SHM stuff the story continues

Transcription

1 SHM stuff the story continues Siple haronic Motion && + ω solution A cos t ( ω + α ) Siple haronic Motion + viscous daping b & + ω & + Viscous daping force A e b t Viscous daped aplitude Viscous daped freq ω o cos ω Siple haronic Motion + dry friction daping && + ω g µ frictional daping force Aplitude relationship not as clear fro this but it wors out that it loses 4µ g/ in aplitued each cycle b µ g cosωot + 1 µ t + α (Note this will only hold for first half-cycle) 1 req sae as undaped g

2 Wee 5 Lecture 3: Probles 34,35,36 orced Oscillations (rench chapter 4, Courseware pg 5 and IP7) Previously we considered free or natural oscillations where the syste oscillates at ω deterined by and. ω This is a specific value set by and. However we can force a syste to oscillate at a different frequency by using an eternal driving force that is also sinusoidal. The equation describing otion becoes (for no daping) (add the forcing ter to the SHM eq) driving force && + ω driving force aplitude - in Newtons cosωt cosωt ω is the forcing frequency (note both ω and ω in here) Coplicating actor: Initially, in real systes the otion is the superposition of ω and ω but the free oscillations will die out due to daping forces leaving only forced ones (see net page). Transient Period: both ω and ω eist (See IP7 also) Steady State: ω dies away due to daping and we are only left with ω.

3 Two Different Eaples of Transient Behaviour During orced Oscillations ree oscillation This is what you see in IP7- the third bloc orced oscillation ~Steady state ree+forced with no daping ree + forced with daping Steady state-forcing frequency only Because it is a lot easier to treat atheatically, we will consider first the case where we have a forced, oscillating syste that is at steady state. Later we will see what the equations loo lie for the transient 3 case.

4 Steady State orced Oscillations (no daping) need the solution to: since we are at steady state, the frequency will be that of the driving force (this hugely siplifies things) Possible solution Evaluating C: differentiate ties sub bac into top equation gives Note this is a bit unrealistic because you can t get to steady state without daping but we will ignore that for now. so now && + C cosωt && ω C cosωt cosωt (steady state because no ω only ω) ( ω C ωt) C cosωt + cos t C cos or ω ω ω C cosωt So we now what ω is, all we have to do is find C ω (since ω ) ω ω cosωt Solution for orced, Undaped, SHM at steady state Aplitude C note it is a function of ω! Another Note: even though ω is not in the frequency ter, it does affect the aplitude! 4

5 C(ω) Plotting the Aplitude C as a function of ω / ω (forcing frequency) ω When ω approaches ω the oscillation aplitude increases draatically. This is RESONANCE ω resonance frequency! The diagra above has a negative value of aplitude for ω>ω. What does this ean? That the driving frequency ω is 18 (or π) out of phase with the natural one ω (consider a swing push fro other side!). e.g. hold a eter stic and gently ove it side to side driving direction sae as at botto A B Move end of stic fro A to B. A B driving direction opposite to that at botto Go slow Go fast ω < ω ω > ω 5

6 get Now for the previous diagra we really want both sides of ω to give a positive C value do this by introducing a phase constant (α). C(ω) ( ω + α ) C cos t for ω α < ω α π for ω < ω / ω ω α π phase lag of displaceent wrt driving force ω ω 6

7 Other Resonance Eaples: Objects ade of bits and pieces (cars, boats, etc.) every bit has a different natural resonance frequency. That s why the ashtray rattles (vibrates) annoyingly when you hit a certain speed the window rattles (vibrates) when your radio plays a certain note Self Ecited Vibrations: Can initiate and sustain vibration of certain systes even though energy source itself is not oscillating. e. violin string after a bow has been pulled across it Self ecited vibrations can also occur due to streas of fluid: e. flute: blowing across the outh hole causes fluid to shed vortices (turbulent flow) pressure fluctuations cause the air colun to vibrate 7

8 The ost faous eaple of this is the Tacoa Narrows Bridge (video). Across Puget Sound near Seattle, Washington. 8 ft long, 39 ft wide, with 8 ft tall stiffening girders on either side. Resonance frequency in torsional oscillation:.5hz, T 4s. Opened July 1, 194 Becae faous for oscillating whenever the wind blew. This was due to vortices on the top and botto of the bridge. Started to oscillate, then the alternating angle of the bridge enhanced the pressure difference between top and botto. These pressure fluctuations occurred as the dec rose and fell, thus atching the natural vibrational frequency. Noveber 7, 194 Wind 4-45 ph, span rippling at 36 cycles/inute. 1: North cable broe loose, allowing the bridge dec to begin oscillating in a twisting resonant ode (.Hz) around centre line. 11: Bridge torn into shreds. 8

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10 Specific Eaple of orced Oscillation (Vibration): orced Vibration Caused by Unbalanced orces e. odel the vibration in a front loading washing achine Assuptions: ignore lateral otion, consider only vertical oveent assue unbalanced ass can be described by ass (which is included in ass ) e ω orcing function is a result of the radial force due to the unbalance ass, and is eω radial force v vertical coponent of eω eω cos ωt ro before we had the forcing function (t) cos ωt so e ω actually a lot of springs, but can odel as a single spring with an overall +ve v radial force eω ωt e g radial accel n g at equilibriu (no rotation) Now since C ( ) ω ω ω e ( ω ω ) cos ( ωt) Note: if the rotating ass is a sall fraction of the total ass then the vibration aplitude is sall this is why any rotating achines (washing achines, dryers) are attached 1 to large blocs.