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Kinematics of Rigid Bodies

Kinematics of Rigid Bodies. Contents. Absolute and Relative Acceleration in Plane Motion Analysis of Plane Motion in Terms of a Parameter Sample Problem 15.6 Sample Problem 15.7 Sample Problem 15.8 Rate of Change With Respect to a Rotating Frame Coriolis Acceleration Sample Problem 15.9

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Kinematics of Rigid Bodies

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  1. Kinematics of Rigid Bodies

  2. Contents Absolute and Relative Acceleration in Plane Motion Analysis of Plane Motion in Terms of a Parameter Sample Problem 15.6 Sample Problem 15.7 Sample Problem 15.8 Rate of Change With Respect to a Rotating Frame Coriolis Acceleration Sample Problem 15.9 Sample Problem 15.10 Motion About a Fixed Point General Motion Sample Problem 15.11 Three Dimensional Motion. Coriolis Acceleration Frame of Reference in General Motion Sample Problem 15.15 Introduction Translation Rotation About a Fixed Axis: Velocity Rotation About a Fixed Axis: Acceleration Rotation About a Fixed Axis: Representative Slab Equations Defining the Rotation of a Rigid Body About a Fixed Axis Sample Problem 5.1 General Plane Motion Absolute and Relative Velocity in Plane Motion Sample Problem 15.2 Sample Problem 15.3 Instantaneous Center of Rotation in Plane Motion Sample Problem 15.4 Sample Problem 15.5

  3. Applications A battering ram is an example of curvilinear translation – the ram stays horizontal as it swings through its motion.

  4. Applications How can we determine the velocity of the tip of a turbine blade?

  5. Applications Planetary gear systems are used to get high reduction ratios with minimum weight and space. How can we design the correct gear ratios?

  6. Applications Biomedical engineers must determine the velocities and accelerations of the leg in order to design prostheses.

  7. translation: • rectilinear translation • curvilinear translation • rotation about a fixed axis • general plane motion • motion about a fixed point Introduction • Kinematics of rigid bodies: relations between time and the positions, velocities, and accelerations of the particles forming a rigid body. • Classification of rigid body motions: • general motion

  8. Consider rigid body in translation: • direction of any straight line inside the body is constant, • all particles forming the body move in parallel lines. • For any two particles in the body, • Differentiating with respect to time, All particles have the same velocity. • Differentiating with respect to time again, All particles have the same acceleration. Translation

  9. Velocity vector of the particle P is tangent to the path with magnitude • The same result is obtained from Rotation About a Fixed Axis. Velocity • Consider rotation of rigid body about a fixed axis AA’

  10. Concept Quiz What is the direction of the velocity of point A on the turbine blade? w • → • ← • ↑ • ↓ A y L x

  11. Differentiating to determine the acceleration, • Acceleration of P is combination of two vectors, Rotation About a Fixed Axis. Acceleration

  12. Velocity of any point P of the slab, • Acceleration of any point P of the slab, • Resolving the acceleration into tangential and normal components, Rotation About a Fixed Axis. Representative Slab • Consider the motion of a representative slab in a plane perpendicular to the axis of rotation.

  13. Concept Quiz What is the direction of the normal acceleration of point A on the turbine blade? w • → • ← • ↑ • ↓ A y L x

  14. Recall • Uniform Rotation, a = 0: • Uniformly Accelerated Rotation, a = constant: Equations Defining the Rotation of a Rigid Body About a Fixed Axis • Motion of a rigid body rotating around a fixed axis is often specified by the type of angular acceleration.

  15. Sample Problem 5.1 • SOLUTION: • Due to the action of the cable, the tangential velocity and acceleration of D are equal to the velocity and acceleration of C. Calculate the initial angular velocity and acceleration. • Apply the relations for uniformly accelerated rotation to determine the velocity and angular position of the pulley after 2 s. Cable C has a constant acceleration of 9 in/s2 and an initial velocity of 12 in/s, both directed to the right. Determine (a) the number of revolutions of the pulley in 2 s, (b) the velocity and change in position of the load B after 2 s, and (c) the acceleration of the point D on the rim of the inner pulley at t = 0. • Evaluate the initial tangential and normal acceleration components of D.

  16. Sample Problem 5.1 • SOLUTION: • The tangential velocity and acceleration of D are equal to the velocity and acceleration of C. • Apply the relations for uniformly accelerated rotation to determine velocity and angular position of pulley after 2 s.

  17. Magnitude and direction of the total acceleration, Sample Problem 5.1 • Evaluate the initial tangential and normal acceleration components of D.

  18. Magnitude and direction of the total acceleration, Group Problem Solving • Evaluate the initial tangential and normal acceleration components of D.

  19. Group Problem Solving • SOLUTION: • Using the linear velocity and accelerations,calculate the angular velocity and acceleration. • Using the angular velocity, determine the normal acceleration. A series of small machine components being moved by a conveyor belt pass over a 6-in.-radius idler pulley. At the instant shown, the velocity of point A is 15 in./s to the left and its acceleration is 9 in./s2 to the right. Determine (a) the angular velocity and angular acceleration of the idler pulley, (b) the total acceleration of the machine component at B. • Determine the total acceleration using the tangential and normal acceleration components of B.

  20. Group Problem Solving at= 9 in/s2 v= 15 in/s Find the angular velocity of the idler pulley using the linear velocity at B. B Find the angular velocity of the idler pulley using the linear velocity at B. What is the direction of the normal acceleration of point B? Find the normal acceleration of point B. Downwards, towards the center

  21. Group Problem Solving at= 9 in/s2 B Find the total acceleration of the machine component at point B. an= 37.5 in./s2 Calculate the magnitude Calculate the angle from the horizontal at= 9 in/s2 Combine for a final answer an= 37.5 in/s2

  22. Golf Robot A golf robot is used to test new equipment. If the angular velocity of the arm is doubled, what happens to the normal acceleration of the club head? Not ours – maybe Tom Mase has pic? If the arm is shortened to ¾ of its original length, what happens to the tangential acceleration of the club head? If the arm is shortened to ¾ of its original length, what happens to the tangential acceleration of the club head? If the speed of the club head is constant, does the club head have any linear accelertion ? w,a

  23. Example – General Plane Motion The knee has linear velocity and acceleration from both translation (the runner moving forward) as well as rotation (the leg rotating about the hip).

  24. Displacement of particles A and B to A2and B2 can be divided into two parts: • translation to A2and • rotation of about A2 to B2 General Plane Motion • General plane motion is neither a translation nor a rotation. • General plane motion can be considered as the sum of a translation and rotation.

  25. Absolute and Relative Velocity in Plane Motion • Any plane motion can be replaced by a translation of an arbitrary reference point A and a simultaneous rotation about A.

  26. Assuming that the velocity vA of end A is known, wish to determine the velocity vB of end B and the angular velocity w in terms of vA, l, and q. • The direction of vB and vB/A are known. Complete the velocity diagram. Absolute and Relative Velocity in Plane Motion

  27. Absolute and Relative Velocity in Plane Motion • Selecting point B as the reference point and solving for the velocity vA of end A and the angular velocity w leads to an equivalent velocity triangle. • vA/B has the same magnitude but opposite sense of vB/A. The sense of the relative velocity is dependent on the choice of reference point. • Angular velocity w of the rod in its rotation about B is the same as its rotation about A. Angular velocity is not dependent on the choice of reference point.

  28. Assuming that the velocity vA of end A is known, wish to determine the velocity vB of end B and the angular velocity w in terms of vA, l, and q. • The direction of vB and vB/A are known. Complete the velocity diagram. Absolute and Relative Velocity in Plane Motion

  29. Absolute and Relative Velocity in Plane Motion • Selecting point B as the reference point and solving for the velocity vA of end A and the angular velocity w leads to an equivalent velocity triangle. • vA/B has the same magnitude but opposite sense of vB/A. The sense of the relative velocity is dependent on the choice of reference point. • Angular velocity w of the rod in its rotation about B is the same as its rotation about A. Angular velocity is not dependent on the choice of reference point.

  30. The velocity for any point P on the gear may be written asEvaluate the velocities of points B and D. Sample Problem 15.2 • SOLUTION: • The displacement of the gear center in one revolution is equal to the outer circumference. Relate the translational and angular displacements. Differentiate to relate the translational and angular velocities. The double gear rolls on the stationary lower rack: the velocity of its center is 1.2 m/s. Determine (a) the angular velocity of the gear, and (b) the velocities of the upper rack R and point D of the gear.

  31. y • SOLUTION: • The displacement of the gear center in one revolution is equal to the outer circumference. • For xA > 0 (moves to right), w < 0 (rotates clockwise). x Differentiate to relate the translational and angular velocities. Sample Problem 15.2

  32. For any point P on the gear, Velocity of the upper rack is equal to velocity of point B: Velocity of the point D: Sample Problem 15.2

  33. SOLUTION: • Will determine the absolute velocity of point D with • The velocity is obtained from the given crank rotation data. • The directions of the absolute velocity and the relative velocity are determined from the problem geometry. • The unknowns in the vector expression are the velocity magnitudeswhich may be determined from the corresponding vector triangle. • The angular velocity of the connecting rod is calculated from Sample Problem 15.3 The crank AB has a constant clockwise angular velocity of 2000 rpm. For the crank position indicated, determine (a) the angular velocity of the connecting rod BD, and (b) the velocity of the piston P.

  34. SOLUTION: • Will determine the absolute velocity of point D with • The velocity is obtained from the crank rotation data. The velocity direction is as shown. • The direction of the absolute velocity is horizontal. The direction of the relative velocity is perpendicular to BD. Compute the angle between the horizontal and the connecting rod from the law of sines. Sample Problem 15.3

  35. Determine the velocity magnitudes from the vector triangle. Sample Problem 15.3

  36. Group Problem Solving In the position shown, bar AB has an angular velocity of 4 rad/s clockwise. Determine the angular velocity of bars BD and DE. Which of the following is true? • The direction of vB is ↑ • The direction of vD is → • Both a) and b) are correct

  37. The velocity for any point P on the gear may be written asEvaluate the velocities of points B and D. Group Problem Solving • SOLUTION: • The displacement of the gear center in one revolution is equal to the outer circumference. Relate the translational and angular displacements. Differentiate to relate the translational and angular velocities. In the position shown, bar AB has an angular velocity of 4 rad/s clockwise. Determine the angular velocity of bars BD and DE.

  38. Group Problem Solving y Determine the angular velocity of bars BD and DE. x How should you proceed? Determine vB with respect to A, then work your way along the linkage to point E. wAB= 4 rad/s Write vB in terms of point A, calculate vB. Does it make sense that vB is in the +j direction?

  39. Group Problem Solving y Determine vD with respect to B. x wAB= 4 rad/s Determine vD with respect to E, then equate it to equation above. Equating components of the two expressions for vD

  40. Plane motion of all particles in a slab can always be replaced by the translation of an arbitrary point A and a rotation about A with an angular velocity that is independent of the choice of A. • The same translational and rotational velocities at A are obtained by allowing the slab to rotate with the same angular velocity about the point C on a perpendicular to the velocity at A. Instantaneous Center of Rotation in Plane Motion • The velocity of all other particles in the slab are the same as originally defined since the angular velocity and translational velocity at A are equivalent. • As far as the velocities are concerned, the slab seems to rotate about the instantaneous center of rotation C.

  41. If the velocity at two points A and B are known, the instantaneous center of rotation lies at the intersection of the perpendiculars to the velocity vectors through A and B . • If the velocity vectors at A and B are perpendicular to the line AB, the instantaneous center of rotation lies at the intersection of the line AB with the line joining the extremities of the velocity vectors at A and B. Instantaneous Center of Rotation in Plane Motion • If the velocity vectors are parallel, the instantaneous center of rotation is at infinity and the angular velocity is zero. • If the velocity magnitudes are equal, the instantaneous center of rotation is at infinity and the angular velocity is zero.

  42. The instantaneous center of rotation lies at the intersection of the perpendiculars to the velocity vectors through A and B . • The trace of the locus of the center of rotation on the body is the body centrode and in space is the space centrode. Instantaneous Center of Rotation in Plane Motion • The velocities of all particles on the rod are as if they were rotated about C. • The particle at the center of rotation has zero velocity. • The particle coinciding with the center of rotation changes with time and the acceleration of the particle at the instantaneous center of rotation is not zero. • The acceleration of the particles in the slab cannot be determined as if the slab were simply rotating about C.

  43. Instantaneous Center of Rotation in Plane Motion At the instant shown, what is the approximate direction of the velocity of point G, the center of bar AB? G

  44. Sample Problem 15.4 • SOLUTION: • The point C is in contact with the stationary lower rack and, instantaneously, has zero velocity. It must be the location of the instantaneous center of rotation. • Determine the angular velocity about C based on the given velocity at A. • Evaluate the velocities at B and D based on their rotation about C. The double gear rolls on the stationary lower rack: the velocity of its center is 1.2 m/s. Determine (a) the angular velocity of the gear, and (b) the velocities of the upper rack R and point D of the gear.

  45. Determine the angular velocity about C based on the given velocity at A. • Evaluate the velocities at B and D based on their rotation about C. Sample Problem 15.4 • SOLUTION: • The point C is in contact with the stationary lower rack and, instantaneously, has zero velocity. It must be the location of the instantaneous center of rotation.

  46. Sample Problem 15.5 • SOLUTION: • Determine the velocity at B from the given crank rotation data. • The direction of the velocity vectors at B and D are known. The instantaneous center of rotation is at the intersection of the perpendiculars to the velocities through B and D. The crank AB has a constant clockwise angular velocity of 2000 rpm. For the crank position indicated, determine (a) the angular velocity of the connecting rod BD, and (b) the velocity of the piston P. • Determine the angular velocity about the center of rotation based on the velocity at B. • Calculate the velocity at D based on its rotation about the instantaneous center of rotation.

  47. SOLUTION: • From Sample Problem 15.3, • Determine the angular velocity about the center of rotation based on the velocity at B. • Calculate the velocity at D based on its rotation about the instantaneous center of rotation. Sample Problem 15.5 • The instantaneous center of rotation is at the intersection of the perpendiculars to the velocities through B and D.

  48. Instantaneous Center of Zero Velocity What happens to the location of the instantaneous center of velocity if the crankshaft angular velocity increases from 2000 rpm in the previous problem to 3000 rpm? What happens to the location of the instantaneous center of velocity if the angle b is 0?

  49. Group Problem Solving In the position shown, bar AB has an angular velocity of 4 rad/s clockwise. Determine the angular velocity of bars BD and DE.

  50. Group Problem Solving What is the velocity of B? What direction is the velocity of B? What direction is the velocity of D? wAB= 4 rad/s vB b Find b vD

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