{"id":8395,"date":"2020-07-11T14:29:18","date_gmt":"2020-07-11T18:29:18","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?page_id=8395"},"modified":"2024-10-05T18:09:20","modified_gmt":"2024-10-05T22:09:20","slug":"impact-drop-tests","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/impact-drop-tests\/","title":{"rendered":"Impact drop tests"},"content":{"rendered":"

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Particle P is dropped from rest from a height of h<\/em>1<\/sub> above a fixed floor. Immediately prior to impact with the floor, P has a speed of v<\/em>P2<\/sub>. After impact, P has a rebound speed of v<\/em>P3<\/sub> eventually reaches a maximum height of h<\/em>4<\/sub> before falling back down.<\/p>\n

Energy analysis<\/em><\/strong>
\nUsing conservation of energy, we can show that:\u00a0v<\/em>P2<\/sub> = –\u00a0\u221a(2gh<\/em>1<\/sub>). In a similar way, conservation of energy gives us that: v<\/em>P3<\/sub> = \u221a(2gh<\/em>4<\/sub>).<\/p>\n

Determination of the coefficient of restitution (COR)<\/em><\/strong>
\nBased on the definition of the COR, we can write:<\/p>\n

e =-v<\/em>P3<\/sub>\/ v<\/em>P2<\/sub>\u00a0= \u221a(h<\/em>4<\/sub>\/h<\/em>1<\/sub>)<\/p>\n

The above result says that we can extract the COR for an impact during a drop test by taking the square root of the ratio of the rebound height to the initial drop height.<\/p>\n

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Drop test results<\/em><\/strong>
\nConsider the animation of a drop test shown below. The ball is dropped from a height of h<\/em>1<\/sub> = 10 inches. The maximum height after the first impact is seen to be h<\/em>4<\/sub> = 4.95 inches. What is the COR for this impact?<\/p>\n

<\/p>\n<\/div>\n

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Influence of COR on rebound height<\/em><\/strong>
\n\"\"<\/p>\n<\/div>\n<\/div>\n

Six balls are dropped from the same initial height h<\/em>1<\/sub>. These balls have varying values of the COR, ranging from e<\/em> = 0 to e<\/em> = 1. The above shows an animation of the simulation results for these drop tests. Recall from above that the rebound height h<\/em>4<\/sub> is given by: \u00a0 h<\/em>4<\/sub>\u00a0= e<\/em>2<\/sup> h<\/em>1<\/sub>. From this, we see that the rebound height decreases in a quadratic fashion as the COR is decreased. Low values of COR show rapid reduction of the rebound heights.<\/p>\n

Questions<\/em><\/span>:<\/p>\n