The member O A of the industrial robot telescopes and pivots...

The member $O A$ of the industrial robot telescopes and pivots about the fixed axis at point $O$. At the instant shown, $\theta=60^{\circ}, \dot{\theta}=1.2 \mathrm{rad} / \mathrm{s}, \ddot{\theta}=0.8$ $\operatorname{rad} / \mathrm{s}^{2}, \quad \overline{O A}=0.9 \mathrm{m}, \quad \dot{\overline{O A}}=0.5 \mathrm{m} / \mathrm{s},$ and $\overrightarrow{O A}=$ $-6 \mathrm{m} / \mathrm{s}^{2} .$ Determine the magnitudes of the velocity and acceleration of joint $A$ of the robot. Also, sketch the velocity and acceleration of $A$ and determine the angles which these vectors make with the positive $x$ -axis. The base of the robot does not revolve about a vertical axis.

The member $O A$ of the industrial robot telescopes and pivots about the fixed axis at point $O$. At the instant shown, $\theta=60^{\circ}, \dot{\theta}=1.2 \mathrm{rad} / \mathrm{s}, \ddot{\theta}=0.8$
$\operatorname{rad} / \mathrm{s}^{2}, \quad \overline{O A}=0.9 \mathrm{m}, \quad \dot{\overline{O A}}=0.5 \mathrm{m} / \mathrm{s},$ and $\overrightarrow{O A}=$
$-6 \mathrm{m} / \mathrm{s}^{2} .$ Determine the magnitudes of the velocity and acceleration of joint $A$ of the robot. Also, sketch the velocity and acceleration of $A$ and determine the angles which these vectors make with the positive $x$ -axis. The base of the robot does not revolve about a vertical axis.

Key Concepts

Angular Kinematics

Angular kinematics involves the study of rotational motion through parameters such as angular displacement, angular velocity, and angular acceleration. It is essential here because the robot member rotates about a fixed point, and these quantities determine the tangential components of velocity and acceleration that result from the rotational motion.

Polar Coordinates in Kinematics

Using polar coordinates allows one to separate the motion of a point on a rotating link into radial and tangential components. In problems like this, the motion due to the telescopic extension (radial motion) is treated separately from the motion due to rotation (tangential motion) about the fixed pivot. This framework simplifies finding the total velocity and acceleration vectors.

Vector Decomposition and Addition

Vector decomposition is the process of breaking a vector into its components, typically along mutually perpendicular directions, such as radial and tangential directions in polar coordinates. Once the individual velocity or acceleration components from different effects (extension and rotation) are determined, vector addition is used to obtain the overall magnitude and direction of the joint’s velocity and acceleration.

Radial and Tangential Accelerations

When a point moves along a rotating arm with variable length, its acceleration can be decomposed into radial (centripetal) and tangential components. The radial acceleration, often given by –r(angular velocity)^2 plus the linear acceleration along the arm, accounts for the change in speed due to direction change, while the tangential acceleration, given by r*(angular acceleration) plus the Coriolis term (2*radial velocity*angular velocity), accounts for the change in the magnitude of the velocity along the tangent to the path.

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