1. A PPLIED M ECHANICS Lecture 02 Slovak University of Technology Faculty of Material Science and Technology in Trnava

2. ENERGY METHODS OF APPLIED MECHANICS 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 APPLIED MECHANICS 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 APPLIED MECHANICS Potential energy - the potential energy of mechanical systems E p is often stored in “springs” - (remember that for a spring F = kx 0 ) Kinetic energy - the kinetic energy of mechanical systems E k is 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) Let us consider that the particle follows during the time interval [ t 1, t 2 ] a motion trajectory distinct from the real one u i. This allows us to define the virtual displacement of the particle the relationship where u i represents the displacement of the particle, X i are forces.

6. Principle of virtual work The virtual work principle for the system of particles Multiplying of equation of motion by associated virtual displacement and sum over the components „The virtual work produced by the effective forces acting on the particle during a virtual displacement  u i is 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 u ik,  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 configuration x ik,  instantaneous configuration determined by  ik (t) = x ik + u ik (t) The kinematical constraints are divided on:  holonomic constraints - defined by f(  ik (t)) = 0  Non-holonomic constraints

8. Principle of virtual work The kinematical constraints are divided on:  holonomic constraints - defined by f(  ik, t) = 0 scleronomic - constraints not explicitly dependent on time rheonomic – constraints explicitly dependent on 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 - s generalized coordinates, noted in terms of which the 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 X ik can 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 the integral remains stationary with respect to any compatible virtual displacement arbitrary between both instants t 1, t 2 but 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  q s is 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 to variation 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 where C ik is a constant f k (v k ) is the function expressing velocity dependence, v k is 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 2 nd order of non-conservative systems with rheonomic constraints