MECH 5505: Stability Theory and Applications

Fundamental concepts and characteristics of modern stability definitions. Sensitivity and variational equations; linear variational equations; phase space analysis; Liapunov’s direct method. Autonomous and non-autonomous systems; stability in first approximation; the effect of force type on stability; frequency method.

Lectures Outline:

  1. Review of fundamentals of Analytical Mechanics:  Hamilton=s principle, Lagrange=s   equations of motion, Hamilton=s canonical equations, motion in the phase space.
  2. Fundamentals and common characteristics of stability definitions:  equilibrium and asymptotic stability; stability in the large; conditional stability; equilibrium and equations of perturbed motion.
  3. The direct Liapunov method for autonomous systems:  Sylvester=s criterion; Liapunov functions; Liapunov=s theorem of stability; asymptotic stability; theorems of instability; methods to obtain Liapunov functions; applications.
  4. Equilibrium states and stationary motions of conservative systems:  Lagrange=s Theorem and its invertibility; cyclic coordinates; the Routh transform; stability of stationary motion; applications.
  5. Stability in first approximation:  general formulation of the problem; theorems of stability in first approximation; Hurwitz=s criterion; applications.
  6. Linear autonomous systems:  matrices and matrix operations; elementary divisors; stability of autonomous linear systems; stability of resonance.
  7. Direct Liapunov method and stability of control systems:  governing differential equations of perturbed motion of automatic control systems; canonical equations of perturbed motion; Liapunov functions; absolute stability.
  8. The frequency method of stability analysis:  transfer functions and frequency characteristics; Nyquist stability criterion; nonlinear systems; applications.

Suggested References: (on reserve at Carleton University Library)

  1. Introduction to the Theory of Stability,  by: D. R. Merkin, Springer 1997
  2. Methods of Analytical Dynamics, by: Leonard Meirovitch, McGraw-Hill 1970
  3. Stability Theory, by: H.H.E. Leipholz, Academic Press, 1970
  4. Introduction to Dynamics and Control, by: Leonard Meirovitch, John Wiley and Sons,1985
  5. Matrix Methods in Stability Theory, by: S. Barnett, C. Storey, Thomas Nelson & Sons Ltd., 1970.
  6. Introduction to Perturbation Techniques, by: A.H. Nayfeh, John Wiley & Sons, 1981