Page 721 - Automotive Engineering Powertrain Chassis System and Vehicle Body

CHAP TER 2 1. 1 Interior noise: Assessment and control

Substituting equation (E21.1.16) into equation Consider what happens to equation (E21.1.15) when
(E21.1.15) yields there is a truly rigid termination to tube, i.e. jZ ML j/N
Z MO r þ ix þ i tan kL
¼ (E21.1.17) Z MO Z R þ i tan kL Z ML
r c 0 S 1 þ i½ðr þ ixÞ tan kLŠ ¼ ¼ Z R
0
r c 0 S 1 þ iZ R tan kL r c 0 S
0
0
Now separate the real and the imaginary parts. Take
Z MO ðZ R þ i tan kLÞð1 iZ R tan kLÞ
the denomintor of equation (E21.1.17) first. Then ¼
2
r c 0 S 1 þ iZ tan kL
2
0
R
D ¼ 1 þ ir tan kL x tan kL
(E21.1.18) Z MO 2 2
D ¼ð1 x tan kLÞþ ir tan kL r c 0 S ¼ Z R iZ tan kL þ i tan kL þ Z R tan kL
R
0

2
2
multiplying equation (E21.1.18) by its complex conju- Z MO Z R 1 tan kL þ ið1 Z Þtan kL
R
gate yields r c 0 S ¼ 1 þ Z tan kL
2
2
0
R
2
*
2
D D ¼½ð1 x tan kLÞþ ir tan kLŠ Z MO Z R 1 tan kL ið1 Z Þtan kL
R
¼ 2 þ 2
2
2
½ð1 x tan kL ir tan kLފ r c 0 S 1 þ Z tan kL 1 þ Z tan kL
0
R
R
2
2
2
*
D D ¼ð1 x tan kLÞ þr tan kL (E21.1.22)
2
2
2
*
D D ¼ 1 2x tan kL þ x tan kL þ r tan kL When Z R /N the reactance /0 when
2
2
2
*
D D ¼ðx þ r Þtan kL 2x tan kL þ 1
1
(E21.1.19) i / 0 i:e: cot kL ¼ 0
tan kL
i:e:
Multiplying the numerator of equation (E21.1.17)by
the complex conjugate of equation (E21.1.18) gives: for k n L ¼ð2n 1Þ p n ¼ 1; 2; 3; 4; .
2
2pf ð2n 1Þp 2n 1 c
*
N D ¼½r þ iðx þ tan kLފ c L ¼ 2 f ¼ 4 L
½ð1 x tan kLÞ ir tan kLŠ (E21.1.23)

2
*
N D ¼ðr rx tan kLÞ ir tan kL Under these conditions, flow reactance is zero and the
input mechanical impedance is a function of flow re-
2
2
þ i x x tan kL þ tan kL x tan kL
sistivity only.
2
þ xr tan kL þ r tan kL

2
2
2
*
N D ¼ r þ r tan kL þ i x x tan kL x tan kL
Appendix 21.1F: The derivation of
2
þ tan kL r tan kL
the linearised mass conservation
2
2
*
2
N D ¼ r tan kL þ 1 þ i x x þ r tan kL equation (after Fahy and Walker,

2
x tan kL þ tan kL 1998)

*
2
N D ¼ r tan kL þ 1
The net rate of mass inflow into a 1-D control volume of
2
2
2
2
þ i x x þ r 1 tan kL x tan kL cross-sectional area S (m ) length dx (m) is:
(E21.1.20)
" #
vðr TOT uÞ
So equation (E21.1.17) becomes (using equations r TOT uS r TOT u þ dx S
(E21.1.20) and (E21.1.19) for numerator and de- vx
nominator respectively): vðr TOT uÞ
¼ vx dxS (F21.1.1)
2
Z MO rðtan kL þ 1Þ
¼
2
2
r c 0 S ðx þ r Þtan kL 2x tan kL þ 1 This net inflow of mass must be balanced by the in-
2
0
2
2
2
i½x ðx þ r 1Þtan kL x tan kLŠ crease in mass in the control volume (the principle of
þ conservation of mass) which is given by:
2
2
2
ðx þ r Þtan kL 2x tan kL þ 1 Rate of increase in mass within control volume ¼
(E21.1.21) vr TOT dxS
vt
732