A PPLIED M ECHANICS

1. Slovak University of Technology Faculty of Material Science and Technology in Trnava APPLIED MECHANICS Lecture 02

2. ENERGY METHODS OF APPLIEDMECHANICS • Energy Methods • An alternative way to determine the equation of motion and an alternative way to calculate the natural frequency of a system • Useful if the forces or torques acting on the object or mechanical part are difficult to determine

3. ENERGY METHODS OF APPLIEDMECHANICS • Quantities used in these methods are scalars - scalar dynamics • Method provides a very powerful tool for two main reason: • It considerably simplifies the analytical formulation of the motion equations for a complex mechanical system • It gives rise to approximate numerical methods for the solution for both discrete and continuous systems in the most natural manner

4. ENERGY METHODS OF APPLIEDMECHANICS • Potential energy - the potential energy of mechanical systems Epis often stored in “springs” - (remember that for a spring F = kx0) • Kinetic energy - the kinetic energy of mechanical systems Ekis due to the motion of the “mass” in the system • Conservation of mechanical energy - for a simply, conservative (i.e. no damper), mass spring system the energy must be conserved.

5. Principle of virtual work • Dynamic equilibrium of the particle (d’Alembert) whereui represents the displacement of the particle, Xiare forces. • Let us consider that the particle follows during the time interval [t1, t2] a motion trajectory distinct from the real one ui. This allows us to define the virtual displacement of the particle the relationship

6. Principle of virtual work • Multiplying of equation of motionby associated virtual displacement and sum over the components • The virtual work principle for the system of particles „The virtual work produced by the effective forces acting on the particle during a virtual displacement duiis equal to zero“. „If the virtual work equation is satisfied for any virtual displacement compatible with the kinematical constraints, the system is in dynamic equilibrium“.

7. Principle of virtual work • The kinematical constraints • the state of the system would be completely defined by the 3N displacement components uik, • the particles are submitted to kinematic constraints which restrain their motion, • they define dependency relationship between particles, • they represent the instantaneous configuration, • starting from the reference configurationxik, • instantaneous configurationdetermined by xik(t)= xik + uik(t) • The kinematical constraints are divided on: • holonomic constraints - defined byf(xik(t)) = 0 • Non-holonomic constraints

8. Principle of virtual work • The kinematical constraints are divided on: • holonomic constraints - defined by f(xik, t) = 0 • scleronomic -constraintsnot explicitly dependenton time • rheonomic – constraintsexplicitly dependenton time • non-holonomic constraints - defined by These equations are generally not integrable.

9. Principle of virtual work • Generalized coordinates and displacements • If s holonomic constraints exist between the 3N displacements of the system, the number of DOF is then reduced to 3N - s. It is then necessary to define n = 3N - sgeneralizedcoordinates, noted in terms of whichthe displacements of the system of particles are expressed in the form • The virtual displacement compatible with the holonomic constraints may be expressed in the form

10. Principle of virtual work • Virtual work equation becomes where is the generalized force

11. Hamilton´s Principle • Hamilton´s principle- time integrated form of the virtual work principle obtained by transforming the expression • Applied forces Xikcan be derived from the potential energy - virtual work is expressed in the form • The generalized forces are derived from the potential energy

12. Hamilton´s Principle • The term associated with inertia forces • The kinetic energy • Then, time integrated form of the virtual work principle

13. Hamilton´s Principle • In terms of generalized coordinates is expressed • “Hamilton´s principle: The real trajectory of the system is such as theintegral remains stationary with respect to any compatible virtual displacement arbitrary betweenboth instants t1, t2but vanishing at the ends of the interval where L – is a kinetic potential or Lagrangian

14. Lagrange´s Equations of 2nd Order • Using expression in equation for Hamilton´s principle The second term can be integrate by parts

15. Lagrange´s Equations of 2nd Order Taking into account the boundary conditions the following is equivalent to Hamilton´s principle The variation dqsis arbitrary on the whole interval and the equations of motion result in the form obtained by Lagrange

16. Lagrange´s Equations of 2nd Order • Classification of generalized forces • Internal forces • Linking force - connection between two particles • Elastic force - elastic body - body for which any produced work is stored in a recoverable form - giving rise tovariation of internal energy with the generalized forces of elastic origin

17. Lagrange´s Equations of 2nd Order • Dissipative force - remains parallel and in opposite direction to the velocity vector and is a functions of its modulus. Dissipative force acting of a mass particle k is expressed by whereCik is a constant fk(vk)is the function expressing velocity dependence, vkis the absolute velocity of particle k The dissipative force The dissipative function

18. Lagrange´s Equations of 2nd Order • External conservative force - conservative - their virtual work remains zero during a cycle generalized force is expressed • External non-conservative force generalized force is expressed General form of Lagrange equation of 2nd order of non-conservative systems with rheonomic constraints