Example 3 (continued) – Virtual Work

In vector form δy_B = L cos θ δθ j and the external force and moment that do virtual work are F_s = - F_s j and M_A = M_A k

Next calculate the force in the spring noting that the value remains constant during a virtual displacement δθ.

F_s = K (extension) = K L sin θ so F_s = - K L sin θ j

4. Calculate the dot product of the external forces with the virtual displacements to establish the virtual work performed during the virtual displacement.

δW = F_s · δy_B + M_A · δθ = - F_s j · L cos θ δθ j + ( M_A k · δθ k )

δW = - K L² sin θ cos θ δθ + M_A δθ

5. Set the virtual work equal to zero and solve for the desired unknown (or unknowns).

δW = 0 = - K L² cos θ δθ + M_A δθ

[ - K L² sin θ cos θ + M_A ] δθ = 0

Since the virtual displacement, δθ, is arbitrary

[ - K L² sin θ cos θ + M_A ] δθ = 0 implies that - K L² sin θ cos θ + M_A = 0

sin θ cos θ = M_A / K L² (final result; what are limitations on M_A, K, and L ?)

Note: sin 2θ = 2 M_A / K L²

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