Homework6_Su_2010_Solution

Homework6_Su_2010_Solution - ME 562 Advanced Dynamics...

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Unformatted text preview: ME 562 Advanced Dynamics Summer 2010 HOMEWORK # 6 Due: July 23, 2010 Q1. Two wheels, each of mass m, are connected by a massless axle of length 1. Each wheel is considered to have its mass concentrated as a particle at its hub. The wheels roll without slipping on a horizontal plane. The hub of wheel A is attached by a spring of stiffness k and unstressed length l to a fixed point 0. Use r, 0, and (1) as generalized coordinates. Then, find: (a) the constraints satisfied by these three variables; (b) the relationships between the virtual displacements in the three variables; (0) if the constraints are holonomic or non-holonomic. (see Problem 6-25 in the text for a figure). Q2. (see Problem 6-7 in the text for a figure). A double pendulum consists of two massless rods of length l and two particles of mass m which can move in the vertical plane. Assume frictionless joints and define the configuration of the system using the coordinates 9 and at). Recall that the system is in the vertical plane. (i) Derive the generalized forces for the generalized coordinates 9 and (1) corresponding to the weights forces of the two particles. (ii) Then, use Lagrange’s equations for holonomic systems and derive the differential equations of motion for the system. Q3. (see Problem 6-13 in the text for a figure). A smooth tube in the form of a circle of radius r is pinned at O and rotates in its vertical plane with a constant angular velocity (1). The position of a particle of mass m that slides inside the tube is given by the relative coordinate ti). (1) is the angle that the line joining the center of the ring/tube (0’) to the particle makes with 00’. Use Lagrange’s equations for holonomic systems to derive the differential equation for (1), the only generalized coordinate. Note that 19 = a) is constant and is specified, thus it is not a generalized coordinate. Q4. (see Problem 6-22 in the text for a figure). A dumbbell is composed of two particles, each of mass m, connected by a massless rod of length 1. One particle of the dumbbell is connected by a pin to the edge of a disk of radius r, which is massless except for a particle of mass m at its center. The disc can roll without slipping on a horizontal surface. Assume frictionless joints and define the configuration of the system using the coordinates 8 and (1) which are absolute rotation angles. The system is in the vertical plane. 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Homework6_Su_2010_Solution - ME 562 Advanced Dynamics...

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