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© Sharif University of Technology - CEDRA

ENERGY PRINCIPLESPurpose:‣ To Study Energy Principles of Dynamics.Topics:‣ Kinetic Energy‣ Work‣ Leibniz (Leibnutz) Equation of Motion‣ Conservative Force Field© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Work: The work done for individual paths of particles:UU( e)N 1vf( e)N 1vf(10.14)Theorem-41: For a Rigid Body, the work done can beequated to the sum of the work done by the resultantforce in displacing any convenient point “A” in the rigidbody and that done by the resultant moment about thesame point in rotating the rigid body.AA v f M (10.18)RUA(e)f ρ βm β© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Virtual WorkDefinition: Virtual Displacement of a system ishypothetical but admissible in accordance with themechanical constraints. Its value is infinitesimalbut arbitrary and definitely independent of time.In a virtual movement, generalized coordinates ofthe system are considered to be incremented byinfinitesimal amounts “q i ” from the values theyhave at an arbitrary instant with time held constant.a) For individual paths of particles:UN 1fr (10.22)© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

) For a Rigid Body:AU f r M (10.23): Virtual (Imaginary) NotationRAExample: A half cylinder of mass“m” rocks without slipping on aflat level surface subjected to aforce that is perpendicular to itsdiametrical plane at all time. Whatis the virtual work in the coordinateof angular position?rCx 2ARPx 1UfRrAMA© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

UfRrAMAfR mge2P(sine1cose2)Ne2ffe1 rARe1, e3mgx 2PMA ( fU [( Psin) R f[fffR Rmgrsin) e3mgrsin]fR]rCARx 1U (mgrPR)sinNf f(f f and N: none-working forces)© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Leibniz Equation of MotionLeibniz Principle: The change of kinetic energy ofa system is equal to the work done on the system.U T, dU dT , U T, (10.23)U TX2Example: A double pendulum ofgiven masses (m 1 and m 2 ) andgiven lightweight rigid links(L 1 and L 2 ) is set in motion.Determine the equations ofmotion?X1L 1 1m 1Δr 1 L 2 2m 2Δr 2© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

The work done by twogravitational forces fromthe fixed equilibrium positionto the arbitrary position( 1 , 2 ) is:RL 1 1m 1 gΔr 1 L 2 2Δr 2m 2 gUU m gL1 (m11(1 m2cos) ) gL11sing[(L(1 sin( Virtual Symbol Differencial Operator )m1211m2gLcos) 212L22(1 cos)]2© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

© Sharif University of Technology - CEDRA By: Professor Ali Meghdari)]cos(2)[(212121)cos(2)sin(cos)sin(cos)sin(cos122121222222212121222211122121222221212221212122122221111122111111 LLmLmLmmTm Vm VTLLLLVLVeeLeeLVeeLVComputation of Kinetic Energy:

T [( m1[m22 m2) L 11 m2L1L 22cos( 21) m2L1L 2 12sin( 21)] 1L m L L cos( ) m L L sin( )] 22221212121212212For small 1 and 2 , the sinusoidal functions are firstexpanded in power series. Then, when ( , , ) areconsidered of the same order and terms of order O( 2 ) inthe brackets are omitted, we have:TT [( m1[m2 [( m1 m2) L 11 m2L1L 22] 1[m L m L L ] 222 m2) L 11 m2L1L 22] 1L m L L ] 2222Then, the Virtual change in Kinetic Energy is:2222112211© Sharif University of Technology - CEDRA By: Professor Ali Meghdari22

Applying the Leibniz Principle we have:[( m1m [ m2T U2) L 11 m2L1L 22 ( m1 m2) gL11]12L m L L m gL ] 022221212222Since 1 and 2 are independent coordinates, 1 and 2are independent and arbitrary. Therefore, the coefficientsof i must vanish:( m1m22m2) L 11 m2L1L 22 ( m1L m L L m gL 2222121222m2 0) gL 11 0D.E.M.© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Conservative Force FieldIn computing work, it is noticed that some of the workintegral depends only on the initial and final positions ofthe path. Yet some are path or time dependent, perhapseven both.The Force Field is said to be conservativeif the work done on a particle in the fielddepends on the initial and final positionsonly. Otherwise, the Force Field is saidto be non-conservative.dr 1 2FB© Sharif University of Technology - CEDRA By: Professor Ali MeghdariA

In order for the work integral to depend only on the limitsof integration (i.e., to be independent of the path takenfrom A to B) it is necessary and sufficient that “F.dr” bean exact differential, which is denoted by “dU” .F.dr =dU (10.24)Definition: A Force Field “F” is conservative whenthe work done on a particle which travels a closedB path in the field is zero. 1 2F dr 0(10.25)drF© Sharif University of Technology - CEDRA By: Professor Ali MeghdariA

Theorem-42: In order for “F.dr” in Equation (10.24) to bean exact differential “dU”, it is necessary and sufficientthat × F = 0. Or, in other words, a force “F” is said tobe conservative if and only if × F = 0. (Curl of F =0)Proof: 1. Consider “F.dr” to be an exact differential.Then working in rectangular coordinates, we may write:Fdrwhere:dU xUxidxyjUyzdyk.Uzdz( U). dr(10.26)F Ugrad(U)(10.27)© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

F Ugrad( U)(10.27)FxUx,FyUy,FzUz(Rectangular componentsof the force)We thus conclude that for a conservative system, theforce is expressible as the gradient of a scalarpotential function, “U”. Therefore, the force is afunction of spatial coordinates only.Finally, we can conclude from Eq. (10.27) that if F isconservative, (i.e., if F⋅dr = dU is a perfect differential)then:F ( )U 0(10.28)© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

2. Conversely, consider that for a particular force “F” ithas been shown that “ × F = 0”.Definition: The Potential Energy “V” is a scalar functionwhich is effectively the work one would perform totransport the mechanical system from a datum state tothe existing state while holding the system in equilibriumwith the external force and internal resilience.dV dU V V 2211Fdr(10.29)© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Example: Given a force system characterized byF x = 2xy + y and F y = x 2 – y 2 + x, is this a conservativesystem?If this is a conservative system, then:which yields the requirement that;(xFxyxi( x2yFyxyj)20( Fxi x)yFyj)(2xy0y)2xF 012x10© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

© Sharif University of Technology - CEDRA

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