Transmission Factor

Page 1

5. Force transmitted to the system constraint Let us consider the SDOF system illustrated in Figure 5.1. 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 wi th respect to initial time. Therefore, the force can be expressed as: () oF( ) F itte βΩ+= (5.1) Figure 5.1 Forced SDOF system Figure 5.2 Force transmitted to the system constraint The steady state response is given by: ( ) ( ) () () ()o o22 1F( ) =X UX 12 i it it it nn xt e e e ekih ϕβ β β ωω Ω+ Ω+ Ω+= = −Ω + Ω  (5.2) That is: ( ) ( )( ) 2222o X1 X 12 nn hωω = −Ω + Ω (5.3) Now we are interested in evaluating the dynamic force, TF() t , transmitted by the system to the constraint. This force is obviously the sum of the elastic and damping forces. That is: TF() = F() + F()edt t t kx cx = +  (5.4) In the steady state condition this force becomes: ( ) () () TTF() = F() + F() X Fi it it edt t t k ic e e e ϕβ β Ω+ Ω+= +Ω =  (5.5) Then, the maximum magnitude of the force TF() t is given by: ( ) ( ) ( )( ) 22 o22 T 2222 F () F =X ( ) 12 nn kk c kc hωω +Ω + Ω= −Ω + Ω  (5.6) Eq.(5.6) can be rewritten as: ( )( ) ( ) ( )( ) 2 T 2222o 12F Tr =F 12 n n nn h h ω ω ωω +ΩΩ = −Ω + Ω  (5.7) The dimensionless function Tr, that is called Transmission Factor, gives the magnitude of the force transmitted to the constraint, as a function of the dimensionless frequency. Eq.(5.5) can also be written as: ( )( ) ( ) ( ) ( ) o () () To 22 F1 2 F ( ) = Tr F 12 n it it n nn ih t ee ih ββω ωωω Ω+ Ω++Ω = Ω −Ω + Ω (5.8) Figure 5.3 shows the magnitude and phase of the Transmission Factor as a function of the dimensionless frequency nωΩ . 0 0.5 1 1.5 2 -180 -135 -90 -45 0 Dimensionless frequency Phase [degree] Transmission Factor ( FT /F0 ) h: 0.04 h: 0.05…