Mechanical Vibrations Pennsylvania State （机械振动宾夕法尼亚州立）.pdfVIP

Mechanical Vibrations A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position of the system). Let u(t) denote the displacement, as a function of time, of the mass relative to its equilibrium position. Recall that the text ook’s convention is that downward is positive. Therefore, u 0 means the spring is stretched beyond its equilibrium length, while u 0 means that the spring is compressed. The mass is then assumed to be set in motion (by any one of several means). © 2008 Zachary S Tseng B3 1 The equations that govern a mass-spring system At equilibrium: (by Hooke’s Law) mg = kL While in motion: m u″ + γ u′ + k u = F(t) This is a second order linear differential equation with constant coefficients. It usually comes with two initial conditions: u(t ) = u , and u′(t ) = u′ . 0 0 0 0 Summary of terms: u(t) = displacement of the mass relative to its equilibrium position. m = mass (m 0) γ = damping constant (γ ≥ 0) k = spring (Hooke’s) constant (k 0) g = gravitational constant L = elongation of the spring caused by the weight F(t) = Externally applied forcing function, if any u(t0) = initial displacement of the mass u′(t ) = initial velocity of the mass 0 © 2008 Zachary S Tseng