SDOF forced vibrations

Pagina 1

2. Forced motion of Single Degree Of Freedom Systems (SDOF) Let us consider the SDOF system illustrated in Figure 2.1. The mass m is subjected to a harmonic force of frequency 2f π= Ω . The maximum amplitude of the force is F o while β is the phase with respect to initial time. Therefore, the force can be expressed as: oF( ) F cos ( )tt β= Ω+ (2.1) Figure 2.1 Forced SDOF system Let us denote as x the absolute displacement of the mass with respect to the s tatic equilibrium posi tion. The equation of motion is given by: ooF cos ( ) F cos ( )mx cx kx t βϑ+ + = Ω+ =  (2.2) The phase β indicates that the external force may not reach its maximum level at time t = 0. In accordance with Euler’s notation it is possible to write: [ ] () ooF F c o s () s i n ()ite t itβ ββΩ+ = Ω+ + Ω+ (2.3) Therefore, eq.(2.3) can be rewritten as: () oRe F itmx cx kx e βΩ+++ =   (2.4) In case of a sinusoidal excitation the motion equation is: ooF sin ( ) F sin ( )mx cx kx t βθ+ + = Ω+ =  (2.5) Eq.(2.5) can be rewritten as: () oIm F itmx cx kx e βΩ+++ =   (2.6) In a more general case we can write: ( ) () ooFF it imx cx kx e e βϑΩ+++ = =  (2.7) where we implicitly assume to consider the real or the imaginary part of the complex term at the right side of eq.(2.7) depending on the type of the harmonic excitation ( )F t (cosine or sine function). The solution of this differential equation is: () () () gsxt x t x t= + (2.8) The term ()gxt is the general solution. It coincides with the solution of eq.(2.7), that is the system response of the free motion. The second tem, ()sxt , is the steady state solution. It only depends on the system excitation. In case of an underdamped system the contribution of the term ()gxt nullifies after a suitably long time interval. For linear systems…