Notes on mechanical vibrations NYU Courant（在机械振动纽约大学报》）.pdf

Notes on mechanical vibrations Stephen Childress April 13, 2005 1 Masses and springs- the linear oscillator These notes will be related to sections 1-29 of the text. I will indicate between brackets [..] references to the sections closest to the discussion here. We are will study mechanical vibrations for two reasons: on the math side, it will introduce second-order ODEs (involving two differentiations) in the context of Newton’s equations of motion. We will revisit the idea of a phase plane in two variables, as we have used it in the study of two-species population models, but now with a physical meaning derived from Newtonian physics. The centerpiece of Newtonian physics is the second law [1-3,5] , stating that a body of mass m will accelerate when a force F is applied to it, in such a way that 2 d x m dt2 = F. (1) Here x(t) is the position of the mass- for simplicity think of a uniform sphere with x denoting the position of its center. (In general x will denote the center of mass of the body.). A better way to state this law is in terms of momentum. The momentum of the mass is mv where v = dx is the velocity of the body (of dt it’s center of mass). Then Newton’s second law becomes dmv = F, (2) dt i.e. it states that the rate of change of the momentum of the body is equal to the instantaneous force applied to the body. In addition to a mass, we also want to introduce a spring with sp