Viscous Damping Summary Sheet No Damping Case: Damped behaviour depends on the relative size of ω o and b/2m 3 Cases: 1.

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1 Viscous Daping: && + & + ω Viscous Daping Suary Shee No Daping Case: & + ω solve A ( ω + α ) Daped ehaviour depends on he relaive size of ω o and / 3 Cases:. Criical Daping Wee 5 Lecure solve sae BC s se BC s ecoes ω A e c ω cos ω ecoes syse doesn oscillae and decays o in he shores possile ie + α. Overdaping > ω again no oscillaions u i aes longer o decay o 3. Ligh Daping T (higher han for undaped) A e < ω A e soluion is eac for his apliude decreases eponenially wih ie cos ω angular frequency is consan wr ie u saller han ω + α

2 Solving viscous daping proles Noe ha alhough he sandard equaion is: os of he ie for proles we jus use he: Apliude: or Daped Frequency: + α ω A e cos e A A ' ω ω c ω fro his ge

3 A furher noe on he previous prole How uch would e ω change if daping was sall?, say c ~.? Again for IP5 condiions, his eans This is a ore noral case..6.6 Loo a our daped frequency again ( ω ') ω (.5) ω ο 3.3 rad/s Noe ha here ω is very close o ω 3.5 rad/s. This is acually he case for los of proles you will encouner. You igh e eped o jus use ω o u if you do you will have ars reoved I wan you o calculae he value for ω o o prove ha i is sall. 3

4 Wee 5 Lecure Proles 3,3,33 Energy Consideraions During Viscous Daping Recall ha oal energy in SHM: E A Apliude for viscous daping: Coining: E () A e A () A() A e or γ Ae γ E E e E e The Qualiy Facor Daping is coon in any differen syses echanical, elecrical, aoic and nuclear. In general we use he qualiy facor o characerize how heavily daped a syse is. The higher Q he lower he daping he higher he qualiy of he syse! Q ω γ γ e γ E he iniial energy efore oving ω ο / diensionless! Unis [rad/s][s/g][g] (undaped) (γ /) ω angular frequency (undaped) γ / also he reciprocal of he ie for oscillaions o decrease o /e of he original energy 4

5 Dry Fricion Daping and SHM (Courseware pg 8 and Ineracive Physics Deo IP6) + equiliriu The dry fricion daping force is independen of speed and opposie o velociy (see W4L3 lecure) Therefore, i always has he sae agniude u reverses sign when he direcion changes. We need o sar wih an equaion of oion, hen deerine he soluion (), lie we did when we had viscous daping. However ecause of he sign change we need o develop differen equaions. v lef v righ Noe ha we didn need o do his righ and lef hing for he viscous daping case && v since he daping force (v) changes direcion when v changes direcion For he dry fricion daping case he daping er doesn conain a v er so i doesn auoaically change direcion when he loc does. 5

6 For loc going lef SHM eq n is: && + µ g v f + or && + µ g To solve wan in for ha loos lie our sandard SHM eq n (no consan) co-ordinae ransforaion ehod for solving diff eqns le so This has our usual SHM soluion! u µ g q + + µ g & q& so q q or Bac ino Acos + B sin Acos + Bsin µ g q + q&& q µ g + Now we need soe Boundary Condiions o find A and B & a a gives B and A released fro res a eension µ g soluion for v lef: µ g g µ cos + 6

7 Les see how his is going o loo for a few poins y ploing he (see plo on ne page) Saring a + and leing go ie ino cycle (ω ο ) dry fricion daped posiion (fricionless) undaped posiion, ω π ω (/4 cycle afer sar) µ g Sop! ω π (/ cycle afer sar) ω π (full cycle afer sar) + µ g 4µ g - + Before doing his one we need o consider differen equaions ecause fricion direcion has changed! 4 We need a differen SHM equaion for loc going righ Now we have && µ g f v + Solve his as efore u wih new BC s gives he resuls shown in he final row of he ale aove 7

8 If No Daping: O Wih Dry Fricion Daping: µ g µ g O wih each ¼ cycle we lose µ g 4µ g!!! 8

9 Poins o Noe: 4µ g. For each full cycle we lose in apliude. Therefore, a plo of vs shows a linear (no eponenial) decrease in apliude (see plo elow).. Noe ha (unlie in viscous daping) he frequency is unalered y he daping (since daping force does no depend on velociy). 3. Bloc coes o res afer each half cycle (a a ) and will say a res if µ g > fricion force spring force a or < µ g (see plo elow) Noe here we have o assue ha µ µ µ s 4µg slope µ g π µg (linear no log decrease in apliude) π π 3π 4π 5π 6π 7π 8π ω 9

10 Saced LEAF SPRINGs: A good eaple of dryfricion daped oscillaions