Analytical Compliance Equations of Generalized Elliptical-Arc-Beam Spherical Flexure Hinges for 3D Elliptical Vibration-Assisted Cutting Mechanisms
Abstract
:1. Introduction
2. Analytical Compliance Equations of Elliptical-Arc-Beam Spherical Flexure Hinges
2.1. Compliance Equations of Generalized Spatial Flexure Hinges
2.2. The Notch Profile of Generalized Elliptical-Arc-Beam Spherical Flexure Hinges
2.3. Analytical Equations of the Factors in the Compliance Matrix
3. Results
3.1. Compliance Factors of Spherical Elliptical-Arc Flexure Hinges
3.2. Simulation Validation by FEA
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
Fi | External load vector acts on node i at the free end of the flexure hinge |
Fik | External force component of Fi with subscript k denoting its direction, k = x, y, z |
Mik | External moment component of Fi with subscripts k denoting its direction, k = x, y, z |
∆i | Displacement vector of node i at the free end of the flexure hinges resulted by Fi |
∆ik | Translation component of ∆i with subscript k denoting its direction |
θik | Rotation component of ∆i with subscript k denoting its direction |
x | Position coordinate along the flexure hinge |
L | Length of the flexure hinge |
A(x) | Section area of the flexure hinge, a function of x |
E | Young’s modulus of material |
G | Shearing modulus of material |
μ | Shearing coefficient for a short beam with circular section |
Fx(x) | Axial force along x-axis of the flexure hinge, a function of x |
Fk(x) | Shear force along k-axis of the flexure hinge, a function of x, k = y, z |
Mk(x) | Moment around k-axis of the flexure hinge, a function of x, k = x, y, z |
U | Strain energy of the flexure hinge acted by Fi |
Ci | Compliance matrix of the flexure hinge at node i |
Cm-n | Compliance in the direction of m caused by the external load n, m = ∆ix, ∆iy, ∆iz, θix, θiy, θiz and n = Fix, Fiy, Fiz, Mix, Miy, Miz |
D(x) | Diameter variation of circular section, a function of x |
a | Length of semi-major axis of ellipse |
b | Length of semi-minor axis of ellipse |
θ | Eccentric angle of ellipse |
xp | Horizontal coordinate of point P on the ellipse |
yp | Vertical coordinate of point P on the ellipse |
Dmin | Diameter of the middle beam |
l | Notch length of the middle beam part |
c | Notch length of the elliptical -arc part |
θm | Maximum eccentric angle |
ζ | Intermediate variable, ζ = Dmin/2b |
Nj | Intermediate variables for integral simplification, j = 1, 2, 3, 4 |
Appendix A
References
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Hinge No. | a (mm) | b (mm) | c (mm) | θm (°) | Dmin (mm) | Notch Type |
---|---|---|---|---|---|---|
1 | 8.25 | 4 | 8.25 | 90 | 0.5 | Elliptical |
2 | 5 | 7 | 5 | 90 | 0.4 | Elliptical |
3 | 7 | 4 | 6.76 | 75 | 0.5 | Elliptical-arc |
4 | 4.5 | 6 | 3.90 | 60 | 0.4 | Elliptical-arc |
5 | 4 | 4 | 3.86 | 75 | 0.5 | Circular |
6 | 6 | 6 | 5.20 | 60 | 0.4 | Circular |
7 | 4 | 4 | 4 | 90 | 0.5 | Right-circular |
8 | 7 | 7 | 7 | 90 | 0.4 | Right-circular |
9 | 1 | 0.1 | 1 | 90 | 0.5 | Elliptical |
10 | 1 | 0.5 | 1 | 90 | 0.6 | Elliptical |
11 | 0.875 | 0.3 | 0.875 | 90 | 0.75 | Elliptical |
Hinge No. | CΔix-Fix (×10−8 m/N) | CΔiy-Fiy (10−5 m/N) | Cθix-Mix (rad/Nm) | CΔiy-Miz (10−3 m/N) | Cθiy-Miy (rad/Nm) |
---|---|---|---|---|---|
1 | 11.19 | 32.23 | 6.01 | 38.10 | 4.62 |
2 | 7.32 | 11.77 | 6.04 | 23.20 | 4.65 |
3 | 9.49 | 18.41 | 5.10 | 26.50 | 3.92 |
4 | 7.08 | 7.00 | 5.87 | 17.60 | 4.52 |
5 | 5.42 | 3.44 | 2.91 | 8.70 | 2.24 |
6 | 9.45 | 16.56 | 7.83 | 31.3 | 6.02 |
7 | 5.42 | 3.68 | 2.91 | 9.00 | 2.24 |
8 | 10.25 | 32.27 | 8.46 | 45.60 | 6.51 |
9 | 4.40 | 0.32 | 3.23 | 2.50 | 2.48 |
10 | 2.28 | 0.09 | 1.00 | 0.76 | 0.76 |
11 | 1.53 | 0.04 | 0.46 | 0.31 | 0.36 |
Hinge No. | a (mm) | b (mm) | c (mm) | θm (°) | Dmin (mm) | l (mm) | Notch Type |
---|---|---|---|---|---|---|---|
1 | 10 | 6 | 10 | 90 | 1 | 0 | Elliptical |
2 | 6 | 10 | 6 | 90 | 1 | 0 | Elliptical |
3 | 10 | 10 | 10 | 90 | 1 | 0 | Right-circular |
4 | 10 | 6 | 8.66 | 60 | 1 | 0 | Elliptical-arc |
5 | 6 | 10 | 5.20 | 60 | 1 | 0 | Elliptical-arc |
6 | 10 | 10 | 8.66 | 60 | 1 | 0 | Circular |
7 | 10 | 6 | 10 | 90 | 1 | 2 | Elliptical-beam |
8 | 6 | 10 | 6 | 90 | 1 | 2 | Elliptical-beam |
9 | 10 | 10 | 10 | 90 | 1 | 2 | Circular-beam |
10 | 10 | 6 | 8.66 | 60 | 1 | 2 | Elliptical-arc-beam |
11 | 6 | 10 | 5.20 | 60 | 1 | 2 | Elliptical-arc-beam |
12 | 10 | 10 | 8.66 | 60 | 1 | 2 | Circular-arc-beam |
13 | 10 | 6 | 10 | 90 | 1 | 4 | Elliptical-beam |
14 | 6 | 10 | 6 | 90 | 1 | 4 | Elliptical-beam |
15 | 10 | 10 | 10 | 90 | 1 | 4 | Circular-beam |
16 | 10 | 6 | 8.66 | 60 | 1 | 4 | Elliptical-arc-beam |
17 | 6 | 10 | 5.20 | 60 | 1 | 4 | Elliptical-arc-beam |
18 | 10 | 10 | 8.66 | 60 | 1 | 4 | Circular-arc-beam |
Hinge No. | CΔix-Fix (×10−8 m/N) | CΔiy-Fiy (10−5 m/N) | Cθix-Mix (rad/Nm) | CΔiy-Miz (10−3 m/N) | Cθiy-Miy (rad/Nm) |
---|---|---|---|---|---|
1 (Analytical) | 3.75 | 4.04 | 5.01 | 3.90 | 3.91 |
1 (FEA) | 3.84 | 4.09 | 5.05 | 3.96 | 3.96 |
2 (Analytical) | 1.78 | 0.67 | 2.34 | 1.10 | 1.83 |
2 (FEA) | 1.94 | 0.71 | 2.42 | 1.15 | 1.92 |
3 (Analytical) | 2.97 | 3.11 | 3.90 | 3.00 | 3.05 |
3 (FEA) | 3.08 | 3.17 | 3.95 | 3.10 | 3.10 |
4 (Analytical) | 3.73 | 3.06 | 5.01 | 3.40 | 3.91 |
4 (FEA) | 3.74 | 3.09 | 5.05 | 3.42 | 3.95 |
5 (Analytical) | 1.78 | 0.51 | 2.34 | 0.95 | 1.83 |
5 (FEA) | 1.94 | 0.54 | 2.42 | 0.99 | 1.92 |
6 (Analytical) | 2.96 | 2.34 | 3.90 | 2.60 | 3.05 |
6 (FEA) | 3.07 | 2.40 | 3.95 | 2.69 | 3.10 |
7 (Analytical) | 4.99 | 7.40 | 7.54 | 6.50 | 5.89 |
7 (FEA) | 5.07 | 7.47 | 7.58 | 6.53 | 5.94 |
8 (Analytical) | 3.02 | 1.93 | 4.87 | 2.70 | 3.81 |
8 (FEA) | 3.18 | 1.98 | 4.95 | 2.73 | 3.90 |
9 (Analytical) | 4.20 | 6.24 | 6.43 | 5.50 | 5.03 |
9 (FEA) | 4.31 | 6.32 | 6.49 | 5.59 | 5.08 |
10 (Analytical) | 4.97 | 5.77 | 7.54 | 5.70 | 5.89 |
10 (FEA) | 4.97 | 5.82 | 7.57 | 5.72 | 5.92 |
11 (Analytical) | 3.01 | 1.52 | 4.87 | 2.40 | 3.81 |
11 (FEA) | 3.18 | 1.57 | 4.95 | 2.41 | 3.90 |
12 (Analytical) | 4.20 | 4.85 | 6.43 | 4.90 | 5.03 |
12 (FEA) | 4.30 | 4.92 | 6.48 | 4.91 | 5.08 |
13 (Analytical) | 6.23 | 11.9 | 1.00 | 9.40 | 7.87 |
13 (FEA) | 6.31 | 12.01 | 1.01 | 9.50 | 7.91 |
14 (Analytical) | 4.25 | 3.89 | 7.40 | 4.60 | 5.78 |
14 (FEA) | 4.42 | 3.94 | 7.43 | 4.68 | 5.86 |
15 (Analytical) | 5.44 | 10.40 | 8.96 | 8.40 | 7.00 |
15 (FEA) | 5.55 | 10.50 | 9.02 | 8.47 | 7.06 |
16 (Analytical) | 6.20 | 9.48 | 1.007 | 8.40 | 7.87 |
16 (FEA) | 6.21 | 9.54 | 1.01 | 8.42 | 7.91 |
17 (Analytical) | 4.25 | 3.18 | 7.40 | 4.20 | 5.78 |
17 (FEA) | 4.42 | 3.24 | 7.49 | 4.23 | 5.87 |
18 (Analytical) | 5.43 | 8.32 | 9.00 | 7.50 | 7.00 |
18 (FEA) | 5.54 | 8.41 | 9.02 | 7.52 | 7.06 |
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Wang, H.; Wu, S.; Shao, Z. Analytical Compliance Equations of Generalized Elliptical-Arc-Beam Spherical Flexure Hinges for 3D Elliptical Vibration-Assisted Cutting Mechanisms. Materials 2021, 14, 5928. https://doi.org/10.3390/ma14205928
Wang H, Wu S, Shao Z. Analytical Compliance Equations of Generalized Elliptical-Arc-Beam Spherical Flexure Hinges for 3D Elliptical Vibration-Assisted Cutting Mechanisms. Materials. 2021; 14(20):5928. https://doi.org/10.3390/ma14205928
Chicago/Turabian StyleWang, Han, Shilei Wu, and Zhongxi Shao. 2021. "Analytical Compliance Equations of Generalized Elliptical-Arc-Beam Spherical Flexure Hinges for 3D Elliptical Vibration-Assisted Cutting Mechanisms" Materials 14, no. 20: 5928. https://doi.org/10.3390/ma14205928