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For rotation about a fixed axis caused by a constant applied moment M α =M/I ω = ωO + (M / I) t θ = θO + ωO t + (M / 2I) t2 the distance from the center of rotation to the center of the mass of the body. θ = the angle between the roadway surface and the horizontal, v = the velocity of the vehicle, and r = the radius of the curve. FREE VIBRATION • The change in kinetic energy of rotation is the work done in accelerating the rigid body from ωO to ω. I O ω2 2 − I O ωO2 2 = òθθO Mdθ Kinetic Energy The kinetic energy of a body in plane motion is ( kδst + kx ) 2 2 T = m v xc + v yc 2 + I c ω2 2 The equation of motion is mx = mg − k ( x + δ st ) Instantaneous Center of Rotation The instantaneous center of rotation for a body in plane motion is defined as that position about which all portions of that body are rotating.

Press. 55 Specific Volume m3/kg Sat. Sat. 982 Internal Energy kJ/kg Sat. Sat. Evap. liquid vapor ufg uf ug Enthalpy kJ/kg Sat. liquid hf Evap. hfg Entropy kJ/(kg·K) Sat. vapor hg Sat. liquid sf Evap. sfg Sat.

N = GJ IL x(t) = x0 cos ωnt, = the torsional spring constant = GJ/L, the initial angle of rotation and The undamped circular natural frequency of torsional vibration is x(t) = x0 cos ωnt + (v0 /ωn) sin ωnt kt ) θ = θ0 cosωnt + θ 0 ωn sinωn t , where ωn = g δ st 28 DYNAMICS (continued) Figure y Area & Centroid A = bh/2 xc = 2b/3 h C yc = h/3 x b y A = bh/2 h C xc = b/3 yc = h/3 x b y A = bh/2 h C a xc = (a + b)/3 yc = h/3 x b 29 y C h b rx2c = h 2 18 I yc = b 3h/36 ry2c = b 2 18 I xc yc = Abh 36 = b 2 h 2 72 rx2 = h 2 6 ry2 = b 2 2 I xy Ix = bh3/12 Iy = b3h/4 I yc = b 3h/36 Ix = bh3/12 Iy = b3h/12 I xc = bh 3 36 [ ( 2 2 C h b x b a x A = ab sin θ xc = (b + a cos θ)/2 yc = (a sin θ)/2 I xy rx2c = h 2 18 )] 36 ( rx2c = h 2 12 [ ( )] [ ( I yc Ix I xc yc = [Ah(2a − b )] 36 ) 18 I xy =h 6 = b 2 + ab + a 2 6 ( 2 ) 2 ) [ ] [ ] = bh 2 (2a − b ) 72 = [Ah(2a + b )] 12 = bh 2 (2a + b ) 24 I xc yc = 0 I xy = Abh 4 = b 2 h 2 4 ) rx2c = (a sinθ) 12 I y = ab sinθ(b + a cosθ) ]3 ) 2 ( 2 2 2 2 ( h 2 a 2 + 4ab + b 2 18(a + b ) 2 h (3a + b ) rx2 = 6(a + b ) rx2c = 3 ( = Abh 12 = b 2 h 2 24 rp2 = b 2 + h 2 12 ( ) = [ab sinθ(b + a cos θ)] 12 = (a b sin θ) 3 [ 2 2 ( )] ( 3 = b − ab + a rx2 = h 2 3 ry2 = b 2 3 h 3 a 2 + 4ab + b 2 36(a + b ) 3 h (3a + b ) Ix = 12 I xc = 2 ry2c = b 2 12 I xc = a 3b sin 3θ 12 C rx2 = h 2 6 ry2 = b 2 6 I xc = b h 3 12 3 I x = bh 3 3 I y = b 3h 3 y I xc yc = − Abh 36 = − b 2 h 2 72 I x = bh 12 I y = bh b 2 + ab + a 2 12 I yc = bh b − ab + a xc = b/2 A = h(a + b ) 2 h(2a + b ) yc = 3(a + b ) ry2c = b 2 18 ry2c rx2 ry2 I yc = b 3 h 12 = Abh 4 = b 2 h 2 8 rx2c = h 2 18 I x c = bh 3 /36 J = bh b 2 + h 2 12 a y Product of Inertia I x c = bh 3 /36 A = bh yc = h/2 x (Radius of Gyration)2 Area Moment of Inertia ) ry2c = b 2 + a 2 cos 2 θ 12 rx2 = (a sinθ) 3 2 ) − a b sinθcosθ 6 ry2 = (b + a cosθ) 3 − (ab cosθ) 6 2 Housner, George W.