Chapter 7
Contributions and Future Research
In this research, we have developed new methods to improve the accuracy and increase the efficiency of rapid prototyping processes. The major contributions of this research are summarized as follows:
• A biarc curve fitting of RP slicing contour data has been developed to smooth cross-sectional contours. The mathematics formulation and a Max-Fit algorithm have been developed to find the biarc curves of the STL slicing data points. Max-Fit biarc fitting algorithm progresses through all the points on the slicing contour data to find an efficient biarc fitting. The results show that rough cross sectional contours can be smoothed with the newly developed method. Therefore, less strict requirements on the STL triangulation tolerance can be used when STL is generated for rapid prototyping.
• Non-uniform offsetting and hollowing by using biarcs fitting has been developed to increase the efficiency of the rapid prototyping process. The developed method can reduce the area that needs to be built so the build time can be reduced significantly. Constant wall thickness is obtained to avoid non-constant shrinkage and warpage.
Reducing the number of layers can improve the total build time for the fabrication of a RP part. A new tool path plan to machine the ruled layers with 5-axis CNC machine using a taper-end mill has been developed
For future research, following are several possible directions:
• If the RP machine controllers based on parametric curves such as Bezier or NURBS become more common, the biarc curve fitting method can be replaced with a Bezier or NURB curve fitting for more accurate and results,
• Extending the proposed methods for hollowing STL models be used in die and mold design,
• Expanding the ruled layer fitting to complex contours (layers with several contours and complex shapes) or cloud of data, especially for medical image processing and reverse engineering,
• Developing methods for combining the traditional 2D RP process and material removal process,
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Appendices
I.Minimizing The Difference of Curvatures
To construct the biarc between two knots k1 and k2 with the best local smoothness, we can find the angles ψ1 and ψ2 from Equation (3.6) using the minimum difference of curvature criterion. From Equation (3.6), the difference of the two curvatures can be represented as follows:
1
2 1 2
2 2
2 1 1
1 2
1 sin
) sin(
) sin(
2 sin
) sin(
) sin(
2 1 1
ψ
ψ ψ ψ
θ ψ
ψ ψ ψ
θ
⋅
− −
+ ⋅
− −
− = −
l l
r
r (A.1)
In Figure 3.6;
θ1 = β1 + ψ1 and θ2 = β2 + ψ2 (A.2)
Since o1k1 ⊥k1M and o1k ⊥PQ (from conditions 1 and 2 in Chapter 3.3 on page 19) and o1k1 =o1k =r1 as shown in Figure 3.6, the triangle ∆k1Pk must be an isosceles triangle. Same thing can be shown for the triangle ∆k2Qk. Considering the angle ∠k1kk2 in Figure 3.6:
2 1 2 1
2 1 2
1 2
1 ( ) ( )
ψ ψ β β
ψ ψ π β β π
+ − = + −
− − = − − = ∠k kk
(A.3)
As mentioned earlier in Chapter 3.4, page 21, a positive angle, such as β1 or 1
ψ runs clockwise from the x-axis while a negative angle, such as β2 or ψ2runs clockwise.
From Equation (A.2) and (A.3), we can write:
(
)
2
2 1 2 1
θ θ β
β − = − and
(
)
2
2 1 2 1
θ θ β
β = + − (A.4)
Also from Equations (A.3) and (A.4):
(
)
2
2 1 2 1
θ θ ψ
ψ − = − (A.5)
1 2 1 2 2 2 2 1 1 1 2 1 sin ) sin( ) sin( 2 sin ) sin( ) sin( 2 1 1 ψ ψ ψ ψ θ ψ ψ ψ ψ θ ⋅ − − − ⋅ − − = − l l r
r (A.6)
From (A.5), we have:
2 2 1 ) ( 2 1 2 1 2 1 2 1 2 2 2 2 2 1 1 θ θ β ψ θ β θ θ ψ θ ψ + + = − − = − = − (A.7)
Inserting (A.7) into (A.6), we can rewrite (A.6) as follows:
B A 2 sin 2 2 sin 2 sin sin 2 ) sin( 2 sin 2 sin 2 1 1 2 1 2 1 2 2 1 2 2 2 2 1 2 1 2 2 1 ⋅ − = − + − ⋅ − − − ⋅ − + − = − l l l r r θ θ θ θ β θ θ β β θ θ θ θ θ β (A.8) where, ) sin( sin 2 sin 2 sin
A 1 2 2 2 2
2 2
1
2 β θ β
θ θ β θ θ β − − − + + + −
= (A.9)
− + + − = 2 sin ) sin(
B 1 2
2 2 2 θ θ β β
θ (A.10)
( )
( )
( 1 2)
Substituting A and B in Equations (A.11) and (A.12) into Equation (A.8), we can get the difference of curvature as follows:
(
)
l r
r
1
2 cos 2
3 2
cos
cos cos
2 sin 2 1
1
2 1 2
1 2
2 1
2 1
2 1
⋅
− −
+ −
−
− =
−
θ θ θ
θ β
θ θ
θ θ
(A.13)
To minimize the difference of curvatures in Equation (A.13),
+ −
2 3 2
cos 1 2
2
θ θ
β has
to equal 1. Therefore, β2can be calculated as
4 3 2
1 2
θ θ
β = + . Substituting β2into
Equation (A.2), we get the angles ψ1 and ψ2 as follows:
− = − =
4 2 1 2 1
θ θ ψ
