AE331AE331

Virtual WorkVirtual Work(Energy Principles)(Energy Principles)

Solution of problems in mechanicsSolution of problems in mechanics

1.1. Choose system of interest.Choose system of interest.2.2. Isolate the system and draw the free Isolate the system and draw the free

body diagram.body diagram.3.3. Write equilibrium equations.Write equilibrium equations.4.4. Study the deformation and write the Study the deformation and write the

conditions of geometric fit.conditions of geometric fit.5.5. Write force-deformation relationsWrite force-deformation relations

Steps 4 and 5 not needed for statically Steps 4 and 5 not needed for statically determinate systemsdeterminate systems

Solution of problems in mechanics Solution of problems in mechanics (Energy Approach)(Energy Approach)

Based on Conservation of Energy Based on Conservation of Energy principleprinciple

Energy Approach- ExampleEnergy Approach- Example Dennis the Menace is a Dennis the Menace is a

daily syndicated daily syndicated newspaper comic strip.newspaper comic strip.

Dennis Mitchell is a five-Dennis Mitchell is a five-year-old boy with a year-old boy with a penchant for mischief.penchant for mischief.

The comic strip was The comic strip was started on March 12, started on March 12, 1951. It was originally 1951. It was originally created, written and created, written and illustrated by Hank illustrated by Hank Ketcham(1920-2001).Ketcham(1920-2001).

Some characters in Dennis lifeSome characters in Dennis life

Principal of minimum total potential energy

For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium extremize the total potential energy . If the extremum condition is a minimum , the equilibrium state is stable.

The total potential energy of an elastic body , is defined as the sum of total strain energy (U) and the work potential (WP) .

Example

Variational Calculus

Variational calculusDifferential Calculus operates on

functions of one or more variables.Example: Use derivative to find the

minimum or maximum

Some problems involve a functional. A functional is the function of a function

Example: work defined on a path; path is a function in space

Varaitional CalculusA trajectory y in space is a parametric function.y(,x) = y(0,x) + (x)Continuous variation (x)End points (x1) = (x2) = 0

Define a function f in space.

Minimize the integral J.If y is varied J must increase

J= f ( y ( x ) ,y' ( x ) ;x )dx

Finding extremum of J

Finding extremum of J

We note that (x)=0 at x=x1 and x2. Hence, we can write:

Euler's equation

Since we have Arbitrary variation in (x). hence The quantity in brackets must vanish.

This is Eulers equation.

Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16