HW1_10_solns - ME 563 HOMEWORK # 1 (Solutions) Fall 2010...

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ME 563 HOMEWORK # 1 (Solutions) Fall 2010 PROBLEM 1: (40%) Derive the equations of motion for the three systems given using Newton-Euler techniques (A, B, and C) and energy/power methods (A and B only). System (A) Our assumptions in this problem are the following: 1. There is no slip between the pulley and the cable on which it rolls (i.e., each point of contact on pulley in contact with cable is instantaneously at rest); 2. The springs are all initially undeformed when the disk is released; 3. The angle of rotation of the wheel is relatively small on the order of < 0.1 radians; 4. Gravity is acting vertically downward. We begin by describing the kinematics of the wheel as it rotates and translates in the diagram shown below. Note there is a single degree-of-freedom because of the no slip constraint.
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2 Newton/Euler: The next step is to draw the free body diagram of the wheel as shown below. Euler’s equation can then be applied about point O to minimize the number of equations needed. Note that the moment arms for the two spring forces are computed assuming a small angle, θ . The scalar equation of motion can then be expressed as follows in terms of the x (or θ ) coordinate only using the kinematic rolling constraint and then dividing both sides of the equation by R: At this point, we check our final equation of motion to make sure it is reasonable. The inertia coefficient is positive; the stiffness coefficient is also positive. The units are also correct. It may often help to check for special sets of system parameters. For example, when b=R and a=0, the only stiffness that resists the motion for small rotations is K 1 +K 3 at the center of mass of the wheel; the equation of motion correctly describes this scenario. x i j Mg f N K 1 (x+a θ )+ K 3 (x+a θ ) K 2 (x–b θ )+ K 4 (x–b θ ) O k Note: FBD is shown slightly exaggerated with the assumption that the angle of rotation is small x x R θ Constraint: x = R θ #DOFs=2(x, θ ) – 1 = 1 Points a and b: x a =x+a sin θ x b =x-b sin θ i j a b k
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3 Energy/Power: To derive the equation of motion for system A using energy methods, we first compute the kinetic and potential energy expressions and substitute the kinematic rolling constraint: Then we apply the first law power equation and differentiate as necessary noting that for general oscillations: This equation matches the equation of motion (EOM) we obtained using Euler’s method. System (B) Our assumptions in this problem are the following: 1. There is no slip between the wheel and the surface on which it rolls (i.e., point of contact on wheel is instantaneously at rest); 2. The springs are all initially undeformed ( x 1u =undeformed length of K 1 and x 2u =undeformed length of K 2 ) when the system is released; 3. The cable in inextensible (i.e., its length cannot change) and is always in tension; 4. Gravity is acting vertically downward. We begin by describing the kinematics of the pulley as it rotates and translates in the diagram shown
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HW1_10_solns - ME 563 HOMEWORK # 1 (Solutions) Fall 2010...

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