ANALYTICAL DYNAMICS

VARIATIONAL PRINCIPALS FOR DYNAMICS

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

Outlines

• Introduction to Calculus of Variation • Variational Principal • The Variational Indicator method • Solving a problem using v

ariational method Sharif University of Technology Int’l Campus 2

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 ‘J’ could represent for a path between two distinct points in space.

is stationary (that is maximum or minimum). The Variational solution is derived from Euler-Lagrange equation Sharif University of Technology Int’l Campus H. Adldoost 3

Calculus of Variation

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: Sharif University of Technology Int’l Campus H. Adldoost 4

Calculus of Variation

Example 1 (cont’d)

The general solution is, Where a and b are determined by two fixed end points.

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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.

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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

f1 f f: external force f1: spring force

Dynamic-force equation Increment of work

(under admissible variation) Sharif University of Technology Int’l Campus H. Adldoost 7

Admissible motion



increment of work

Example 2 (Cont’d)

f1 f Substitute Sharif University of Technology Int’l Campus H. Adldoost 8

Example 2 (Cont’d)

f1 f 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, Increment in: Kinetic C0-Energy Potential Energy Work of external force Sharif University of Technology Int’l Campus H. Adldoost 9

Example 2 (Cont’d)

f1 f Sharif University of Technology Int’l Campus H. Adldoost 10

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 .

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4. Solving a problem using

V.I.

Find the Dynamic Eq.

y

Solution:

Restrictions: R-r x 2r 1DOF system Sharif University of Technology Int’l Campus H. Adldoost 12

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

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

Thank You

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