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Lecture Notes copyright Brian Feeny 2023
Viscously Damped^ Free (^) Vibration (^) (2.6) IF = mj. m FBD lix
mi (^) + kx +F= 0 damping
x
T I element linear^ mass- (^) damping (N) (^) (i)(m/s) spring^ fore Linear viscous (^) damping I =cX^ = - mi +(x^ +kx^ = 0 x e (1) where (^) wasm 25w=E Ur^3 = zwn=s Lin: Use (^) me, c, ke as (^) effective mass, (^) damping, stiffness coefts in (^) more complicated (^) systems line, (^) use coeftsof, x, x when defining wa,^33 Units ofC^ are^ N/mss =^ Ns/m
kg/s 3 = .rs is msionless Crad (^) is a math unit) I called the^ gratio, afactor (^2) is called the^ damping coefficent Find soln^ to^ (1). Seek^ x= Aest^ plug into (1) x = Asest = Asest
Ast +^ 2wnAset+ wiket
- o (^3) + 23WnS + W, = 0 characteristic equ Roots:11,2 =^ - 3w = Wn General (^) soln (S, FS2): x = A,esit + (^) Acest = A,el- 3w^
- wri)t +Anetzwerwilt Note:Three cases 331, S,^ Se^ are^ real 31, S., Se are^ complex (^) conjugates 3 = 1, S,^ =^ S2^ repeated^ real (^) roots 331, overdumped x = A,emi)t +^ Anist 5, O (^) 5249, 40 #let
m
#est (20 here) t Note: est decays more^ quickly than^ est est (^) defines the speed ofthe^ system as^ it returns to^ zero (^) (equilib).
Wd=frequencyofdamped^ oscillation.^ Wa=WVSI Footnotes &can solve for (^) A., Az in (^) terms ofC, (^) (real)
I(, +i),Az^ =^ t(c,-i() C, (^) real IX, preal) **Initial conditions x(0) = xo, XCU) = vo. Apply to x(t) =^2
swnt[C,coswat t? (^) sinwat] x(0) =^ eo(C,(r^
xo x(0) =^ - 5w,4 (^) +WaCz =Vo => = No c = WnXo real
(Formula (^) good for^ free^ vibration^ only!)
Free (^) Vibration, linear viscous damping Had Y^ +23WnX + (^) WX = 0 X = Aest => S,2 =^ -^ fun!Wn-
- (^) overdamped, exponential response (^37) underdumped, decaying oscillatoryresponse.^ We^ Wars" S1,2 = -3WntiWgt, x^ = Ne-Swntcos(wattY) 3 = 1 critically damped^ S,^ = S2 =^ -^ TWn^ = - Wn XI) = A,c- Wnt +Azte
- Wat (^) X I Response "looks"exponential en Root (^) Locus:sketch roots, S,,2^ as^3 varies
t Im 0x3; (^) · iWn S
Re 3 - 1 3 = 1 C I 9 =^0 S =^ - 5wnE wa- · (^) - iwn 3 = (^0) Sz = liwn x(t) = Asin(Wnt+) undamped,^ studied earlier (^321) S,2 =^ - zwn1iwngr on circle 3 =1^ S,,2 =^ - We (^331) Se =^ - 3w =w-^ real est (^) real (^) partof S (^) determines decay rate im (^) partof^ s (^) determines oscillation (^) freg. (Ifunderdamped "optimal"decay rate^ ("fastest")^ when^ 3=
From (^) data (bar (^) only), solve (^) for a (^) (and (^) xi) Need (^) Io = iumb"
m(z
5)=... = km y (a) Solve^ for^ C=>
c =
ato: N2) (sm2) (5m) = 1257kg/s Need (^) Ki = We lo (only) = (2π- 10 ris)"(I kgm2)^
4386(Nm) I ↑ withoutM (b) find 5 with^ added^ M = Fobar+Frm
kgm
(=in)
5 kym New (^) in = I-swY = (^20) s 12 = 2 =4.47 (^) H() ca
- -^ (()(5m) 2 Now Y^ =^ 0.
(^2) Is Wa 2(=kym2) (28.145) Note:C doesn't (^) change when M is added. Moral:damping factor (ration (^) depends on a (^) and all (^) other (^) parameters (m, (^) M, a,k)
3 = let Effective coeffs^ from^ ODE -- I.+ca^
- k, 0 =^0 w left me +Cefft^
kerfO = 0 I (^) - melt.
