| Multi-rigid-body dynamics has grown in the past four decades to be an important tool in the design and simulation of complex mechanical systems . This is especially true in the area of aerospace dynamics and control , robotics , biomechanics , and road and rail vehicle . At present , the modeling methods of multi-rigid-body system can be concluded to two kinds : Absolute coordinate method and relative coordinate method . In the absolute coordinate method , the position and orientation of every rigid body in the system are represented by the absolute coordinates with respect to the reference base of the system , the array of absolute coordinates of the system consists of the absolute coordinates of all rigid bodies . Owing to the existence of constraints in the system, so its absolute coordinates aren't independent completely . Apply Lagrange equation of the first kind to the system , and get a set of the differential - algebraic equations (DAEs) of its absolute coordinates . By the absolute coordinate method , the handle to the constrain equations of the system is fairly simpler , but a number of governing equation of the system is most . In addition , the numerical methods solved DAE aren't perfect like the initial value problem of pure differential equations , and its intrinsic ill-condition can lead to many difficulties in the numerical calculation .In the relative coordinate method , the array of the generalized coordinates of the system is formed by the relative coordinates of all rigid bodies . The principle of virtual power and Newton – Euler equations are applied to the system , and arrive at a set of second order differential equations of the generalized coordinates . By this method , the number of the dynamic equations is least , but it is necessary that the constraint relations are established between the velocity variation of every rigid body and the generalized velocity variation , the relations are often complicated . Moreover , for the treelike system with prescribed motion and the non-treelike system with closed – loop , the generalized coordinates aren't independent completely . Only the dynamical equations of the system and the constrain equations resulting from the prescribed motion and the closed – loop are combined to form DAEs , it can be solved .In this paper , the modeling theory of multi – rigid – body dynamics was thoroughly investigated in order to make up the shortcoming in the above modeling method . The major research achievements are as follows :Based on the matrix of direction cosine between the body – fixed base and the reference base , and Poisson equation of a rigid body rotating , the coordinate transformation method for the modeling of treelike multi – rigid – body dynamics is derived . The method is very routinization and easy to use the computer to carry on the symbol derivation . The dynamic equations of the system obtained by the method are pure differential eguations , and convenient for the numerical calculation .A method is presented to study the dynamic problems of treelike multi – rigid – body system with described motion . This method bases on the dynamic equations derived by the coordinate transformation method , and handles the additional constraints using Lagrange multiplier method and coordinate partitioning method . In the numerical calculation of the dynamic equations , the theory of generalized inverse of matrices is applied in the modified constraint violation . This method also is applicable to study non – treelike multi – rigid – body system .The state equations of a inertial vibrating machine are derived by the coordinate transformation method , and its electromechanical coupling model is established by combining its state equations with the one of its motor . The starting behavior of the machine is studied by way of numerical simulation to this model .Based on the research to the starting behavior of the inertial vibrating machine , a new type of inertial exciter is developed , and has applied for a invention patent to...
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