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appigures. Assume that the rod is massless, perfectly rigid, and pivoted at point P. When the rod is perfectly horizontal, the angle = 0, the displacement y = 0, and the springs are in neither tension nor compression. Gravity acts on the system (e.g. on mass M). We assume that y is a small displacement. A mass M is attached at the end of the rod. Your tasks: 0 3k a M A Derive an equation of motion for the system in terms of the angular displacement, and its derivatives (you should not have y or its derivatives in this equation.) ( B Derive an equation of motion for the system in terms of the displacement y, and its derivatives (you should not have 0 or its derivatives in this equation. C Assuming there is no external actuator force F acting on the system, write down the total energy H of the system in terms of 0,0 and element constants. Derive an expression for the time derivative H of the total energy. (4 D Transform the equation from part B, which is in y, to another variable, z, which is zero at the static equilibrium position. ("Gravity Trick") E Assume now the system has aged and the bearings have worn out, so that you have a damping torque applied at the origin, B0 in the direction opposite of theta. Give the new equation of motion in z. You may derive it again from scratch or use another approach, but you must justify your method

appigures. Assume that the rod is massless, perfectly rigid, and pivoted at point P. When the rod is perfectly horizontal, the angle = 0, the displacement y = 0, and the springs are in neither tension nor compression. Gravity acts on the system (e.g. on mass M). We assume that y is a small displacement. A mass M is attached at the end of the rod. Your tasks: 0 3k a M A Derive an equation of motion for the system in terms of the angular displacement, and its derivatives (you should not have y or its derivatives in this equation.) ( B Derive an equation of motion for the system in terms of the displacement y, and its derivatives (you should not have 0 or its derivatives in this equation. C Assuming there is no external actuator force F acting on the system, write down the total energy H of the system in terms of 0,0 and element constants. Derive an expression for the time derivative H of the total energy. (4 D Transform the equation from part B, which is in y, to another variable, z, which is zero at the static equilibrium position. ("Gravity Trick") E Assume now the system has aged and the bearings have worn out, so that you have a damping torque applied at the origin, B0 in the direction opposite of theta. Give the new equation of motion in z. You may derive it again from scratch or use another approach, but you must justify your method

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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spec Figure 2. Assume that the rod is massless, perfectly rigid, and pivoted at point P. When the rod is perfectly horizontal, the angle 0 = 0, the displacement y 0, and the springs are in neither tension nor compression. Gravity acts on the system (e.g. on mass M). We assume that y is a small displacement. A mass M is attached at the end of the rod. = Your tasks: k a 0 3k a a T y M A Derive an equation of motion for the system in terms of the angular displacement 0, and its derivatives (you should not have y or its derivatives in this equation.) B Derive an equation of motion for the system in terms of the displacement y, and its derivatives (you should not have or its derivatives in this equation.) C Assuming there is no external actuator force F acting on the system, write down the total energy H of the system in terms of 0,0 and element constants. Derive an expression for the time derivative H of the total energy. (4) D Transform the equation from part B, which is in y, to another variable, z, which is zero at the static equilibrium position. ("Gravity Trick") E Assume now the system has aged and the bearings have worn out, so that you have a damping torque applied at the origin, B0 in the direction opposite of theta. Give the new equation of motion in z. You may derive it again from scratch or use another approach, but you must justify your method
Transcribed Image Text:spec Figure 2. Assume that the rod is massless, perfectly rigid, and pivoted at point P. When the rod is perfectly horizontal, the angle 0 = 0, the displacement y 0, and the springs are in neither tension nor compression. Gravity acts on the system (e.g. on mass M). We assume that y is a small displacement. A mass M is attached at the end of the rod. = Your tasks: k a 0 3k a a T y M A Derive an equation of motion for the system in terms of the angular displacement 0, and its derivatives (you should not have y or its derivatives in this equation.) B Derive an equation of motion for the system in terms of the displacement y, and its derivatives (you should not have or its derivatives in this equation.) C Assuming there is no external actuator force F acting on the system, write down the total energy H of the system in terms of 0,0 and element constants. Derive an expression for the time derivative H of the total energy. (4) D Transform the equation from part B, which is in y, to another variable, z, which is zero at the static equilibrium position. ("Gravity Trick") E Assume now the system has aged and the bearings have worn out, so that you have a damping torque applied at the origin, B0 in the direction opposite of theta. Give the new equation of motion in z. You may derive it again from scratch or use another approach, but you must justify your method
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