VECTOR AND SCALAR EQUATIONS TYPES TO SOLVE PROBLEMS OF THE DYNAMICS OF THE STEWART PLATFORM
The Stewart platform on the stationary body are considered. It consists of body, which rigidly connected to the ground, platform and six identical suspension devices for platform. Each suspension device consists of three bodies - the base, the guide and the rod. The solution of the dynamics equations and formulas of calculating the dynamic suspension reactions are discussed in earlier articles of authors. In this article the formulas are used in the derivation of equation for Stewart platform. The first assertion is proved a formula for expansion related to the axis of the guide forces of dynamic reactions in ball joints for connecting the platform to the ends of the rods. The second assertion is proved formulas for calculating the driving forces of rods. The third assertion is proved formulas for calculating the dynamic reactions in joints suspensions. All received formulas contains geometric, kinematic and inertial parameters of moving bodies. This allows to solve the problem of synthesis parameters to achieve the desired Stewart platform dynamic properties. Explicit types of vector and scalar systems of differential-algebraic equations for solving the second problem of the dynamics are prepared for re-use in mathematical models of Stewart platform on the mobile (carrier) body. These models are needed to investigate the influence of movements of the body carried by supporting body. Three examples are considered in order to verify the absence of errors in the calculation formulas. The particular cases of motion of the bodies of the system are analyzed in these examples. The formulas to calculate the driving forces and the obvious equation of geometric relations are obtained as a special case of the proven formulas for the motion of the platform on three hangers in the vertical plane. The equations of the dynamics for two hangers, connected at the ends of the rods with rotational hinge, and the corresponding apparent equation of geometric relations are obtained in the second example of these formulas, as a special case. The obvious equation for movement along a horizontal line rods that follows from Newton's second law are obtained in the third example.