C_4 - Chapter 4 Moments of Inertia 4.1 Introduction A system of n particle Pi i = 1 2 n is considered The mass of the particle Pi is mi as shown in

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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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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 ,
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4 4 Moments of Inertia r x O d dm y z Δ u Δ θ Fig. 4.3 Rigid body and an arbitrary axis Δ
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This note was uploaded on 08/29/2011 for the course MECH 2110 taught by Professor Clark,b during the Spring '08 term at Auburn University.

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C_4 - Chapter 4 Moments of Inertia 4.1 Introduction A system of n particle Pi i = 1 2 n is considered The mass of the particle Pi is mi as shown in

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