RESPONSE UNDER A GENERAL PERIODIC FORCE When he exernl force F() is periodic wih periodτ / ω,i cn be expnded in Fourier series F( ) o α ω α b ω () where τ F( ) ω d, τ,,,... ()
nd b τ F( ) ω d, τ,,... (3) The equion of moion for spring, mss dmper sysem cn be expressed s.. m x. c x kx F( ) ω b ω (4)
The righ-hnd side of his equion is consn plus sum of hrmonic funcions. Ug he principle of superposiion, he sedy se soluion of Eq. (4) is he sum of he sedy se soluions of he following equions:.. m x.. m x. c x. c x kx kx ω (5) (6).. m x. c x kx b ω (7)
Noing h he soluion of Eq.5 is given by x p ( ) k nd we cn express he soluion of Eqs.(6) nd (7) respecively, s x p ( ) ( / k) ( ) r ( ζr) (8) ( ω φ ) (9) x p ( ) ( b / k) ( ) r ( ζr) ( ω φ ) ()
where () n r r ζ φ nd () n r ω ω Thus he complee sedy se soluion of Eq. (4) is given by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) (3 / / ) ( p r r k b r r k k x φ ω ζ φ ω ζ
EXAMPLE : Periodic Vibrion of Hydrulic Vlve In he sudy of vibrions of vlves used in hydrulic conrol sysems, he vlve nd is elsic sem re modeled s dmped spring mss sysem, s shown in Fig (). In ddiion o he spring force nd dmping force, here is fluid pressure force on he vlve h chnges wih he moun of opening or clog of he vlve. Find he Fourier series erms of he vlve when he pressure in he chmber vries s indiced in Fig.. Assume k 5 N/m,c N-s/m, nd m.5kg.
Soluion: The vlve cn be considered s mss conneced o spring nd dmper on one side nd subeced o forcing funcion F() on he oher side. The forcing funcion cn be expressed s F()Ap() (E.)
Fig. Periodic vibrion of hydrulic vlve
where A is he cross secionl re of he chmber, given by A 5) 4 ( 65 mm.65 m ( E.) nd p() is he pressure cing on he vlve ny insn. Since p() is periodic wih period τ seconds nd A is consn seconds nd A is consn, F() is lso period τ seconds. The frequency of he forcing funcion is ω ( / τ ) rd / s. F() cn be expressed in Fourier series s
F( ) ω ω... b ω b ω... ( E.3) where nd b re given by Eqs. () nd (3). Since he funcion F() is given by τ 5 A for F( ) ( E.4) τ 5 A( ) for τ
he Fourier coefficiens nd b cn be compued wih he help of Eqs. () nd (3): 5 A d 5 A( ) d 5 A ( E.5) 5 A d 5 A( ) d 5 A ( E.6)
.9) ( ) ( 5 5.8) ( ) ( 5 5.7) ( ) ( 5 5 E d A d A b E d A d A E d A d A b.) ( 9 3 ) ( 5 3 5 5 3 E A d A d A
b 3 5 A 3 d 5 A( ) 3 d ( E.) Likewise, we cn obin 4 6. b 4 b 5 b 6. By considering only he firs hree hrmonics, he forcing funcion cn be pproximed: 5 ( A F ) 5 A ω 9 5 A ω ( E.)
Response Under Periodic Force of Irregulr Form In some cses, he force cing on sysem my be quie irregulr nd my be deermined only experimenlly. Exmples of such forces include wind nd erh quke induced forces. In such cses, he forces will be vilble in grphicl form nd no nlyicl expression cn be found o describe F(). Someimes, he vlue of F() my be vilble only number of discree poins,,.., N. In ll hese cses, i is possible o find he Fourier coefficiens by ug numericl inegrion procedure.
If F, F,,F N denoe he vlues of F(),,,.., N respecively, where N denoes n even number of equidisn poins in one ime period τ ( τ N ), s shown in Fig., he pplicion of rpezoidl rule gives N N F i i () N Fi N i τ i,,,... ()
b N F N i i τ i,,,... (3) Once he Fourier coefficiens o,, nd b re known, he sedy se response of he sysem cn be found ug Eq.(3) wih r τωn
Sedy Se Vibrion of Hydrulic Vlve Find he sedy se response of he vlve in Exmple if he pressure flucuions in he chmber re found o be periodic. The vlues of pressure mesured. second inervls in one cycle re given below. Time, i (seconds) p i p( i ) (kn/m )...3.4.5.6.7.8.9... 34 4 49 53 7 6 36 6 7
Fig. An irregulr forcing funcion
Soluion: Since he pressure flucuions on he vlve re periodic, he Fourier nlysis of he given d of pressures in cycle gives p( ) 3483.3 6996. 5.36 837.7 5.36 46.7 4.7 368.34.7 5833.3 57.8 333.3 57.8... N / m