tailieunhanh - Dynamics of Mechanical Systems 2009 Part 5

Tham khảo tài liệu 'dynamics of mechanical systems 2009 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 182 Dynamics of Mechanical Systems By eliminating SA and SB between these three equations we have SA SB 0 Now suppose that SA is represented by a single force say FA passing through some common point C of A and B or A and B extended together with a couple with torque TA. Similarly let SB be represented by a single force FB passing through C together with a couple with torque TB. Then because SA and SB taken together form a zero system Eq. the resultant of SA and SB and the moment of SA and SB about C must be zero. That is Fa FB 0 or Fa -Fb and Ta TB 0 or Ta -Tb Equations and or the equivalent wording represent the law of action and reaction. First Moments Consider a particle P with mass m or alternatively a point P with associated mass m as depicted in Figure . Let O be an arbitrary reference point and let p be a position vector locating P relative to O. The first moment of P relative to O ỘP O is defined as PPO mp Consider next a set S of N particles Pị i 1 . N having masses as in Figure where O is an arbitrary reference point. The first moment of S for O ỘS O is defined as the sum of the first moments of the individual particles of S for O. That is N ỳS O m1P1 m2P2 K mNPn X miPi i 1 Observe that in general ỘS O is not zero. However if a point G can be found such that the first moment of S relative to G ỘS G is zero then G is defined as the mass center of S. Using this definition the existence and location of G can be determined from Eq. . Specifically if G is the mass center and if ri locates pi relative to G as in Figure then the first moment of S relative to G may be expressed as fG ịn 0 i 1 Forces and Force Systems 183 FIGURE A particle P and reference point O. P m FIGURE A set S of particles and a reference point O. FIGURE A set S of particles with mass center G. From Figure we have S Pi Pg ri or ri Pi - Pg Hence by substituting into Eq.

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