Dynamic Force Analysis - Chapter 4 Dynamic Force Analysis...

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Chapter 4 Dynamic Force Analysis 4.1 Equation of Motion for General Planar Motion The friction effects in the joints are assumed to be negligible. Figure 4.1 shows an arbitrary body with the total mass m . The body can be divided into n particles, the i th particle having mass, m i , and the total mass is m = n i = 1 m i . A rigid body can be considered as a collection of particles in which the number of particles approaches in±nity and in which the distance between any two points remains constant. As N approaches in±nity, each particle is treated as a differential mass element, and the summation is replaced by integration over the body m = ± dm . The position of the mass center of a collection of particles is de±ned by r i m i f kj f jk r C C O x ı j k y z P i m v C P k P j P l i +1 P Fig. 4.1 Rigid body divided into particles 109
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110 4 Dynamic Force Analysis r C = 1 m n i = 1 m i r i or r C = 1 m ± r dm , (4.1) where r i = r OP i = r P i is the position vector from the origin O to the i th particle. The time derivative of Eq. 4.1 gives N i = 1 m i d 2 r i dt 2 = m d 2 r C 2 = m a C , (4.2) where a C is the acceleration of the mass center. Any particle of the system is acted on by two types of forces: internal forces (exerted by other particles that are also part of the system) and external forces (exerted by a particle or object not included in the system). Let f ij be the internal force exerted on the j th particle by the i th particle. Newton’s third law (action and reaction) states that the j th particle exerts a force on the i th particle of equal magnitude, and opposite direction, and collinear with the force exerted by the i th particle on the j th particle f ji = f , j ± = i . Newton’s second law for the i th particle must include all of the internal forces exerted by all of the other particles in the system on the i th particle, plus the sum of any external forces exerted by particles, objects outside of the system on the i th particle j f + F ext i = m i d 2 r i 2 , j ± = i , (4.3) where F ext i is the external force on the i th particle. Equation 4.3 is written for each particle in the collection of particles. Summing the resulting equations over all of the particles from i = 1to N the following relation is obtained i j f + i F ext i = m a C , j ± = i . (4.4) The sum of the internal forces includes pairs of equal and opposite forces. The sum of any such pair must be zero. The sum of all of the internal forces on the collection of particles is zero (Newton’s third law) i j f = 0 , j ± = i . The term i F ext i is the sum of the external forces on the collection of particles i F ext i = F . The sum of the external forces acting on a closed system equals the product of the mass and the acceleration of the mass center m a C = F . (4.5) Equation 4.5 is Newton’s second law for a rigid body and is applicable to planar and three-dimensional motions.
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This note was uploaded on 10/28/2009 for the course MECH 320 taught by Professor D.a during the Spring '09 term at American University of Beirut.

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Dynamic Force Analysis - Chapter 4 Dynamic Force Analysis...

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