Page 703 - Automotive Engineering Powertrain Chassis System and Vehicle Body

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

The effect of the porosity is to introduce a modified 21.1.9.6 The modified 1-D linear plane
bulk modulus wave equation

k 0
k ¼ (21.1.96) The modified equation of motion becomes:
h
vp sr vu 0
so that ¼ 0 ru 0 (21.1.101)
vx h vt
vu 0 1 vp 0
¼ (21.1.97) Reminder: u is the volume flow rate/unit cross section
vx k vt area not the flow velocity in the pores.
The derivation of the linearised mass conservation For simple harmonic motion of frequency u
equation is given in Appendix 21.1F.
vp sr 0 r vu 0
¼ þ (21.1.102)
vx h iu vt
21.1.9.5 The structure factor This is a modified Euler equation with the complex
terms in the brackets being the effective density (Fahy
The structure factor (s) expresses the influence of the and Walker, 1998). The derivation of the 1-D Euler
geometric form of the structure on the effective density equation is given in Appendix 21.1G.
of the fluid (Fahy and Walker, 1998). Now, if this equation is combined with the linearised
There are four mechanisms for this: mass conservation equation (21.1.92)
1. Side pockets within the material (outside the flow vu 0 1 vp
stream) reduce the effective bulk modulus of the vx ¼ k vt (21.1.92)
0
fluid. This produces a smaller acoustic pressure for a modified wave equation is obtained in this manner:
a given strain gradient in the pore (see above equa-
tion) which leaves the impression that the effective Differentiate both sides of the modified Euler equa-
density of the fluid is higher in the material than in tion with respect to x
free space. 2 h i 0
v p sr 0 r v vu
2. Non-uniform pores causing sudden expansions that vx 2 ¼ h þ iu vx vt
leave the impression of added mass.
3. Non-axial pore orientation. The in viscid momentum Now take the time derivative of the linearised mass
equation normal to the orientation of the pore is: conservation equation

vp vu n 2 0 0
¼ r 0 (21.1.98) 1 v p v vu v vu
vh vt ¼ ¼
k vt 2 vt vx vx vt
0
So, the combination gives:
or for orientation angle q to the surface of the material: " #
2
2
v p h sr 0 r i 1 v p
vp vu 0 1 ¼ þ
cos q ¼ r 0 vx 2 h iu k vt 2
0
vx vt h cos q
i 2
2
vp r 0 vu 0 v p h sr 0 r v p
¼ k 0 ¼ þ
2
vx h cos q vt vx 2 h iu vt 2
vp sr vu 0 2 2
0
¼ (21.1.99) 2 v p sr 0 r v p
vx h vt r c 2 ¼ þ 2
0 0
vx h iu vt
1
sfa (21.1.100) 2 2 2
2
cos q c v p ¼ 1 þ r v p
0
0
The effective density is increased by factor (s). For s vx 2 h ir su vt 2
a random orientation of pores s ¼ 3. Generally s is in the
range of 1.2–2.0 (Fahy and Walker, 1998). Finally the modified wave equation is
4. The velocity profile across each pore acts to reduce 2 2 2
volume acceleration produced by a given gradient c 2 1 v p ¼ 1 v p þ r vp c ¼ c 0 (21.1.103)
2
1
and hence contributes to s. vx 2 h vt 2 r s vt s
0
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