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