To determine The magnitude of acceleration at point P Explanation Draw the schematic diagram of the industrial robot as shown in figure (1). Refer the figure (1), Write the expression for the radial distance r . r = 300 cos β + 200 sin β (I). Here, angle made by the OA link with the horizontal is β . Differentiate the equation (I) to get the velocity of the arm. r ˙ = ( − 300 sin β + 200 cos β ) β ˙ (II). Again differentiate the equation (II) to get the acceleration of the arm. r ¨ = ( − 300 cos β − 200 sin β ) β ˙ 2 + ( − 300 sin β + 200 cos β ) β ¨ (III). Refer the figure (1), Write the expression for the vertical distance z . z = h + 300 sin β − 200 cos β (IV). Differentiate the equation (I) to get the velocity of the arm. z ˙ = ( 300 cos β + 200 sin β ) β ˙ (V). Again differentiate the equation (II) to get the acceleration of the arm. z ¨ = ( − 300 sin β + 200 cos β ) β ˙ 2 + ( 300 cos β + 200 sin β ) β ¨ (VI). Write the expression for the component of acceleration in the radial direction. a r = r ¨ − r θ ˙ 2 (VII). Write the expression for the component of acceleration in the tangential direction. a θ = r θ ¨ + 2 r ˙ θ ˙ (VIII). Write the expression for the magnitude of acceleration at point P . a = a r 2 + a θ 2 + a z 2 (IX). Conclusion: Substitute 30 ° for β in equation (I). r = 300 cos ( 30 ° ) + 200 sin ( 30 ° ) = 359.8 mm Substitute 30 ° for β and 10 ° for β ˙ in equation (II). r ˙ = ( − 300 sin ( 30 ° ) + 200 cos ( 30 ° ) ) ( 10 ° ) r ˙ = ( − 300 sin ( 30 ° ) + 200 cos ( 30 ° ) ) ( 10 ° × π 180 ) r ˙ = ( − 300 sin 30 ° + 200 cos 30 ° ) 0.1745 = 4.05 mm/s Substitute 30 ° for β , 10 ° for β ˙ , and 20 ° for β ¨ in equation (III). r ¨ = ( − 300 cos ( 30 ° ) − 200 sin ( 30 ° ) ) ( 10 ° ) 2 + ( − 300 sin ( 30 ° ) + 200 cos ( 30 ° ) ) ( 20 ° ) r ¨ = ( − 300 cos ( 30 ° ) − 200 sin ( 30 ° ) ) ( 10 ° × π 180 ) 2 + ( − 300 sin ( 30 ° ) + 200 cos ( 30 ° ) ) ( 20 ° × π 180 ) r ¨ = ( − 300 cos 30 ° − 200 sin 30 ° ) 0

8th Edition

ISBN: 9781118885840

Author: James L. Meriam, L. G. Kraige, J. N. Bolton

Publisher: WILEY

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Chapter 2.7, Problem 175P

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The magnitude of acceleration at point P

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Chapter 2.7, Problem 174P

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