MECH 1401 Study Notes

In the scope of General Planar motion; ! For orientation to be measured, two points on an object are required (to make a line). ! Pure Translation o All lines on body do not change orientation. (do not rotate) o All points on body follow parallel paths and have the same v and a. o Hence; only one point needs to be tracked on the body to fully define motion. ! Fixed Axis Rotation o All lines on body rotate with the same 𝜔 and ∝. o All points on body travel in concentric circles about the axis of rotation o Can use circular motion equations to describe the motion of any point on the body ! General Planar Motion o A combination of translation and rotation. o Planar motion has 3 degrees of freedom i.e. ! 2 linear (translation) and 1 rotation What is the absolute motion method (method used to analyse general planar motion)? ! Define geometry and then differentiate to find velocity and acceleration What are the general equations whilst analysing a point B (OB = rB) on a non slipping wheel? ! ! Position: 𝑥! = 𝑠 − 𝑟! 𝑠𝑖𝑛𝜃 and 𝑦! = 𝑅 − 𝑟! 𝑐𝑜𝑠𝜃 ! Velocity: 𝑥! = 𝑣! − 𝑟! 𝜔𝑐𝑜𝑠𝜃 and 𝑦! = 𝑟! 𝜔𝑠𝑖𝑛𝜃 ! ! Acceleration: 𝑥! = 𝑎! − 𝑟! 𝛼𝑐𝑜𝑠𝜃 + 𝑟! 𝜔 𝑠𝑖𝑛𝜃 and 𝑦! = 𝑟! 𝛼𝑠𝑖𝑛𝜃 + 𝑟! 𝜔 ! 𝑐𝑜𝑠𝜃 ! If wheel rolls without slipping o 𝑠 = 𝑅𝜃 o 𝑣 = 𝑅𝜔 o 𝑎 = 𝑅𝛼 What is the analysis on the special case where point C is in contact with the ground (i.e. 𝜃 = 0) ! Velocity: 𝑥! = 𝑣! − 𝑟! 𝜔𝑐𝑜𝑠𝜃 and 𝑦! = 𝑟! 𝜔𝑠𝑖𝑛𝜃 o 𝑥! = 𝑣! − 𝑟! 𝜔𝑐𝑜𝑠0 and 𝑦! = 𝑟! 𝜔𝑠𝑖𝑛0 o 𝑥! = 𝑣! − 𝑟! 𝜔 and 𝑦! = 0 o 𝒗! = (𝑣! − 𝑟! 𝜔)! o 𝑵𝑩: if 𝑣! = 𝜔𝑅 if no slipping, then 𝒗! = 0 ! Acceleration: 𝑥! = 𝑎! − 𝑟! 𝛼𝑐𝑜𝑠𝜃 + 𝑟! 𝜔 ! 𝑠𝑖𝑛𝜃 and 𝑦! = 𝑟! 𝛼𝑠𝑖𝑛𝜃 + 𝑟! 𝜔 ! 𝑐𝑜𝑠𝜃 o 𝑥! = 𝑎! − 𝑟! 𝛼𝑐𝑜𝑠0 + 𝑟! 𝜔 ! 𝑠𝑖𝑛0 and 𝑦! = 𝑟! 𝛼𝑠𝑖𝑛0 + 𝑟! 𝜔 ! 𝑐𝑜𝑠0 o 𝑥! = 𝑎! − 𝑟! 𝛼 and 𝑦! = 𝑟! 𝜔 ! o 𝒂! = (𝑎! − 𝑟! 𝛼)! + (𝑟! 𝜔 ! )! o 𝑵𝑩: if 𝑎! = 𝛼𝑅 if no slipping, then 𝒂! = (𝑟! 𝜔 ! )!