112
Joseph A. C. Humphrey et al.
2.3 Evaluation of Physically Approximate Analytical Solution
2.3.1 Filiform Hairs in Air
Prior to using the physically-approximate analytical results given by Eqs. (19a,
19b) and (20a, 20b) to discuss and interpret the dependencies of Wres(£JJ• B,.., Wres(VJ•
and V,.. on the physical parameters that affect them, we first establish their
physical accuracy. For this, we perform calculations corresponding to the
experimental conditions of Barth et al. (1993). These authors measured maximum
deflection angles and resonance frequencies for a number of individual hairs of
four selected groups of trichobothria on the spider Cupiennius salei under
controlled oscillating flow conditions. In particular, for the MeDl group of hairs
reported they found the following fits for maximum deflection angle and
resonance frequency: Bres = 3.3 + 7 10 3 L andfres((J) = 4.25 10- 3 Dl.3 16 , where Bres is
in degrees,fres((J) in Hz and L in m. They also determined values for the damping
constant R and the torsional restoring constant S for two of these metatarsal hairs.
For a short hair they obtained: L = 250 J.lm, d = 5 J.lm, R = 0.27 10- 15 N m s rad·\
and S = 0.62 10- 12 N m rad" 1 ; and, for a long hair: L = 750 J.lm, d = 7 J.lm, R = 2.20
10" 15 N m s rad- 1 , and S = 5.77 IO - 12 N m rad- 1 . (Although the long hair had an
additional curved portion 250 J.lm long, it was not considered in the calculations of
their Fig. 19 with which we compare here.) Curve fits of these data to the formy =
aLb yield: d = 6.343 10- 5 L 0 " 306 , R = 2.031 10- 9 Ll. 909 , and S = L272 10- 5 L 203 ,
where L is in m. Because these fits are based on only two points each, they are
subject to (unknown) uncertainties. Similar fits for R and S have been obtained
from larger data sets by Shimozawa et al. ( 1998) for the filiform hairs of crickets.
In spite of the care taken by these authors to obtain accurate measurements, the
scatter in the data ranges from ± 20 to ± 80% for S and ± 7 to ± 38% for R,
approximately, over the range of hair length explored.
Except for where otherwise noted, present calculations for hairs in air based on
the physically approximate solution assume d = 7 J.lm, R = I I o- 15 N m s rad- 1 , and
S = 4 10 - 12 N m rad- 1 regardless of hair length. For hairs in water the same value
of dis used, but with different values for RandS, as discussed in Section 2.3.2.
The calculations for hairs in air based on the physically exact solution given by
Eq. (10) use the above fits ford, R, and S. All calculations assume fluid medium
physical properties at 27 oc: air (p = L177 kg m- 3 , p = 1.846 10- 5 kg m- 1 s- 1 ),
water (p = 995.8 kg m- 3 , J.i = 8.6 10- 4 kg m- 1 s- 1 ). The hair density used is Ph= 1100
kg m- 3 and the far field velocity amplitude is set to U0 = 50 mm s- 1 • All
calculations are performed in SI units. To solve Eq. ( 1 0) directly for Wres(£JJ we
proceed as follows. A value of w is fixed in the equation and t is varied until the
maximum deflection angle, Bmax. is found. Sequentially varying w yields values of
Bmax from which the value of Wres(£JJ• corresponding to the largest value of Bmax (=
B, •• ), can be obtained. This approach retains all the OJ dependencies embedded in
the P, Q, 11 and R1 terms which the physically approximate analytical solution
neglects. When applied to the time derivative of Eq. (10), the same approach
yields Wres(V) and Vres· To solve Eq. (19a) for Wres(IJ) and Eq. (20a) for Wres(V), we
Joseph A. C. Humphrey et al.
2.3 Evaluation of Physically Approximate Analytical Solution
2.3.1 Filiform Hairs in Air
Prior to using the physically-approximate analytical results given by Eqs. (19a,
19b) and (20a, 20b) to discuss and interpret the dependencies of Wres(£JJ• B,.., Wres(VJ•
and V,.. on the physical parameters that affect them, we first establish their
physical accuracy. For this, we perform calculations corresponding to the
experimental conditions of Barth et al. (1993). These authors measured maximum
deflection angles and resonance frequencies for a number of individual hairs of
four selected groups of trichobothria on the spider Cupiennius salei under
controlled oscillating flow conditions. In particular, for the MeDl group of hairs
reported they found the following fits for maximum deflection angle and
resonance frequency: Bres = 3.3 + 7 10 3 L andfres((J) = 4.25 10- 3 Dl.3 16 , where Bres is
in degrees,fres((J) in Hz and L in m. They also determined values for the damping
constant R and the torsional restoring constant S for two of these metatarsal hairs.
For a short hair they obtained: L = 250 J.lm, d = 5 J.lm, R = 0.27 10- 15 N m s rad·\
and S = 0.62 10- 12 N m rad" 1 ; and, for a long hair: L = 750 J.lm, d = 7 J.lm, R = 2.20
10" 15 N m s rad- 1 , and S = 5.77 IO - 12 N m rad- 1 . (Although the long hair had an
additional curved portion 250 J.lm long, it was not considered in the calculations of
their Fig. 19 with which we compare here.) Curve fits of these data to the formy =
aLb yield: d = 6.343 10- 5 L 0 " 306 , R = 2.031 10- 9 Ll. 909 , and S = L272 10- 5 L 203 ,
where L is in m. Because these fits are based on only two points each, they are
subject to (unknown) uncertainties. Similar fits for R and S have been obtained
from larger data sets by Shimozawa et al. ( 1998) for the filiform hairs of crickets.
In spite of the care taken by these authors to obtain accurate measurements, the
scatter in the data ranges from ± 20 to ± 80% for S and ± 7 to ± 38% for R,
approximately, over the range of hair length explored.
Except for where otherwise noted, present calculations for hairs in air based on
the physically approximate solution assume d = 7 J.lm, R = I I o- 15 N m s rad- 1 , and
S = 4 10 - 12 N m rad- 1 regardless of hair length. For hairs in water the same value
of dis used, but with different values for RandS, as discussed in Section 2.3.2.
The calculations for hairs in air based on the physically exact solution given by
Eq. (10) use the above fits ford, R, and S. All calculations assume fluid medium
physical properties at 27 oc: air (p = L177 kg m- 3 , p = 1.846 10- 5 kg m- 1 s- 1 ),
water (p = 995.8 kg m- 3 , J.i = 8.6 10- 4 kg m- 1 s- 1 ). The hair density used is Ph= 1100
kg m- 3 and the far field velocity amplitude is set to U0 = 50 mm s- 1 • All
calculations are performed in SI units. To solve Eq. ( 1 0) directly for Wres(£JJ we
proceed as follows. A value of w is fixed in the equation and t is varied until the
maximum deflection angle, Bmax. is found. Sequentially varying w yields values of
Bmax from which the value of Wres(£JJ• corresponding to the largest value of Bmax (=
B, •• ), can be obtained. This approach retains all the OJ dependencies embedded in
the P, Q, 11 and R1 terms which the physically approximate analytical solution
neglects. When applied to the time derivative of Eq. (10), the same approach
yields Wres(V) and Vres· To solve Eq. (19a) for Wres(IJ) and Eq. (20a) for Wres(V), we