Dynamics of Bass Reflex Loudspeaker Systems (3)

Transcription

1 MCAP5E Dynaics of Bass Refle Loudspeaker Systes (3) Deriving Equations of Motion of Bass Refle Speaker Systes 3. Deriving Equations of Motion Shigeru Suzuki Released in May 5th, 8 in Japanese Released in Feb. 7th, in English We have seen classification of bass refle loudspeaker systes in the previous paper. We will review how to derive those equations of otion in this chapter. 3. Deriving Characteristic Frequency of Helholtz's Cavity Resonator Bass refle loudspeaker cabinet is an application of Helholtz's cavity resonator. It has characteristic frequencies. Refer to MCAPE for ore details. State equation of ideal gas under adiabatic condition is epressed in equation (). P γ =constant P : Pressure of air in chaber [N ] : olue of chaber [ 3 ] () γ : Ratio of specific heats Arranging equation (), we get d(p γ )= γ dp+p γ γ d = C p C v. Multiplying both ebers of above equation by γ + Therefore, dp Pd =. dp= γ Pd Hooke's low is epressed in equation (3). df= kd k : spring constant of air in chaber [N /] F : restoring force by air spring [N ],we get (). (3) Copyright 8 Shigeru Suzuki

2 There is the following relationship epressed in equation (4). { df=adp d =ad a : cross sectional area of duct [ ] Substituting equation (4) to equation (3), we get df= γ ap d = γ a P d. MCAP5E Hence spring constant of air in the chaber under adiabatic condition is epressed as follows: k= γ a P If we assue isotheral condition, we have equation (5)'. k= a P As a result, we can calculate characteristic frequency of Helholtz's cavity resonator as following equations: Adiabatic condition: f D = Isotheral condition: f D = = ρal k = ap l ap l [Hz ] : Mass of air involved in duct [g] ρ : Density of air at roo teperature [kg/ 3 ] l : Length of duct [] (4) (5) (5)' [Hz ] (6) (6)' Assuing adiabatic condition is theoretically correct; however, assuing isotheral condition resulted in ore reasonable result, based on author's eperience. This discrepancy ay be due to other reasons. Copyright 8 Shigeru Suzuki

3 MCAP5E 3. Deriving Equations of Motion of MCAS CR (Multiple Chaber Aligned in Series Cavity Resonator) Equation (5) shows that spring constant if a function of cross sectional area. Therefore, we use spring constant for reference cross sectional area. Definitions of sybols are given in Table 3 of MCAP4E. Table 3 of MCAP4E Definitions of Sybols Syb ol Definition Note a Reference area [ ] Effective area of vibrating ebrane a Cross sectional area of each duct [ ] r Ratio of areas a /a l P k Mass of bulk air involved in each duct [g] Density of air at roo teperature [kg/ 3 ] Effective length of each duct [] Atospheric pressure at roo teperature [ Pa] Spring constant of chaber for reference area [N /] =.[kg / 3 ] P=,3[Pa ] k = a P [N /] olue of each chaber [ 3 ] Fig.3 in MCAP4E shows scheatics of MCAS CR with n chabers. Applied force to asses through n n is function of volues of chabers net to the ass and displaceent of the ass and net asses. Fig.6 focuses on through n n. Fig.6 Coss sectional area of is a. Forces fro k side and fro k side, while oves by, are respectively F and F in the following equations: Copyright 8 Shigeru Suzuki

4 MCAP5E F = k a F = k a a = k r a = k r where we suppose other asses do not ove. Effects of other asses will be considered later superposing effects. Net let displaceent of is zero ( = ) and displaceents of and + are, respectively, and +. Let F ' and F ' be forces fro side k and k +, then ' a F a a a a a a a r r F ' a a + a a a + a + a a r r + + Superposing above equations since these are linear differential equations, we get equation (7) as follows: videlicet, d dt +k r +k r k r r k r r + + = d dt k r r +(k +k )r k r r + + = (7) We have got equation of otion for asses of both ends.. Then we will derive equations of otion of We focus on and in Fig.3 of MCAP4E then draw Fig. 7. Fig.7 Mass is equivalent ass of ebrane. This ass is forced to vibrate by power aplifier. We note that direction of is different fro other asses' then we derive equation (8) as in the sae anner above. Copyright 8 Shigeru Suzuki

5 MCAP5E videlicet, a d dt +k +k +k u a =f (t ) d dt +(k +k ) +k r u =f (t ) (8) In the sae anner, we get equation of ass as below: d dt +k r +(k +k )r k r = (9) At last, we derive equations of otion of ass n. Let us note that n+ does not eist, then equation of otion becoes as equation (). Also we note that volue of roo is uch greater than the chaber: k n =. d n n dt k r r +(k +k )r n n n n n n n n = () Equation of otion of all asses in the MCAS CR syste is epressed in atri for as follows: M d X dt =[ M +K=f 3 4 5] =[ 3 f t ],, k r k (k +k ) k r ] k r (k +k )r k r K=[ku+k. k r (k +k 3 ) k 3 k 3 (k 3 +k 4 ) k 4 k 4 k 5 4 5] f =[ We ay use these equations for single and double bass refle loudspeaker systes as well, where n= for single and n= for double bass refle systes. Stiffness atri K is tri diagonal atri so that it is not difficult to calculate characteristic frequencies. () Copyright 8 Shigeru Suzuki

6 MCAP5E 3.3 Deriving Equation of Motion of Standard MCAP CR (Multiple Chaber Aligned in Parallel Cavity Resonator) Standard MCAP CR is typical application of ultiple degree of freedo cavity resonators as well as MCAS CR. Equations of otion under free vibration condition were already described in MCAPE. Now we include ass of vibration ebrane and discuss forced vibration equations. Fig.8 shows structure of standard MCAP CR. Nuber of sub chabers theoretically unliited, though we see only three in Fig.8. n is Fig.8 Structure of Standard MCAP CR Copyright 8 Shigeru Suzuki

7 MCAP5E Total restoring force to ass is derived adding restoring forces k,..., k n of equation (8), so that we get equation of otion of ass as follows: d dt k k k r k r... k r = f t u n n (). Equation of otion of ass is derived in the sae anner above. videlicet, d dr k r a a k r a a... k r a a... k r a n a n k r a n n a = d dt k r r k r r... k k r... k r n r n k r k r n n = (3). Equation if otion of ass n MCAS CR: is epressed in equation (4), as sae anner as d n n k dt r r n k r n n = (4) =[ Equation () applies to MCAP CR sae as MCAS CR. For MCAP CR, we get f t, K=[ A ] B,, C D] M =[ 6] A=[k u+k k k r k k (k +k ) k r k ], k r k r (k +k )r k r k k k r (k +k 3 ) ] f =[ k B=[ k r k 3 r 6] k C=[ k r 6], k 3 r 4 D=[kr k k 3 r 6 ]. Copyright 8 Shigeru Suzuki

8 MCAP5E 3.4 AICC CR (Arbitrary Inter Chaber Connection Cavity Resonator) Syste AICC CR is the ost cople syste so that it is not easy to show typical application. We derive equations for Fig.5c of MCAP4E. Fig.5c is siilar to standard MCAP CR with two sub chabers where sub chabers are connected each other. Therefore equation of otion for is sae as equation () where n=. Equations of otion for and are as follows: d dt +k r +k r +k r k 3 k 5 (5) d dt +k r +k r +k r r k r 5 k r r 6 6 (6). Equations of otion for 3 and 4 are as follows: d 5 3 dt +k r 3 +k 5 k = (7) d 5 4 dt +k r 4 4 +k 5 k r 4 = (8). Equation of otion for M 5 is as follows: d 5 5 dt = (k +k ) r 5 +k ( a 3 Arranging above equation, we get a 5 3 a a 5 ) +k r 5 ( a a 5 a 4 a 5 4) =. d 5 5 dt k r r +k r r +k r r k r r (k +k ) r 5 5 = (8). Deriving equations of otion for general AICC CR application is not practical, since it is too cople. On the other hand, we can derive equations for specified AICC CR application as seen above. AICC CR is, at this tie, not well analyzed. We will later ake soe typical AICC CR applications practical enough. End of this report Copyright 8 Shigeru Suzuki