整合連體力學與熱動力學之變分原理(2/3)

In dealing with the mechanics of material bodies modeled by particles, rigid body, or
deformable body subject to various types of constraint, variational principles can usually
provide a suitable path to obtain the solutions. The main objective of this project is to seek
additional applications and possible extensions on the basis of the Principle of Virtual
Power, and to explore whether the proposed principle can be unified with the law of
thermodynamics to form a unified variational principle. After nearly two-year term of
endeavor, we are able to obtain the desired Principle by choosing the velocity and the
velocity gradient as fundamental variables in mechanics, and the entropy and the hear flux
as the fundamental quantities in theomodynamics. The Principle indeed unify the theory of
thermodynamics and continuum mechanics, from which a variational equation can be
established. The Cauchy’s laws of linear and angular momentum and the newly developed
variational constitutive law are then deduced. By selecting appropriate internal energy
function and dissipating function, we have derived the constitutive equations for elastic
body or viscous fluid in mechanics, and Fourier law and Maxwell-Catteneo Equation in the
first year. In the second year, we established the constitutive laws for the thermo-viscous
fluid and micro-polar material. See Part I of this report for more details. On the other hand,
in the aspect of tracking control, we have used the Reduced Appell Equation to construct
equations of motion of the privileged coordinates, and found that the equations are
decoupled from the non-privileged coordinates in the first year. The fuzzy controller and
sliding mode controller were then used to design the tracking control loop. This year, we
adopt the backstepping method applied to the chained form of the kinematic equation to
compute the compensated values for the privileged variables. Simulation results on the
tracking control of wheeled vehicle show that the performance is excellent. The results are
shown in Part II of this report.