《航空航天动力学》英文版 lecture8 Examples Using Lagrange's Equations

16.61 Aerospace Dynamics Spring 2003 Lecture #8 Examples Using Lagrange's Equations Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002

16.61 Aerospace Dynamics Spring 2003 Lecture #8 Examples Using Lagrange's Equations Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 2

16.61 Aerospace Dynamics Spring 2003 Example Given: Catapult rotating at a constant rate(frictionless, in the horizontal plane) Find the eom of the particle as it leaves the tube y x Massachusetts Institute of Technology C How-Deyst 2003(Based on Notes by Blair 2002

16.61 Aerospace Dynamics Spring 2003 Example Given: Catapult rotating at a constant rate (frictionless, in the horizontal plane) Find the EOM of the particle as it leaves the tube. ω x y Massachusetts Institute of Technology © How-Deyst 2003 (Based on Notes by Blair 2002) 1

16.61 Aerospace Dynamics Spring 2003 Derivatives d/aT dT =nmro dr dt dr External forces: None Lagrange's equation gives the equation of motion as i-ro=0 What do we get if we solve this via Newtons method? Massachusetts Institute of Technology C How-Deyst 2003(Based on Notes by Blair 2002

r r 16.61 Aerospace Dynamics Spring 2003 Derivatives: ∂T = mrD, d   ∂T   = mrDD, ∂T = mrω2 ∂ D dt  ∂ D  ∂r External forces: None Lagrange’s equation gives the equation of motion as r r CC− ω2 = 0 What do we get if we solve this via Newton’s method? Massachusetts Institute of Technology © How-Deyst 2003 (Based on Notes by Blair 2002) 3

16.61 Aerospace Dynamics Spring 2003 上 Xample Mass particle in a frictionless spinning ring Ring spins at constant rate a Spherical coordinate set (2-11) Two holonomic constraints o r= constant ·φ=ot+ po which gives the spin rate of the tube So only 1 doF- use 0 as the generalized coordinate Massachusetts Institute of Technology C How-Deyst 2003(Based on Notes by Blair 2002

16.61 Aerospace Dynamics Spring 2003 Example Mass particle in a frictionless spinning ring. Ring spins at constant rate ω m θ g r ω m θ g r ω Spherical coordinate set (2-11) Two holonomic constraints • r = constant • φ = ωt+φ0 which gives the spin rate of the tube So only 1 DOF � use θ as the generalized coordinate Massachusetts Institute of Technology © How-Deyst 2003 (Based on Notes by Blair 2002) 1

16.61 Aerospace Dynamics Spring 2003 Example le System of 3"" suspended by pulleys Neglect mass of pulleys.) y Massachusetts Institute of Technology C How-Deyst 2003(Based on Notes by Blair 2002)

16.61 Aerospace Dynamics Spring 2003 Example System of 3 “particles” suspended by pulleys. (Neglect mass of pulleys.) g m1 m2 m3 l h y1 y2 s1 s2 s3 g Massachusetts Institute of Technology © How-Deyst 2003 (Based on Notes by Blair 2002) 1

16.61 Aerospace Dynamics Spring 2003 Example le 2 particles in a frictionless tube held by springs. Assume that s=0 and a=0 g m, 70=const Motor Elevator Massachusetts Institute of Technology C How-Deyst 2003(Based on Notes by Blair 2002

16.61 Aerospace Dynamics Spring 2003 Example 2 particles in a frictionless tube held by springs. Assume that s = 0 and a = 0 Elevator ω = const. a g s k1 k2 k3 m1 m2 Elevator Motor ω = const. a g s k1 k2 k3 m1 m2 Massachusetts Institute of Technology © How-Deyst 2003 (Based on Notes by Blair 2002) 1