Course Title: Theoretical Mechanics
Course Attribute: Specialized Elective Course
Hours/credits: 40 hours/2 credits
Pre-courses: Calculus Linear algebra Analytic Geometry
Teaching goal and requirement:
As a fundamental and core course, Theoretical Mechanics (Dynamics + Vibration) provides not only an in-depth learning on Nature but also a powerful tool for the future engineers. This course will provide a rigorous, comprehensive and compact presentation on the rigid body dynamics and vibration. The students are required to proficiently master the methods of both vectorial and analytical mechanics, and understand the basics of the relativistic mechanics.
Course summary:
Chapter 1 Fundamental Concepts
1-1 Newton’s three laws, Euler’s two laws, nonrelativistic space and time.
1-2 Statics, three factors of a force, moment, force couple, balance of force, friction
1-3 Reference frame, coordinate system
Chapter 2 Kinematics of a particle
2-1 Trajectory/motion, velocity, acceleration, invariance of Galilean transformation
2-2 Intrinsic coordinate, intrinsic equation, osculating plane, principle normal, binormal
Chapter 3 Kinematics of rigid body
3-1 Translation, rotation, Chasles’theorem
3-2 Rotation about a point and an axis
3-3 Pseudo-vector property of angular velocity and its independence on the choice of base point, Euler angles.
3-4 Rigid body motions: relative velocity/acceleration, entrainment velocity/ acceleration, Coriolis acceleration.
3-5 Application of the rigid body kinematics in astronomy: Precession, Ptolemy’s geocentric and Copernicus’s heliocentric views on the solar system.
Chapter 4 Dynamics of rigid body
4-1 Momentum, angular momentum, moment of inertia
4-2 Dynamics of the toy “yo-yo”: The application of Koenig theorem.
4-3 Dynamics of the variable mass system, Tsiolkovsky formula, related dynamics problems in a spacecraft launch.
4-4 Dynamics in a non-inertial frame, D’Alembert Principle
4-5 Foucault pendulum, monsoon, cannonball trajectory in battle of the Falkland islands during World War I
Chapter 5: Gravity
5-1 Kepler problems
5-2 The three escape velocities
5-3 Application: The trajectory problem encountered by the Appollo 13, Einstein’s explanation on the Mercury precession
Chapter 6 Relativistic mechanics
6-1 Equivalence postulate, invariance of the light speed, Minkowski space, Galilean transformation, Lorentz transformation.
6-2 Newton’s second law in relativistic mechanics, Minkowski force, Thomas precession frequency
Chapter 7 Analytical Mechanics
7-1 Virtual displacement/work, stationary value
7-2 Least “action” principle, geodesics, brachistochrone
7-3 Euler-Lagrange equation, Hamilton Principle and their equivalence to the Newton’s laws.
7-4 Constraint, Lagrange multiplier
Chapter 8 Vibration
8-1 Degree of freedom (DOF), one DOF system vibration, natural frequency, damping, frequency response
8-2 Two/multiple DOF system vibration, mode, eigenvalue, eigenvector
8-3 Vibration of the continuous system, transverse vibration of a string, longitudinal vibration of a bar, separation of variables, D’Alembert solution, wave velocity, dispersion.
8-4 Orthogonality of modes, self-adjoint operator, Sturm-Liouville problem, Schrödinger equation
8-5 Applications: Harmonics of the string/violin vibration
8-6 Introduction to nonlinear vibration: Isochrone curve of pendulum, Duffing equation, phase diagram, attractor, strange attractor, Lyapunov stability
8-7 Applications: The inverse problems in the mass resonators.
Assessment methods: Homework 10%, project 10%, midterm exam: 20%, Final exam: 60%.
Course Text: Zhu Zhaoxuan et al., Theoretical Mechanics, Peking University Press,1982.
Relevant Texts:
Teacher's Profile: Dr. Yin Zhang Rm 244, main building of Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190.
