EG260 Lagrange

# EG260 Lagrange - A.K Slone EG-260 Dynamics(1 EG-260...

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 1 of 19 EG-260 DYNAMICS I – Lagrange’s Equation 1. Introduction ...................................................... 2 2. An overview of the procedure . ........................ 3 3. Useful energy expressions ................................ 4 4. A mass-spring example .................................... 4 5. A trolley mounted pendulum example ........... 7 6. Two pendulum example ................................. 13 7. Damping .......................................................... 16 8. Advantages of Lagrange’s method ............... 19

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 2 of 19 1. Introduction Lagrangian methods use an energy approach to obtain the equations of motion in terms of generalised co- ordinates. The equation of motion in terms of the kinetic energy (T), potential energy (U) and generalised co- ordinated q i for an n degree of freedom system is given by ( 29 n ,....... , i Q q U q T q T dt d n i i i i 2 1 = = + - & (1) where t q q i = & is the generalised velocity and Q i (n) is the non-conservative force corresponding to the generalised co-ordinates q i .The forces represented by Q i (n) may be either damping, or dissipative forces or other external forces. If F xk , F yk and F zk represent the external forces acting on the k th mass if the system in the x , y and z directions, respectively, the generalised force is given by: ( 29 + + = i k zk i k i k n i q z F q y F q x F Q (2) where x k , y k , z k are the displacements of the k th. mass in the x , y and z directions, respectively. For a torsional system the force F xk is replaced by the moment M xk about the x axis and the displacement x k is replaced by the angular displacement q xk about the x axis
A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 3 of 19 If the Lagrangian, L, is defined as L = T- U, then equation (1) may be expressed as: i i i Q q L q L dt d = - & (3) For a conservative system Q i (n) = 0, so that equation (1) becomes: n ,....... , i q U q T q T dt d i i i 2 1 0 = = + - & (4) 2. An overview of the procedure . An overview of the procedure is given below. Determine a set of generalized co-ordinates, denoted by q , which are an independent set of variables to record position. Express the kinetic and potential energy in terms of the generalised co-ordinates q . Express the work done by forces and moments. Derive the equation of motion using equations of the type expressed by equations (1) or (3).

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 4 of 19 3. Useful energy expressions KE of a mass 2 2 1 x m T & = (5) KE of body rotating about centre of gyration 2 2 1 q & G I T = (6) PE of a spring ( 29 2 2 1 extension k U = (7) PE due to gravity mgh U = (8) 4. A mass-spring example k 1 k 2 k 3 Figure 1 MDOF mass-spring system m 1 m 2
A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 5 of 19 Figure 1 shows a 2 degree of freedom mass-spring system, the kinetic energy is given by: 2 2 2 2 1 1 2 1 2 1 q m q m T & & + = (9) The potential energy is given by: ( 29 2 2 3 2 1 2 2 2 1 1 2 1 2 1 2 1 q k q q k q k U + - + = (10)

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EG260 Lagrange - A.K Slone EG-260 Dynamics(1 EG-260...

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