Physics 5153 Classical Mechanics. Principle of Virtual Work1


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1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal system, and s fundamental for the later development of analytcal mechancs (Lagrangan and Hamltonan methods). The concept of vrtual work s centered on the dea of calculatng the amount of work done on a system of partcles through a vrtual dsplacement. We wll start by defnng what we mean by a vrtual dsplacement, then dscuss vrtual work, state the prncple of vrtual work, and consder an example of ts use. 1.1 Vrtual Dsplacement To defne what we mean by a vrtual dsplacement, let s consder a system composed of N partcles, possbly subject to some set of constrants, defned by 3N Cartesan coordnates (x ) relatve to an nertal frame. Let s assume that at some nstant of tme the system undergoes nfntesmal dsplacements that are vrtual n the sense that they occur wthout the passage of tme (nstantaneous), do not necessarly conform to the constrants. Ths change, δ x, n the confguraton of the system s known as a vrtual dsplacement. In the usual case, a vrtual dsplacement conforms to the nstantaneous constrants, that s, movng constrants are assumed stopped durng the dsplacement. For example, consder a system subject to n holonomc constrants f (x 1,..., x 3N, t) = 0 (1) A total dervatve corresponds to a nfntesmal dsplacement of the system and s gve by df = j f dx j + f dt = 0 (2) x t notce that ths gves both a spacal and a temporal dsplacement. In the case of a vrtual dsplacement, we assume that the temporal dsplacement s zero, therefore the constrant changes by δf = f δx j = 0 (3) x j It s mportant to note the dfference, the dsplacement occurs n zero tme. One queston that may be asked, are there any condtons under whch a real and vrtual dsplacement are the same? The answer can be seen by comparng Eqs. 2 and 3. If the constrant equaton s scleronomc, a vrtual dsplacement s the same as a real dsplacement. Therefore, n the general case vrtual and real dsplacements are not the same, but n the scleronomc case they are. Prncple of Vrtual Work1
2 P. Guterrez Before concludng our dscusson of vrtual dsplacements, let s consder the nonholonomc case where the constrant s gven n terms of dervatves. Assume n constrant equatons on a system of 3N degrees of freedom a j dx + a jt dt = 0 (4) where j corresponds to the j th constrant. Based on our defnton of a vrtual dsplacement consstent wth the constrants, a vrtual dsplacement for nonholonomc constrants s gven by a j δx = 0 (5) Ths equaton wll become mportant later when we dscuss calculatng forces of constrant through the Lagrange multpler method. So far we have consdered vrtual dsplacements n terms of Cartesan coordnates. Vrtual dsplacements n terms of generalzed coordnates are also possble. Smply transform the Cartesan constrant equatons to the generalzed coordnates. The form of the constrant equaton s gven by a j dq + a jt dt = 0 (6) where replacng the a wth f j f j and (7) q t gves the holonomc constrant. For a vrtual dsplacement, the constrant equaton becomes a j δq = 0 (8) Therefore the form s the same usng any set of coordnates. 1.2 Vrtual Work Let s agan consder a system of N partcles wth 3N degrees of freedom whose confguraton s gven by the Cartesan coordnates x 1... x 3N. In addton, suppose that the forces F 1... F 3N are actng on the partcles at the correspondng coordnates n a postve sense. The vrtual work s gven by δw = F δx = F δ r (9) The second equalty mples that the vrtual work s ndependent of coordnates used. The equaton can be transformed as follows to any set of generalzed coordnates δw = ( ) x F δq j (10) q j j From ths equaton, we defne the generalzed force as ( ) x Q j = F q j δw = j Q j δq j (11) Prncple of Vrtual Work2
3 P. Guterrez where we note that the generalzed force does not have to have unts of a force, just lke the generalzed coordnates do not have to have unts of a length. But, the product of generalzed force and coordnates has the unts of work (energy). In the expresson for vrtual work, the forces are assumed to reman constant throughout the vrtual dsplacement. Ths s true even f the forces vary drastcally over a nfntesmal dsplacement. A sudden change of force wth poston can occur n certan nonlnear systems. Now assume that the system s subject to constrants. The force can be separated nto appled forces F a and constrant forces F c. The vrtual work of the constrant forces n terms of generalzed coordnates s gven by δw c = Q c δq (12) If the dsplacement s consstent wth the constrant, the vrtual work s zero snce the force does not act n the drecton of the force δw c = Q c δq = 0 (13) whch s referred to as a workless constrant. These wll be the type of constrant that we wll deal wth most often. If the constrants are workless, then the total vrtual work on the system s gven by the appled forces δw = Q a δq (14) 1.3 Prncple of Vrtual Work One of the mportant applcatons of the dea of vrtual work arses n the study of statc equlbrum of mechancal system. Assume a scleronomc system of N partcles. If the system s n statc equlbrum, then Newton s laws for each of the N partcles gve The vrtual work for ths system s gven by F a + F c = 0 (15) δw = F a δ r + F c δ r = 0 (16) If we now assume that the constrants are workless, and the vrtual dsplacements reversble (one can replace δ r wth δ r), then the condton for statc equlbrum s δw = F a δ r = 0 δw = Q a δq = 0 (17) where the second equaton s gven usng generalzed coordnates. A very mportant pont to note here s that unlke the Newtonan approach, we do not need to know what the constrant forces are. We only need to know the appled forces. Now assume that the system s ntally motonless, but not n equlbrum. Then one or more of the partcles has a net appled force on t, and n accord wth Newton s laws, t wll start to move n the drecton of the force. Snce any moton must be compatble wth the constrants, the Prncple of Vrtual Work3
4 P. Guterrez vrtual dsplacements can be chosen to be n the drecton of the actual moton at each pont. In ths case the vrtual work s postve δw = F a δ r + F c δ r > 0 (18) Snce the constrants are workless, the condton becomes δw = F a δ r > 0 (19) If the vrtual dsplacements are reversed, then the vrtual work s negatve. Nonetheless, f the system s not n equlbrum, one can fnd a set of vrtual dsplacements that wll result n the vrtual work beng nonzero. These results can be summarzed n the prncple of vrtual work: The necessary and suffcent condton for the statc equlbrum of an ntally motonless scleronomc system that s subject to workless constrants s that zero vrtual work be done by the appled forces n movng through an arbtrary vrtual dsplacement satsfyng the constrants. 1.4 Example As a smple consder the system descrbed n Fg. 1,where we want to determne the force F that wll keep the system n equlbrum. If we use the Newtonan approach, we requre 3 equatons to solve the problem Fx = 0 N 1 F = 0 (20) Fy = 0 N 2 2mg = 0 τ = 0 mgl cos θ N1 l sn θ = 0 From ths pont t s farly straght forward to solve the problem. One fnds F = mg cot θ. Usng the prncple of vrtual work, we set up the equaton as follows mgδy F δx = 0 (21) N 1 m PSfrag replacements mg l θ m F N 2 Fgure 1: Two blocks on frctonless surfaces constraned by by a rod to move together. Prncple of Vrtual Work4
5 P. Guterrez wth the constrant between x and y } { } x = l cos θ δx = δθl sn θ y = l sn θ δy = δθl cos θ δx cos θ δy sn θ = 0 (22) snce I have already assumed drectons for δx and δy n Eq. 21, the sgn here s dropped between the mddle and fnal equatons. Combnng the two equatons (mg cot θ F )δx = 0 (23) Snce the dsplacement s arbtrary, and ths equaton must hold for all possble vrtual dsplacements, the quantty nsde the parenthess must be zero F = mg cot θ (24) the same as the Newtonan method. The pont of ths example s not to show that one method s superor to the other, but that dfferences n the two methods. In the Newtonan method, we requred the constrant force and a set of 3 equatons to specfy the problem. Usng the method of vrtual work we need only two equatons, one descrbng the work done and the second descrbng the constrants. We don t need the constrant forces. Prncple of Vrtual Work5
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P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
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