Based on the equivalence between transformation equation and constrained equation consisting of times explicitly,the unified description of criteria for scleronomic constraint and scleronomic system is proposed.

Kinetic equations of multibody systems with scleronomic constraints are projected along the feasible and unfeasible directons of the constiaints respectively,and generalized accelerations of the systems are decomposed along the two directons using the bases of constraint matrix and its orthocomplement.

Starting with Lagrange equations of the first kind, pure differential equations of multibody systems with rheonomic constraints are established.

Furthermore, the restriction index Ri, the ratio between real and theoretical averaged cross-sectional solid holdups in the particle back-mixing region section, was defined to quantitatively show the restric.

By analyzing the deforming process of conical cold forging, this paper finds out the object function and constrained functions and gets the optimum parameters of conical forging die by using optimization melhod.

With terminal function of spring minimal quality,with optimization parameters of the wire diameter,the mean spring diameter and the number of active coils,and with constraint function of shear stress,maximum deflection and index ect.

Because there are many infeasible chromosomes,the genetic algorithm was mended to provide a constraint function for operating the infeasible chromosomes.

In the basis of the characteristic of complex trusses, the constraint functions are separated into local constraints and global constraints, and a simple method for working constraint functions is p resented in this paper.