6161103 11.6 potential energy criterion for equilibrium

11.6 Potential-Energy Criterion for Equilibrium


System having One Degree of Freedom

� When the displacement of a frictionless connected system is infinitesimal, from q to q + dq

dU = V (q) – V (q + dq)

Or dU = -dVOr dU = -dV

� If the system undergoes a virtual displacement δq, rather than an actual displacement dq,

δU = -δV

� For equilibrium, principle of work requires δU = 0, provided that the potential function for the system is known, δV = 0



System having One Degree of FreedomdV/dq = 0

� When a frictionless connected system of rigid bodies is in equilibrium, the first variation or change in V is zerozero

� Change is determined by taking first derivative of the potential function and setting it to zero

Example� To determine equilibrium position

of spring and block



System having One Degree of Freedom

dV/dy = -W + ky = 0

� Hence equilibrium posyion y = yeq

y = W/kyeq = W/k

� Same results obtained by applying ∑Fy = 0 to the forces acting on the FBD of the block



System having n Degree of Freedom� When the system of n connected bodies has n

degrees of freedom, total potential energy stored in the system is a function of n independent coordinates qn, V = V (q1, q2, … , qn) coordinates qn, V = V (q1, q2, … , qn)

� In order to apply the equilibrium criterion, δV = 0, determine change in potential energy δV by using chain rule of differential calculus

δV = (∂V/∂q1)δq1 + (∂V/∂q2)δq2 + … + (∂V/∂qn)δqn

� Virtual displacements δq1, δq2, … , δqn are independent of one another



System having n Degree of Freedom

� Equation is satisfied

∂V/∂q1 = 0, ∂V/∂q2 = 0, ∂V/∂qn = 0

It is possible to write n independent � It is possible to write n independent equations for a system having n degrees of freedom

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6161103 11.6 potential energy criterion for equilibrium