Yu Liu

Date of Award

Spring 2013

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mechanical Engineering

Committee Director

Gene Hou

Committee Member

Duc Nguyen

Committee Member

Miltiadis Koyinis

Committee Member

Jennifer Michaeli


Research in computational dynamics has tremendously developed in the recent years because of the demand for analysis and simulation of various multi-body systems in the growing bio-medical, mechanical and aerospace industries. These multi-body systems are made of individual bodies that are interconnected via mechanical joints. Mathematically, these joints that connect the bodies can be described as constraint equations imposed upon the motions of the involved free bodies. This process will result in an equation of motion expressed in the form of a differential-algebraic equation (DAE). This is one of the main difficulties when dealing with the multi-body system because these constraints must be satisfied all the time.

The main objective of this dissertation is to develop an efficient and accurate solution algorithm to solve the DAE resulting from flexible multi-body dynamics. The principle of virtual work and D'Alembert's principle are used in this dissertation to formulate the equation of motion for a general three dimensional (3D) multi-body dynamic system that involves rigid as well as flexible bodies. The elastic mode shapes and modal coordinates are used to convert the time-variant integrals associated with elastic deformations in the mass matrix into time-invariant ones. In addition, the transient stress distribution is obtained directly in terms of the linear combination of the modal element stress and corresponding modal coordinates solved from DAE. Euler parameters and the matrix exponential method are used to calculate the time-dependent transformation matrix for a general 3D problem. The projection method with constraints correction is proposed in this dissertation to solve the DAE modeling the motion of a constrained multi-body dynamic system. The mixed order technique with the additional Euler parameters method is proposed to solve a general 3D flexible multi-body system. Two examples are studied in this dissertation: a planar slider-crank mechanism and a 3D flexible moving craft in irregular waves.

The planar slider-crank mechanism is used in this dissertation to demonstrate the application of the integrals calculated as time-invariants and the proposed projection method with displacement and velocity constraints correction. The flexibility of the connecting rod of the slider-crank mechanism is included in the formulation. The numerical results obtained by the projection method will be compared with those by the commonly used coordinate partitioning method. The results show the validation and efficiency of the proposed constraints correction method.

For the 3D flexible craft dynamics, the pressure distribution reconstruction algorithm is carried out to construct the hydrodynamics pressure on the wetted surface based upon the test pressure data. Then the nodal pressure loads are converted to the equivalent nodal force as the external loads for the flexible craft. Both Euler parameters and angular velocities are treated as the generalized coordinates in the equation of motion to model the rotation motion of the craft. Hence, the second order of Euler parameters is not incorporated in the equation of motion. It means that only constraints on first order time derivation of Euler parameter are needed. The results from the proposed Euler parameter methods are compared with the matrix exponential based Newmark method. It shows that the proposed Euler parameter method is non-sensitive to the time steps and has good accuracy. Finally, the least square error optimization method is used to find the Von Mises stress at each node. Thus, the time history nodal stress can be obtained directly from the modal element stress and modal coordinates solved from DAE. Hence, it doesn't need to rerun the dynamic analysis under the nodal displacement to obtain the node stress.