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C_4 - Chapter 4 Moments of Inertia 4.1 Introduction A...

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Chapter 4 Moments of Inertia 4.1 Introduction A system of n particle P i , i = 1 , 2 ,..., n is considered. The mass of the particle P i is m i as shown in Fig. 4.1 P i ( m i ) r i x i y i z i x y z P 1 ( ) m 1 P 2 ( ) m 2 O Fig. 4.1 Particle P i with the mass m i The position vector of the particle P i is r i = x i ı + y i j + z i k . The moments of inertia of the system about the planes xOy , yOz , and zOx are I xOy = i m i z 2 i , I yOz = i m i x 2 i , I zOx = i m i y 2 i . (4.1) The moments of inertia of the system about x , y , and z axes are I xx = A = i m i ( y 2 i + z 2 i ) , 1
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2 4 Moments of Inertia I yy = B = i m i ( z 2 i + x 2 i ) , I zz = C = i m i ( x 2 i + y 2 i ) . (4.2) The moment of inertia of the system about the origin O is I O = i m i ( x 2 i + y 2 i + z 2 i ) . (4.3) The products of inertia of the system about the axes xy , yz and zx are I yz = D = i m i y i z i , I zx = E = i m i z i x i , I xy = F = i m i x i y i . (4.4) Between the different moments of inertia one can write the relations I O = I x 0 y + I y 0 z + I z 0 x = 1 2 ( I xx + I yy + I zz ) , and I xx = I y 0 z + I z 0 x . For a continuous domain D , the previous relations become I xOy = Z D z 2 dm , I yOz = Z D x 2 dm , I zOx = Z D y 2 dm , I xx = Z D ( y 2 + z 2 ) dm , I yy = Z D ( x 2 + z 2 ) dm , I zz = Z D ( x 2 + y 2 ) dm , I O = Z D ( x 2 + y 2 + z 2 ) dm , I xy = Z D xydm , I xz = Z D xzdm , I yz = Z D yzdm . (4.5) The infinitesimal mass element dm can have the values dm = ρ v dV , dm = ρ A dA , dm = ρ l dl , where ρ v , ρ A and ρ l are the volume density, area density and length density. 4.2 Moment of Inertia of a Rigid Body For a rigid body with mass m , density ρ , and volume V , as shown in Fig. 4.2, the moments of inertia are defined as follows
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4.2 Moment of Inertia of a Rigid Body 3 r x y z dV O Fig. 4.2 Rigid body in space with mass m and differential volume dV I xx = Z V ρ ( y 2 + z 2 ) dV , I yy = Z V ρ ( z 2 + x 2 ) dV , I zz = Z V ρ ( x 2 + y 2 ) dV , (4.6) and the products of inertia I xy = I yx = Z V ρ xydV , I xz = I zx = Z V ρ xzdV , I yz = I zy = Z V ρ yzdV . (4.7) The moment of inertia given in Eq. (4.6) is just the second moment of the mass distribution with respect to a cartesian axis. For example, I xx is the integral of sum- mation of the infinitesimal mass elements ρ dV , each multiplied by the square of its distance from the x axis. The effective value of this distance for a certain body is known as its radius of gyration with respect to the given axis. The radius of gyration corresponding to I j j is defined as k j = r I j j m , where m is the total mass of the rigid body, and where the symbol j can be replaced by x , y or z . The inertia matrix of a rigid body is represented by the matrix [ I ] = I xx - I xy - I xz - I yx I yy - I yz - I zx - I zy I zz . Moment of inertia about an arbitrary axis Consider the rigid body shown in Fig. 4.3. The reference frame x , y , z has the origin at O . The direction of an arbitrary axis Δ through O is defined by the unit vector u Δ u Δ = cos α ı + cos β j + cos γ k where cos α , cos β , cos γ are the direction cosines. The moment of inertia about the Δ axis, for a differential mass element dm of the body is by definition I Δ = Z D d 2 dm ,
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4 4 Moments of Inertia r x O d dm y z Δ u Δ θ Fig. 4.3
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