The elements of the mechanism for deployment of a spacecraft...

The elements of the mechanism for deployment of a spacecraft magnetometer boom are shown. Determine the angular velocity of the boom when the driving link $O B$ crosses the $y$ -axis with an angular velocity $\omega_{O B}=0.5 \mathrm{rad} / \mathrm{sec}$ if $\tan \theta=4 / 3$ at this instant.

Key Concepts

Angular Velocity

Angular velocity is a measure of how fast an object rotates or revolves relative to another point, typically specified in radians per second. It is a fundamental concept in rotational kinematics that describes the rate of change of the angular position of a rotating body, and it is key when determining how different parts of a mechanism influence each other’s motion.

Relative Motion Analysis in Mechanisms

Relative motion analysis in mechanisms involves determining the interdependent movements of various connected components. It uses the principles of kinematics to relate the angular velocity of one link to the angular or linear velocities of other connected links, often through vector relationships and geometric constraints.

Vector Relationships in Rotational Motion

In rotational motion, the velocity of any point on a rotating object can be determined by taking the cross product of the angular velocity vector and the position vector of the point relative to the axis of rotation. This relationship is crucial when analyzing the motion within mechanical linkages, as it provides a systematic way to relate rotational speeds to the linear velocities of points on a mechanism.

Trigonometric Relationships in Kinematics

Trigonometric relationships, such as those involving tangent, sine, or cosine functions, are used in kinematics to relate the angles within a mechanism to its geometric and velocity parameters. These relationships help translate the geometry of a system into equations that describe the motion of the system, thereby linking angular quantities to linear dimensions and speeds.

Transcript

00:03 First we calculate the velocity vector at b, that is vb, vector with given angular velocity of link ob.

00:11 So we can write vb equals to vo plus v omega -ob cross r -b vectors.

00:19 Okay? so as v -0 is equal to 0, so v -b will be equal to 0 plus omega -ob is equal to 0 .5 k cap cross r -o -b equals to 6 j -cap.

00:32 So from here, vb comes out to be minus 3i meter per second.

00:37 After having vb, we can calculate va equals to vb vector plus omega ab vector cross rba vector.

00:50 Okay? so substituting the values, we get va vector equals to vb is minus 3i plus omega ab is omega ab is omega ab k cap omega ab k cap cross r a b is 6 plus 10 k k cap plus 10 xx theta minus 6 j cap okay so this is the value of r b so from here v a vector comes out to be minus 3 i plus omega ab omega ab 6 plus 10 kosaeta j cap j cap okay and minus omega ab 10 sine theta minus 6 i cap okay so this is the so now we can also obtain the value of v a from omega ac that is va vector equals to vc vector plus omega ac vector cross r vector ca so substituting the values we get va vector equals to this vc is equal to 0 plus omega ac k cap plus rca is 10 cost theta i cap plus sine theta j cap okay so now performing the cross product we get this is plus not cross okay substituting the after substituting the values and performing the cross product va vector will be equals to minus 10 omega ac sine theta i cap plus 10 omega ac cost theta j cap now comparing this equation with this equation as both are equal to va vector.

03:12 So comparing icap with icap and j cap with j cap, we get minus 10 omega ac sine theta equals to minus 3, minus 10 omega ab sine theta plus six omega ab...

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