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Homework5_Su_2010_Solution

# Homework5_Su_2010_Solution - ME 562 Advanced Dynamics...

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Unformatted text preview: ME 562 Advanced Dynamics Summer 2010 HOMEWORK # 5 Due: July 16, 2010 Q1. (see Problem 6—1 in the text for the ﬁgure). A ﬁxed smooth rod makes an angle of 30° with the ﬂoor. A small (negligible radius) ring (say P.) of mass m can slide on the rod and supports a string which has one end connected to a point on the ﬂoor; the other end is attached to a particle (say P2) of mass 2m. Assume that the String and the rod lie in the same vertical plane. Since there are two objects with mass, one can use their Cartesian coordinates with respect to a (x,y) system with origin at the lower left corner. Let 1 be the.length of the string. Since the ring is constrained to move along the rod, and I is constant, the four coordinates (xbyl) and (x2,y2) are constrained. Also, let a be the coordinate of the point on the ﬂoor where the string is attached, and b be the height of the point where the rod is attached to the vertical wall. Then, (a) Show that the coordinates (xhyl) and (xz,y2) satisfy the constraints: y.=—x1/J§+b V(x1_a)2+y12 “l” (xz—x1)2+(y2_y1)2 =1 (b) Derives constraints on velocities by differentiating the constraints in (a) with respect to time; (c) Derive the relations in virtual displacements in the four coordinates (x1,yl) and (xbyz). Q2. Use the devc10pments in Q1. and the principle of virtual work to ﬁnd the static equilibrium position of the system. Note that this will give you four expressions for the coordinates (xhyl) and (x232), the variables that have been used to deﬁne the conﬁguration of the system at any given instant of time. Two of the equations are the constraint equations in Q1. and two equations will come additionally from using the principle of virtual work. Finally, use the deﬁnition of angle w, as given in the ﬁgure, to express the equilibrium position. Q3. (see Problem 6-3 in the text for a ﬁgure). One end of a thin uniform rod of mass m and length 31' rests against a smooth vertical wall. The other end of the rod is attached by a string of length l to a ﬁxed point 0 which is located at a distance 21 from the wall. The rod and the string remain in the same vertical plane perpendicular to the wall. Let the position of the rod be deﬁned by the coordinates (xG,yG) of the center of mass (denoted as G) and its angular orientation 0. The origin of the coordinate system is located at the ﬁxed point 0. Then, (a) Show that the coordinates (xG,yG) and the angular orientation 9 satisfy the constraints: x6 +3lsint9/2= 21 (XS —3l’sint9/2)2 +(yG +3ICOSH/2)2 :12 (b) Now, use the principle of virtual work to ﬁnd the static equilibrium position of the system. Note that this will give you expressions for the coordinates (x5,yG) and the angular orientation 0, the variables that have been used to deﬁne the conﬁguration of the system at any given instant of time. Q4. (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 B and ti). The system is in the vertical plane. Then, use the D’Alembert’s principle and derive the differential equations of motion for the system. Assume now that the angles 0 and 4), as well as their time derivatives, remain small during the motion. Then, obtain the linearized equations of motion for small 9 and (ii. 61) Conﬁmmﬁ w. m’: coon!" [Ta {(3, 5 £761? .-. 9:7 \$ (1‘2? W) C) rdaﬂons m w‘r‘ha/ d{5[)/a(€i'F'F?/i%5 m #6 four" (coral/naféj‘ {%;23;;552J2) ® a/gwﬂéwﬁo @1?_ ("KGB—L- (762-76!) 4- iﬁ __ (My!) ("T zéﬁfjgg’ + (962:?!)5762 % + {512* 3,}592 ﬂ 2 (XXX!) 'f/HZH U!) (3624901)?! 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Homework5_Su_2010_Solution - ME 562 Advanced Dynamics...

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