AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Analytical Dynamics of Discrete Systems These slides are based on the recommended textbook M Ge radin and D Rixen Mechanical Vibrations Theory and Applications to Structural Dynamics Second Edition Wiley John Sons Incorporated ISBN 13 9780471975465 AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Outline 1 Principle of Virtual Work for a Particle 2 Principle of Virtual Work for a System of N Particles 3 Hamilton s Principle for Conservative Systems and Lagrange Equations 4 Lagrange Equations in the General Case AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a Particle Particle mass m Particle force force vector f f1 f2 f3 T force component fi i 1 3 Particle displacement displacement vector u u1 u2 u3 T displacement component ui i 1 3 motion trajectory u t where t denotes time AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a Particle Particle virtual displacement arbitrary displacement u can be zero virtual displacement u u u arbitrary by definition family of arbitrary virtual displacements defined in a time interval t1 t2 and satisfying the variational constraints u t1 u t2 0 Important property d d du dui ui ui ui i u i u i u i dt dt dt dt d d dt dt commutativity AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a Particle Equilibrium strong form mu f 0 mu i fi 0 i 1 3 weak form u uT mu f 0 3 X mu i fi ui 0 i 1 mu i fi ui 0 T i 1 3 T u mu f mu f u is homogeneous to a work virtual work W Virtual work principle The virtual work produced by the effective forces acting on a particle during a virtual displacement is equal to zero AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles N particles k 1 N Equilibrium mu k fk 0 k 1 N Family of virtual displacements uk u k uk satisfying the variational constraints uk t1 uk t2 0 1 Virtual work mu k fk 0 N X k 1 uT k mu k fk N X mu k fk T uk 0 k 1 AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles Conversely uk compatible with the variational constraints 1 N X uT k mu k fk 0 N X 3 X mk u i k fik uik 0 2 k 1 i 1 k 1 If 2 is true uk compatible with 1 2 is true for uk 1 0 0 T uk 0 1 0 T and uk 0 0 1 T t t1 t2 N X mk u i k fik 0 i 1 3 k 1 If the virtual work equation is satisfied for any displacement compatible with the variational constraints the system of particles is in dynamic equilibrium AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles Major result dynamic equilibrium virtual work principle AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles Kinematic Constraints In the absence of kinematic constraints the state of a system of N particles can be defined by 3N displacement components uik i 1 3 k 1 N Instantaneous configuration ik xik uik x t 3N dofs However most mechanical systems incorporate some sort of constraints holonomic constraints non holonomic constraints AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles Kinematic Constraints Holonomic constraints rheonomic constraints defined by f ik t 0 no explicit dependence on any velocity scleronomic constraints defined by f ik 0 a holonomic constraint reduces by 1 the number of dofs of a mechanical system example rigidity conservation of length 3 P i2 i1 2 l 2 i 1 AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles Kinematic Constraints Non holonomic constraints defined by f ik ik t 0 example no slip speed of point P 0 x 1 0 r cos y 1 0 r sin in addition x2 x1 r sin cos z1 r x 1 r cos 0 y 1 r sin 0 y2 y1 r sin sin z2 z1 r cos AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles Kinematic Constraints hence this system has 8 variables x1 y1 z1 x2 y2 z2 4 holonomic constraints 2 non holonomic constraints in general f ik ik t 0 is not integrable and therefore non holonomic constraints do not reduce the number of dofs of a mechanical system therefore the above mechanical system wheel has 8 4 4 dofs 2 translations in the rolling plane 2 rotations AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles Generalized Displacements Let n denote the number of dofs of a mechanical system for example for a system with N material points and R holonomic constraints n 3N R The generalized coordinates of this system are defined as the n configuration parameters q1 q2 qn in terms of which the displacements can be expressed as uik x t Uik q1 q2 qn t If the system is not constrained by any non holonomic constraint then the generalized coordinates q1 q2 qn are independent they can vary arbitrarily without violating the kinematic constraints AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles Generalized Displacements Example 2 2 holonomic constraint HC1 11 21 l12 holonomic constraint HC2 12 11 2 22 21 2 l22 4 2 2dofs one possible choice of q1 q2 is 1 2 11 l1 cos 1 21 l1 sin 1 12 l1 cos 1 l2 cos 1 2 22 l1 sin 1 l2 sin 1 2 AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles Generalized Displacements Virtual displacements uik x t Uik q1 q2 qn t uik n X Ui s 1 Virtual work equation N 3 n X XX s 1 k 1 i 1 Uik mk u i k fik qs qs 0 n P Qs qs where s 1 3 N X X fik k 1 i 1 qs Second term in above equation can be written as Qs k qs Uik qs is the generalized force conjugate to qs AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Hamilton s Principle for Conservative Systems and Lagrange Equations Sir William Rowan Hamilton 4 August 1805 2 September 1865 Irish physicist astronomer and mathematician contributions classical mechanics optics and algebra inventor of quaternions and most importantly reformulation of Newtonian mechanics now called Hamiltonian mechanics impact modern study of electromagnetism development of quantum mechanics AA242B MECHANICAL VIBRATIONS AA242B MECHANICAL VIBRATIONS Hamilton s Principle for Conservative Systems and Lagrange Equations Hamilton s principle Z t2 t1 R t2 t1 virtual work principle N 3 XX mk u i k fik uik dt 0 k