatural gas 1951 38827 100 1783 38826 100

119865

Sale price of natural gas forCNG stations as advised byFederal Government underSection 8(3) of OGRAordinance includingGDScross-subsidization

3519 70032 38 3214 70002 38

119866Operating cost of gasstations 2080 11380 22 2080 45301 25

119867Profit of CNG stationowners 1119 22260 12 1059 23061 13

119868 GIDC 1325 26357 14 918 20000 11119869 GST (25 of price 119865 + 1) 1210 24069 13 1033 22500 12

119870Differential margin forregion II mdash 150 3267 2

119871Total CNG consumer price(119865 minus 119870) 9253 184098 100 8454 184131 100

0

1

TT6 T2

minus1

minus12

12

Figure 7 Hat type force

In Table 1 the equilibrium price was raised up by addingsome factors and in Table 2 it was lowered down by deletingsome factors So when the authority needs money it raisedup the prices and when it wants to give relieve to people itlowers down the prices

A number of more examples can be found Every con-sumer has to pay Rs 35 as ptv (Pakistan television) fee and tosupport Neelum Jhelum project they are bearing the cost ofdelay and inefficiency and are paying a surcharge of 10 paisa

per unit in their electricity bills Moreover different slabs ofusage units have different rates of billing

Appendices

Here some more periodic forces with zero mean are given

A Hat Force

The first force is rectangular hat type (see Figure 7) definedby

119871119888(119905) =

1

2if 0 le 119905 lt 1

6119879

1 if 16119879 le 119905 lt

1

3119879

1

2if 13119879 le 119905 lt

1

2119879

minus1

2if 12119879 le 119905 lt

2

3119879

minus1 if 23119879 le 119905 lt

5

6119879

minus1

2if 56119879 le 119905 lt 119879

(A1)


Table 2 CNG consumer price detailed break up

S number Components of price Region I Region II

1

Cost of production(a) Cost of gas billed 3109 2840(b) Cost of compression 546 546total cost of production 3655 3386

2 (10ndash12) Retailer fixed profit 4386 4063

3

Taxes(a) GIDC 13 13(b) GST 21 cost of gas 6529 5964

19529 18964Total (1 + 2 + 3) 60465 56887

5 Consumer retail price (consumerwillingness to pay) 61 57

Its mean value about its period is zero Then by Fourierexpansion in place of (A1)

119886119896= 0

119887119896=1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)

(A2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962[1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)]2

= 025119860 (09208)

= 02302119860

(A3)

This force also lowered down the equilibrium price ascompared to harmonic force but raised up the triangular typeforce

B Trapezoidal Force

Thenext force is trapezoidal type force119879119898(119905) = 119879

119898(119905+119879) (see

Figure 8) given by

119879119898(119905) =

8119905

119879if 0 le 119905 lt 119879

8

1 if 1198798le 119905 lt

3119879

8

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt

7119879

8

8 (119905 minus 119879)

119879if 71198798le 119905 lt 119879

(B1)

since 1198860= 0 rArr 119879

119898= 0

Next the Fourier coefficients of (B1) are119886119896= 0

119887119896=16

12058721

1198962sin 119896120587

4

(B2)

0

1

TT8

minus1

3T8 5T8 7T8

Figure 8 Trapezoidal type force


Δ119901lowast= 025119860

256

1205874

infin

sum119896=1

1

1198966sin2119896120587

4

= 025119860 (13571)

= 03393119860

(B3)

Due to this force the equilibrium price has been raised up ascompared to hat type force

C Quadratic Force

The next force is quadratic type force 119876119888(119905) = 119876

119888(119905 + 119879) (see

Figure 9) given by

119876119888(119905) =

1 if 0 le 119905 lt 31198798

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt 119879

(C1)

with the same property 119876119888= 0 as 119886

0= 0 Then by Fourier

expansion in the place of (C1)

119886119896= 0

119887119896= (

2

119896120587+

8

12058721198962sin 119896120587

4)

(C2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962(2

119896120587+

8

12058721198962sin 119896120587

4)2

= 025119860 (15426)

= 03857119860

(C3)

It is observed that applying this external force the equilib-rium price has more raised up

All these results with conditional stable points are givenin Table 3

By applying a different force an increasedecrease inequilibrium price can be made


0

1

T

minus1

3T8 5T8

Figure 9 Quadratic type force

Table 3 Shift in price equilibrium by fast oscillation

Force typeSum

infin

sum119896=1

(1198862119896+ 1198872119896)

1198962

Stabilitycondition at 120601 = 120587

Shift in priceEquilibrium Δ119901lowast

Sin 1 1205962 gt 2119892119897 025119860

Triangular 0658 1205962 gt 30396119892119897 0164119860

Linear hat 09208 1205962 gt 2172119892119897 02302119860

Trapezium 13571 1205962 gt 14736119892119897 03393119860

Quadratic 15426 1205962 gt 12967119892119897 03857119860

Rectangular 1645 1205962 gt 12159119892119897 0411119860

Conflict of Interests

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

References

[1] R N Mantegna and H E Stanley An Introduction to Econo-physics Correlations and Complexity in Finance CambridgeUniversity Press Cambridge UK 1999


[3] J-P Bouchaud ldquoAn introduction to statistical financerdquo PhysicaA vol 313 no 1-2 pp 238ndash251 2002

[4] M G Daniels J D Farmer L Gillemot G Iori and E SmithldquoQuantitative model of price diffusion and market frictionbased on trading as a mechanistic random processrdquo PhysicalReview Letters vol 90 Article ID 108102 2003

[5] R Friedmann and W G Sanddorf-Kohle ldquoVolatility clusteringand nontrading days in Chinese stock marketsrdquo Journal ofEconomics and Business vol 54 no 2 pp 193ndash217 2002

[6] G Bonanno D Valenti and B Spagnolo ldquoMean escape time ina system with stochastic volatilityrdquo Physical Review E vol 75Article ID 016106 2007

[7] D Eliezer and I I Kogan ldquoScaling laws for the marketmicrostructure of the interdealer broker marketsrdquo SSRN eLi-brary 1998

[8] X Yiping R Chandramouli and C Cordeiro ldquoPrice dynamicsin competitive agile spectrum access marketsrdquo IEEE Journal onSelected Areas in Communications vol 25 no 3 pp 613ndash6212007

[9] DValenti B Spagnolo andG Bonanno ldquoHitting time distribu-tions in financial marketsrdquo Physica A vol 382 no 1 pp 311ndash3202007

[10] H Mizuta K Steiglitz and E Lirov ldquoEffects of price signalchoices on market stabilityrdquo Journal of Economic Behavior andOrganization vol 52 no 2 pp 235ndash251 2003

[11] G Bonanno D Valenti and B Spagnolo ldquoRole of noise ina market model with stochastic volatilityrdquo European PhysicalJournal B vol 53 no 3 pp 405ndash409 2006

[12] P L Kapitza ldquoDynamic stability of a pendulum with anoscillating point of suspensionrdquo Journal of Experimental andTheoretical Physics vol 21 pp 588ndash597 1951

[13] L D Landau and E M Lifshitz Mecanics PergamonPressButterworth Oxford UK 3rd edition 2005

[14] J A Hołyst and W Wojciechowski ldquoThe effect of Kapitzapendulum and price equilibriumrdquo Physica A vol 324 no 1-2pp 388ndash395 2003

[15] B Ahmad and S Borisenok ldquoControl of effective potentialminima for Kapitza oscillator by periodical kicking pulsesrdquoPhysics Letters A vol 373 no 7 pp 701ndash707 2009

[16] httpsupremecourtgovpkwebuser filesFileCONSTP33-34-2005pdf

[17] ldquoCRCP House Islamabadrdquo Islamabad Pakistan httpwwwcrcporgpk

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VLSI Design



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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Propagation




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DistributedSensor Networks



O

f(x t)

l

g

120601 = 0

120601

Figure 1 Vertical modulation

Also the Fourier coefficient 1198860is

1198860(119909) =

2

119879int119879

0

1198912(119909 119905) 119889119905 (4)

Since we are choosing a force with zero mean then from (3)and (4) it follows that

119891 cong 1198860= 0 (5)

In (2) 119886119896and 119887119896are the Fourier coefficients given by

119886119896(119909) =

2

119879int119879

0

1198912(119909 119905) cos 119896120596119905 119889119905

119887119896(119909) =

2

119879int119879

0

1198912(119909 119905) sin 119896120596119905 119889119905

(6)

Due to (1) and (2) the equation of motion is

119898 = 1198911(119909) + 119891

2(119909 119905) (7)

Equation (7) represents that the system has two motions at atime one along a smooth path and the other small but fastoscillations So the path can be written as

119909 (119905) = 119883 (119905) + 120585 (119905) (8)

Here 119883(119905) represents smooth path and 120585(119905) represents fastoscillations By averaging procedure the effective potentialenergy function is [15]

119880eff = 119880 +1

41198981205962

infin

sum119896=1

(1198862119896+ 1198872119896)

1198962 (9)

21 Stability Kapitza pendulum modulated vertically byarbitrary periodic force 119891(119909 119905) is illustrated in Figure 1Where 119892 is acceleration due to gravity 119897 is the length of themassless string The system is stabilized by minimizing (9)The position 120601 = 0 is always stable (see Figure 2(a)) whilethe position 120601 = 120587 is stable if 1205962 gt 05119892119897suminfin

119896=1(1198962(1198862

119896+ 1198872119896))

(see Figure 2(b)) The sum suminfin

119896=1((1198862119896+ 1198872119896)1198962) is associated

with external force It follows that due to fast oscillation theinverted position may also be stable [15]

The concept of fast oscillation is used in the marketThen another equilibrium price with an increase can beselected This increase can be controlled by varying theexternal periodic force

3 Effect of Fast Oscillation onEquilibrium Price

Almost every commodity hasmore price than its actual priceas some external factors are involved in that increase Inthis paper a mathematical approach is presented how anauthority can increasedecrease the price of the commodity

Let 119901(119905) be the commodity price at any time 119905 The timerate of change of 119901(119905) can be assumed proportional to thedifference between the demand 119863(119901) and the supply 119878(119901)functions that is

= 120573 [119863 (119901) minus 119878 (119901)] (10)

where 120573 is proportionally constant Let 119901 = 119901lowast be theequilibrium price then

119863(119901lowast) = 119878 (119901

lowast) (11)

Next a fast oscillating periodic forcewith zeromean is appliedto the dynamics of the market then (10) takes the form

= 120573 [119863 (119901) minus 119878 (119901)]

+infin

sum119896=1

(119886119896cos 119896120596119905 + 119887

119896sin 119896120596119905)

(12)

Here 120596 = 2120587119879 ≫ 1205960= 2120587119879

0is the frequency of market

perturbation with 1205960which is the approaching speed of (10)

towards equilibrium price In (12)

119886119896= 119886119896(119901)

119887119896= 119887119896(119901)

(13)

are price dependent amplitudesLike Kapitza method the price 119901(119905) can be split into its

slow 120601(119905) and fast 120585(119905) components as

119901 (119905) = 120601 (119905) + 120585 (119905) (14)

and its time derivative is

= 120601 + 120585 (15)

We assume that 119886119896= 119886119896(120601) and 119887

119896= 119887119896(120601) Using above

transformations defined in (14) and (15) (12) becomes

120601 + 120585 = 120573 [119863 (120601 + 120585) minus 119878 (120601 + 120585)]

+infin

sum119896=1

(119886119896cos 119896120596119905 + 119887

119896sin 119896120596119905)

(16)

Next using Taylorrsquos series expansion up to 2nd order term(16) can be rewritten as

120601 + 120585 = 119865 (120601) + 1205851198651015840(120601) +

1

2120585211986510158401015840(120601)

+infin

sum119896=1

(119886119896cos 119896120596119905 + 119887

119896sin 119896120596119905)

(17)

where 119865(120601) = 120573[119863(119901) minus 119878(119901)]


minus120587 120587

120601

00

Uef

fmgl

(a)

0minus120587 120587

120601

0

Uef

fmgl

(b)



120585 =infin

sum119896=1

(119886119896cos 119896120596119905 + 119887

119896sin 119896120596119905) (18)


component is

120585 =1

120596

infin

sum119896=1

1

119896(119886119896sin 119896120596119905 minus 119887

119896cos 119896120596119905) (19)


119901 = 120601

= 120601(20)



1205852 =1

21205962

infin

sum119896=1

(1198862119896+ 1198872119896

1198962) (21)


120601 = 119865 (120601) +1

41205962

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)11986510158401015840(120601) (22)

or

= 120573([119863 minus 119878] (120601) +1

41205962

infin

sum119896=1

(1198862119896+ 1198872119896

1198962) [11986310158401015840minus 11987810158401015840] (120601))

(23)


= 120573( [1198631015840(120601) minus 119878

1015840(120601)] (120601 minus 120601

lowast)

+1

41205962[11986310158401015840(120601) minus 119878

10158401015840(120601)]infin

sum119896=1

(1198862119896+ 1198872119896

1198962))

(24)


0 = 120573( [1198631015840(120601) minus 119878

1015840(120601)] (120601 minus 120601

lowast)

+1

41205962[11986310158401015840(120601) minus 119878

10158401015840(120601)]infin

sum119896=1

(1198862119896+ 1198872119896

1198962))

(25)

consequently


4120596211986310158401015840 (120601) minus 11987810158401015840 (120601)

1198781015840 (120601) minus 1198631015840 (120601)

infin

sum119896=1

(1198862119896+ 1198872119896

1198962) (26)





4120596211986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)

(27)




11986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)gt 0 (28)


11986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)= 1198622gt 0 (29)


Δ119901lowast=

1

412059621198622

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)

= 025119860119878119896

(30)

where 119860 = 11986221205962




119891 (119905) = sin120596119905 (31)




119886119896= 0

119887119896=

0 119896 = 1

1 119896 = 1

(32)

1

05

0

minus05

minus1

2120587120587

t

1205874 1205872 31205874 51205874 31205872 71205874

f(t)


Quantity

Pric

e

0


D

Q

P1

P2

S



Δ119901lowast= 025119860 (33)



119904(119905) = 119877

119904(119905 + 119879)

119877119878(119905) =

4119905

119879if 0 le 119905 lt 119879

4

4

119879(119879

2minus 119905) if 119879

4le 119905 lt

3119879

4

4 (119905 minus 119879)

119879if 31198794le 119905 lt 119879

(34)




119886119896= 0

119887119896=

4

11989621205872[1 minus (minus1)

119896]

(35)

or

119887119896=

0 119896 is even8

11989621205872119896 is odd

(36)


Δ119901lowast= 025119860

64

1205874

infin

sum119896=1

1

(2119896 minus 1)6

=16

12058741205876

960119860

= 0164119860

(37)



119897(119905) = 119877


119877119897(119905) =

1 0 le 119905 le119879

2

minus1119879

2le 119905 le 119879

(38)



119886119896= 0

119887119896=

0 119896 is even4

119896120587119896 is odd

(39)

or

1198872119896minus1

=4

(2119896 minus 1) 120587 (40)


Δ119901lowast= 025119860

16

1205872

infin

sum119896=1

1

(2119896 minus 1)4

= 025119860 (1645)

= 0411119860

(41)


0

1

TT4 3T4

minus1


0

1

TT2

minus1


5 Conclusions










1804 35887 92 1648 35887 92



1804 35887 92 1648 35887 92



119865


3519 70032 38 3214 70002 38



119868 GIDC 1325 26357 14 918 20000 11119869 GST (25 of price 119865 + 1) 1210 24069 13 1033 22500 12



0

1

TT6 T2

minus1

minus12

12





Appendices


A Hat Force


119871119888(119905) =

1

2if 0 le 119905 lt 1

6119879

1 if 16119879 le 119905 lt

1

3119879

1

2if 13119879 le 119905 lt

1

2119879

minus1

2if 12119879 le 119905 lt

2

3119879

minus1 if 23119879 le 119905 lt

5

6119879

minus1

2if 56119879 le 119905 lt 119879

(A1)




1



3


19529 18964Total (1 + 2 + 3) 60465 56887



119886119896= 0

119887119896=1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)

(A2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962[1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)]2

= 025119860 (09208)

= 02302119860

(A3)


B Trapezoidal Force


119898(119905+119879) (see

Figure 8) given by

119879119898(119905) =

8119905

119879if 0 le 119905 lt 119879

8

1 if 1198798le 119905 lt

3119879

8

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt

7119879

8

8 (119905 minus 119879)

119879if 71198798le 119905 lt 119879

(B1)

since 1198860= 0 rArr 119879

119898= 0


119887119896=16

12058721

1198962sin 119896120587

4

(B2)

0

1

TT8

minus1

3T8 5T8 7T8



Δ119901lowast= 025119860

256

1205874

infin

sum119896=1

1

1198966sin2119896120587

4

= 025119860 (13571)

= 03393119860

(B3)


C Quadratic Force


119888(119905 + 119879) (see

Figure 9) given by

119876119888(119905) =

1 if 0 le 119905 lt 31198798

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt 119879

(C1)




119886119896= 0

119887119896= (

2

119896120587+

8

12058721198962sin 119896120587

4)

(C2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962(2

119896120587+

8

12058721198962sin 119896120587

4)2

= 025119860 (15426)

= 03857119860

(C3)





0

1

T

minus1

3T8 5T8



Force typeSum

infin

sum119896=1

(1198862119896+ 1198872119896)

1198962



Sin 1 1205962 gt 2119892119897 025119860

Triangular 0658 1205962 gt 30396119892119897 0164119860

Linear hat 09208 1205962 gt 2172119892119897 02302119860

Trapezium 13571 1205962 gt 14736119892119897 03393119860

Quadratic 15426 1205962 gt 12967119892119897 03857119860

Rectangular 1645 1205962 gt 12159119892119897 0411119860



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus120587 120587

120601

00

Uef

fmgl

(a)

0minus120587 120587

120601

0

Uef

fmgl

(b)



120585 =infin

sum119896=1

(119886119896cos 119896120596119905 + 119887

119896sin 119896120596119905) (18)


component is

120585 =1

120596

infin

sum119896=1

1

119896(119886119896sin 119896120596119905 minus 119887

119896cos 119896120596119905) (19)


119901 = 120601

= 120601(20)



1205852 =1

21205962

infin

sum119896=1

(1198862119896+ 1198872119896

1198962) (21)


120601 = 119865 (120601) +1

41205962

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)11986510158401015840(120601) (22)

or

= 120573([119863 minus 119878] (120601) +1

41205962

infin

sum119896=1

(1198862119896+ 1198872119896

1198962) [11986310158401015840minus 11987810158401015840] (120601))

(23)


= 120573( [1198631015840(120601) minus 119878

1015840(120601)] (120601 minus 120601

lowast)

+1

41205962[11986310158401015840(120601) minus 119878

10158401015840(120601)]infin

sum119896=1

(1198862119896+ 1198872119896

1198962))

(24)


0 = 120573( [1198631015840(120601) minus 119878

1015840(120601)] (120601 minus 120601

lowast)

+1

41205962[11986310158401015840(120601) minus 119878

10158401015840(120601)]infin

sum119896=1

(1198862119896+ 1198872119896

1198962))

(25)

consequently


4120596211986310158401015840 (120601) minus 11987810158401015840 (120601)

1198781015840 (120601) minus 1198631015840 (120601)

infin

sum119896=1

(1198862119896+ 1198872119896

1198962) (26)





4120596211986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)

(27)




11986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)gt 0 (28)


11986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)= 1198622gt 0 (29)


Δ119901lowast=

1

412059621198622

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)

= 025119860119878119896

(30)

where 119860 = 11986221205962




119891 (119905) = sin120596119905 (31)




119886119896= 0

119887119896=

0 119896 = 1

1 119896 = 1

(32)

1

05

0

minus05

minus1

2120587120587

t

1205874 1205872 31205874 51205874 31205872 71205874

f(t)


Quantity

Pric

e

0


D

Q

P1

P2

S



Δ119901lowast= 025119860 (33)



119904(119905) = 119877

119904(119905 + 119879)

119877119878(119905) =

4119905

119879if 0 le 119905 lt 119879

4

4

119879(119879

2minus 119905) if 119879

4le 119905 lt

3119879

4

4 (119905 minus 119879)

119879if 31198794le 119905 lt 119879

(34)




119886119896= 0

119887119896=

4

11989621205872[1 minus (minus1)

119896]

(35)

or

119887119896=

0 119896 is even8

11989621205872119896 is odd

(36)


Δ119901lowast= 025119860

64

1205874

infin

sum119896=1

1

(2119896 minus 1)6

=16

12058741205876

960119860

= 0164119860

(37)



119897(119905) = 119877


119877119897(119905) =

1 0 le 119905 le119879

2

minus1119879

2le 119905 le 119879

(38)



119886119896= 0

119887119896=

0 119896 is even4

119896120587119896 is odd

(39)

or

1198872119896minus1

=4

(2119896 minus 1) 120587 (40)


Δ119901lowast= 025119860

16

1205872

infin

sum119896=1

1

(2119896 minus 1)4

= 025119860 (1645)

= 0411119860

(41)


0

1

TT4 3T4

minus1


0

1

TT2

minus1


5 Conclusions










1804 35887 92 1648 35887 92



1804 35887 92 1648 35887 92



119865


3519 70032 38 3214 70002 38



119868 GIDC 1325 26357 14 918 20000 11119869 GST (25 of price 119865 + 1) 1210 24069 13 1033 22500 12



0

1

TT6 T2

minus1

minus12

12





Appendices


A Hat Force


119871119888(119905) =

1

2if 0 le 119905 lt 1

6119879

1 if 16119879 le 119905 lt

1

3119879

1

2if 13119879 le 119905 lt

1

2119879

minus1

2if 12119879 le 119905 lt

2

3119879

minus1 if 23119879 le 119905 lt

5

6119879

minus1

2if 56119879 le 119905 lt 119879

(A1)




1



3


19529 18964Total (1 + 2 + 3) 60465 56887



119886119896= 0

119887119896=1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)

(A2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962[1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)]2

= 025119860 (09208)

= 02302119860

(A3)


B Trapezoidal Force


119898(119905+119879) (see

Figure 8) given by

119879119898(119905) =

8119905

119879if 0 le 119905 lt 119879

8

1 if 1198798le 119905 lt

3119879

8

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt

7119879

8

8 (119905 minus 119879)

119879if 71198798le 119905 lt 119879

(B1)

since 1198860= 0 rArr 119879

119898= 0


119887119896=16

12058721

1198962sin 119896120587

4

(B2)

0

1

TT8

minus1

3T8 5T8 7T8



Δ119901lowast= 025119860

256

1205874

infin

sum119896=1

1

1198966sin2119896120587

4

= 025119860 (13571)

= 03393119860

(B3)


C Quadratic Force


119888(119905 + 119879) (see

Figure 9) given by

119876119888(119905) =

1 if 0 le 119905 lt 31198798

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt 119879

(C1)




119886119896= 0

119887119896= (

2

119896120587+

8

12058721198962sin 119896120587

4)

(C2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962(2

119896120587+

8

12058721198962sin 119896120587

4)2

= 025119860 (15426)

= 03857119860

(C3)





0

1

T

minus1

3T8 5T8



Force typeSum

infin

sum119896=1

(1198862119896+ 1198872119896)

1198962



Sin 1 1205962 gt 2119892119897 025119860

Triangular 0658 1205962 gt 30396119892119897 0164119860

Linear hat 09208 1205962 gt 2172119892119897 02302119860

Trapezium 13571 1205962 gt 14736119892119897 03393119860

Quadratic 15426 1205962 gt 12967119892119897 03857119860

Rectangular 1645 1205962 gt 12159119892119897 0411119860



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












4120596211986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)

(27)




11986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)gt 0 (28)


11986310158401015840 (119901) minus 11987810158401015840 (119901)

1198781015840 (119901) minus 1198631015840 (119901)= 1198622gt 0 (29)


Δ119901lowast=

1

412059621198622

infin

sum119896=1

(1198862119896+ 1198872119896

1198962)

= 025119860119878119896

(30)

where 119860 = 11986221205962




119891 (119905) = sin120596119905 (31)




119886119896= 0

119887119896=

0 119896 = 1

1 119896 = 1

(32)

1

05

0

minus05

minus1

2120587120587

t

1205874 1205872 31205874 51205874 31205872 71205874

f(t)


Quantity

Pric

e

0


D

Q

P1

P2

S



Δ119901lowast= 025119860 (33)



119904(119905) = 119877

119904(119905 + 119879)

119877119878(119905) =

4119905

119879if 0 le 119905 lt 119879

4

4

119879(119879

2minus 119905) if 119879

4le 119905 lt

3119879

4

4 (119905 minus 119879)

119879if 31198794le 119905 lt 119879

(34)




119886119896= 0

119887119896=

4

11989621205872[1 minus (minus1)

119896]

(35)

or

119887119896=

0 119896 is even8

11989621205872119896 is odd

(36)


Δ119901lowast= 025119860

64

1205874

infin

sum119896=1

1

(2119896 minus 1)6

=16

12058741205876

960119860

= 0164119860

(37)



119897(119905) = 119877


119877119897(119905) =

1 0 le 119905 le119879

2

minus1119879

2le 119905 le 119879

(38)



119886119896= 0

119887119896=

0 119896 is even4

119896120587119896 is odd

(39)

or

1198872119896minus1

=4

(2119896 minus 1) 120587 (40)


Δ119901lowast= 025119860

16

1205872

infin

sum119896=1

1

(2119896 minus 1)4

= 025119860 (1645)

= 0411119860

(41)


0

1

TT4 3T4

minus1


0

1

TT2

minus1


5 Conclusions










1804 35887 92 1648 35887 92



1804 35887 92 1648 35887 92



119865


3519 70032 38 3214 70002 38



119868 GIDC 1325 26357 14 918 20000 11119869 GST (25 of price 119865 + 1) 1210 24069 13 1033 22500 12



0

1

TT6 T2

minus1

minus12

12





Appendices


A Hat Force


119871119888(119905) =

1

2if 0 le 119905 lt 1

6119879

1 if 16119879 le 119905 lt

1

3119879

1

2if 13119879 le 119905 lt

1

2119879

minus1

2if 12119879 le 119905 lt

2

3119879

minus1 if 23119879 le 119905 lt

5

6119879

minus1

2if 56119879 le 119905 lt 119879

(A1)




1



3


19529 18964Total (1 + 2 + 3) 60465 56887



119886119896= 0

119887119896=1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)

(A2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962[1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)]2

= 025119860 (09208)

= 02302119860

(A3)


B Trapezoidal Force


119898(119905+119879) (see

Figure 8) given by

119879119898(119905) =

8119905

119879if 0 le 119905 lt 119879

8

1 if 1198798le 119905 lt

3119879

8

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt

7119879

8

8 (119905 minus 119879)

119879if 71198798le 119905 lt 119879

(B1)

since 1198860= 0 rArr 119879

119898= 0


119887119896=16

12058721

1198962sin 119896120587

4

(B2)

0

1

TT8

minus1

3T8 5T8 7T8



Δ119901lowast= 025119860

256

1205874

infin

sum119896=1

1

1198966sin2119896120587

4

= 025119860 (13571)

= 03393119860

(B3)


C Quadratic Force


119888(119905 + 119879) (see

Figure 9) given by

119876119888(119905) =

1 if 0 le 119905 lt 31198798

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt 119879

(C1)




119886119896= 0

119887119896= (

2

119896120587+

8

12058721198962sin 119896120587

4)

(C2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962(2

119896120587+

8

12058721198962sin 119896120587

4)2

= 025119860 (15426)

= 03857119860

(C3)





0

1

T

minus1

3T8 5T8



Force typeSum

infin

sum119896=1

(1198862119896+ 1198872119896)

1198962



Sin 1 1205962 gt 2119892119897 025119860

Triangular 0658 1205962 gt 30396119892119897 0164119860

Linear hat 09208 1205962 gt 2172119892119897 02302119860

Trapezium 13571 1205962 gt 14736119892119897 03393119860

Quadratic 15426 1205962 gt 12967119892119897 03857119860

Rectangular 1645 1205962 gt 12159119892119897 0411119860



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119886119896= 0

119887119896=

4

11989621205872[1 minus (minus1)

119896]

(35)

or

119887119896=

0 119896 is even8

11989621205872119896 is odd

(36)


Δ119901lowast= 025119860

64

1205874

infin

sum119896=1

1

(2119896 minus 1)6

=16

12058741205876

960119860

= 0164119860

(37)



119897(119905) = 119877


119877119897(119905) =

1 0 le 119905 le119879

2

minus1119879

2le 119905 le 119879

(38)



119886119896= 0

119887119896=

0 119896 is even4

119896120587119896 is odd

(39)

or

1198872119896minus1

=4

(2119896 minus 1) 120587 (40)


Δ119901lowast= 025119860

16

1205872

infin

sum119896=1

1

(2119896 minus 1)4

= 025119860 (1645)

= 0411119860

(41)


0

1

TT4 3T4

minus1


0

1

TT2

minus1


5 Conclusions










1804 35887 92 1648 35887 92



1804 35887 92 1648 35887 92



119865


3519 70032 38 3214 70002 38



119868 GIDC 1325 26357 14 918 20000 11119869 GST (25 of price 119865 + 1) 1210 24069 13 1033 22500 12



0

1

TT6 T2

minus1

minus12

12





Appendices


A Hat Force


119871119888(119905) =

1

2if 0 le 119905 lt 1

6119879

1 if 16119879 le 119905 lt

1

3119879

1

2if 13119879 le 119905 lt

1

2119879

minus1

2if 12119879 le 119905 lt

2

3119879

minus1 if 23119879 le 119905 lt

5

6119879

minus1

2if 56119879 le 119905 lt 119879

(A1)




1



3


19529 18964Total (1 + 2 + 3) 60465 56887



119886119896= 0

119887119896=1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)

(A2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962[1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)]2

= 025119860 (09208)

= 02302119860

(A3)


B Trapezoidal Force


119898(119905+119879) (see

Figure 8) given by

119879119898(119905) =

8119905

119879if 0 le 119905 lt 119879

8

1 if 1198798le 119905 lt

3119879

8

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt

7119879

8

8 (119905 minus 119879)

119879if 71198798le 119905 lt 119879

(B1)

since 1198860= 0 rArr 119879

119898= 0


119887119896=16

12058721

1198962sin 119896120587

4

(B2)

0

1

TT8

minus1

3T8 5T8 7T8



Δ119901lowast= 025119860

256

1205874

infin

sum119896=1

1

1198966sin2119896120587

4

= 025119860 (13571)

= 03393119860

(B3)


C Quadratic Force


119888(119905 + 119879) (see

Figure 9) given by

119876119888(119905) =

1 if 0 le 119905 lt 31198798

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt 119879

(C1)




119886119896= 0

119887119896= (

2

119896120587+

8

12058721198962sin 119896120587

4)

(C2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962(2

119896120587+

8

12058721198962sin 119896120587

4)2

= 025119860 (15426)

= 03857119860

(C3)





0

1

T

minus1

3T8 5T8



Force typeSum

infin

sum119896=1

(1198862119896+ 1198872119896)

1198962



Sin 1 1205962 gt 2119892119897 025119860

Triangular 0658 1205962 gt 30396119892119897 0164119860

Linear hat 09208 1205962 gt 2172119892119897 02302119860

Trapezium 13571 1205962 gt 14736119892119897 03393119860

Quadratic 15426 1205962 gt 12967119892119897 03857119860

Rectangular 1645 1205962 gt 12159119892119897 0411119860



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1804 35887 92 1648 35887 92



1804 35887 92 1648 35887 92



119865


3519 70032 38 3214 70002 38



119868 GIDC 1325 26357 14 918 20000 11119869 GST (25 of price 119865 + 1) 1210 24069 13 1033 22500 12



0

1

TT6 T2

minus1

minus12

12





Appendices


A Hat Force


119871119888(119905) =

1

2if 0 le 119905 lt 1

6119879

1 if 16119879 le 119905 lt

1

3119879

1

2if 13119879 le 119905 lt

1

2119879

minus1

2if 12119879 le 119905 lt

2

3119879

minus1 if 23119879 le 119905 lt

5

6119879

minus1

2if 56119879 le 119905 lt 119879

(A1)




1



3


19529 18964Total (1 + 2 + 3) 60465 56887



119886119896= 0

119887119896=1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)

(A2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962[1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)]2

= 025119860 (09208)

= 02302119860

(A3)


B Trapezoidal Force


119898(119905+119879) (see

Figure 8) given by

119879119898(119905) =

8119905

119879if 0 le 119905 lt 119879

8

1 if 1198798le 119905 lt

3119879

8

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt

7119879

8

8 (119905 minus 119879)

119879if 71198798le 119905 lt 119879

(B1)

since 1198860= 0 rArr 119879

119898= 0


119887119896=16

12058721

1198962sin 119896120587

4

(B2)

0

1

TT8

minus1

3T8 5T8 7T8



Δ119901lowast= 025119860

256

1205874

infin

sum119896=1

1

1198966sin2119896120587

4

= 025119860 (13571)

= 03393119860

(B3)


C Quadratic Force


119888(119905 + 119879) (see

Figure 9) given by

119876119888(119905) =

1 if 0 le 119905 lt 31198798

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt 119879

(C1)




119886119896= 0

119887119896= (

2

119896120587+

8

12058721198962sin 119896120587

4)

(C2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962(2

119896120587+

8

12058721198962sin 119896120587

4)2

= 025119860 (15426)

= 03857119860

(C3)





0

1

T

minus1

3T8 5T8



Force typeSum

infin

sum119896=1

(1198862119896+ 1198872119896)

1198962



Sin 1 1205962 gt 2119892119897 025119860

Triangular 0658 1205962 gt 30396119892119897 0164119860

Linear hat 09208 1205962 gt 2172119892119897 02302119860

Trapezium 13571 1205962 gt 14736119892119897 03393119860

Quadratic 15426 1205962 gt 12967119892119897 03857119860

Rectangular 1645 1205962 gt 12159119892119897 0411119860



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1



3


19529 18964Total (1 + 2 + 3) 60465 56887



119886119896= 0

119887119896=1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)

(A2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962[1

119896120587(1 minus cos 119896120587 + 2 cos 119896120587

3)]2

= 025119860 (09208)

= 02302119860

(A3)


B Trapezoidal Force


119898(119905+119879) (see

Figure 8) given by

119879119898(119905) =

8119905

119879if 0 le 119905 lt 119879

8

1 if 1198798le 119905 lt

3119879

8

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt

7119879

8

8 (119905 minus 119879)

119879if 71198798le 119905 lt 119879

(B1)

since 1198860= 0 rArr 119879

119898= 0


119887119896=16

12058721

1198962sin 119896120587

4

(B2)

0

1

TT8

minus1

3T8 5T8 7T8



Δ119901lowast= 025119860

256

1205874

infin

sum119896=1

1

1198966sin2119896120587

4

= 025119860 (13571)

= 03393119860

(B3)


C Quadratic Force


119888(119905 + 119879) (see

Figure 9) given by

119876119888(119905) =

1 if 0 le 119905 lt 31198798

8

119879(119879

2minus 119905) if 3119879

8le 119905 lt

5119879

8

minus1 if 51198798le 119905 lt 119879

(C1)




119886119896= 0

119887119896= (

2

119896120587+

8

12058721198962sin 119896120587

4)

(C2)


Δ119901lowast= 025119860

infin

sum119896=1

1

1198962(2

119896120587+

8

12058721198962sin 119896120587

4)2

= 025119860 (15426)

= 03857119860

(C3)





0

1

T

minus1

3T8 5T8



Force typeSum

infin

sum119896=1

(1198862119896+ 1198872119896)

1198962



Sin 1 1205962 gt 2119892119897 025119860

Triangular 0658 1205962 gt 30396119892119897 0164119860

Linear hat 09208 1205962 gt 2172119892119897 02302119860

Trapezium 13571 1205962 gt 14736119892119897 03393119860

Quadratic 15426 1205962 gt 12967119892119897 03857119860

Rectangular 1645 1205962 gt 12159119892119897 0411119860



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1

T

minus1

3T8 5T8



Force typeSum

infin

sum119896=1

(1198862119896+ 1198872119896)

1198962



Sin 1 1205962 gt 2119892119897 025119860

Triangular 0658 1205962 gt 30396119892119897 0164119860

Linear hat 09208 1205962 gt 2172119892119897 02302119860

Trapezium 13571 1205962 gt 14736119892119897 03393119860

Quadratic 15426 1205962 gt 12967119892119897 03857119860

Rectangular 1645 1205962 gt 12159119892119897 0411119860



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









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