Wuhan University of TechnologyChapter16Variational formulation of theeguations ofmotion16-1
16-1 Wuhan University of Technology Chapter 16 Variational formulation of the equations of motion
Wuhan University of TechnologyContents16.1 Generalized coordinates16.2 Hamilton's principle16.3 Lagrange's equations of motion16-2
16-2 Wuhan University of Technology 16.1 Generalized coordinates 16.2 Hamilton's principle 16.3 Lagrange's equations of motion Contents
Wuhan University of Technology16.1 Generalized coordinatesInformulatingthevariationalMDOFtechnigue,extensiveusewill bemadeofgeneralized coordinates, and in this development a precise definition of theconceptisneededratherthanthesomewhatlooseterminologythathassufficeduntil now.Thus,generalizedcoordinatesforasystemwithNdegreesoffreedomaredefinedhereasanysetofNindependentquantitieswhichcompletelyspecifythe position of every point within the system.Being completely independent, generalized coordinates must not be related inanywaythroughgeometricconstraintsimposedonthesystem.16-3
16-3 Wuhan University of Technology 16.1 Generalized coordinates In formulating the variational MDOF technique, extensive use will be made of generalized coordinates, and in this development a precise definition of the concept is needed rather than the somewhat loose terminology that has sufficed until now. Thus, generalized coordinates for a system with N degrees of freedom are defined here as any set of N independent quantities which completely specify the position of every point within the system. Being completely independent, generalized coordinates must not be related in any way through geometric constraints imposed on the system
Wuhan Universityof Technology16.1 Generalized coordinatesxAy1y2mXi202m2X2FIGURE16-1Doublependulumwithhingesupport16-4
16-4 Wuhan University of Technology 16.1 Generalized coordinates FIGURE 16-1 Double pendulum with hinge support
Wuhan University of Technology16.1 Generalized coordinatesIntheclassical doublependulum shown inFig.161, thepositionof thetwomasses mi and m2 could be specified using the coordinates Xi, Y1, X2, Y2;however,twogeometricconstraintconditionsmustbeimposedonthesecoordinates, namely,c+y-Li=0(2 - a1)2 + (y2- 91)2- L= 0Suppose, on the other hand, the angles 1 and 2 were specified as thecoordinates to be used in defining the positions of masses mi and m2:Clearlyeitherofthesecoordinates canbechangedwhileholdingtheotherconstant;thus,theyare seento becompletelyindependentandthereforeasuitablesetofgeneralizedcoordinates.16-5
16-5 Wuhan University of Technology 16.1 Generalized coordinates In the classical double pendulum shown in Fig. 161, the position of the two masses m1 and m 2 could be specified using the coordinates x1, y1, x 2, y 2; however, two geometric constraint conditions must be imposed on these coordinates, namely, Suppose, on the other hand, the angles and were specified as the coordinates to be used in defining the positions of masses m1 and m 2. Clearly either of these coordinates can be changed while holding the other constant; thus, they are seen to be completely independent and therefore a suitable set of generalized coordinates