Modeling of vibration systems

Transcription

1 Modeling of vibration systes Atual syste Mae design deision Choose physial paraeters, hange or augent syste if neessary Physial odeling Mae siple approiations based on engineering judgeent Physial odel Matheatial odeling Apply physial laws to obtain equation of otion Math odel Tests Atual response Analysis Solve EOM to predit dynai harateristis and tie response Predited response

2 Modeling eaple () rider rider vehile strut wheel tire

3 Modeling eaple () Helial gear pair Equation of otion M C K( t)( e( t)) = W θ W θ J M r b J r b K T T e C C K(t) Physial odel Physial odel

4 Modeling eaple (3) There is no unique physial odels for one partiular hardware

5 Eaple: An autoobile An autoobile oving over a rough road an be odeled onsidering (a) weight of the ar body, passengers, seats, front wheels, and rear wheels; (b) elastiity of tires, suspension; () daping of seats, front and rare suspensions.

6 Eaple: Reiproating engine A reiproating engine is ounted on a foundation as shown. The unbalaned fores and oents developed in the engine are transitted to the frae and the foundation. An elasti pad is plaed between the engine and the foundation blo to redue the transission of vibration. Develop the vibration odel.

7 Engineering judgeent Modeling requires good engineering judgeent and eperienes with hardware. Purposes of odel Assuptions Modeling Model opleity liitations Analysis tehniques

8 Degree of freedo () Degree of freedo (DOF): The iniu nuber of independent oordinates required to deterine opletely the positions of all parts of a syste at any instant of tie. Single degree of freedo systes

9 Degree of freedo () Single degree of freedo systes Two degree of freedo systes

10 Degree of freedo (3) Three-degree of freedo systes

11 Degree of freedo (4) Infinite-nuber-of-degrees-of-freedo systes (ontinuous or distributed systes) Inreasing nuber of degrees of freedo More aurate result More opleity

12 Equations of otion Proedures () Geoetry Define oordinates and their positive diretions Note degrees of freedo (DOF) Write geoetri onstraints and opatibility () Kineatis Write neessary ineati relations (3) Fore equations Draw free-body diagra Apply Newton s nd law on the free body (4) Cobine all relations

13 Eaple : A spring-ass syste () Unstrethed Position Δ Δ (Δ) Stati equilibriu Position g g () Geoetry = ass position easured fro equilibriu position DOF, only EOM required () Kineatis position, veloity, and aeleration are,,

14 Eaple : A spring-ass syste () Unstrethed Position Δ Δ g (Δ) g Stati equilibriu Position (3) Fore equations [ = 0] F At equilibriu g Δ = 0 ; During vibration[ F = a] g ( Δ ) g = = Δ (3) Cobine all relations EOM: = 0

15 Eaple : A spring-ass syste (3) Unstrethed Position Δ Δ g (Δ) g Stati equilibriu Position What if is easured fro the other positions? What if there are the other fores applied to the syste? What if a daper is added to the syste?

16 Eaple : -- systes (DOF) () f(t) l l () Geoetry l, l = positions of and easured when both springs are unstrethed, = positions of and easured fro their unstrethed positions DOFs, EOMs required () Kineatis,,,, and for ass and

17 Eaple : -- systes (DOF) () l l f(t) ( ) ( ) f(t) ( ) ( ) (3) Fore equations [ ] a F = ) ( ) ( = ) ( ) ( ) ( t f = In atri for, EOM is = ) ( t f

18 Eaple : -- systes (DOF) (3) In atri for, EOM is = ) ( t f or siply )(tfkcm= where M is ass or inertia atri C is daping atri K is stiffness atri is position vetor F is input vetor

19 Eaple 3 Draw the free-body diagra and derive the EOM using Newton s seond law of otion