PHYS1002 Physics 1 FUNDAMENTALSModule 3OSCILLATIONS & WAVESTextPhysics by HechtChapter 10OSCILLATIONSDamping and Forced OscillationsResonance ProblemsSections:10.8 Examples: 10.11CHECKLISTThere are usually external forces acting on an oscillator in addition to the restoring force. The motion can be damped so that the oscillations die away. Friction, damping, viscous damping, drag force Underdamped, critically damped, overdamping.An external force can drive the oscillations. An external source can supply energy to the vibrating system so that the system vibrates at the same frequency as the external source. The oscillations can grow in amplitude if the driving frequency approaches thenatural frequency of oscillation. This phenomena is known as resonance andthe system vibrates at its resonance frequency with large amplitude. Self-excited vibrations can occurthe vibrations are initiated and sustained by an energy source that is not oscillatory. a!"#p$#waves#waves1008.doc $$%&" A'$!T"S#ampin$(n real oscillating systems, mechanical energy is lost from the system due to frictionalor damping forces acting. The oscillation die away with time and the system comes to rest.)hen the amplitude of the oscillation decays away very slowly, the system is said to be underdamped. For example, when a tuning fork is set vibrating, the sound of vibrations persists for quite sometime.A car shock absorber uses viscous dampin$ *frictional force proportional to the speed+. )hen a car hits a pothole, the piston is ,erked away from its equilibrium position. -ecause of the large damping, the piston returns to its equilibrium position without sustained oscillations. This prevents the car from bobbing up.and.down for a long time after hitting a bump. A sports car has a rigid suspension and oscillations maybe damped out in less than a cycle. A luxury car often has a soft suspension and there maybe a few cycles of the oscillations before they die out. /ou can change the characteristics of the suspension system in some expensive cars.)hen non.vibratory motion occurs in the shortest time interval, the system is said to be critically damped. The spring system in a moving coil meter is critically damped and also the mechanism on electronic scales to measure mass.)hen non.vibratory motion occurs and it takes a long time for the system to come to rest at its equilibrium position, the system is said to be overdamped. 0eavy public doors on some building are overdamped to prevent them closing too quickly, giving time for people to enter and so that the doors are do not slam shut. The doors have some hydraulic dashpot *type of shock absorber+ to provide the damping.The figures below show the motion for increasing the damping *damping coefficient b where damping force F1 2 . b v +.a!"#p$#waves#waves1008.doc $$%&" A'3a!"#p$#waves#waves1008.doc $$%&" A'"0 5 10 15 20-0.1-0.0500.050.1b = 6 position x (m)timet(s)0 5 10 15 20-0.1-0.0500.050.1b = 0 position x (m)timet(s)0 5 10 15 20-0.1-0.0500.050.1b = 2 position x (m)timet(s)a!"#p$#waves#waves1008.doc $$%&" A'4a!"#p$#waves#waves1008.doc $$%&" A'&0 5 10 15 20-0.0200.020.040.060.080.1b = 16 position x (m)timet(s)0 5 10 15 20-0.0200.020.040.060.080.1b = 24 position x (m)timet(s)0 5 10 15 2000.020.040.060.080.1b = 100 position x (m)timet(s)a!"#p$#waves#waves1008.doc $$%&" A'5%nterest articlehttp%66physicsweb.org6article6news6&6$!6$&-uildings and bridges may be among the structures to benefit from a proposed shock absorber that could reduce the force of an impact by up to 789. :ura,it :en and colleagues at the :tate University of ;ew /ork at -uffalo demonstrated the effect with computer simulations, which also showed that it should be possible to turn the absorbed energy into heat. :imilar devices could even harness the energy from naturalimpacts such as ocean waves *: :en et al 3!!$ e, the shock waves are not always dispersed effectively. (nstead, :en?s team simulated a shock wave travelling along a chain of several hundred spherical elastic beads of ever.decreasing si>e. The beads at one end of the chain were around ten centimetres in diameter, and became progressively smaller. After the shock wave has passed through the large sphere at the beginning of the chain, it proceeds to the next . slightly smaller . sphere. -ut the wave cannot be transmitted symmetrically into this sphere. To ensure that its energy is conserved, the wave is forced to stretch out. (ts leading edge accelerates away from its trailing edge and this effect occurs every time the wave moves from one bead to the next. As the beads get smaller, the energy of the impulse is distributed and successive beads carry less and less kinetic energy. :en?s group found that the smallest bead at the other end of the chain feels the initial large impact as a long series of very small shocks. The amplitudes of these mini.shocks are less than $!9 of the original impulse. @This very simple system demonstrates that theoretically, any si>e shock can be absorbed with assemblies of appropriately tapered chains@, explains :en. &orced !scillations and resonanceForced oscillations occur through the application of an external force that adds energyto a system. For example% noises in the homeplumbing, refrigerators, air conditions.A system responds by oscillating at the same frequency as the drivin$ frequency. 'hen the drivin$ frequency approaches a natural frequency of vibration( the resultin$ oscillations dramatically increase in amplitude. )esonance occurs whenthe drivin$ frequency matches the natural frequency and the amplitude of the oscillation reaches a maximum value.At resonance, most of the energy is added to the mechanical energy of the vibrating system, very little energy is returned to the driving source. The smaller the damping, than the greater the amplitude of vibration. a!"#p$#waves#waves1008.doc $$%&" A'ABesonance phenomena occur widely in natural and in technological applications%Cmission D absorption of lightEasersTuning of radio and television sets'obile phones'icrowave communications'achine, building and bridge design'usical instruments'edicinenuclear magnetic resonance, F.rays0earinguclear ma$netic resonance scana!"#p$#waves#waves1008.doc $$%&" A'80 0.5 1 1.5 2 2.5 300.050.10.150.20.250.30.350.4amplitude A (m)d /ob = 2 b = 8 b = 10 A different resonance phenomena is when the driving energy source is not vibratory. The response of the system itself produces the alternations in the applied force to giveself-exited vibrations. There are many examples of self.excited vibrations%:inging-lowing across the mouth of a flutecauses vortices to peel off periodically, creating a fluctuating pressure.'usical glassesCarthquakesbuilding resonances-ridgessoldiers break step, Tacoma ;arrows *;ov A, $74!+.*athematical modellin$ for harmonic motion;ewtonGs :econd Eaw can be applied to the oscillating system33* + d x tF ma mdt = =rrr F 2 restoring force H damping force H driving force F*t+ 2 . k x*t+.b v*t+ HFd*t+ 3d3* + * + $* + * + !d x t b dx t kx t F tm dt m mdt+ + =For a harmonic driving force at a single frequency Fd*t+ 2 Fmaxcos*t H +. This differential equation can be solved to give x*t+, v*t+ and a*t+.a!"#p$#waves#waves1008.doc $$%&" A'7b + 0b + ,a!"#p$#waves#waves1008.doc $$%&" A'$!0 2 4 6 8 10 12 14 16 18 20-0.100.10 2 4 6 8 10 12 14 16 18 20-0.200.20 2 4 6 8 10 12 14 16 18 20-0.200.2-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1-0.200.20 2 4 6 8 10 12 14 16 18 20-0.100.10 2 4 6 8 10 12 14 16 18 20-0.200.20 2 4 6 8 10 12 14 16 18 20-0.200.2-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1-0.200.2b 2 !tttxxvavtxvaxvxttta!"#p$#waves#waves1008.doc $$%&" A'$$0 2 4 6 800.020.040.060.080.10.12b = 0 energy K U E (J)timet(s)KEPE E 0 2 4 6 800.020.040.060.080.10.12b = 6 energy K U E (J)timet(s)KE PE E Sinusoidal drivin$ force d - o + 0.1Sinusoidal drivin$ force d - o + 1Sinusoidal drivin$ forced - o + .%mpulsive force / constant force applied for a short time interval.a!"#p$#waves#waves1008.doc $$%&" A'$30 20 40 60 80 100-1-0.500.51b = 2 position x (m)timet(s)0 20 40 60 80 100-1-0.500.51b = 2 position x (m)timet(s)0 20 40 60 80 100-1-0.500.51b = 2 position x (m)timet(s)0 20 40 60 80 100-1-0.500.51b = 2 position x (m)timet(s)0roblemA spring is hanging from a support without any ob,ect attached to it and its length is &!! mm. An ob,ect of mass 3&! g is attached to the end of the spring. The length of the spring is now 8&! mm.*a+ )hat is the spring constantIThe spring is pulled down $3! mm and then released from rest.*b+ 1escribe the motion on the ob,ect attached to the end of the spring.*c+ )hat is the displacement amplitudeI*d+ )hat are the natural frequency of oscillation and period of motionIAnother ob,ect of mass 3&! g is attached to the end of the spring.*e+ Assuming the spring is in its new equilibrium position, what is the length of the springI*f+ (f the ob,ect is set vibrating, what is the ratio of the periods of oscillation for the two situationsISolutionSetupL! 2 &!! mm 2 !.&!! m g 2 7.8 m.s.3m$ 2 3&! g 2 !.3&! kg m3 2 !.&!! kgL$ 2 8&! mm 2 !.8&! mL3 2 I mx$max 2 $3! mm 2!.$3! m x3max 2 A3 2 I mf$ 2 I 0>T$ 2 I sT3 6 T$ 2 I0ookeGs Eaw D :0'%F 2 k xo$3kfm =a!"#p$#waves#waves1008.doc $$%&" A'$"equilibrium $ 3Action*a+ spring constant;ewtonGs Eaws% m g 2 k s k 2 m g 6 ss$ 2 L$L! 2 *!.8&!!.&!!+ m 2 !."&! mk 2 m$ g 6 s$ 2 *!.3&!+*7.8+6*!."&+ 2 A.! ;.m.$*c+ amplitude% spring pulled down $3! mmx$max 2 A$ 2 !.$3! m*d+ frequency and period *does not depend upon amplitude+$$ $ A!.84 0>3 3 !.3&kfm = = =T$ 2 $ 6 f$ 2 $.3 s*e+;ewtonGs Eaws% m g 2 k s s2 m g 6 ks3 2 m3 g 6 k 2 *!.&!!+*7.8+ 6 *A.!+ 2 !.A! mL2 2 s3 H L! 2 !.A! H !.&! 2 $.3! m*f+$$3mTk =333mTk =3 3$ $!.&!!3 $.4!.3&!T mT m= = = =0roblemAn 8.!! kg stone is resting on a spring. The spring is compressed $!! mm by the stone. *a+ )hat is the spring constantIThe stone is pushed down an additional "!! mm and released. *b+ )hat is the potential energy of the stonespring system ,ust before the releaseI *c+ )hat is the speed of the stone assuming it is released as the spring moves past its equilibrium position. *d+ 0ow high above the release position will the stone riseISolutionm 2 8.!! kg s$ 2 $!! mm 2 !.$! mk 2 I ;.m.$s3 2 "!! mm 2 !."!! m