A mechanism consists of a slotted disk of mass m and radius R that rotates at...

 

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A mechanism consists of a slotted disk of mass m and radius R that rotates at a constant rate about fixed point A. As shown in the figure below left, a rod of mass m and length L is arranged so end B rides in the slot of the disk and end C is pinned to a collar also having mass m that translates on a horizontal guide. If the disk rotates at constant rate = 2, the system has 1 degree of freedom. However, we choose to model this system as detached bodies as shown below right. The choice of generalize coordinates is q1 = xc, q2 = 0, and q3 = TB. You may assume that the slotted disk can be approximated as a uniform disk with its mass center at point A. Gravity acts downward. (a) Obtain the velocity constraints for this system and write them in the standard matrix form [a]{q} + {b} = {0}. Use the generalized coords q1 = xc, q2 = 0, and q3 = TB. (b) Find the kinetic and potential energies of the system when detached as shown below right. (c) Find the right-hand sides of the 3 Lagrange's equations for this system. M(t) A TB B R G Xc C m ALL M(t) Body 1 TB R Xc Body 2 C2₂ m ALU A mechanism consists of a slotted disk of mass m and radius R that rotates at a constant rate about fixed point A. As shown in the figure below left, a rod of mass m and length L is arranged so end B rides in the slot of the disk and end C is pinned to a collar also having mass m that translates on a horizontal guide. If the disk rotates at constant rate = 2, the system has 1 degree of freedom. However, we choose to model this system as detached bodies as shown below right. The choice of generalize coordinates is q1 = xc, q2 = 0, and q3 = TB. You may assume that the slotted disk can be approximated as a uniform disk with its mass center at point A. Gravity acts downward. (a) Obtain the velocity constraints for this system and write them in the standard matrix form [a]{q} + {b} = {0}. Use the generalized coords q1 = xc, q2 = 0, and q3 = TB. (b) Find the kinetic and potential energies of the system when detached as shown below right. (c) Find the right-hand sides of the 3 Lagrange's equations for this system. M(t) A TB B R G Xc C m ALL M(t) Body 1 TB R Xc Body 2 C2₂ m ALU