Name: crane boom motion on rotating axes
Uploaded: 2024-03-07T13:22:24.000Z
Duration: 1 min 29 s
Channel: JoVE

The crane&#39;s telescopic boom rotates with an angular velocity of 0.04 rad/s and angular acceleration of 0.02 rad/s2.
Simultaneously the boom extends with a constant speed of 5 m/s, measured relative to point C. At the given instant, the distance between points C and D is 60 meters.
Determine the magnitudes of the velocity and acceleration of point D at the same instant.
Here, point D is translating and rotating with respect to point C, so relative motion analysis can be done using a rotating frame of reference.
At the instant, the linear velocity and the linear acceleration of point C are zero. The angular velocity and angular acceleration of point D are along the negative z-axis.
Writing the relative velocity equation for point D in the rotating frame of reference, and substituting the known quantities, gives the magnitude of the velocity of point D.
Similarly, using the relative acceleration equation in a rotating frame and substituting the known quantities gives the acceleration of point D.

relative motion analysis using rotating axes-problem solving

Relative Motion Analysis using Rotating Axes-Problem Solving

Consider a crane whose telescopic boom rotates with an angular velocity of 0.04 rad/s and angular acceleration of 0.02 rad/s2. Along with the rotation, the boom also extends linearly with a uniform speed of 5 m/s. The extension of the boom is measured at point D, which is measured with respect to the fixed point C on the other end of the boom. For the given instant, the distance between points C and D is 60 meters.
Here, in order to determine the magnitude of velocity and acceleration for point D, relative motion analysis is used. For the given situation, point D is translating and rotating with respect to point C, so the relative motion analysis can be done using a rotating frame of reference. At the considered instant, the linear velocity and linear acceleration of point C are zero. At the same time, the angular velocity and angular acceleration of point D are along the negative z-axis as the boom is rotating in the clockwise direction.
The relative velocity and relative acceleration equations for the rotating frame of reference for point D are used to calculate the magnitudes of linear velocity and linear acceleration of point D.
<img alt="Velocity vector equation diagram; vD, vC, &#969;CD&#215;rD/C, vD/C XYZ; physics mechanics analysis." src="/files/ftp_upload/15541/15541_PT_Equation_001.svg" style="text-align: center; height: 20px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
<img alt="Rotational dynamics equation (a&#8345; = a_c + &#945;_c&#8345;&#215;r)/c); vector analysis, physics formula." src="/files/ftp_upload/15541/15541_PT_Equation_002v2.svg" style="text-align: center; height: 22px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
Substituting the known values, magnitudes of linear velocity and linear acceleration of point D are found to be 5.55 m/s and 1.4 m/s2, respectively.

The crane&#39;s telescopic boom rotates with an angular velocity of 0.04 rad/s and angular acceleration of 0.02 rad/s2.
Simultaneously the boom extends ...

relative-motion-analysis-using-rotating-axes-problem-solving

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Mechanical Engineering

Unit 1

Planar Kinematics of a Rigid Body

KEY TERMS AND CONCEPTS

crane boom motion on rotating axes

Crane boom motion on rotating axes combines rotation and extension in one kinematics problem. A telescopic boom rotates while it also extends outward. Relative motion analysis helps find the velocity and acceleration of the moving end of the boom.

In the setup, the boom rotates with an angular velocity of 0.04 rad/s and an angular acceleration of 0.02 rad/s². It also extends linearly at a constant speed of 5 m/s. The extension is measured at point D with respect to the fixed point C at the other end of the boom.

At the instant considered, the distance between C and D is 60 m. Point D is both translating and rotating relative to point C, so a rotating frame of reference is used. The linear velocity and linear acceleration of point C are zero at that moment.

The boom turns clockwise, so the angular velocity and angular acceleration of point D point along the negative z-axis. The relative velocity and relative acceleration equations for a rotating frame are then applied to point D. Using the given values, the magnitude of the linear velocity of point D is 5.55 m/s, and the magnitude of its linear acceleration is 1.4 m/s².

Crane Boom Motion on Rotating Axes

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Crane boom motion on rotating axes combines rotation and extension in one kinematics problem. A telescopic boom rotates while it also extends outward. Relative motion analysis helps find the velocity and acceleration of the moving end of the boom. In the setup, the boom rotates with an angular velocity of 0.04 rad/s and an angular acceleration of 0.02 rad/s². It also extends linearly at a constant speed of 5 m/s. The extension is measured at point D with respect to the fixed point C at the...

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key terms and concepts

Analyze a telescopic crane boom that rotates and extends. Use relative motion in a rotating frame to find point D's velocity and acceleration.

Crane Boom Motion on Rotating Axes | JoVE K12

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Crane Boom Motion on Rotating Axes

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