Chapter 18 KINETICS OF RIGID BODIES IN THREE DIMENSIONS. The two fundamental equations for the motion of a system of particles .

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1 hapter 18 KINETIS F RIID DIES IN THREE DIMENSINS The to fundamental equations for the motion of a sstem of particles ΣF = ma ΣM = H H provide the foundation for three dimensional analsis, just as the do in the case of plane motion of rigid bodies The computation of the angular momentum H and its derivative H, hoever, are no considerabl more involved The rectangular components of the angular momentum H of a rigid bod ma be epressed in terms of the components of its angular velocit and of its centroidal moments and products of inertia: H = +I ω - I ω - I ω H = -I ω + I ω - I ω H = -I ω - I ω + I ω If principal aes of inertia are used, these relations reduce to H H = I ω H = I ω H = I ω

2 H The sstem of the momenta of the particles forming a rigid bod ma be reduced to the vector mv attached at and the couple H nce these are determined, the angular momentum H of the bod about an given point ma be obtained b riting H = r mv + H In general, the angular momentum H and the angular velocit do not have the same direction The ill, hoever, have the same direction if is directed along one of the principal aes of inertia of the bod H r mv In the particular case of a rigid bod constrained to rotate about a fied point, the components of the angular momentum H of the bod about ma be obtained directl from the components of its angular velocit and from its moments and products of inertia ith respect to aes through H H = +I ω - I ω - I ω H = -I ω + I ω - I ω H = -I ω - I ω + I ω

3 The principle of impulse and momentum for a rigid bod in threedimensional motion is epressed b the same fundamental formula used for a rigid bod in plane motion Sst Momenta 1 + Sst Et Imp 1 2 = Sst Momenta 2 The initial and final sstem momenta should be represented as shon in the figure and computed from or H = +I ω - I ω - I ω H = -I ω + I ω - I ω H = -I ω - I ω + I ω H = I ω H = I ω H r mv H = I ω The kinetic energ of a rigid bod in three-dimensional motion ma be divided into to parts, one associated ith the motion of its mass center, and the other ith its motion about Using principal aes,,, e rite T = mv 2 + (I ω + I ω + I ω ) here v = velocit of the mass center = angular velocit m = mass of rigid bod I, I, I = principal centroidal moments of inertia

4 In the case of a rigid bod constrained to rotate about a fied point, the kinetic energ ma be epressed as T = 2 (I ω + I ω + I ω ) here,, and aes are the principal aes of inertia of the bod at The equations for kinetic energ make it possible to etend to the three-dimensional motion of a rigid bod the application of the principle of ork and energ and of the principle of conservation of energ The fundamental equations ΣF = ma ΣM H = H can be applied to the motion of a rigid bod in three dimensions We first recall that H represents the angular momentum of the bod relative to a centroidal frame of fied orientation and that H represents the rate of change of H ith respect to that frame s the bod rotates, its moments and products of inertia ith respect to change continuall It is therefore more convenient to use a frame rotating ith the bod to resolve into components and to compute the moments and products of inertia hich are used to determine H

5 H ΣF = ma ΣM = H H represents the rate of change of H ith respect to the frame of fied orientation, therefore H = (H ) + W H here H = angular momentum of the bod ith respect to the frame of fied orientation (H ) = rate of change of H ith respect to the rotating frame W = angular velocit of the rotating frame H ΣF = ma ΣM = H H = (H ) + W H Substituting H above into Σ M, ΣM = (H ) + W H If the rotating frame is attached to the bod, its angular velocit W is identical to the angular velocit of the bod Setting W =, using principal aes, and riting this equation in scalar form, e obtain Euler s equations of moton

6 H In the case of a rigid bod constrained to rotate about a fied point, an alternative method of solution ma be used, involving moments of the forces and the rate of change of the angular momentum about point ΣM = (H ) + W H here ΣM = sum of the moments about of the forces applied to the rigid bod H = angular momentum of the bod ith respect to the frame (H ) = rate of change of H ith respect to the rotating frame W = angular velocit of the rotating frame φ D ψ D When the motion of groscopes and other aismmetrical bodies are considered, the Eulerian angles φ,, and ψ are introduced to define the position of a groscope The time derivatives of these angles represent, respectivel, the rates of precession, nutation, and spin of the groscope The angular velocit is epressed in terms of these derivatives as = -φ sin i + j + (ψ + φ cos )k

7 φ D ψ D = -φ sin i + j + (ψ + φ cos )k The unit vectors are associated ith the frame attached to the inner gimbal of the groscope (figure to the right)and rotate, therfore, ith the angular velocit W = -φ sin i + j + φ cos k φk j ψk φ D ψ Denoting b I the moment of inertia of the groscope ith respect to its spin ais and b I its moment of inertia ith respect to a transverse ais through, e rite φk j ψk H = -I φ sin i + I j + I(ψ + φ cos )k W = -φ sin i + j + φ cos k ΣM = (H ) + W H Substituting for H and D leads to the differential equations defining the motion of the groscope into

8 φk ψk ΣM In the particular case of the stead precession of a groscope, the angle, the rate of precession φ, and the rate of spin ψ remain constant Such motion is possible onl if the moments of the eternal forces about satisf the relation ΣM = (Iω - I φ cos )φ sin j ie, if the eternal forces reduce to a couple of moment equal to the right-hand member of the equation above and applied about an ais perpendicular to the precession ais and to the spin ais