Acceleration Analysis
Acceleration analysis Objective: Compute accelerations (linear and angular) of all components of a mechanism
Acceleration of link in pure rotation AtPA P APA Length of link: p AnPA , A Magnitude of tangential component = p, magnitude of normal component = p 2
Graphical acceleration analysis AtBA B AtB 3 A AtA Clockwise acceleration of crank 4 2 AnA 1
Problem definition: given the positions of the links, their angular velocities and the acceleration of the input link (link 2), find the linear accelerations of A and B and the angular accelerations of links 2 and 3. Solution: Find velocity of A Solve graphically equation: Find the angular accelerations of links 3 and 4
Graphical solution of equation AB=AA+ABA AtB AtBA 2 3 4 1 AtA AnA B A AtB AA AtBA AnBA -AnB Steps: Draw AA, AnBA, -AtBA Draw line normal to link 3 starting from tip of –AnB Draw line normal to link 4 starting from origin of AA Find intersection and draw AtB and AtBA.
Guidelines Start from the link for which you have most information Find the accelerations of its points Continue with the next link, formulate and solve equation: acceleration of one end = acceleration of other end + acceleration difference We always know the normal components of the acceleration of a point if we know the angular velocity of the link on which it lies We always know the direction of the tangential components of the acceleration
Algebraic acceleration analysis 2 3 4 1 B A a b c R4 R2 R1 1 Given: dimensions, positions, and velocities of links and angular acceleration of crank, find angular accelerations of coupler and rocker and linear accelerations of nodes A and B
Loop equation Differentiate twice: This equation means:
General approach for acceleration analysis P, P’ (colocated points at some instant), P on slider, P’ on bar Acceleration of P = Acceleration of P’ + Acceleration of P seen from observer moving with rod+Coriolis acceleration of P’
Coriolis acceleration APslip: acceleration of P as seen by observer moving with rod VPslip P APcoriolis AP’n AP’t APslip O AP
Coriolis acceleration Coriolis acceleration=2Vslip Coriolis acceleration is normal to the radius, OP, and it points towards the left of an observer moving with the slider if rotation is counterclockwise. If the rotation is clockwise it points to the right. To find the acceleration of a point, P, moving on a rotating path: Consider a point, P’, that is fixed on the path and coincides with P at a particular instant. Find the acceleration of P’, and add the slip acceleration of P and the Coriolis acceleration of P. AP=acceleration of P’+acceleration of P seen from observer moving with rod+Coriolis acceleration=AP’+APslip+APCoriolis
Application: crank-slider mechanism B2 on link 2 B3 on link 3 These points coincide at the instant when the mechanism is shown. When 2=0, a=d-b Unknown quantities marked in blue . B2, B3 normal to crank Link 2, a 3, 3, 3 2, 2 2 O2 Link 3, b d
Velocity analysis VB3= VB2+ VB3B2 VB3B2 // crank VB3 ┴ B2 on crank, rocker B2 on crank, B3, on slider . crank rocker VB2 ┴ crank O2
Rolling acceleration First assume that angular acceleration, , is zero O O R (absolute) C (absolute) C r P No slip condition: VP=0
Find accelerations of C and P -(R-r)/r (Negative sign means that CCW rotation around center of big circle, O, results in CW rotation of disk around its own center) VC= (R-r) (Normal to radius OC) AnC=VC2/(R-r) (directed toward the center O) AnP=VC2/(R-r)+ VC2 /r (also directed toward the center O) Tangential components of acceleration of C and P are zero
Summary of results R AC, length VC2/(R-r) C VC, length (R-r) r AP, length VC2/(R-r)+ VC2 /r P VP=0
“MOTION OF ONE PART LEAD TO THEMOTION OF OTHER PARTS”
Relative-Motion Analysis : Relative Velocity Instantaneous Center of Zero Velocity Relative Acceleration
Position Rotating axes
Velocity VC = Velocity of the Collar, measured from the X, Y, Z reference VO = Velocity of the origin O of the x,y,z reference measured from the X,Y,Z reference (VC/O)xyz = relative velocity of “C with respect to O” observer attached to the rotating x,y,z reference = angular velocity of the x,y,z reference, measured from the X,Y,Z reference rC/O = relative position of “C with respect to O”
Coriolis acceleration Whenever a point is moving on a path and the path is rotating, there is an extra component of the acceleration due to coupling between the motion of the point on the path and the rotation of the path. This component is called Coriolis acceleration. First measured by the French engineer G.C. Coriolis Important in studying the effect “force and acceleration” of earth rotation on the rockets and long-range projectiles
Recall cylindrical coordinate Rearrange Compare !
Recall – Cylindrical coordinate Coriolis acceleration
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