16.61 Aerospace Dynamics Spring 2003 Lecture #7 Lagrange's equations Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 1 Lecture #7 Lagrange's Equations
16.61 Aerospace Dynamics Spring 2003 Lagrange's equations Joseph-Louis lagrange 1736-1813 http://www-groups.dcs.st-and.ac.uk/-history/mathematicians/lagranGe.html Born in Italy. later lived in berlin and paris Originally studied to be a lawyer Interest in math from reading halleys 1693 work on algebra in optics If I had been rich, I probably would not have devoted myself to mathematics Contemporary of euler Bernoulli Leibniz. D Alembert Laplace, Legendre (Newton 1643-1727) e Contributions o Calculus of variations o Calculus of probabilities o Propagation of sound o Vibrating strings o Integration of differential equations Orbits o Number theory whatever this great man says, deserves the highest degree of consideration, but he is too abstract for youth student at ecole polytechnique Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 1 Lagrange’s Equations Joseph-Louis Lagrange 1736-1813 • http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html • Born in Italy, later lived in Berlin and Paris. • Originally studied to be a lawyer • Interest in math from reading Halley’s 1693 work on algebra in optics • “If I had been rich, I probably would not have devoted myself to mathematics.” • Contemporary of Euler, Bernoulli, Leibniz, D’Alembert, Laplace, Legendre (Newton 1643-1727) • Contributions o Calculus of variations o Calculus of probabilities o Propagation of sound o Vibrating strings o Integration of differential equations o Orbits o Number theory o … • “… whatever this great man says, deserves the highest degree of consideration, but he is too abstract for youth” -- student at Ecole Polytechnique
16.61 Aerospace Dynamics Spring 2003 Why Lagrange(or why NOT Newton) Newton -Given motion deduce forces Rotating launcher FBD of projectile N g Or given forces-solve for motion Spring mass system F m Great for simple systems Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 2 Why Lagrange (or why NOT Newton) • Newton – Given motion, deduce forces ω Rotating Launcher N mg FBD of projectile • Or given forces – solve for motion Spring mass system m1 m2 x1 x2 F x2 t Spring mass system m1 m2 x1 x2 F x2 t Great for “simple systems
16.61 Aerospace Dynamics Spring 2003 What about real systems? Complexity increased by Vectoral equations-difficult to manage Constraints- what holds the system together? No general procedures Lagrange provides: Avoiding some constraints Equations presented in a standard form Termed analytic Mechanics Originated by leibnitz(1646-1716) Motion(or equilibrium)is determined by scalar equations Big picture Use kinetic and potential energy to solve for the motion No need to solve for accelerations(Ke is a velocity term) Do need to solve for inertial velocities Lets start with the answer, and then explain how we get there Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 3 What about “real” systems? Complexity increased by: • Vectoral equations – difficult to manage • Constraints – what holds the system together? • No general procedures Lagrange provides: • Avoiding some constraints • Equations presented in a standard form Termed Analytic Mechanics • Originated by Leibnitz (1646-1716) • Motion (or equilibrium) is determined by scalar equations Big Picture • Use kinetic and potential energy to solve for the motion • No need to solve for accelerations (KE is a velocity term) • Do need to solve for inertial velocities Let’s start with the answer, and then explain how we get there
16.61 Aerospace Dynamics Spring 2003 Define: Lagrangian Function .L=T-V(Kinetic-Potential energies) Lagrange’ s Equation For conservative systems d/aLdL 0)g Results in the differential equations that describe the equations of motion of the system Key point: Newton approach requires that you find accelerations in all 3 directions, equate F=ma, solve for the constraint forces and then eliminate these to reduce the problem to characteristic size Lagrangian approach enables us to immediately reduce the problem to this"characteristic size, we only have to solve for that many equations in the first place The ease of handling external constraints really differentiates the two approaches Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 4 Define: Lagrangian Function • L = T – V (Kinetic – Potential energies) Lagrange’s Equation • For conservative systems 0 i i dL L dt q q ∂ ∂ − = ∂ ∂ • Results in the differential equations that describe the equations of motion of the system Key point: • Newton approach requires that you find accelerations in all 3 directions, equate F=ma, solve for the constraint forces, and then eliminate these to reduce the problem to “characteristic size” • Lagrangian approach enables us to immediately reduce the problem to this “characteristic size” we only have to solve for that many equations in the first place. The ease of handling external constraints really differentiates the two approaches
16.61 Aerospace Dynamics Spring 2003 Simple example · Spring- mass system ass syster Linear spring Frictionless table m · Lagrangian L=T-V L=T-Ⅴ=Dmx · Lagranges equation d aLaL aq, a q e Do the derivatives aL d aL dL -x mmX dt( dq Put it all together d dL dL n元+kx=0 dt(d qi dc Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 5 Simple Example • Spring – mass system Spring mass system • Linear spring • Frictionless table m x k Spring mass system • Linear spring • Frictionless table m x k • Lagrangian L = T – V 1 1 2 2 L = T V 2 2 −= − mx kx • Lagrange’s Equation 0 i i dL L dt q q ∂ ∂ − = ∂ ∂ • Do the derivatives i L mx q ∂ = ∂ , i d L mx dt q ∂ = ∂ , i L kx q ∂ = − ∂ • Put it all together 0 i i dL L mx kx dt q q ∂ ∂ − = += ∂ ∂
16.61 Aerospace Dynamics Spring 2003 Consider the mgr problem with the mass oscillating between the two springs only i degree of freedom of interest here so take gi = R qi oIR+R 0|+|0 0 o(R+r 0 0 (1)(4)=(R2+02(R+R) =2-R2 L=T-V=m(R2+03(R+R)2)-kR dL tar dL mo(R+R)-2kR dR So the equations of motion are: mR-mo (r+r)+2kR=o 2k r+ or=Ro2 which is the same as on(3-4) Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 6 Consider the MGR problem with the mass oscillating between the two springs. Only 1 degree of freedom of interest here so, take qi=R D D D ( ) (D ) (D ) D ( ) D ( ) D DD ( ) r R RR R R R T m r r m R RR V k R LTV m R R R kR d dt L R mR L R m R R kR M I o o M I T M I o o o = L N M M M O Q P P P + L N M M M O Q P P P L + N M M M O Q P P P = + L N M M M O Q P P P = =++ = =−= + + − ∂ ∂ F H G I K J = ∂ ∂ = +− × 0 0 0 0 0 0 0 2 2 2 2 2 2 22 2 2 22 2 2 2 ω ω ω ω ω c h c h So the equations of motion are: mR m R R kR R k m R R o o DD ( ) DD − ++ = + − F H I K = ω ω ω 2 2 2 2 0 2 or which is the same as on (3- 4)
16.61 Aerospace namIcs Spring 2003 Degrees of Freedom dOF e DOF=n-m o n= number of coordinates o m= number of constraints Critical point The number of dof is a characteristic of the system and does not depend on the particular set of coordinates used to describe the configuration Example 1 o Particle in space X Coordinate sets: x,y,: or r, e,o DOF=n-m=3 Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 7 Degrees of Freedom (DOF) • DOF = n – m o n = number of coordinates o m = number of constraints Critical Point: The number of DOF is a characteristic of the system and does NOT depend on the particular set of coordinates used to describe the configuration. Example 1 o Particle in space n = 3 Coordinate sets: x, y, z or r, θ, φ m = 0 DOF = n – m = 3 θ φ r x y z θ φ r x y z
16.61 Aerospace Dynamics Spring 2003 E ample z o Conical pendulum r=L X Cartesian Coordinates pherical coordinates ,y, n=2(6,c m=1(x2+y2+z=R 0 DOF=2 DOF=2 Example 3 o two particles at a fixed distance( dumbbell) Coordinates n EOC'S DOF Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 8 Example 2 o Conical Pendulum θ φ r = L x y z θ φ r = L x y z Cartesian Coordinates Spherical Coordinates n = 3 (x, y, z) n = 2 (θ, φ) m = 1 (x2 + y2 + z2 = R2 ) m = 0 DOF = 2 DOF = 2 Example 3 o Two particles at a fixed distance (dumbbell) Coordinates: n = m = EOC’s = DOF =
16.61 Aerospace Dynamics Spring 2003 Generalized Coordinates No specific set of coordinates is required to analyze the stem Number of coordinates depends on the system and not the set selected Any set of parameters that are used to represent a system are called generalized coordinates Coordinate Transformation Often find that the "best set of generalized coordinates used to solve a problem may not provide the information needed for further analysis e Use a coordinate transformation to convert between sets of generalized coordinates Example: Work in polar coordinates, then transform to rectangular coordinates, e.g x=rsin 0 cos y=rsin Asin g z=rcos e Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 9 Generalized Coordinates • No specific set of coordinates is required to analyze the system. • Number of coordinates depends on the system, and not the set selected. • Any set of parameters that are used to represent a system are called generalized coordinates. Coordinate Transformation • Often find that the “best” set of generalized coordinates used to solve a problem may not provide the information needed for further analysis. • Use a coordinate transformation to convert between sets of generalized coordinates. Example: Work in polar coordinates, then transform to rectangular coordinates, e.g. sin cos sin sin cos x r y r z r θ φ θ φ θ = = =
