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2.2.1 Newton's Second Law of Motion Applied to Mechanical Systems Modeling Newton's second law of motion states that the acceleration of a particle ( a = x ¨ = d 2 x ( t ) / d t 2 ) is proportional to the force applied to it and is in the direction of this force. The proportionality constant is the mass m for a constant-mass particle or a rigid body undergoing translatory motion whose inertia can be reduced to a point. In cases where several external forces act on the particle in the motion direction, the acceleration is proportional to the algebraic sum of external forces fj (2.34) m x ¨ = ∑ j = 1 n f j An equation similar to Eq. (2.34) can be derived from Newton's second law of motion (which is formulated for translation), expressing the dynamic equilibrium of a body of constant mass moment of inertia J rotating about a fixed axis: (2.35) J θ ¨ = ∑ j = 1 n m t j where mtj are the torques acting on the rotating body and the rotary acceleration is θ ¨ = d 2 θ ( t ) / d t 2 (the symbol mt has been used for moments to avoid confusion with the mass symbol m). Equation (2.35) is also valid for a particle rotating about an external axis. Equation (2.34) for translation and Eq. (2.35) enable deriving mathematical models formed of the differential equation(s) connecting the input and the output through system component parameters. Instrumental to correctly applying these equations is the free-body diagram, which isolates a mechanical element from its system using external forcing as well as reaction forces and moments from the elements that have been separated from the studied element. The equations based on Newton's second law of motion and the free-body diagrams are utilized for the remainder of this chapter, as well as in Chapter 3, to determine the free and forced responses of mechanical systems PHEW thank GOD for copy and paste- Take what you will from this
