Lecture13-Prof.Ju

Lecture13-Prof.Ju - CEE M237A MAE M269A Lecture 13...

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CEE M237A / MAE M269A Lecture 13: Hamilton’s Principle on Dynamic Structural Systems I Professor J. Woody Ju

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Hamilton’s Principle y Hamilton’s principle is a potent structural dynamics tool for the formulation of the equations of motion of a complex structural system from a statement of its energies y The full account of the energies of the system appears in the form of a Lagrangian functional L , which is also referred to as the Hamiltonian . y The functional L is defined as the difference between two scalar potentials representing the kinetic energy T and total potential energy V , i.e., L = T - V . 2
Hamilton’s Principle (Cont’d) y This approach is in stark contrast to Newtonian methodology , which is based on force and moment balances of vectorial variables y Within a single statement, Hamilton’s principle enables the complete set of governing equations and associated boundary conditions for any given structural member and/or system to be set forth. y It involves a variation of the integral of the Lagrangian functional L , the mathematical details in this procedure belongs to the subject of the calculus of variations 3

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Hamilton’s Principle (Cont’d) y Consider a system of n rigid particles : 4 {} * * For rigid particl where is e, Bernou a diagonal mass m lli's virtual work atrix is an array of pa principle rticle accelerations where is any kinematical i ly admissi 0 s 0 : δ −= ±± T FM u M u u uF M u For a system in dynamic equilibrium, the virtual work of the forces traveling through a kinematically admissible virtual displacement i lbe s z virtual displ. ero. ⎛⎞ ⎜⎟ ⎝⎠
Hamilton’s Principle (Cont’d) 5 {} * int int * deformable linkages conservative syst If there are between these particles For a , the internal virtual work appears as a change in its strain energy Assume can be em T n W U WU δδ ⇒• = ⇒= uF M u F ±± * derived from a potential function 0 E T T E E V UV V = + += uM u

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Hamilton’s Principle (Cont’d) 6 {} principle of minimum total potential energ If the term, , was omitted the where y 0 total potentia inertial is the
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Lecture13-Prof.Ju - CEE M237A MAE M269A Lecture 13...

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