This paper makes an investigation on geometric relationships among nodes of the valuated binary trees, including parallelism, connection and penetration. By defining central lines and distance from a node to a line, some intrinsic connections are discovered to connect nodes between different subtrees. It is proved that a node out of a subtree can penetrate into the subtree along a parallel connection. If the connection starts downward from a node that is a multiple of the subtree’s root, then all the nodes on the connection are multiples of the root. Accordingly composite odd integers on such connections can be easily factorized. The paper proves the new results with detail mathematical reasoning and demonstrates several numerical experiments made with Maple software to factorize rapidly a kind of big odd integers that are of the length from 59 to 99 decimal digits. It is once again shown that the valuated binary tree might be a key to unlock the lock of the integer factorization problem.

Abstract. The article demonstrates that the energy method utilized in computational mechanics is a special form of the variational method through mathematical deduction and comparison to the two methods. The result discloses a basic fact that mechanical computations performed by different methods are intrinsically related in some way. The article also points out that a deep study on this intrinsic relation is necessary and helpful to guarantee loyalty of the computational results to the computed objectives.

To reduce the weight of vehicles and save on fuel consumption, the forming analysis and process study for a 6082 aluminum alloy automobile control arm were conducted, and the results are reported here. First, the parameters of the constitutive equation of the 6082 aluminum alloy are determined, and the forming process and the dies of the stretch rolling, bending, and die-forging are proposed and discussed according to the given contours of the control arm. Then, the control arm is simulated by means of Deform-3D, the experiment results of the automobile control arm are compared with the simulation results, and the tensile strength, yield strength, and elongation of the test bars obtained from the control arm can meet the mechanical performance requirements based on the tensile test rig. Finally, the relationship of the die load, die wear, workpiece damage, and workpiece equivalent stress with different processing parameters are analyzed, and the effects of different solid solution times on the strength of control arm are discussed. The results show that when the friction coefficient is 0.3, the die temperature is 250°C, the forging velocity is 230 mm/s, the strain rate is 1s−1 and the solid solution time is 30 min, the die load, die wear, workpiece damage, workpiece equivalent stress, and the mechanical strength are then in a good state. The results of this research provide a reliable basis and analysis method for the forging of aluminum alloy automobile control arms.

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