Equations of Motion in Cylindrical Coordinates
Equations of motion in cylindrical coordinates can be expressed in terms of components or scalar equations. Practical applications include analyzing motion in various engineering systems. In-class practice problems involve dealing with spring forces and applying the chain rule to determine forces in specific positions.
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13.6 Equation of Motion: Cylindrical Coordinates Equations of Motion in Cylindrical Coordinates The equations of motion can also be written along cylindrical coordinates. F F F F + + = a m In terms of components: = + + u u u u u u ma ma ma r r z z r r z z Or as scalar equations: = = = F F F ma ma ma r r (EQ 13-9) z z ( ( ) = = = + F m r r 2 r ) or 2 F F m r mz r z
In-Class Practice Problem 1 Can you do this in n-t-b? Can you do this in r- -z? Apart from amusement park rides, what are some practical applications of this problem?
In-Class Practice Problem 2 13-85 The spring-held follower AB has a weight of 0.75 lb and moves back and forth as its end rolls on the contoured surface of the cam, where r = 0.2 ft and z = 0.1*sin(2 ) ft. If the cam is rotating at constant rate of 6 rad/sec, determine the force at the end A when = 45 . In this position the spring is compressed 0.4 ft. Neglect friction at the bearing C. Find FA when ? = 45 An actual r- -z problem! How to deal with spring force? Need to take time derivatives of r, ?, and z r = constant ? = constant ? = ?(?) Chain Rule!
13.4 Common External Forces Spring Forces Spring force is related to the deformation (deflection) of the spring by the equation: ( ) = s l l = = = = = s F ks spring stiffness force unit length stretch or deflection of spring deformed length undeformed length k s l l 0 0
In-Class Practice Problem 2 13-85 The spring-held follower AB has a weight of 0.75 lb and moves back and forth as its end rolls on the contoured surface of the cam, where r = 0.2 ft and z = 0.1*sin(2 ) ft. If the cam is rotating at constant rate of 6 rad/sec, determine the force at the end A when = 45 . In this position the spring is compressed 0.4 ft. Neglect friction at the bearing C. ??= ? ? ?0 When ? = 45 , ? ?0 = 0.4 ?? Want to get ? ??? ?0 in terms of z ? 45 = 0.1 sin 90 = 0.1 This means ? ?0 = z + 0.3 Thus, ??= ? ? + 0.3
In-Class Practice Problem 2 13-85 The spring-held follower AB has a weight of 0.75 lb and moves back and forth as its end rolls on the contoured surface of the cam, where r = 0.2 ft and z = 0.1*sin(2 ) ft. If the cam is rotating at constant rate of 6 rad/sec, determine the force at the end A when = 45 . In this position the spring is compressed 0.4 ft. Neglect friction at the bearing C. ? = ?(?) Chain Rule! ?? ??=?(??????) ? ??????? ?? ?? ??= 0.1 2 ? cos(2?) Evaluating with = 45 and ? = 0
In-Class Practice Problem 2 13-85 The spring-held follower AB has a weight of 0.75 lb and moves back and forth as its end rolls on the contoured surface of the cam, where r = 0.2 ft and z = 0.1*sin(2 ) ft. If the cam is rotating at constant rate of 6 rad/sec, determine the force at the end A when = 45 . In this position the spring is compressed 0.4 ft. Neglect friction at the bearing C.
