Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_1

⚡ Mecademy AIENG정역학 · ch7  Problem Statement Determine the moment applied to the lower link through its shaft which is necessary to support the load in terms of the angle . Neglect the weights of the parts. Problem 7/1 (a) Determination of required moment using the Principle of Virtual Work 1. Present Final Formula: The principle of virtual work states that for a system in equilibrium, the total virtual work done by all external active forces and couples for any virtual displacement is zero: 2. Substitute Values: Let the vertical height of the point where force is applied be , measured from the fixed base hinge. Based on the geometry of the two links, each of length and at an angle with the horizontal: The virtual displacement of the load corresponding to a virtual rotation of the lower link is: The moment acts in the direction of increasing (counter-clockwise), while the force acts downward (opposite to the direction of increasing ). 3. Partial Operations: Differentiate the height expression with respect to : Substitute this into the virtual work equation: M Pθ M δU δθ δU=M δθ + ∑ ii F δs = ∑ jj 0 Py rθ y=rsinθ+rsinθ=2rsinθ Pδθ δy= δθ = dθ dy (2 r sin θ ) δθ dθ d Mθ Py δU=Mδθ−Pδy=0 θ = dθ dy 2rcosθ Mδθ−P(2rcosθδθ)=0 Factor out the common virtual displacement : 4. Final Calculation: Since , the term inside the parentheses must be zero for equilibrium: ● Final Conclusion: The required moment to support the load is . As increases, the moment arm for force relative to the verti