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LECTURE+17+COE-3001-A

# centrifuge rotang in a horizontal

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Unformatted text preview: at r=b: w↵ ((L + b)2 b2 ) 2g w ↵L = (L + 2b) (max contribu=on from arm 2g at r=L+b) Vmax = Vmax W Now, for the shear force contribu=on Vmass = (L + b + c) ↵ from the mass on the end of the arm: g Maximum shear force: V = w ↵L W ( L + 2 b) + (L + b + c)↵ 2g g Finally, subs=tute W=2w, b=L/9 and c=L/10 to obtain: 2/18/13 Vmax 91wL2 ↵ at x=L+b where tangen=al = accelera=on is maximum 30g M. Mello/Georgia Tech Aerospace 14 PROBLEM 4.3- 15 “CENTRIFUGE ROTATING WITH CONSTANT ANGULAR ACCELERATION ” (cont.) BENDING- MOMENT CALCULATION: V (r ) = M (r ) = dM dr Z L+b V ( ⇠ ) d⇠ r w↵ M (r ) = 2g ↵ b Free- body diagram: c V dm ↵ r M L+b V (r ) = w↵ ((L + b)2 2g 2/18/13 r2 ) Z ²་  Integrate the shear force within the arm in order to determine the bending- moment at any radial loca=on (r) within the arm. L+b [(L + b)2 ⇠ 2 ] d⇠ r L+b w↵ ⇠3 2 M (r ) = ( L + b) ⇠ 2g 3r ✓ ◆ w↵ ( L + b) 3 M (r ) = ( L + b) 2 ( L + b ) 2g 3 w↵ M (r ) = 2(L + b)3 3(L + b)2 r + r3 6g ✓ ( L + b) 2 r r3 3 ◆ ²་  Arm contribu=on to bending- moment at any radial loca=on (r) M. Mello/Georgia Tech Aerospace 15 Maximum bending- moment at r=b w↵ Set r=b in M (r) = 2(L + b)3 6g 3(L + b)2 r + r3 w↵L2 arm Leads to: Mmax = (2L + 3b) (Arm contribu=on to maximum 6g bending moment at r=b) Next, calculate bending- moment contribu=on from mass on the end of the arm to the maximum bending moment at r=b: W↵ Mmass = (L + b + c)(L + c) g w↵L2 W↵ (2L + 3b) + (L + b + c)(L + c) Maximum bending moment at r=b: Mmax = 6g g Finally, subs=tute: W = 2w, b = L/9, c = L/10 Mmax 2/18/13 229wL3 ↵ = 75g M. Mello/Georgia Tech Aerospace 16 And one last thing… dv FINALLY, RECALL THAT dx = q (r ) (no concentrated loads within a distributed load) w↵ In this case we have: V (r) = ((L + b)2 r2 ) (shear force contribu=on from arm) 2g dV wr↵ = And so, dr g wr↵ Hence, q (r) = eﬀec=ve triangular load distribu=on which would deﬂect a sta=c g can=lever beam in the exact same way! q (r ) = wr↵ g ↵ EQUIVALENT STATIC LOAD PROBLEM! (ignoring the weight (W) on the end of the arm) 2/18/13 M. Mello/Georgia Tech Aerospace 17...
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