MEEN Cheat Sheet for SDOF motion

Size: px

Start display at page:

Download "MEEN Cheat Sheet for SDOF motion"

Della Leonard
5 years ago
Views:

1 MEEN 67 - Cheat Sheet for SDOF motion EOM: Me yke yce yf() t Luis San Anres - January 03 Pages an (only) allowe in exams Given a mechanical system with equivalent parameters (Me) mass, (Ke) stiffness, (Ce) viscous amping coefficient. efine natural frequency an amping ratio as: n M e C e M e n C e M e C e n K e TRANSIENT RESPONSE system to STEP Force F(t)Fo for Unerampe system only, < system response is: Y ss for : see page ) Y a bt e nt C cos t Yt () e nt C cos t C Y 0 Y ss C sin t C sin t + initial conitions B a A C e Y ss B ; b ; M e Y C e t t Y Y 0 Y0 ( ) C Y 0 a Y t Y M e Y C e t t Y Y A Bt C b n Y 0 a at t0 Ao ln n An Ao an An are peak motion amplitues separate by n perios 4 LOG DEC (): formula to estimate amping ratio () from a free response TRANSIENT RESPONSE system to Force F(t)A+Bt n Y 0 Y ss C an n ampe natural frequency Transient response of overampe system, step force Foconstant M e Y C e t t Y Y Yt () A e s t A e s t + initial conitions Y ss Y 0 Y0 ( ) t Y at t0 s n s n A Y ss A Y 0 Y ss K s, s < 0 e A s A s Solve for A an A Transient response of critically ampe system, step force Foconstant M e Y C e t t Y Y Yt () e st A ta Y ss + initial conitions Y ss A Y 0 Y ss Y 0 A s A s Y0 ( ) n t Y at t0

2 STEADY RESPONSE of system to PERIODIC LOADS with frequency Case: perioic force of constant magnitue Ft () sint Define operating frequency ratio: System perioic response: Yt () shr () sin t r F o Hr () r care with angle, range: 0 to -80eg s tan r r r n Case: base motion of constant amplitue Me Y C e t t Y Y KY B C e t Y B Y B ( t) Acos t System perioic response: Yt () AGr () sin t Gr () r tan r r r 3 4 r Case: response to mass imbalance System perioic response: Yt () e m Jr ( ) sin t M e Ft () me sin t uimbalance (offset center of mass) isplacement me M e u M e M m Jr () r r r tan r r care with angle, range: 0 to -80eg OTHER USEFUL formulas: (program them in a calculator) Unerampe system <: step force response Given Yt () e nt C cos t C sin t Y ss constants in formulas fin velocity Vt () t Y Vt () e nt D cos t D sin t C C D n C D n C fin acceleration at () t V at () e nt E cos t E sin t D D E n D E n D

3 A case your memory shoul retain forever TRANSIENT RESPONSE of M-K system to STEP Force F(t)Fo NO DAMPING, C0 Ns/m Unampe system, response is: M e Y t Y an velociy an acceleration: Yt () C cos n t + initial conitions Y 0 Y0 ( ) t Y at t0 C sin n t Y ss Y ss C Y 0 Y ss C n MOTION never ies since there is no issipation action (no amping) Vt () t Y Vt () D cos n t D sin n t D C n D C n at () t V at () E cos n t E sin n t E C n E C n Note that the velocity an acceleration superimpose a cos & a sin functions. Thus, the maximum values of velocity an acceleration equal 0 V max D D V max n C C 0 a max D D a max n C C n V max since Y ss C Y 0 Y ss C n a max n Y 0 Y ss n Note: the function xt ( ) acos t bsin t can be written as xt ( ) ccos t c a b tan b a

4 OTHER important information given a function f() t fin its maximum value The maximum or mimimum values are obtaine from For example, for the unerampe response, <, the system response for a step loa is t f 0 Yt () e nt C cos t C sin t Y ss n Y 0 Y ss an Y ss C Y 0 Y ss C n when oes Y(t) peak (max or min)? ampe natural frequency from the formulas sheet Vt () e nt D cos t D sin t A peak value occurs at time t when Y/tV0, i.e. C C D n C D n C e n 0 for most times; hence 0 D cos 0 e n D cos D sin D sin tan D D solve this equation to fin there are an infinite # of time values () satisfying the equation above. Select the lowest as this will probaly will give you the largest peak. M e Y C e t t Y Y A Bt Example: Obtain constants C an C for case of force F(t)A+Bt - unerampe response Given Yt () e nt C cos t C sin t a bt Vt () t Y Vt () e nt D cos t C D sin t D n C D n C at t0, Y 0 Y0 ( ) t Y at t0 satisfy initial conitions: Y 0 Y0 ( ) Y 0 C a b C C Y 0 a t Y C D b n C b n Y 0 a C b b n Y 0 a C

5 FREQUENCY RESPONSE FUNCTIONS for PERIODIC LOAD with frequency Case: perioic force of constant magnitue M e Y C e t t Y Y sin t Define operating frequency ratio: Displacement: System perioic responses: velocity: r n Yt () shr () sin t Vt () t Y shr () sin t acceleration: at () Y s Hr () sint Y t harmonic response s Hr () r r tan r r care with angle, range: 0 to -80eg Define imensionless amplitues of frequency response function for isplacement Y s Hr () velocity acceleration V n s a n s rhr () r Hr () Jr () Disclaimer: The list of formulas liste above (solutions to ODE) WILL AID you in the analysis of system response The list is NOT exhaustive. The formulas o NOT answer the question: Which formula o I use? One must use jugement (knowlege of the problem - how it works) to select the appropriate formula (response) Dr. San Anres oes NOT guarantee that all formulae given above are type CORRECTLY. It is the stuent's responsibility to verify the formulas above. ONLY PAGES ONE & TWO of this DOCUMENT ALLOWED in ME67 exams

6 GRAPHS for AMPLITUDE OF TRANSFER functions Constant amplitue force Hr r r formula for amping ratio r a H a r a Given esire H an freq r H a r a. H Amplitue, H 6 4 Hr a frequency ratio, r H Acceleration or Imbalance r Jr J r r 6 formula for amping ratio r a r a 4 J a r a Given esire J an freq r J a r a 0.9 Amplitue, J 4 Jr a frequency ratio, r J 4.69

7 Given esire T an freq r Transmitte force to base or founation r Tr T * 3 0 r r r T T r T r r r r a r a 0.75 T a.0 Tr a 0.86 r a T a T a 0.86 Amplitue Tr a T frequency ratio, r.864

a) Identify the kinematical constraint relating motions Y and X. The cable does NOT slip on the pulley. For items (c) & (e-f-g) use

a) Identify the kinematical constraint relating motions Y and X. The cable does NOT slip on the pulley. For items (c) & (e-f-g) use EAMPLE PROBLEM for MEEN 363 SPRING 6 Objectives: a) To erive EOMS of a DOF system b) To unerstan concept of static equilibrium c) To learn the correct usage of physical units (US system) ) To calculate