Elastic Element showing low Stiffness loss at large Deflections
We present a curved hinge flexure, which is an elastic element with only a limited stiffness decrease at large rotational deflections. In terms of degrees of freedom the curved hinge flexure might be compared to a leaf-spring. A leaf-spring ideally constrains 3 degrees of freedom if it is straight. The three stiff directions are the longitudinal elongation and the two bending modes in-plane of the leaf-spring. When twisting or bending the leaf-spring in the out-of-plane direction the 3 stiff directions loss stiffness. The curved hinge flexure also constrains 3 DOFs, however if deflected by an in-plane bending moment the stiffness loss is less compared to the stiffness loss of a single leaf-spring.
In its basic form the curved hinge flexure consists out of two curved leaf-springs. In the low stress or neutral position the leaf-springs are curved and contribute both to for example the longitudinal stiffness. The longitudinal stiffness however is not at its maximum due to the curved shapes of the leaf-springs. When bending the curved hinge flexure in the out-of-plane direction, one leaf-spring will straighten while the other becomes more curved. The longitudinal stiffness of the more straightened leaf-spring will increase faster than longitudinal stiffness of the more bended leaf-spring will decrease. Therefore the overall longitudinal stiffness, will increase. The overall longitudinal stiffness increases until one of the leaf-springs is exactly straight, after which the overall longitudinal stiffness will drop. The described theory about stiffness in the longitudinal direction applies in more or less the same way to the stiffness of the two in-plane bending modes.
For example, a curved hinge flexure has been optimized in such a way that during an increased external bending moment at a certain instant one of the leaf-springs becomes exactly straight at 10 degrees. When comparing a single leaf-spring with dimensions 50 x 50 x 0.5 mm3 with the curved hinge flexure with leaf-spring dimensions 50 x 50 x 0.35 mm3, over a +/- 15 degrees Rz stroke, the worst case Cx, Cz and CRy stiffness of the single leaf-spring is respectively 12x, 8x and 4x less (Figure 2, 3 and 4.) The strain of the curved hinge flexure in the example rises up to 0.30%, which can be a point of concern.
The Curved Hinge Flexure: http://www.youtube.com/watch?v=sBFB_5SQpak
Figure 1. The curved hinge flexure consists of 2 pre-shaped leaf-springs. In this case the dimensions of the leaf-springs are 50 x 50 x 0.35 mm3.
Figure 2. The pre-curvature of the leaf-springs is optimized in such a way that if the suspended body is loaded by a bending moment around the z-axis, one of the leaf-springs becomes exactly straight. The corresponding rotation of the suspended body is chosen 10 degrees in this case. The coordinate system representing the stiffness directions is fixed to the suspended body. The linearized stiffnesses are calculated in 6 DOFs for this rotation Rz.
Figure 3. Graph showing the translational stiffnesses Cx, CY and Cz of a single leaf-spring flexure with dimensions 50 x 50 x 05. mm3 and 4 curved hinge flexure with each leaf-spring having the dimensions 50 x 50 x 0.35 mm3. The 4 curved hinge flexures are optimized for 3, 6, 10 and 15 degrees deflection in Rz respectively.
Figure 4. Graph showing the rotational stiffnesses CRx, CRy and CRz of a single leaf-spring flexure with dimensions 50 x 50 x 05. mm3 and 4 curved hinge flexure with each leaf-spring having the dimensions 50 x 50 x 0.35 mm3. The 4 curved hinge flexures are optimized for 3, 6, 10 and 15 degrees deflection in Rz respectively.
For more information:
D.M. Brouwer, J.P. Meijaard, J.B. Jonker, Elastic element showing low stiffness loss at large deflection, Proceedings of the 24th annual meeting of the American Society of Precision Engineering, 4-9 Oct 2009, Monterey, pp30-33, ISBN 978-1-887706-51-3
S.E. Boer, R.G.K.M. Aarts, D.M. Brouwer, J.B. Jonker, Multibody modelling and optimization of a Curved Hinge Flexure, The 1st Joint International Conference on Multibody System Dynamics, May 25–27, 2010, Lappeenranta, Finland, Submitted.