Forced Mechanical Oscillations

Transcription

1 9 Carl von Ossietzky University Oldenurg Faculty V - Institute of Physics Module Introductory Laoratory Course Physics Part I Forced Mechanical Oscillations Keywords: HOOKE's law, haronic oscillation, haronic oscillator, eigenfrequency, daped haronic oscillator, resonance, aplitude resonance, energy resonance, resonance curves. Measuring progra: Measureent of the aplitude resonance curve and the phase curve for strong and weak daping. References: // EMTRÖER, W.: Eperientalphysik Mechanik und Wäre, Springer-Verlag, Berlin aong others. // TIPLER, P.A.: Physik, Spektru Akadeischer Verlag, Heidelerg aong others. Introduction It is the oject of this eperient to study the properties of a haronic oscillator in a siple echanical odel. Such haronic oscillators will e encountered again in different fields of physics, for eaple in electrodynaics (see eperient Electroagnetic resonant circuit ) and atoic physics. Therefore it is very iportant to understand this eperient, especially the iportance of the aplitude resonance and phase curves. Theory. Undaped Haronic Oscillator Let us oserve a set-up according to Fig., where a sphere of ass K is vertically suspended (-direction) on a spring. Let us neglect the effects of friction for the oent. When the sphere is at rest, there is an equiliriu etween the force of gravity, which points downwards, and the dragging resilience which points upwards; the centre of the sphere is then in the position =. A deflection of the sphere fro its equiliriu position y causes a proportional dragging force F R opposite to : () FF RR ~ The proportionality constant (elastic or spring constant or directional quantity) is denoted, and Eq. () ecoes the well-known HOOKE s law : () FR = Following deflection and release the dragging force causes an acceleration a of the sphere. According to Newton s second law (3) FR = Ka In coination with Eq. () we therefore otain: (4) d Ka= K = K = (t: tie) d t the three ters on the left side erely representing different ways to write the relation force = ass acceleration. Eq. (4) is the iportant differential equation (also called the equation of otion), y eans of which all systes can e descried which react with a dragging force on a deflection fro their position of rest or equiliriu that is proportional to the degree of deflection. Such systes will e encountered very often in different fields of physics. We are interested in learning which oveent the sphere akes when it is deflected fro its position at rest and then released, its initial velocity v at the oent of release eing zero. So we look for the function ROBERT HOOKE (635 73)

2 9 - k + Fig. : Mass/spring syste. (t), which is a solution of the differential equation (4) under the condition v(t = ) =. Note that apart fro ultiplicative factors, this function ust e equal to its second tie derivative. Hence, we attept to solve the differential equation with a function (t), which descries a so-called haronic oscillation: (5) t ( ) = ( t+ ϕ ) cos is the aplitude, (t + ϕ) the phase, ϕ the initial phase and the angular eigenfrequency of the oscillation (cf. Fig. ). Inserting Eq. (5) into Eq. (4) and perforing differentiation twice with respect to tie t, we find: (6) cos( t + ϕ) = cos( t + ϕ) K Fro this follows the value of, for which Eq. (5) is a solution of Eq. (4): (7) = : = K Thus, the sphere perfors oscillations with the angular eigenfrequency when it is released. Since we assue that there is no friction, the aplitude of the oscillation reains constant. as well as the initial phase ϕ are free paraeters which have to e chosen such that Eq. (5) is adjusted to the process to e descried, i.e. that Eq. (5) reflects the oserved otion with the correct aplitude and initial phase. Equation (7) is only valid if the ass of the spring, F, is negligile copared to the ass K of the sphere. If this is not true, we have to consider that the spring s different eleents of ass also oscillate following its deflection and release. The oscillation aplitudes of these eleents of ass, however, are very different: They increase fro zero at the point of suspension of the spring to a value at the end of the spring. An eact calculation shows that the oscillation of the single eleents of ass with different aplitudes equals the oscillation of one third of the whole spring ass with the aplitude. Therefore, the correct equation for the angular eigenfrequency reads: (8) = : = with : = K + 3 K + F 3 In the eperient to e perfored the sphere is not directly fied to the spring ut y a ar S, with an attached reflective plate R (Fig. 8). In that case, K in Eq. (8) has to e replaced y the total ass: (9) G = K + S + R S and R eing the asses of S and R. An eaple illustrates the descried relationships. According to Fig. we oserve a sphere of the ass K =. kg suspended y the ar and reflective plate ( S + R =.7 kg) on a spring with the spring constant = 8 kg/s and the ass F =. kg. The sphere is deflected y =.5 downwards fro F See for eaple ALONSO, M., FINN, E. J.: Fundaental University Physics, Vol. : Mechanics, Addison-Wesley Pulishing Copany, Reading (Mass.) aong others.

3 93 its position at rest. Then we release the sphere and it perfors oscillations with the aplitude and the eigenfrequency f = /(π).9 Hz (Eq. (8)). If we start to record the otion (t) of the sphere eactly when it has achieved its aiu upward deflection, the cosine according to Eq. (5) starts at an initial phase of ϕ = π = 8 (ind the sign of in Fig.!). This situation is represented in Fig.. ϕ / (t) T t Fig. : efinition of the aplitude, period length T = π/ and initial phase ϕ of a haronic oscillation. The phase ϕ ust e divided y for the presentation on the t-ais. A syste according to the arrangeent considered here (also called ass/spring syste) that perfors haronic oscillations is called a haronic oscillator. The haronic oscillator is characterized y a dragging force proportional to the deflection leading to a typical equation of otion in the for of (4) with a solution in the for of (5). Equally characteristic of the haronic oscillator is the paraolic ehaviour of its potential energy E p as a function of the position (Fig. 3): () Ep = E p Fig. 3: - + Course of the potential energy E p as a function of displaceent for the haronic oscillator.. aped Haronic Oscillator Now we oserve the ore realistic case of a ass/spring syste under the influence of friction. We start fro the siple case where, in addition to the restoring force F R = -, a frictional force F proportional to the velocity v is acting on the syste. For F we can write: d () F = v = d t eing a constant of friction, which represents the agnitude of the friction. Question : - Which unit does have? Why is there a inus sign in Eq. ()? In this case the equation of otion (4) takes on the for: () d d = dt dt Usually, this differential equation is written in the for: (3) d d d + + = t dt

4 94 Here again, it is interesting to know what type of otion the sphere perfors after eing deflected once fro its position at rest and then released with an initial velocity of zero. Thus, we are once again searching for the function (t) which resolves the differential equation (3) under the condition v(t = ) =. As a consequence of daping, we epect a decreasing aplitude of the oscillation and therefore try a solution with an eponentially decreasing aplitude (cf. Fig. 4): α t (4) ( t ϕ) = e cos + (α : daping constant) We insert Eq. (4) into Eq. (3), perfor the differentiations, and find that Eq. (4) represents a solution of Eq. (3) if the following is true for the paraeters α and : (5) (6) α = and = (t) e α t t Fig. 4: aped haronic oscillation. We will now interpret this result. First we note that the aplitude of the oscillation decreases ore rapidly the larger the daping constant (the daping coefficient) α is. In the case of invariale ass this eans according to Eq. (5) that the aplitude of the oscillation decreases ore rapidly the larger the constant of friction is - which is plausile. Fro Eq. (6) we can read how the angular frequency of this daped haronic oscillation changes with the constant of friction. We study the following different cases: (i) = = In the case of vanishing friction ( = ) we have the case of the undaped haronic oscillator as discussed in Chapter..The sphere perfors a periodic oscillation at the angular eigenfrequency. (ii) (/()) = = This is the case of critical daping in which the sphere does not perfor a periodic oscillation any ore. It is therefore called the case of critical daping. The sphere only returns to its starting position eponentially (cf. rearks). (iii) (/()) > iaginary In the case of supercritical daping there is no periodic oscillation either. This case is called aperiodical case or over daped case. Here again, the sphere only returns to its starting position, however, with additional daping, i.e., ore slowly (cf. rearks). (iv) < < < This ost general case, the oscillation case, leads to a periodic oscillation at a angular frequency (eq. (6)), which is slightly lower than the angular eigenfrequency of the undaped haronic oscillator.

5 95 Rearks: Under the conditions discussed aove (v(t = ) = ) there is no considerale difference etween the case of critical daping and supercritical daping: In oth cases the sphere returns to its starting position along an eponential path; in the case of supercritical daping there is only a stronger daping. We find a different situation in the case v(t = ). If we do not only release the sphere, ut push it thus giving it a certain starting velocity, it is possile in the case of critical daping that the sphere oscillates eyond its position at rest once, and only then returns to its starting position along an eponential path. In the case of supercritical daping such an oscillation eyond that position does not occur. The sphere always returns to its position at rest along an eponential path. etailed calculations (solution of the differential equation (3) under the conditions (ii) and (iii)) confir these relationships..3 Forced Haronic Oscillations In Chapters. and. we have oserved how the sphere oscillates if we deflect it once fro its position at rest and then release it. Now we will investigate which oscillations the sphere perfors if the syste is suject to a periodically changing eternal force F e (Fig. 5), for which the following is true: (7) F = F sin ( t) e F is the aplitude of the eternal force and its angular frequency. The sign is chosen such that the forces directed downwards are counted as positive and upward forces are counted as negative. - Fig. 5: + Oscillation generation of a ass/spring syste with an eternal force F e, eing the ass according to Eqs. (8) and (9). F e The eternal force F e additionally acts on the spring. The equation of otion thus takes the for (cf. Eqs. () and (3)): d d (8) = + F e dt dt and hence d d F t dt dt (9) + + = sin ( ) It is epected that the otion of the sphere following a certain transient tie, i.e., after the end of the transient otion, occurs at the sae frequency as does the change of the eternal force. There would e no plausile eplanation for another frequency. However, a phase shiftφ etween the stiulating force and the deflection of the sphere could e assued. We ay epect the oscillation aplitude to reain constant upon copletion of the transient otion since the syste is provided with new eternal energy again and again. Based on these considerations the following ansatz is suggested for the differential equation (9): () = sin ( t+ φ) In this case φ is the phase shift etween the deflection (t) and the eternal force F e. For φ< the deflection lags ehind the stiulating force. By inserting Eq. () into Eq. (9) we find that Eq. () represents a

6 96 solution of Eq. (9) if the following is true for the aplitude and the phase shift φ (derivation cf. Appendi chapter 4): F () = ( ) + () π φ = arctan Contrary to the cases discussed in Chapters. and., the aplitude and the phase φ are no longer freely selectale paraeters, rather they are definitely deterined y the quantities F,,, and = /. Eq. () shows that the aplitude of the sphere's oscillation, the so called resonance aplitude, depends on the frequency of the stiulating force. Plotting over, we otain the aplitude resonance curve. Fig. 6 (top) shows soe typical aplitude resonance curves for different values of the friction constant. In the stationary case, i.e. for =, we otain the aplitude known fro HOOKE's law fro Eq.(): F = : = = (3) ( ) This is the value y which the sphere is deflected if it is affected y a constant force F.Sustituting F fro Eq. (3) into Eq. (), one otains for the resonance aplitude : (4) = ( ) + The position of the aiu of as a function of is found y eans of the condition d /d =. Fro Eq. (4) follows: (5) =,a for = Ecept for the case =, the aiu of the aplitude resonance curve is thus not found at the angular eigenfrequencies, ut at a slightly lower angular frequencies <. The lower part of Fig. 6 shows the so called phase curves which deterine the developent of the phase shift φ as a function of the angular frequency. Fro Eq. () it follows that φ is always negative, i.e., the deflection of the sphere always lags ehind the stiulating force ecept for the case =. We will now discuss soe special cases: (i) In the case << the aplitude F / is independent of for not too large. The aplitude resonance curve is nearly horizontal for sall ecitation frequencies and the phase shift φ tends to : φ. Thus the otion of the sphere alost directly follows the stiulating force. (ii) In the resonance case ( according to Eq.(5)), the aplitude is aial and given y,a = F 4

7 97 The saller is, the larger,a ecoes; for,,a. In this case the sphere's deflection lags ehind the generating force y 9 (φ = - π/). (iii) In the case >>, F /( ), i.e., the aplitude drops y /. The phase shift is φ = - π in this case, i.e., the sphere's deflection lags ehind the generating force y 8. Fig. 6: Aplitude resonance curves (top) and phase curves (otto) for a daped haronic oscillator. (F =. N, =. kg, = kg/s, in kg/s). Fro the aplitude resonance curves and the special cases discussed in (i) - (iii) the daping ehaviour of a ass-spring-syste can e read, i.e. of a viration isolating tale, which is frequently used in optical precision etrology. The eigenfrequencies of such tales are in the range of aout Hz. If an eternal disturance (e.g. uilding oscillation) has a very low frequency ( ), the aplitude of the perturation is transferred onto the tale without daping. Close to the angular eigenfrequency ( ) it is (unintentionally) aplified, whereas in the range of high frequencies ( >> ) it is daped strongly. The daping ehaviour of such a syste can e influenced y changing the ass. Fig. 7 shows that a larger reduces the angular eigenfrequency with the other paraeters reaining unchanged and that the daping for frequencies aove the angular eigenfrequency can e increased significantly. Thus, oscillation dapening tales often have large asses in the range of 3 kg. Finally we will eaine at which frequency the aial energy transfer occurs fro the generating syste to the oscillating syste. As we know that the aial kinetic energy is equivalent to the aiu velocity, we first calculate the teporal course of the velocity v of the sphere using Eq. (): d dt (6) v= = cos ( t+ φ) : = vcos( t+ φ) With Eq. (4) we thus otain for the velocity v : (7) v = = ( ) + and hence: (8) v = +

8 98 v ecoes aial when the denoinator of Eq. (8) ecoes inial, i.e., if the following is true (for ): (9) = v = v,a Hence it follows: = = v = v (3),a Fig. 7: Aplitude resonance curves for different asses (in kg) with other paraeters reaining unchanged (F =. N, = kg/s, =. kg/s). Thus the velocity and also the kinetic energy ecoe aial (in contrast to the resonance aplitude!) if the syste is stiulated with its angular eigenfrequency. Therefore, this case is called energy resonance, a case in which the generating syste can transfer the aial energy to the oscillating syste. 3 Eperiental Procedure Equipent: Spring ( = (.7 ±.5) kg/s, F = (.575 ± -4 ) kg), sphere on suspension ar with reflective plate ( G needs to e weighed), ecitation syste on stand with otor and light arrier, electronic speed controller for otor, laser distance sensor (type BAUMER OAM U646/S35, easuring range (6 ) ), power supplies (PHYWE ( 5 / 3) V) for otor, light arrier and laser distance sensor, glasses with different glycerine/water itures (,7 kg/s for the ore viscous iture at T = C), desk for the glasses, digital oscilloscope TEKTRONIX TS / B / C / TBS B. 3. escription of Eperiental Set-Up The eperients are perfored in a set-up according to Fig. 8. This allows for contact-free easureent of the aplitude resonance curves and phase curves. This set-up is descried in the following, efore presenting the actual easuring tasks in Chap. 3.: A sphere K of ass K is suspended on a spring y eans of a ar S. The sphere is plunged into a glass B filled with a glycerine/water iture to dap its oscillation. A reflective plate R is fied on the ar. A laser ea fro the laser distance sensor LS (the operating principle was detailed in the eperient Sensors... ) is incident on the reflective plate. The sensor output is a voltage signal U LS(t), which varies linearly with the distance s etween LS and R. The spring is connected to a piston rod P via a joint G with a second ar S which runs in a guide F. The piston rod P is fied on a rotary disk via a joint G. The disk can e rotated at an angular frequency via a otor. Thus, the suspension point of the spring is set in a periodic vertical otion and a periodic driving force F e(t) is eerted on the spring. After the end of the transient otion, the sphere, together with S and R, will also show a periodic vertical otion with aplitude. This causes the laser distance sensor to produce a periodic voltage signal U LS(t) with an aplitude of U ~ and an offset U C which depends on the distance s etween LS and R in the rest position of the sphere. The period T of U LS is given y

9 99 (3) π T = Thus the aplitude resonance curve U ( ) can e easured y varying. Using the caliration factor k of the laser distance sensor for voltage differences (3) k =,96 V/ the aplitude resonance curve ( ) can e deterined. k ay e taken as an error free quantity. The easureent of the phase curve, i.e. the phase shift φ etween the driving force F e(t) and the vertical displaceent (t) of the sphere as a function of the angular frequency can e carried out as follows: With the aid of a arker M and the light arrier LS, which is interrupted y M, an electric pulse U LS(t) is generated every tie the suspension point of the spring reaches its highest position (tie t in Fig. 9). At this tie, the driving force F e(t) = d /dt is at its iniu (keep in ind the sign according to Fig. 5). At a later tie t, the sphere (not the suspension point of the spring!) reaches its highest position and thus the deflection (t) its iniu (- ; here too keep in ind the sign according to Fig. 5). In this position, the distance s etween LS and R and thus also U LS(t) is inial. The phase shift φ etween F e(t) and (t) is then given y (cf. Fig. 9): (33) φ t t t : t T π = = T π = Therefore y variation of, the phase curveφ ( ) can e easured. In practice, the aplitude U ( ) and tie difference t( ) are easured siultaneously for each angular frequency with the aid of an oscilloscope. Finally one reark on the teporal course of the driving force F e(t): Ecept a constant phase shift, it corresponds to the teporal course of the vertical otion of the join G, i.e. the suspension point of the spring. This otion is descried y the quantity y(t) (cf. Fig. ). M LS G LS P S G F s Feder S R B K Fig. 8: Sketch of the eperiental set-up.

10 U T U LS U LS Fig. 9: t t t Teporal course of the output voltages of the light arrier LS (U LS) and the laser distance sensor LS (U LS).Tie t : suspension point of the spring at highest position, driving force F e(t) inial. Tie t : Sphere at highest position, (t) and U LS inial. r θ l ψ G y S Fig. : efinition of quantities for calculating the oveent of the join G (cf. Fig. 8). If the piston rod is ounted on the disk at a distance r fro the ais of rotation, we otain: (34) y = rcosθ + lcosψ and (35) rsinθ = lsinψ r sinψ = sinθ l With (36) cosψ = r sin ψ = sin θ l and (37) θ = t we finally otain: (38) y = rcos( t) + l r sin ( t) The purely haronic otion (r cos( t)) is thus superiposed y a disturance (square root ter in Eq. (38)) which, unfortunately, is also tie-dependent and therefore akes the otion anharonic. Therefore, the driving force F e(t) is not copletely haronic either. If we choose l >> r, however, l >> r sin ( t) and hence (...) l. Instead of a tie-dependent disturance we then have to deal with a erely additive constant l which no longer disturs the harony.

11 3. Aplitude Resonance Curve and Phase Curve for Strong and Weak aping Using the setup according to Fig. 8, for a sphere with suspension ar S and reflective plate Rand a spring with known and F (for data see Equipent) the aplitude resonance curve ( ) and the phase curve φ( ) within the frequency range f = /π etween Hz and appro. 5 Hz are to e easured for two different dapings (glasses containing different glycerine/water itures). The piston rod P of the ecitation syste is fied to the disk in the second hole fro the centre. The anharonic disturance according to Eq. (38) can e neglected in this case. The output signals of the light arrier (U LS) and the laser distance sensor (U LS) are displayed on a digital oscilloscope, which is triggered y the signal U LS. The period tie T of U LS and the peak-peak value (U SS = U ) of U LS are deterined y using the oscilloscope s MESSUNG / MEASURE function. Fro these quantities, the angular frequency and the aplitudes U and, respectively can e deterined. The tie difference t = t t, fro which the phase shift φ can e calculated according to Eq. (33) is easured y using the TIME-CURSOR (cf. Fig. 9). Hint: In order to achieve a ostly unifor otion of the disk, the disk ust always e rotated counter clockwise. For the sae reason, an electronic speed controller (operating voltage V) ust e used for adjusting the revolution nuer of the otor within the frequency range etween Hz and appro..5 Hz, which is ounted etween the power supply and the otor. For frequencies eceeding.5 Hz the otor can e directly connected with the power supply and the nuer of revolutions can e controlled via the operating voltage (increase voltage slowly fro V to a. V). For oth glycerine/water itures, the aplitude U ( ) of the sphere otion, the period duration T, and the tie difference t are easured for as any different values of as possile (at least ), especially near the eigenfrequency. The easureents are perfored after the end of the transient otion. For the case, the aplitude U is deterined y anually turning the ais of the otor (while the otor is switched off) to the positions piston rod up, piston rod down and easuring the corresponding voltages U LS. We plot over for oth itures in one diagra, and φ over likewise in one diagra. The aiu errors of and φ are also entered in the for of error ars (estiate errors fro the fluctuations of the easureents for U SS and T at the oscilloscope). Then freehand regression curves are drawn through the easured values and their course is copared with the theoretical epectations. Rearks: In the vicinity of the angular eigenfrequency the easureent under weak daping ay ecoe difficult, ecause the aplitudes ay e large and the spring (possily even the ount) ay get into uncontrollale otion or the sphere ay even hit the otto of the glass. In that case the spring syste ust e daped anually and rapidly proceeded to the net frequency value. 4 Appendi: Calculation of the Resonance Aplitude and the Phase Shift We want to deonstrate that the resonance aplitude and the phase shift φ can e calculated with a few siple calculation steps, if we change over to cople representation. In cople representation Eq. (9) reads: d d i t (39) + + = F e dt dt In analogy to Eq. () we choose as a cople approach: (4) i( t + ) e i t e i = e = φ φ Following differentiation and division y i t e insertion of Eq. (4) into Eq. (39) yields: (4) iφ iφ iφ F e i e e + + =

12 Hence it follows with the definition of the angular eigenfrequency according to Eq. (8): (4) F i φ e = : = z + i As already deonstrated in the eperient easureent of capacities..., Eq. (4) is one representation for of a cople nuer z, whose asolute value (odulus) z = is given y zz *, with z* eing the conjugate cople quantity of z. Hence it follows: (43) F F = zz = + i i fro which we otain Eq. () y siple ultiplication. For calculating the phase angle we again use (cf. eperient easureent of capacities... ) the second representation of cople nuers, naely z = α + iβ, α eing the real part and β the iaginary part of z. As is generally known, the phase angle φ can e calculated fro these quantities as (44) β + π for α < and β φ = arctan α π for α < and β < In order to convert Eq. (4) into the for α + iβ, we etend the fraction in Eq. (4) with the conjugated cople denoinator: (45) F F F i ( ) i e i φ = = i i + ( ) + fro which we can read off the quantities α and β : (46) F F ( ) α = and β = ( ) ( ) + + which yields y insertion into Eq. (44): (47) φ = arctan { π > } With (48) arctan ( y) it finally yields Eq. (). π = arctan y