VARIATIONAL PRINCIPALS FOR DYNAMICS By Hamed Adldoost Instructor: Prof. Dr. Zohoor ANALYTICAL DYNAMICS 1 Sharif University of Technology, Int’l Campus, Kish, Iran

Outlines Introduction to Calculus of Variation Variational Principal The Variational Indicator method Solving a problem using v ariational method 2Sharif University of Technology Int’l Campus

1.Calculus of Variation Calculus of variation deals with problems to find a function y(x) with specified values at end-point x0 and x1 such that the integral is stationary (that is maximum or minimum). The Variational solution is derived from 3 Sharif University of Technology Int’l Campus H. Adldoost ‘J’ could represent for a path between two distinct points in space. Euler-Lagrange equation

Calculus of Variation 4 Sharif University of Technology Int’l Campus H. Adldoost Example 1 Minimum surface of revolution. A surface of revolution is formed by taking some curve passing between two fixed end points (x1, y1) and (x2, y2), and revolving it about the y-axis. Find the curve, y=y(x), for which the surface area is minimum. Solution: The total surface area is And the integral function is identified as Use Euler-Lagrange equation:

Calculus of Variation 5 Sharif University of Technology Int’l Campus H. Adldoost Example 1 (cont’d) The general solution is, Where a and b are determined by two fixed end points.

6 Sharif University of Technology Int’l Campus H. Adldoost Hamilton’s Principle: The motion of the system (in configuration space) from time t1 to time t2 is such that the line integral (the action or action integral) has a stationary value for the actual path of motion. = Lagrangian of the system Stationary value means I is an extreme. Hamilton principle suggests Nature always minimizes certain quantities when a physical process takes place.

2. Variational Principal for Dynamics The increment of work done by resultant unbalanced force in the direction of increasing x under an admissible variation must vanish identically if the resultant dynamic- force equation is always satisfied. Example 2 Dynamic-force equation Increment of work (under admissible variation) f f1 7 Sharif University of Technology Int’l Campus H. Adldoost f: external force f1: spring force

Admissible motion  increment of work Example 2 (Cont’d) ff1 8 Sharif University of Technology Int’l Campus H. Adldoost Substitute

Example 2 (Cont’d) ff1 9 Sharif University of Technology Int’l Campus H. Adldoost To eliminate time derivative term, integrate over a time interval from t1 to t2. The principle states that V.I. must vanish. If we agree on the end conditions that at t1 and t2, Kinetic C0-Energy Potential Energy Work of external force Increment in:

Example 2 (Cont’d) ff1 10 Sharif University of Technology Int’l Campus H. Adldoost

11 Sharif University of Technology Int’l Campus H. Adldoost 3. The Variational indicator method In general V.I. is a time integral over an interval t1 to t2 of the increments of work done by all forces (including inertia forces) acting on all masses in a geometrically admissible variation. T* is the sum of kinetic coenergies of all the individual mass particles in the system. V is the sum of the potential energies of the individual energy –storage elements. relates to any force f i whose work increments are not accounted for in and.

12 Sharif University of Technology Int’l Campus H. Adldoost 4. Solving a problem using V.I. R-r 2r x y Restrictions: 1DOF system Find the Dynamic Eq. Solution:

13 Sharif University of Technology Int’l Campus H. Adldoost Solution (cont’d)

14 Sharif University of Technology Int’l Campus H. Adldoost or equivalently, Solution (cont’d)

Sharif University of Technology Int’l Campus H. Adldoost 15 Thank You My Homepage: Kish.sharif.edu/~adldoost