EN40: Dynamics and Vibrations. Midterm Examination Tuesday March

Transcription

1 EN4: Dynaics and ibrations Midter Exaination Tuesday Marc 4 14 Scool of Engineering Brown University NAME: General Instructions No collaboration of any kind is peritted on tis exaination. You ay bring double sided pages of reference notes. No oter aterial ay be consulted Write all your solutions in te space provided. No seets sould be added to te exa. Make diagras and sketces as clear as possible, and sow all your derivations clearly. Incoplete solutions will receive only partial credit, even if te answer is correct. If you find you are unable to coplete part of a question, proceed to te next part. Please initial te stateent below to sow tat you ave read it `By affixing y nae to tis paper, I affir tat I ave executed te exaination in accordance wit te Acadeic Honor Code of Brown University. PLEASE WRITE YOUR NAME ABOE ALSO! 1 (1 points). (13 points) 3. (7 points) 4. (1 points) TOTAL (4 points)

2 1. Te engines on an aircraft generate a trust tat decreases wit te aircraft speed according to te relation v FT F 1 v were F, v are constants. Te aircraft as ass. It starts at rest at tie t= and ust reac speed v TO in order to take off. F T 1.1 Use Newton s law to deterine te acceleration of te aircraft and ence deterine an expression for its speed as a function of tie. Air resistance and friction ay be neglected. dv F Newton gives v ax 1 dt v Separate variables and integrate v dv t F F F log(1 / ) 1 exp 1 / dt v v v t v v v v t v 1. Find a forula for te distance traveled by te aircraft as a function of tie. dx F v 1 exp t dt v x t F dx v 1 exp t v F v x vt exp t 1 F v 1.3 Hence find a forula for te iniu lengt of runway necessary for te aircraft to reac take-off speed, in ters of F, v,, and v TO Te aircraft ust reac take-off speed, so v log(1 vto / v ) t F vt F exp t 1 v v and ence v F v vv d vt exp t 1 log(1 v / ) TO To v F v F F [4 POINTS]

3 . Te figure sows a picture of a car inside te Dunlop Deat Loop. Te goal of tis proble is to calculate a forula for te iniu speed at wic te car can drive around te track. R n t.1 Assuing te car drives at constant speed and its center of ass is a eigt above te road, write down a forula for te acceleration of te veicle in ters of, te radius R of te track and. Express your answer as coponents on te noral-tangential basis sown in te figure. a n R [ POINTS]. Draw a free body diagra sowing te forces acting on te veicle on te figure provided below. Assue tat te car as rear weel drive, and tat te front weels roll freely. N B t n T A N A L d g [ POINTS].3 Write down Newton s law of otion and te equation for rotational otion for te veicle. Express your answer in noral-tangential coponents. F ( NA NB g sin ) n ( T g cos ) t n ( R ) T NB( L d) N Ad.4 Hence find forulas for te reaction forces acting at te two weels [ POINTS] Te vector equation sows tat T g cos NA NB g sin ( R ) Substituting for T in te oent balance equation 3

4 NAd NB ( L d ) g cos Multiplying te second equation by (L-d) and adding te tird sows tat NAL ( L d) g ( L d)sin cos R Multiplying te second equation by d and subtracting te tird gives LNB d g d sin cos ( R ) Hence d d NA g (1 ) (1 )sin cos g( R ) L L L N B d d g sin cos ( R )g L L L.4 Hence, calculate a forula for te iniu speed required to ensure tat bot front and rear weels reain in contact wit te track, in ters of, L, g, d and R. To reain in contact wit te road te reactions ust bot be positive for all values of. For te rear weel d g cos d sin ( R ) One can axiize te rigt and side differentiate wrt and set to zero tis sows tat 1 d d sin dcos tan cos sin d d cos d sin d ax,or (ore quickly) note tat we can write d g cos d sin g d cos sin g d cos cos sin sin d d 1 d g d cos( ) tan and te axiu value of cos( ) is 1. Terefore NB ( R )g 1 d Te sae approac for N A sows tat NA ( R )g 1 ( L d ) Te critical velocity is given by te larger of te two results. [4 POINTS] 4

5 3. Te figure sows a design for a icro-electro-ecanical sock sensor. It consists of a proof ass inside a casing. Te ass can ove inside te casing, and is eld in place by a spring wit stiffness k and un-stretced lengt L. Te casing oves orizontally wit a given displaceent y(t) relative to a fixed origin. Te otion of te ass xt () relative to te casing is easured electrically. L / y(t) x(t) k,l 3.1Draw a free body diagra sowing te forces acting on te ass. Gravity ay be neglected F S N [ POINTS] 3. Use Newton s laws of otion to sow tat x(t) satisfies te differential equation Te acceleration of te ass is Te spring force law gives Tus d x L d y k x dt L /4 x dt d x d y. Newton s law gives dt dt d x d y F x s dt dt L /4 x Fs k L /4 x L d x L d y k x dt L /4 x dt 3.3 Re-arrange te equation into a for tat MATLAB could solve We ust re-arrange te second order ODE into two first-order ODEs. Te standard approac gives dx v dt dv dt L /4 x k L d y x dt [ POINTS] 5

6 4. A uniforly carged sperical particle of laser-printer toner experiences a force of attraction to a conductive surface given by F were is a constant and is te distance of te center of te particle fro te surface. 4.1 Sow tat te potential energy of te particle can be expressed as C Were C is a constant. i F R r B 1 1 By definition F dr i ( dxi ) x A and we can coose / A ra c A [ POINTS] 4. A carged toner particle wit ass is launced wit speed v towards a conducting surface fro a point infinitely far fro te surface. Find a forula for te speed v 1 of te particle just before it ipacts te surface. Use energy conservation: 1 1 v C v C v v R R 1 1 [ POINTS] 4.3 Assue tat te ipact between te particle and te surface as restitution coefficient e. Calculate te speed v of te particle just after ipact. vi e( v1 i ) v ev1 [1 POINT] 4.4 Hence, find a forula for te axiu distance fro te surface tat te particle will reac after te rebound, in ters of, R, e, and v. At te point of axiu distance te velocity is zero. Energy is conserved after rebound, so 1 v C C R 1 1 e v (1 e ) e v R R R (1 e ) / R e v / [ POINTS] 4.5 Sow tat te particle will eventually attac to te surface if te initial speed v is less tan a critical value, and find a forula for te critical speed in ters of e,, and R. 6

7 Te particle will escape fro te surface if. For a finite positive it will be attracted back to te surface and lose energy at eac ipact eventually it will coe to rest attaced to te surface. Te critical velocity for satisfies (1 e ) / R e v / (1 e ) v Re 7