Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_11

⚡ Mecademy AIENG정역학 · ch7  Problem Statement The gear train shown is used to transmit motion to the vertical rack . If an input torque is applied to gear , what force is required to establish equilibrium of the system? Gear is keyed to the same shaft as gear . Gears , , and have pitch diameters , , and , respectively. Neglect the weight of the rack. Problem 7/11 (a) Determination of Force F for Equilibrium ● Calculation Process 1. Present Final Formula: Using the Principle of Virtual Work, the equilibrium condition is given by , which relates the input torque and the output force through the system's kinematics. where is a virtual rotation of gear and is the resulting virtual vertical displacement of rack . 2. Substitute Values: We need to express in terms of using the geometric constraints of the gear train. Gear and mesh: Gears and are on the same shaft: Gear and rack mesh: Substituting the expression for : 3. Partial Operations: Substitute back into the virtual work equation: DM AFC BABCd A d B d C δU=0M F δU=Mδθ − A Fδy = D 0 δθ A Aδy D D δy D δθ A ABδθ = B δθ = A ( r B r A )δθ A ( d B d A ) BCδθ = C δθ = B δθ A ( d B d A ) CDδy = D δθ ⋅ C r = C δθ C ( 2 d C ) δθ C δy = D δθ = ( A d B d A )( 2 d C )δθ A 2d B d d AC δy D Mδθ − A Fδθ = ( A 2d B dd AC )0 Divide by the non-zero virtual rotation : 4. Final Calculation: Solve for : ● Final Conclusion: The force required to maintain equilibrium of the system under the applied torque is . ✨ Final Answer Summary (a) Mecademy AI S