Dervao of he Fler Coeffce for he Ramp Ivara Meho a Apple o Bae Excao of a Sgle-egree-of-Freeom Sem Revo B B om Irve Emal: om@vbraoaa.com Aprl, 0 Irouco Coer he gle-egree-of-freeom em Fgure. m &&x k c && Fgure. where m Ma c vcou ampg coeffce k Sffe x abolue placeme of he ma bae pu placeme A free-bo agram how Fgure. &&x m k(-x) c(& x& ) Fgure.
Summao of force he vercal reco Le F mx && () mx && c(& x& ) k ( x) () u x u& x& & && u && x && && x && u && Subug he relave placeme erm o equao () el m(u & && ) cu& ku () m & u cu& ku m & () Dvg hrough b ma el & u (c / m)u& (k / m)u & (5) B coveo, ( c / m) ξ (6) ( k / m) (7) where he aural frequec (raa/ec), a ξ he ampg rao. Subue he coveo erm o equao (5). & u ξ u & u & (8)
Equao (8) oe o have a cloe-form oluo for he geeral cae whch & & a arbrar fuco. A covoluo egral approach mu be ue o olve he equao. Noe ha he mpule repoe fuco embee he covoluo egral. Dplaceme he ampe aural frequec - ξ (9) he placeme equao va a covoluo egral u() ( τ) 0 [ { ξ - τ) }][ ( - τ) ] τ ( (0) he correpog mpule repoe fuco for he placeme ĥ () [ ( ξ )][ ] () Furher eal regarg h ervao are gve Referece. he correpog Laplace raform H () () ξ he Z-raform for he ramp vara mulao Ĥ () ( ) Z L ( ) ξ where he me ep ()
Evaluae he vere Laplace raform. ( ) [ ] ( ) [ ] ( ) ξ ξ ξ ξ co L () he Z-raform become ( ) ( ) [ ] ( ) [ ] ( ) ξ ξ ξ ξ co Z () Ĥ (5) Le [ ] ξ α (6) β ξ (7)
Evaluae he Z-raform. Ĥ () - β ( ) ( ) ( ) ( ) β{ [ ξ ]} { [ ]} co { [ ξ ] }{ co[ ] } { [ ξ ] } ( ) α{ [ ξ ]} { [ ]} { [ ξ ] }{ co[ ] } [ ξ ] { } (8) Ĥ () ( β) ( ) β{ [ ξ ]} { [ ]} co { [ ξ ] }{ co[ ] } { [ ξ ] } ( ) α{ [ ξ ]} { [ ]} { [ ξ ] }{ co[ ] } [ ξ ] { } (9) 5
Ĥ () ( β) ( ) β{ [ ξ ]} { [ ]} α{ [ ξ ]} { [ ]} { [ ]} { [ ]} { [ ]} co ξ co ξ (0) Ĥ () β β β ( ξ ) { α( ) βco( ) } [( ) ] { ( ξ ) }{ co( ) } ( ξ ) { } () Le ψ ( ξ ) { α( ) βco( ) } () { ( ξ ) }{ co( ) } ρ λ ( ξ ) () () η β (5) B ubuo, [ ] β ψ Ĥ () [ β η] ( ) ρ λ (6) ρ λ [ β η ] [ ] β ψ Ĥ () ρ λ ρ λ (7) 6
7 [ ] [ ] [ ] [ ] ρ λ ψ β ρ λ ψ β ρ λ ψ β ρ λ ρ λ η ρ λ ρ λ β () Ĥ (8) ρ λ ψ β ρ λ ψ β ρ λ ψ β ρ λ ηλ ηρ η ρ λ βλ βρ β () Ĥ (9) ( ) ρ λ ψ β ψ β ψ ηλ β ηρ η βλ βρ β () Ĥ (0)
( βρ ψ η β) ( βλ ηρ ψ β) ( ηλ ψ) ( ρ λ) Ĥ () () Solve for he fler coeffce ug he meho Referece. b 0 b b a a [( βρ ψ η β) ( βλ ηρ ψ β) ( ηλ ψ) ] ρ λ / m () Solve for a. ( ξ ) co( ) a ρ () Solve for a. ( ) a λ ξ () Noe ha he a a a coeffce are commo for placeme, veloc a accelerao. Solve for b 0. [ βρ ψ η β ] b / 0 (5) [ ψ η β ( ρ )] b / 0 (6) b0 ( ξ ) [ α ( ) βco( ) ] β β[ ( ξ ) co( ) ] (7) 8
b0 ( ξ ) [ α ( ) β co( ) ] β β[ ( ξ ) co( ) ] (8) b0 ( ξ ) [ ξ ] ( ) ξ co( ) ξ ξ [ ( ξ ) co( ) ] (9) b0 [ ] ( ξ ) [ ξ ] ( ) ξ ( ξ ) co( ) (0) b0 [ ] ( ξ ) [ ξ ] ( ) ξ ( ξ ) co( ) () 9
b0 ξ [ ( ξ ) co( ) ] ( ξ ) [ ξ ] ( ) () Solve for b. b b βλ ηρ ψ β ηρ ψ β ( λ) () () b β ( ξ ) co( ) ( ξ ) [ α( ) βco( ) ] β [ ( ξ ) ] (5) b ( ξ ) co( ) ( ξ ) α ( ) β[ ( ξ ) ] (6) b ( ξ ) co( ) ( ξ ) α ( ) β[ ( ξ ) ] (7) 0
b ( ξ ) co( ) [ ξ ] ( ξ ) ( ) ξ [ ( ξ ) ] (8) b ( ξ ) co( ) ξ ( ξ ) Solve for b. [ ] [ ξ ] ( ξ ) ( ) (9) b ηλ ψ (50) b β ( ξ ) ( ξ ) { α( ) βco( ) } (5) b ξ ( ξ ) ( ξ ) [ ξ ] ( ) ξ co( )
(5) ( ) ( ) [ ] ( ) ( ) co b ξ ξ ξ ξ ξ (5) he gal recurve flerg relaohp for he relave placeme 0 b b b u a u a u && && && (5)
he gal recurve relaohp for he relave placeme hu [ ] [ ] [ ] ( ) ( ) [ ] ( ) [ ] ( ) ( ) ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( ) [ ] ( ) ( ) co co co u u co u ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ && && && (55)
Relave Veloc he mpule repoe fuco for he relave veloc repoe ξ ĥ ξ v () (- ) co (56) he correpog Laplace raform Hv () ξ (57) he Z-raform for he ramp vara mulao ( ) ( ) Ĥ v () Z L (58) ξ ( ) ( ) Ĥ v () Z L (59) ξ Evaluae he vere Laplace raform per Referece a. L ( ξ ) ( ξ ) co( ) ( ) ξ (60) he Z-raform for he ramp vara mulao ( ) ξ Ĥ () Z ( ξ ) co( ) v ( ) (6)
( ) ξ Ĥ v () Z ( ξ ) co( ) ( ) (6) he Z-raform evaluae ug he meho Referece 5. Ĥv () ( ) ξ ( ξ) co( ) [ ( ξ ) co( ) ] [ ( ξ ) ( ) ] ( ξ ) (6) Ĥv () ( ) ξ ( ξ) co( ) [ ( ξ ) co( ) ] [ ( ξ ) ( ) ] ( ξ ) (6) Ĥv () ( ) ( ) ξ ( ξ) co( ) [ ( ξ ) co( ) ] [ ( ξ ) ( ) ] ( ξ ) (65) 5
Ĥv () ( ) ( ) ξ ( ξ ) co( ) ( ξ) co( ) ( ) ( ξ ) (66) Le ξ ψ ( ξ ) co( ) ( ) (67) { ( ξ ) }{ co( ) } ρ (68) λ ( ξ ) (69) ψ Ĥ v () ( ) ( ) ρ λ (70) ψ Ĥ v () ρ λ (7) ( ) ( ) ( ) ( ) ψ( ) Ĥ v () ρ λ (7) ( ) ( ψ ψ ψ) Ĥ v () ρ λ (7) ψ ψ ψ Ĥ v () ( ) ρ λ (7) 6
( ψ) ( ψ) ψ Ĥ v () ( ) ρ λ (75) ( ψ) ( ψ) ψ Ĥ v () ( ) ρ λ (76) ( ψ) ( ψ) ρ λ ψ Ĥ v () ( ) ρ λ ρ λ (77) ( ψ) ( ψ) ρ λ ρ λ ψ Ĥ v () ρ λ ρ λ (78) ( ρ ) ( λ ρ) λ ( ψ) ( ψ) ψ Ĥ v () ρ λ ρ λ (79) ( ψ ρ ) ( λ ρ ψ) ψ λ Ĥ v () ρ λ (80) ( ψ ρ ) ( λ ρ ψ ) ψ λ Ĥ v () ρ λ (8) 7
( ψ ρ ) ( λ ρ ψ ) ψ λ Ĥ v () ρ λ (8) Solve for he fler coeffce ug he meho Referece. c 0 c c a a ( ψ ρ ) ( λ ρ ψ ) ρ λ ψ λ (8) Solve for a. ( ξ ) co( ) a ρ (8) Solve for a. ( ) a λ ξ (85) Solve for c 0. c0 ψ ρ (86) c0 ξ ( ξ ) co( ) ( ) ( ξ ) co( ) (87) c0 ξ ( ξ ) co( ) ( ) (88) 8
Solve for c. λ ρ ψ c (89) ( ξ ) ( ξ ) co( ) ( ξ ) co( ) ( ) c ξ (90) ( ξ ) ( ξ ) ( ) c ξ (9) Solve for c. c ψ λ (9) c ξ ( ξ ) co( ) ( ) ( ξ ) (9) 9
0 he gal recurve flerg relaohp for he relave veloc 0 c c c u a u a u && && && & & & (9) [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ) co( ) ( ) ( ) co( ) ( u u co u ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ && && && & & & (95)
Abolue Accelerao he mpule repoe fuco for he abolue accelerao repoe ( ) ( ) ( ) ( ) ξ ξ ξ co () ĥ a (96) he correpog Laplace raform ξ ξ a () H (97) he Z-raform ( ) ξ ξ a L Z () Ĥ (98) ξ ξ ξ L L (99) ( ) ξ ξ ξ L L (00)
ξ L L ( ξ ) ξ ( ) (0) Evaluae he Z-raform. ( ) Ĥa () Z ( ξ )( ) (0) [ ] [ ] ( ) ξ Ĥ a () (0) ( ) { [ ξ] co[ ] } { [ ξ ] } Ĥa () ( ) [ ξ ] [ ] { [ ξ ] co[ ] } { [ ξ ] } (0) Ĥa () ( ) [ ξ ] [ ] { [ ξ ] co[ ] } { [ ξ ] } (05) Ĥa () [ ξ ] co[ ] [ ξ ] ( ) [ ξ ] [ ] [ ξ ] co[ ] [ ξ ] (06)
Ĥa () [ ξ ] co[ ] [ ξ ] ( ) [ ξ ] [ ] [ ξ ] co[ ] [ ξ ] (07) Ĥa () ( ξ ) ( ) [ ξ ] co( ) [ ξ ] ( ξ ) co( ) ( ) ( ξ ) co( ) ( ξ ) ( ξ ) ( ξ ) ( ) ( ξ ) co( ) ( ξ ) (08)
c 0 c c a a ( ξ ) ( ) [ ξ ] co( ) [ ξ ] ( ξ ) co( ) ( ) ( ξ ) co( ) ( ξ ) ( ξ ) ( ξ ) ( ) ( ξ ) co( ) ( ξ ) (09) Solve for a. Solve for a. ( ξ ) co( ) a ( ) a ξ (0) () Solve for c 0. c0 ( ξ) ( ) () Solve for c. ( ) c ξ ( ) ( ) co ()
Solve for c. c ( ξ) ( ξ) ( ) () he gal recurve flerg relaohp for abolue accelerao && x a&& x a && x c0&& c && c&& (5) && x [ ξ ] co[ ] && x [ ξ ] && x ( ξ ) ( ) && ( ξ ) co( ) ( ) && ( ξ) ( ξ) ( ) && (6) Relave Accelerao he mpule repoe fuco for he relave accelerao repoe 5
ĥ () δ() ( ξ ) ξ co( ) ( ξ ra ) ( ) (7) he correpog Laplace raform Hra ( ) (8) ξ he Z-raform ( ) Ĥ a () Z L (9) ξ ( ) Ĥ a () Z L (0) ξ Evaluae he vere Laplace raform. ( ) L L () ξ ξ L ξ ( ξ ) ( ) () he Z-raform for he ramp vara mulao ( ) Ĥ ( ξ ) ra () Z () 6
( ) Ĥ ( ξ ) ra () Z () he Z-raform evaluae a. ( ) [ ( ξ ) ( ) ] ( ) Ĥ () ra (5) ( ξ) co( ) ξ ( ) [ ( ξ ) ( ) ] ( ) Ĥ () ra (6) ( ξ) co( ) ξ ( ξ ) ( ) ( ) ( ) Ĥ () ra (7) ( ξ) co( ) ξ ( ξ ) ( ) ( ) Ĥ () ra (8) ( ξ) co( ) ξ Solve for he fler coeffce. 0 a a ( ξ ) ( ) ( ) ( ξ) co( ) ξ (9) 7
Solve for a. ( ξ ) co( ) a ρ (0) Solve for a. ( ) a λ ξ () Solve for 0. 0 ( ξ ) ( ) () Solve for. 0 () Solve for. 0 () he gal recurve flerg relaohp for he relave accelerao && u a&& u a && u 0&& && & (5) && u [ ξ ] co[ ] && u [ ξ ] && u ( ξ) ( ) { && && && } (6) 8
Referece. Dav O. Smallwoo, A Improve Recurve Formula for Calculag Shock Repoe Specra, Shock a Vbrao Bulle, No. 5, Ma 98... Irve, he Impule Repoe Fuco for Bae Excao, Vbraoaa, 0... Irve, able of Laplace raform, Revo J, Vbraoaa, 0... Irve, Paral Fraco Shock a Vbrao Aal, Revo I, Vbraoaa, 0. 5. R. Dorf, Moer Corol Sem, Ao-Wele, Reag, Maachue, 980. 9