《结构动力学》课程教学课件（讲稿）16 Variational formulation of the equations of motion

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

Wuhan University of Technology16.2 Hamilton's principle28r(t)12Varied pathmr(t)7Real pathF(t)i,j,kUnitvectorsr(t)F(t)=F(t)i+F(t)j+F.(t)kr(t)=xi+yj+zk8r(t)=8xi+8yj+8zk1kFIGURE16-2Realandvariedmotionsofmassparticlem16-6

16-6 Wuhan University of Technology 16.2 Hamilton's principle FIGURE 16-2 Real and varied motions of mass particle m

Wuhan University of Technology16.2 Hamilton's principleThe virtual work of all forces, including the inertial force, must equal zero asexpressed by[F(t) -mi(t)] &r(t) +[F,(t) -mj(t)] y(t)+[F(t) -m2(t)] z(t) =0Rearranging terms and integrating this equation from time t; to time t2 givet-m[i(t) r(t) +j(t) oy(t) +(t) &z(t)l dt[F(t)Sr(t)+F(t)oy(t)+F(t)Sz(t)] dt=016-7

16-7 Wuhan University of Technology 16.2 Hamilton's principle The virtual work of all forces, including the inertial force, must equal zero as expressed by Rearranging terms and integrating this equation from time t1 to time t 2 give

Wuhan University of Technology16.2 Hamilton's principleIntegrating the first integral (I,) by parts and recognizing that the virtualdisplacement must vanish atthe beginning and theend of thisvaried pathi.e., that Sr(ti)and Sr(t2) equal zero, one obtainsL1m [i(t) Sr(t) +g(t) og(t) +z(t) s2(t)] dtT(t) dtST(t) dt = sJtim [i(t)2 +9(t)2 +2(t)]T(t)Inthisdiscussion,itishelpful to separatetheforcevectorF(t)into itsconservativeandnonconservativecomponentsasrepresentedbyF(t) =Fc(t) +Fnc(t)16-8

16-8 Wuhan University of Technology 16.2 Hamilton's principle Integrating the first integral (I1) by parts and recognizing that the virtual displacement must vanish at the beginning and the end of this varied path, i.e., that and equal zero, one obtains In this discussion, it is helpful to separate the force vector F(t) into its conservative and nonconservative components as represented by

Wuhan University of Technology16.2 Hamilton's principleav(r,y,z,t)=-Fr,e(t)araV(r,y, z,t)= -Fy,c(t)QyV(r,y,z,t)= -Fz,c(t)02-oV(r,y,z,t)dt+Wne(t) dtJti[T(t) -V(t)] dt +oWnc(t)dt=0ts16-9

16-9 Wuhan University of Technology 16.2 Hamilton's principle

Wuhan Universityof Technology16.2 Hamilton's principleEquation (169),which isgenerallyknown as Hamilton'svariational statement ofdynamics,showsthatthesumofthetimevariationsofthedifferenceinkineticandpotentialenergiesandtheworkdonebythenonconservativeforcesoveranytimeintervalt1tot2equalszero.Theapplicationofthisprincipleleadsdirectlytotheequationsofmotionforanygivensystem.16-10

16-10 Wuhan University of Technology 16.2 Hamilton's principle Equation (169), which is generally known as Hamilton's variational statement of dynamics, shows that the sum of the timevariations of the difference in kinetic and potential energies and the work done by the nonconservative forces over any time interval t1 to t2 equals zero. The application of this principle leads directly to the equations of motion for any given system