Equations to Calculate Characteristic Frequencies of Multiple Chamber Aligned in Parallel Cavity Resonator (MCAP-CR)

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1 MCAPE Equations to Calculate Chaacteistic Fequencies of Multiple Chabe Aligne in Paallel Cavity Resonato (MCAP-CR) Shigeu Suzui Mach, (Revise in ovebe, ). Peface It is necessay to solve the equations of otion, in oe to estiate chaacteistic fequencies of Multiple Chabe Aligne in Paallel Cavity Resonato (MCAP-CR). I eplain the equations of otion of MCAP-CR an suggeste solution of the equation in this aticle. Set of equations of otions is well aange; howeve, it is not easy to solve the equations. Suggeste solution will be eplaine in Appeni-B.. Physical Moel of MCAP-CR MCAP-CR consists of ain chabe, sub-chabes, an ucts. Speae unit(s) is installe in the ain chabe, an sub-chabes ae connecte to ain chabe though ucts. One o oe sub-chabe has an open-to-ai uct. ube of sub-chabes (let us efine as ) ust be two o geate. Fig. shows illustation of an MCAP-CR that has fou sub-chabes. The chabe in the ile of Fig. is the ain chabe whee a speae unit is installe. Fou sub-chabes ae connecte to the ain chabe though ucts. All chabes have an open-toai uct. Each chabe acts as ai sping, an ai in each uct acts as ass. In this case this syste configues egees of feeo poble, because thee ae asses. One avantage of this syste is that it coul have twice as any chaacteistic fequencies of nube of sub-chabes. Soe of open-to-ai ucts ay be plugge in oe to change chaecteistic fequencies. ube of sub-chabes ay be two o oe an theoetically aiu nube is infinity; pactical liit will be appoiately eight Oiginal nae of the syste was MCAPSS (MCAP Speae Syste). S. Suzui

2 MCAPE Fig. Scheatic of MCAP-CR (). Basic Equations of Physical Moel fo MCAP-CR We coul say that MCAP-CR is a in of bass-efle syste that has ultiple chaacteistic fequency. It eans basic equations can be etene fo bass-efle equations. I begin with bass-efle equations an eten the equations to MCAP-CR, because govening equations of MCAP-CR ae oe cople. Equations of Single Bass-Refle Syste Single Bass-Refle syste is a cavity esonato. This esonato consists of one chabe an one uct. A chabe acts as ai sping an a uct acts as ass. This is siple vibation poble. Fig. shows cavity esonato an its equivalent oel. Fig. Physical Moel of Single Bass-Refle Syste S. Suzui

3 MCAPE Equations of cavity esonato ae eive using state equation of ieal gas. It is epesse in equation () une aiabatic conition. Whee, γ P constant () P: Absolute ai pessue insie the chabe [Pa] : Capacity of chabe[ ] γ: Ratio of Specific Heats ( γ. fo ai). Total eivative of equation () is epesse as: γ γ γ ( P ) P P γ. ( γ ) Multiplying both tes of above equation by, we get: P γ P γ P P () Hooe's law is epesse as equation () F () whee, : Sping constant of the chabe fo the ass[/] F: Foce acting to ass[]. ow, F a P a () whee, a: Coss sectional aea of the uct[ ]. Substituting equation () fo equation (), we get F a γ P a γ P. Theefoe, sping constant of the chabe is epesse as S. Suzui

4 MCAPE γ a P (5). By the way, sping constant of the chabe une isotheal conition is epesse as a P (5)'. Above equations ae the basic equations of single bass-efle syste. Chaacteistic fequency of the syste une aiabatic conition is epesse by equation (6). f D π π γ a P ρ l [Hz] (6) In the sae anne, chaacteistic fequency of the syste une isotheal conition is epesse by equation (6)'. f D π a P ρ l [Hz] (6)' whee, ρ a l : Mass of ai in the uct[g] ρ: Density of ai[g/ ] l: Equivalent length of uct[] These ae the basic equations of single bass-efle syste. These ae also use to eive equations of MCAP-CR. Equation of Motion of MCAP-CR of Fee ibation In geneal, aiabatic conition shoul be use; howeve, I applie isotheal conition fo MCAP-CR calculation, because isotheal conition ae bette esults than aiabatic conition. Please note this assuption is iffeent fo public unestaning. Fig. efines ais of each otion whee. Aows stan fo positive iection of isplaceent ( - ). - stans fo sping constants of chabes fo efeence coss sectional aea. Equivalent aea of speae con is use as efeence value of coss sectional aea fo siplicity. Following naing ule of paaetes is applie in this calculation. S. Suzui

5 MCAPE A) Subscipt efeences paaete of ain chabe. B) Subscipts though efeence paaetes of sub-chabes. C) Subscipts though efeence fo paaetes of open-to-ai ucts Fig. Definition of Diections of Paaetes of MCAP-CR () Sping constant of each chabe is a function of coss sectional aea of uct, so we use facto of each coss sectional aea of uct ivie by efeence aea. Refeence aea a is efine as effective con aea of speae unit. Let us efine,,..., as capacity of each chabe, then we get sping constant of each chabe fo efeence aea a as epesse in equation (7). a P (7) Let us efine coss sectional aea of each uct as a, a,..., a, a,..., a, then sping constant of each chabe fo each uct is epesse in equation (). Subscipts,,..., efeence factos of ucts between ain chabe an sub-chabes, an subscipts,,..., efeence factos of open-to-ai ucts. * a P () whee, a. a S. Suzui

6 MCAPE Equations of otion of fee vibation of this syste ae epesse in equations (9), whee epesents isplaceent of each ass. ( ) ( ) i i i (9) whee, :,,..., : ube of sub-chabes. An eaple of egees of feeo case () is shown below. Equation of otion in ati foat is epesse in equation (). KX X M () whee, M X.. X D C B A K ( ) ( ) ( ) ( ) A C B D. stans fo ass of involve ai in each uct, i.e. l a ρ whee, ρ an l stans fo espectively ensity of ai an effective length of each uct. S. Suzui

7 MCAPE. Solution of Chaacteistic Fequencies of MCAP-CR We nee calculation to solve equation (9), so let us calculate eigen values of the equation of otion () in ati foat. Eigen values of the equation of otion can be calculate to solve equation (). K λ M () o M K λ E ()'. Since equation () configues a polynoial whose egee is, then we coul get oots. Solution ay inclue ultiple oot an/o cople oots. Chaacteistic fequencies can be calculate afte getting eigen values of this poble as follows. whee, λ f π ω π λ : Eigen values ω : Angula fequency [a/s] f :,,...,. Fequency [Hz] We have got all the poceue to solve chaacteistic fequencies of MCAP-CR as shown above. Since it is vey tough to solve eigen value pobles of egee of thee o geate, we shoul use copute pogas to calculate nueically. Thee ae a nube of algoiths to calculate eigen values. The best way to solve MCAP-CR equations is to use poven coecial pogas; howeve, it ay cost a lot to puchase one. () Relatively siple coes ae to use fae algoith o Jacob's algoith; howeve, thee eain soe pobles. Fae algoith is not suitable to calculate nueically, because calculation eos ay not be negligible. I tie to use fae algoith, but I saw huge eo an its esult was not pactical. I also tie to use Jacob's algoith. It woe fine, but I ha to esign the MCAP-CR to ae sae ass of ai involve in each uct. Pactically the siplest solution was to calculate eteinant values nueically as a function of fequency evey iscete fequencies ove suppose fequency ange. Chaacteistic fequency eists in a ange between two consecutive iscete fequencies whee sign changes negative to positive o positive to negative. This esolution of calculation shoul be pactically enough, because thee ae any oe uncetain factos. S. Suzui

8 MCAPE In case (Double Bass-Refle Syste) Appeni -A MCAP-CR equies o geate, but sae equations can be use whee. This calculation is siple an can analytically solve without copute pogas. Equation (9) is siplifie to equation (A). ( ) (A) Epessing equation (A) in ati foat, then (A) becoes equation (A) o (A). M X KX X M KX (A) (A) Chaacteistic equation of (A) is as follows: M K λ E (A) whee, M K ( ) X E. Let us calculate an eaple of DB- by agaoa. Pincipal iensions of DB- ae given in Table A anb. Table A Main Chabe W[] D[] H[] Capacity[ ] Capacity [l ] Chabe Duct Displaceent of speae unit Total Tetsuo agaoa, " ewest oiginal Speae Caft (Japanese)", pp-6, Ongaunotoo (96) S. Suzui

9 MCAPE Table B Sub-Chabe W[] D[] H[] Capacity[ ] Capacity [l ] Chabe Duct Ribs Total.. Equation (A) is tansfoe as equation (A5). ( λ )( λ ) Aanging equation (A5) to polynoial fo, we get ( ) λ λ no Because it is obvious that neithe λ ( ) ( ) Chaacteistic fequencies ae then calculate as f λ, π ± f π Calculation esults wee f 9.Hz an 5.Hz. λ (A5) (A6)., foula of oot is applicable, then we get (A7). Refeence by agaoa gives appoiate foulae. Using the efeence foulae, we get f9.hz an f5.5hz. Thee esults ae pactically close enough. Please note that y equations oes not assue aiabatic but isotheal conition as I aleay iscusse. In any case y equations ae pactical enough fo ouble bass-efle systes. S. Suzui

10 MCAPE Appeni-B In case ( egees of feeo o geate): Tue MCAP-CR In Appeni-A we have aleay solve the siplest case (: ouble bass-efle); howeve, it is necessay to use nueical calculation to solve geneal cases ( ). We now that it is not siple to calculate eigen values of thee o oe egees of feeo pobles. We will ty the siplest etho to estiate eigen values of ultiple egee of feeo fee vibation pobles. We now that eigen values ae oots of equation (). We consie the function (B). G ( λ ) M K λ E (B) whee, λ ω π f f: Fequency [Hz] ω: Angula fequency [a/sec] We consie g ( λ ) in oe to euce nube of igit of the value, because ( λ ) vey big value. Calculating G ( λ ) will incease nueical eo. (B) G will becoe g L L ( λ ) G( λ ) M K λ E LL LL (B) whee, L. Let us efine iscete value of fequency as fin f (B). then, f λ π f (B5). f in shall be efine by ouselves. It ust be o geate. f shall also by efine by ouselves. It epens on which esolution we nee. f [Hz] will be pactical enough. We calculate the following function one by one. g L ( ) M f K f E π (B6) LL This calculation will let us now whee the oots ae in esolution ange. Fig. B- shows a esult of a case. S. Suzui

MCAPE Fig. B- Eaple of Calculation Result () Hoizontal ais is efines as fequency an vetical ais is efine as g ( f ) fequencies ae in suoune aea by ellipses whee the cuve cosses hoizontal ais.

Fig. B- gives cleae view so that we now whee othe chaacteistic fequencies ae. Chaacteistic fequencies Fig. B- Patially Epane iew of Fig.

11 MCAPE Fig. B- Eaple of Calculation Result () Hoizontal ais is efines as fequency an vetical ais is efine as g ( f ) fequencies ae in suoune aea by ellipses whee the cuve cosses hoizontal ais.. Chaacteistic Chaacteistic fequencies in two ellipses in the ight sie sees clea whee they ae, but egion in the left ellipse oes not give clea view to us. Epane view of this egion is seen in Fig. B-. Fig. B- gives cleae view so that we now whee othe chaacteistic fequencies ae. Chaacteistic fequencies Fig. B- Patially Epane iew of Fig. B- g ( f ) shoul be epesse in the polynoial fo of 6 egees; howeve, these plots sees a little bit iffeent. Etene eseach on this will be necessay; howeve, the cuve loos nice. Fae algoith gave e coefficients of the polynoial, but it i not give e easonable solution, because coefficients becae too big an oun off eo was not negligible. In any way, etho that was given hee will be pactical enough to estiate MCAP-CR chaactes. S. Suzui

Equations to Calculate Characteristic Frequencies of Multiple Chamber Aligned in Parallel Cavity Resonator (MCAP-CR)

Equations to Calculate Characteristic Frequencies of Multiple Chamber Aligned in Parallel Cavity Resonator (MCAP-CR) Shigeru Suzui March 3, 008 (Revised in November 8, 008 1 ) 1. Preface It is necessary