bending in 2 planes parametric form assuming small deflections and no torsion
TRANSCRIPT
Bending in 2 Planes
Parametric form assuming small deflections and no torsion.
Parametric Form
zzzz
yyyy
xxxx
auauauauz
auauauauy
auauauaux
012
23
3
012
23
3
012
23
3
)(
)(
)(
Where
Condensed:
1,0u
012
23
3)( aaaap uuuu
Deflection in Y & Z
Sensor 1 Sensor 2
x1
x2
L = 15cm
Tip Deflection
y
xz
Cross-section at Sensors
y
zx
Known:
0
0,0
0,,)0,0,0()0(
2
2
2
2
11
000
dx
zd
dx
yd
aadx
dz
dx
dy
aaap
zy
zyx
At u = 0:
At u = 1:
Seven unknowns remain.
Curvature InformationAssuming no torsion, the orientation of the local and global coordinate frames are the same.
From each known curvature component, we have an equation with 5 unknowns.
For example, at u = 1:
Therefore, at each sensor two orthogonal components of curvature are known, and 2 equations with 5 unknowns result. (y”(x) a1x, a2x, a3x, a2y, a3y and z”(x) a1x, a2x, a3x, a2z, a3z). Therefore two sets of curvatures are needed.
02662
0
)(
)/(0
21231312
32
2
yxyxxyxx
uuuuuu
u
uuuuuu
u
uuu
aaaaaaaa
yxyx
x
yxyx
x
xy
dx
yd
To Simplify Math
22
33
22
33
)(
)(
)(
uauauz
uauauy
uLux
L
xu
zz
yy
Assume small deflections in Y and Z, such that
Then,
We have rid of 3 more unknowns (x coefficients).
Simplified Derivatives
zzuu
yyuu
uu
zzu
yyu
u
auauzz
auauyy
uxx
uauauzz
uauauyy
Luxx
23
23
22
3
22
3
26)("
26)("
0)("
23)('
23)('
)('
The parametric slope equations are now:
The parametric curvature equations are now:
Parametric Equations in Terms of x
2
2
3
3
2
2
3
3
)(
)(
L
xa
L
xaxz
L
xa
L
xaxy
L
xu
zz
yy
Substitute in for u:
Parametric Derivatives in Terms of x
22
33
2
2
3
3
222
33
2
223
3
26)("
26)("
23)('
23)('
L
ax
L
axz
L
ax
L
axy
xL
ax
L
axz
xL
ax
L
axy
zz
yy
zz
yy
Slope equations:
Curvature equations:
Now, the equations to solve for the four remaining unknown coefficients are simple.
y”(x) and z”(x) are the known orthogonal components of the curvature at the sensor locations (and at the tip).
Model
• Find y’’(x) at sensor locations. • Calculate a set of unknowns for the region
from the base to x2 (second sensor location.)• Estimate y(x2) and y’(x2).• Use y(x2), y’(x2) and y’’(L) to calculate a set of
unknowns from the region from x2 to the tip of the needle.