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Question: statics and dynamics utilizing the virtual work principle derive ...

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STATICS AND DYNAMIcs: ·Utilizing the virtual work principle, derive τ = JTF, where J is the kinematic Jacobian, τ repre- sents joint torques and F denote external forces applied at the end-effector. Make sure to explicitly discuss validity of each of the three conditions under which virtual work principle can be applied. - Using the principle of virtual work, calculate the horizontal force F needed to keep the sliding rod in static equilibrium at the instant depicted in the figure. Assume that the rod is uniform with length L and mass m, gravity acts donwards and all reaction forces are ideal and frictionless. L/2 Figure 6: Sliding Rod -Consider the figure depicting a simple pendulum of mass m and length inside a vertical elevator of mass M. Let a vertical external force R act on the elevator, while a horizontal external force F acts on the pendulum. An external torque τ is also applied to the maseles link of the pendulum. Consider all constraint forces be frictionless and ideal. Taking {r θ as the generalized coordinates for the system, derive the kinetic co-energy, potential energy and generalized forces acting on the system. Note that gravity g acts downwards in the figure. Figure 7: Simple Pendulum in an Elevator -Consider a serial link manipulator with n links. Let the inertia matrix of ilink expressed in ith frame be given asI. Given the angular velocity ot link with respect to the fixed frame expressed in coordinate frame 0, the rotation matrix Rof ith frame with respect to fixed frame and the linear velocityof the center of mass of the ih link expressed in the fixed frame, write down an expression to calculate the kinetic co-energy of ith link.

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