ME357 Problem Set The wheel is a thin homogeneous disk that rolls without slip. sin. The wall moves with a specified motion x t. sin..

Transcription

1 ME357 Proble Set 3 Derive the equation(s) of otion for the systes shown using Newton s Method. For ultiple degree of freedo systes put you answer in atri for. Unless otherwise speified the degrees of freedo are easured fro the syste s equilibriu position. Unless otherwise stated all otion is sall.. The wheel is a thin hoogeneous dis that rolls without slip. The wall oves with a speified otion t X t sin. w w w sin.. The base oves with a speified otion t X t b b b w t R b t 3 bx bsin bt wx w os wt X wsin wt

2 . a. The wheel is a thin unifor dis of ass 3 that rolls without slip. b. The pulley is a thin unifor dis of ass.. What effet does the inlined plane have? Where would it have an effet? Equilibriu 3 3 A R B Equilibriu R ( )

3 3. a. First assue large displaeents for. b. Linearize your result fro a. for sall displaeents. Assue the two rigid lins of length L are assless. The ass is onstrained to ove vertially. The syste is shown in its equilibriu position. a a L L h a. h h ( ) 4 a a a h a a a h b. 4 0 a 0

4 4. Bar OB is rigid and assless. a o A B l a l a l 0

5 5. a. Bar AD with ass 3 and EI with ass 4 are rigid, unifor and pinned at A and I. b. Bar CF is rigid and assless.. Deterine the equation of otion in ters of. A L B L C L D E L L G L L H I F

6 6. a. Bars AC and DG an be onsidered as thin rigid unifor bars with asses and 3 respetively. b. Bar BE is assless and rigid. Deterine the equilibriu position of the two bars as easured fro the horizontal. d. Deterine the equation of otion in ters of as easured fro the horizontal. What does this assue? A L B L C D L L F L G E a. b. eq 3 g 34 l

7 7. a. Bars AC and DF an be onsidered as thin rigid unifor bars with asses and respetively. b. Bar CE is assless and rigid.. Deterine the equation of otion in ters of. A L B C L D 3L E L L F 6 8 L

8 8. a. Bars AD and EG an be onsidered as thin rigid unifor bars with asses 3 and 4 respetively. b. Bar BE is assless and rigid.. Deterine the equation of otion in ters of. A L B L L C D E L F L G

9 9. The pendulu is a rigid thin unifor bar. a. Deterine the equation of otion if the base is fied yb t 0 b. Deterine the equation if y t Y t () sin. Where does the vertial otion enter the b b b equation? What does it ae gravity/stiffness ter behave lie? yb t X a. g 3 0 l b. g Y b 3 b sinbt 0 l l

10 0. The wheel is onstruted of unifor thin diss. The inner dis has a radius R and a ass. and is fied to the outer dis. The outer dis is a rigid unifor dis with radius R and ass. Assue the wheel rolls without slip. Use as the degree of freedo. R R

11 . Deterine the equation of otion in ters of. Assue the wheel rolls without slip. The two asses are onstrained to ove only vertially and the pulleys are assless. The wheel has a ass of and a ass oent of inertia about its enter of ass of 4R. Cord EDCA is elasti and is odeled as a spring/daper. Cords BI and FG are inetensible. D C B H I E R O F A R G

12 . The drive wheel is a thin unifor dis of radius R and ass The driven dis is radius R and ass. Both roll without slip. Bar AB is rigid and assless. Deterine the equation of otion in ters of. Ft R R R A B : F () t 8 4

13 3. Deterine the equation of otion in ters of. Assue sall otion. The pendulu is pinned to the dis whih rotates with a onstant angular veloity.. The rod is rigid and assless and the ass at its tip is a point ass. O R L g R R os L L L t sin t

14 4. The thin hoogeneous dis is pinned at O and is driven by a otor that produes a torque, Mt. A point ass of is pinned to it with assless bar of length L. Deterine the equations of otion in ters of and. O R Mt L

15 5. Deterine the equations of otion in ters of and. Assue sall angles and that the wheel rolls without slip. The ass of the thin hoogeneous large dis of radius R is. The ass of the thin hoogeneous inner dis of radius R is. The rod of length R is assless and rigid. The two pulleys are assless. R O R Mt P R Ft g M t R 0 3 4r F t R R R R r R gr RF t

16 6. Deterine the equations of otion for the syste shown in ters of and assuing sall otion. The blo the dis is sitting on oves with a presribed otion b t. The wall the botto spring is attahed to oves with a presribed otion w t. The load Pt is always direted along the pendulu rod. Both diss of radius R and R are unifor and hoogenous with asses and respetively. The wheel rolls without slip. R P R O b t R w t Pt 9 g P R R P R R R R 0 0 R R gr g P 4 b b w R b R Rb Rb R 8 4 w

17 7. Repeat Prob if the wheel slips. Use and as your degrees of freedo. Assue a oeffiient of frition between the wheel and base of. g sign ( b) 0 R ( ) g sign b b w

18 8. a. Deterine the equations of otion for Eaple 9 of the notes for large. b. Redue your results to the equations of otion for sall. a. losl sin P sin F ( t) l gsin0 l 0 P() t F() t b. l l gl 0

19 9. Coplete Eaple 8 of the notes. M(t ) R 4 M(t ) 3 4 R

20 0. Deterine the equations of otion in ters of and. The pulley is assless. The inner hoogeneous thin dis of radius R has a ass. The outer dis of radius R has ass. The wheel rolls without slip. The pendulu is oposed of a assless rod and a point ass. R R O P l R I R l 3 R R R R R 0 R l l R 0 0 R R R R g l R R R R 0 R l 0 R l gl

21 . Deterine the equations of otion in ters of,, 3. All s ' are easured fro the equilibriu position. Masses and slide on a fritionless surfae. The thin hoogeneous dis has a preload on it suh that there is no slip, between it and the blo 3. The dis is driven by a torque Tt. There is a visous interfae between 3 and and 3 and. Assue the blos do not ollide or and that 3 stays in ontat with and. R T T sin t 0 f fritionless T o sinrt R

22 . The syste below odels a tightly strethed spring of length L and ass. The string is strethed with a tension T. Assue the otion is sall suh that the tension reains onstant along the string. Deterine the equations of otion in ters of the oordinates,,. By observing the pattern write the equations of otion for 5 asses. 3 L 4 L 4 L 4 L T T L L 0 T T T L L L T T L L