Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsToday’s Objectives:Students will be able to:a) Find EOMs using Lagrange’s EquationsMulti-DOF systems and Lagrange’s EquationsPortrait of Joseph-Louis Lagrange0=∂∂−⎥⎦⎤⎢⎣⎡∂∂iiqLqLdtd&Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsMulti-degree of freedom systems[]{}[]{}[]{}{}FxKxCxM=++&&&Linear EOM for a MDOF system:Finite element model of a pressure vesselRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsJoseph-Louis Lagrange (1736-1813) Some of the primary contributors to dynamics and when they lived. 1564781642 851646701667 811707 761717 661736771805 601400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900Galileo NewtonLeibniz BernoulliEulerD'AlembertLagrangeHamiltonRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsJoseph-Louis Lagrange• Viewed himself almost exclusively as a pure mathematician who sought mathematical elegance. In, Mechanique Analytique (1788), he states “No diagram will be found in this work. The methods that I explain in it require neither construction nor geometrical or mechanical arguments, but only the algebraic operations inherent to a regular and uniform process. Those who love Analysis will, with joy, see mechanics become a new branch of it and will be grateful to me for thus having extended its field.”•In Mécanique Analytique, Lagrange – summarized all the work done in mechanics since the time of Newton– laid the foundation for variational dynamics– generalized the Principle of Least Action (which states that nature chooses the most economical path for moving bodies and was first formulated by Pierre de Maupertuis),– presented a new and very powerful tool for deriving equations of motion, now called “Lagrange’s equations”.Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsBackground• Generalized coordinates: qi– Any set of n independent coordinates that completely describes asystemRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsLagrange’s Equations0=∂∂−⎥⎦⎤⎢⎣⎡∂∂iiqLqLdtd&Lagrange’s equations is usually presented for conservative systems as:Wherecoordinate dGeneralizeenergy potential Totalenergy kinetic Totaln)(Lagrangia ===−=iqVTVTLThis can be derived from Newton’s 2ndlaw or visa versaRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsThe form we will use is:iiiiiQqRqVqTqTdtd=∂∂+∂∂+∂∂−⎥⎦⎤⎢⎣⎡∂∂&&function) potential a from derivablenot forces externalother or edissapativ be(may force veconservati-non dGeneralize velocitydGeneralizet)independen andlly geometrica ined(unconstra coordinate dGeneralize)21 (e.g.function edissipativ sRayleigh' )21 (e.g.energy lpotentiona Total motion) planefor 2121 (e.g.energy kinetic Total 2222======+==+=jjjjGGQdtdqqqcvRRmgzkxVVωImvT&Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsGeneral Procedure1. Select generalized coordinates, qj. 2. Write T, V and R in terms of generalized coordinates.3. Determine generalized force, Qj, if required.4. Apply Lagrange’s equation for each of the coordinates.jNjjqQWδδ∑==1Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsFinding the generalized forceCase 1: You can write the work in terms of the generalized coordinates to obtain an expression for the virtual work in terms of the generalized coordinates. Example. Let θ1and θ2be coordinatesjNjjqQWδδ∑==1M1M2θ1θ2Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsGeneralized force (cont.)Case 2: You can write the work in terms of a convenient coordinate, xi, which can be expressed in terms of generalized coordinates.FL1L2θ1θ2()()(