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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 d**is**placing 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 D**is**placement of a system **is**hypothetical but adm**is**sible 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. What**is** 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 van**is**h:( 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. Otherw**is**e, 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 **is**conservative, (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 ex**is**ting 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** th**is** a conservativesystem?If th**is** **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