Curve Optimization of Tapered Cantilever Beams Under Tip Loads

The study adopts the variational method for analyzing the cantilever tapered beams under a tip load as well as a definite end displacement,and further determining the optimized shapes and materials that can minimize the weights.Two types of beams are taken into account,i.e.,the Euler-Bernoulli beam without considering shear deformation and the Timoshenko beam with shear deformation.By using the energy theorem and the reference of isoperimetric problem,the width variation curves and the corresponding minimum masses are derived for both beam types.The optimized curve of beam width for the Euler-Bernoulli beam is found to be a linear function,but nonlinear for the Timoshenko beam.It is also found that the optimized curve in the Timoshenko beam case starts from non-zero at the tip end,but its tendency gradually approaches the one of the Euler-Bernoulli beam.The results indicate that with the increase of the Poisson’s ratio,the required minimum mass of the beam will increase no matter how the material changes,suggesting that the optimized mass for the case of Euler-Bernoulli beam is the lower boundary limit which the Timoshenko case cannot go beyond.Furthermore,the ratio r/E(density against Elastic Modulus)of the material should be as small as possible,while the ratio h2/L4 of the beam should be as large as possible in order to minimize the mass for the case of Euler-Bernoulli beam,of which the conclusion is extended to be applicable for the case of Timoshenko beam.In addition,the optimized curves for Euler-Bernoulli beam types are all found to be power functions of length for both tip point load cases and uniform load cases.

A Path Planning Method for Robotic Belt Surface Grinding

The flexible contact and machining with wide strip are two prominent advantages for the robotic belt grinding system, which can be widely used to improve the surface quality and machining efficiency while finishing the workpieces with sculptured surfaces. There lacks research on grinding path planning with the constraint of curvature. With complicated contact between the contact wheel and the workpiece, the grinding paths for robot can be obtained by the theory of contact kinematics. The grinding process must satisfy the universal demands of the belt grinding technologies, and the most important thing is to make the contact wheel conform to the local geometrical features on the contact area. For the local surfaces with small curvature, the curve length between the neighboring cutting locations becomes longer to ensure processing efficiency. Otherwise, for the local areas with large curvature, the curve length becomes shorter to ensure machining accuracy. A series of planes are created to intersect with the target surface to be ground, and the corresponding sectional profile curves are obtained. For each curve, the curve length between the neighboring cutting points is optimized by inserting a cutter location at the local area with large curvatures. A method of generating the grinding paths including curve length spacing optimization is set up. The validity is completely approved by the off-line simulation, and during the grinding experiments with the method, the quality of surface is improved. The path planning method provides a theoretical support for the smooth and accuracy path of robotic surface grinding.

Optimization of parabolic bars for maximum Bragg resonant reflection of long waves

This paper presents an analytical solution for the problem of the long wave reflection by a series of artificial bars with parabolic configuration in terms of the associated Legendre functions. It is shown that both the reflection and transmission coefficients depend solely upon the number of bars, the dimensionless bar height, the dimensionless bar width and the dimensionless bar distance. Particularly, under the Bragg resonance condition, i.e., the distance between two adjacent bars is about half of the wavelength of the normal incident waves, the analytical solution for the peak Bragg resonant reflection is obtained, which reveals that the peak Bragg resonance depends upon the number of bars, the dimensionless bar height and the dimensionless bar width. Based on this solution, the optimization of the parabolic bars is made to obtain the maximum Bragg resonance and a group of optimal curves, which may be very useful in the design of Bragg breakwaters with parabolic bars.