TOPIC E: OSCILLATIONS SPRING 018 1. Introduction 1.1 Overview 1. Degrees of freedo 1.3 Siple haronic otion. Undaped free oscillation.1 Generalised ass-spring syste: siple haronic otion. Natural frequency and period.3 Aplitude and phase.4 Velocity and acceleration.5 Displaceent fro equilibriu.6 Sall-aplitude approiations.7 Derivation of the SHM equation fro energy principles 3. Daped free oscillation 3.1 The equation of otion 3. Solution for different daping levels 4. Forced oscillation 4.1 Matheatical epression of the proble 4. Static load 4.3 Undaped forced oscillation 4.4 Daped forced oscillation Mechanics Topic E (Oscillations) - 1 David Apsley
1. INTRODUCTION 1.1 Overview Many iportant dynaical probles arise fro the oscillation of systes responding to applied disturbances in the presence of restoring forces. Eaples include: response of a structure to earthquaes; huan-induced structural oscillations (e.g. grandstands at concerts/sports venues); flow-induced oscillations (e.g. chineys, pipelines, power lines); vibration of unbalanced rotating achinery. Most systes when displaced fro a position of equilibriu have one or ore natural frequencies of oscillation which depend upon the strength of restoring forces (stiffness) and resistance to change of otion (inertia). If the syste is left to oscillate without further influence fro outside this is referred to as free oscillation. On the other hand, the eaples above are ainly concerned with forced oscillation; that is, oscillations of a certain frequency are iposed on the syste by eternal forces. If the applied frequency is close to the natural frequency of the syste then there is considerable transfer of energy and resonance occurs, with potentially catastrophic consequences. In general, all systes are subject to soe degree of frictional daping that reoves energy. A freely-oscillating syste ay be under-daped (oscillates, but with gradually diinishing aplitude) or over-daped (so restricted that it never oscillates). When periodic forces act on structures, daping is a crucial factor in reducing the aplitude of oscillation. 1. Degrees of Freedo Many systes have several odes of oscillation. For eaple, a long power line or suspended bridge dec ay gallop (bounce up and down) or it ay twist. Often the geoetric configuration at any instant can be defined by a sall nuber of paraeters: usually displaceents or angles θ. Paraeters which are just sufficient to describe the geoetric configuration of the syste are called degrees of freedo. Here, for siplicity, we shall restrict ourselves to single-degree-of-freedo (SDOF) systes. The equation of otion then describes the variation of that paraeter with tie. Eaples of SDOF dynaical systes and their degree of freedo are: ass suspended by a spring (vertical displaceent); pivoted body (angular displaceent). In both systes, oscillations occur about a position of static equilibriu in which applied forces or oents are in balance. 1.3 Siple Haronic Motion For any systes the forces arising fro a sall displaceent are opposite in direction and proportional in size to the displaceent. The equation of otion is particularly siple and solutions tae the for of a sinusoidal variation with tie. This ubiquitous and iportant type of oscillation is nown as siple haronic otion (SHM). Mechanics Topic E (Oscillations) - David Apsley
. UNDAMPED FREE OSCILLATION The oscillatory otion of a syste displaced fro stable equilibriu and then allowed to adjust in the absence of eternally-iposed forces is tered free oscillation. If there are no frictional forces the otion is called undaped free oscillation..1 Generalised Mass-Spring Syste: Siple Haronic Motion This is a general odel for a linear free-oscillation proble. It doesn t physically have to correspond to asses and springs. A generalised ass-spring syste is one for which the resultant force following displaceent fro equilibriu is a function of the displaceent and of opposite sign (i.e. a restoring force). For ideal springs (or, in practice, for sall displaceents) the relationship is linear: F (1) is called the stiffness or spring constant. It is easured in newtons per etre (N 1 ). The equation of otion is then d () To solve () note that d / ( / ) and that / is a positive constant. Thus, it is coon to write the equation as Siple Haronic Motion (SHM) Equation: d d ω or ω 0 (3) where stiffness ω (4) inertia To solve (3) loo for solutions (t) whose second derivative is proportional to the original function, but of opposite sign. The obvious candidates are sine and cosine functions. 1 The general solution (with two arbitrary constants) ay be written in either of the fors: Asin(ωt ) (5) C sin ωt Dcos ωt Any syste whose degree of freedo evolves sinusoidally at a single frequency is said to undergo siple haronic otion (SHM). ω is called the natural circular frequency and is easured in radians per second (rad s 1 ). 1 In your atheatics classes, when you have covered cople nubers, you will encounter a third for of the general solution involving cople eponentials: = Ae iωt + Be iωt. Mechanics Topic E (Oscillations) - 3 David Apsley
. Natural Frequency and Period One coplete cycle is copleted when ωt changes by π. Hence: π period of oscillation: T (6) ω 1 ω frequency: f in cycles per second or hertz (Hz) (7) T π Iportant note. It is coon in theoretical wor to refer to ω rather than f as the natural frequency, because: (a) it avoids factors of π in the solution; (b) it is ω rather than f which appears in the governing equation. Thus, one should be quite careful about precisely what is being referred to as frequency ; usually it will be obvious fro the notation or units: ω in rad s 1, f in cycles s 1 (Hz). The following eaple deonstrates that SHM can occur without any elastic forces. Eaple 1. (Meria and Kraige) Two fied counter-rotating pulleys a distance 0.4 apart are driven at the sae angular speed ω 0. A bar is placed across the pulleys as shown. The coefficient of friction between bar and pulleys is μ = 0.. Show that, provided the angular speed ω 0 is sufficiently large, the bar ay undergo SHM and find the period of oscillation. 0 g 0 0.4 Mechanics Topic E (Oscillations) - 4 David Apsley
.3 Aplitude and Phase The general solution of the SHM equation (two arbitrary constants) ay be written as either: Asin(ωt ) C sin ωt Dcos ωt The first is called the aplitude/phase-angle for (A = aplitude, = phase). Whichever for is ore convenient ay be used. They are easily interconverted as follows. Epand the aplitude/phase-angle for: Asin( ωt ) A(sinωt cos cosωt sin ) Copare with the second for and equate coefficients of sin ωt and cos ωt to obtain C Acos, D Asin Eliinating and A in turn gives: aplitude A C D phase angle tan 1 ( D/ C) Note that there are two alternative values of (in opposite quadrants) with the sae value of tan. These ust be distinguished by the individual signs of C and D. Eaple. Write the following epressions in aplitude/phase-angle for, A sin( ωt ) : (a) 1cos 3t 5sin 3t (b) 4cos t 3sin t In the general solution the two free constants can be deterined if initial boundary conditions are given for displaceent 0 and initial velocity ( d /) 0. There are two special cases: (1) Start fro rest at displaceent A: Acos ωt () Start fro the equilibriu position with initial velocity v 0 : Asin ωt where v ω 1 cos t 0 A 0 sint p/ p 3p/ t p -1 Mechanics Topic E (Oscillations) - 5 David Apsley
Eaple 3. For the syste shown, find: (a) the equivalent single spring; (b) the natural circular frequency ω; (c) the natural frequency of oscillation f; (d) the period of oscillation; (e) the aiu speed of the cart if it is displaced 0.1 fro its position of equilibriu and then released. 100 N/ 60 N/ 10 g.4 Velocity and Acceleration Fro the general solution in phase-angle for, Asin( ωt ) Differentiation then gives for the velocity: v Aωcos(ωt ) Squaring and adding, using cos θ sin θ 1: v ω A ω Hence, we can find the velocity at any given position in the cycle: v ω ( A ) (8) Note also the aiu displaceent, velocity and acceleration: A a (9) va ωa (10) a ω A (11) a Eaple 4. A particle P of ass 0.7 g is attached to one end of a light elastic spring of natural length 1.5 and odulus of elasticity λ = 90 N. The other end of the spring is fied to a point O on the sooth horizontal surface on which P is placed. The particle is held at rest with OP = 1. and then released. (a) Show that P oves with siple haronic otion. (b) Find the period and aplitude of the otion. (c) Find the aiu speed and aiu acceleration of P. (d) Find the speed of P when it is 1.7 fro O. Mechanics Topic E (Oscillations) - 6 David Apsley
.5 Displaceent Fro Equilibriu For any oscillating systes it is the displaceent fro a position of static equilibriu that is iportant, rather than the absolute displaceent. The equilibriu position and the oscillation about it can be obtained siultaneously by: (1) writing down the equation of otion for any convenient degree of freedo; () identifying the point of equilibriu as the point where the acceleration is zero; (3) rewriting the equation of otion in ters of the displaceent fro this point. Consider a ass suspended by a spring. The equilibriu etension e could easily be obtained by balancing weight and spring forces: g g e e Alternatively, writing down the general equation of otion in ters of g the spring etension: d g g ( ) Fro this, the position of equilibriu can be identified by setting acceleration d / = 0. Change variables to g X e displaceent fro equilibriu and note that, since e is just a constant, d X/ = d /. Then: d X X A constant force (such as gravity) changes the position of equilibriu. A linear-restoring-force syste undergoes SHM about the equilibriu position. Eaple 5. A bloc of ass 16 g is suspended vertically by two light springs of stiffness 00 N 1. Find: (a) the equivalent single spring; (b) the etension at equilibriu; (c) the period of oscillation about the point of equilibriu. = 00 N/ = 00 N/ 16 g Mechanics Topic E (Oscillations) - 7 David Apsley
Eaple 6. A 4 g ass is suspended vertically by a string of elastic odulus λ = 480 N and unstretched length. What is its etension in the equilibriu position? If it is pulled down fro its equilibriu position by a distance 0., will it undergo SHM? Eaple 7. A ass is hung in the loop of a light sooth cable whose two ends are fied to a horizontal support by springs of stiffness and (see figure). Find the period of vertical oscillations in ters of and. Oscillation about a point of equilibriu can also be observed in ulti-spring systes where coponents are already loaded at equilibriu, as in the following eaple. Eaple 8. A particle of ass 0.4 g is confined to ove along a sooth horizontal plane between two points A and B a distance apart by two light springs, both of natural length 0.8. The springs connecting the particle to A and B have stiffnesses L = 50 N 1 and R = 150 N 1 respectively. (a) Write down the equation of otion of the particle in ters of the distance fro A. (b) (c) Find the position of equilibriu. Show that, if released fro rest half way between the walls, the particle undergoes siple haronic otion and calculate: (i) the period of oscillation; (ii) the aiu speed and aiu acceleration. Mechanics Topic E (Oscillations) - 8 David Apsley
.6 Sall-Aplitude Approiations Many oscillatory systes do not undergo eact siple haronic otion (where the restoring force is proportional to a displaceent), but their otion is approiately SHM provided that the aplitude of oscillation is sall. Such sall-aplitude approiations are particularly coon for rotational otion about a fied point (where the restoring torque is often provided by gravity or elasticity) and rely on the approiations sin θ θ cosθ 1 (soeties,1 1 θ ) (1) when θ is easured in radians. These ay be derived forally by power-series epansions, but their essential validity is easily seen geoetrically (see right). sin 1 The following table shows that the approiation for sin θ is accurate to about 1% or better for angles as large as 15. If a structural eber were actually displaced by this uch then oscillation would be the least of your worries! θ (degrees) 5 10 15 0 θ (radians) 0.0873 0.175 0.6 0.349 sin θ 0.087 0.174 0.59 0.34 Iportant warning: duplication of notation When considering rotational oscillations we will run into probles with the sae sybols being used for different quantities. T is used for both torque and period of oscillation (and soeties tension); in this section we will use it to ean torque and write period out in full. ω is used for both natural circular frequency and angular velocity; while dealing with oscillations we shall reserve it to ean the natural circular frequency and write θ or dθ/ for the angular velocity..6.1 Rotational Oscillations Driven by Gravity Copound Pendulu In a siple pendulu all the ass is concentrated at one point. This can be treated using either the oentu or angular-oentu equations. In a copound pendulu the ass is distributed; the syste ust be analysed by rotational dynaics. In the absence of friction, two forces act on the body: the weight of the body (which acts through the centre of gravity) and the reaction at the ais. Only the forer has any oent about the ais. Let the distance fro ais to centre of gravity be L and let the angular displaceent of line AG fro the vertical be θ. A L G The line of action of the weight Mg lies at a distance L sin θ fro the ais of rotation, and hence iparts a torque (oent of force) of agnitude T Mg Lsin θ and in the opposite sense to θ. The rotational equation of otion is Mg Mechanics Topic E (Oscillations) - 9 David Apsley
torque = oent of inertia angular acceleration d θ T I Hence, d θ Mg Lsin θ I For sall oscillations, sin θ θ, so that d θ MgL θ I MgL This is SHM with natural circular frequency ω. I Eaple 9. A unifor circular disc of radius 0.5 is suspended fro a horizontal ais passing through a point halfway between the centre and the circuference. Find the period of sall oscillations. Eaple 10. (Ea 017) A unifor circular dis of ass 3 g and radius 0.4 is suspended fro a horizontal ais passing through a point on its circuference and perpendicular to the plane of the dis. A sall particle of ass g is attached to the other side of the dis at the opposite end of a diaeter. (a) (b) Find the oent of inertia of the cobination about the given ais. Find the period of sall oscillations about the ais. Ais 3 g g Mechanics Topic E (Oscillations) - 10 David Apsley
Eaple 11. A pub sign consists of a square plate of ass 0 g and sides 0.5, suspended fro a horizontal bar by two rods, each of ass 5 g and length 0.5. The sign is rigidly attached to the rods and swings freely about the bar. Find the period of sall oscillations. 0.5 The Dog and Duc 0.5 0.5.6. Rotational Oscillations Driven By Elastic Forces If the rotational displaceent θ is sall then the additional etension of a spring at distance r fro an ais is essentially the length of circular arc: rθ This gives a force of agnitude F (rθ) opposing the displaceent and hence a torque of agnitude Fr acting in the opposite direction to the rotational displaceent: T r θ ais r r Eaple 1. A unifor bar of ass M and length L is allowed to pivot about a horizontal ais though its centre. It is attached to a level plane by two equal springs of stiffness at its ends as shown. Find the natural circular frequency for sall oscillations. L Mechanics Topic E (Oscillations) - 11 David Apsley
.7 Derivation of the SHM equation fro Energy Principles For a body, of ass, subject only to elastic forces with stiffness, the total (i.e. inetic + potential) energy is constant: 1 1 v constant Differentiating with respect to tie gives d ( 1 1 v ) 0 Apply the chain rule to each ter: dv d v 0 Replacing v by d/: d d d 0 Dividing by d/: d 0 Hence, d ω 0 ( ω ) Eaple 13. A sign of ass M hangs fro a fied support by two rigid rods of negligible ass and length L (see below). The rods are freely pivoted at the points shown, so that the sign ay swing in a vertical plane without rotating, the rods aing an angle θ with the vertical. (a) (b) (c) (d) Write eact epressions for the potential energy and inetic energy of the sign in ters of M, L, g, the displaceent angle θ and its tie derivative θ. If the sign is displaced an angle θ = p/3 radians and then released, find an epression for its aiu speed. Find an epression for the total (i.e. inetic + potential) energy using the sall-angle approiations sin θ θ, cos θ 1 1 θ. Show that, for sall-aplitude oscillations, the assuption of constant total energy leads to siple haronic otion, and find its period. L L M Mechanics Topic E (Oscillations) - 1 David Apsley
3. DAMPED FREE OSCILLATION All real dynaical systes are subject to friction, which opposes relative otion and consues echanical energy. For a syste undergoing free oscillation, we shall show that: oderate daping decaying aplitude and reduced frequency; large daping oscillation prevented. The level at which oscillation is just suppressed is called critical daping. 3.1 The Equation of Motion Friction always acts in a direction so as to oppose relative otion. For a frictional force that depends on velocity, the daping force is often odelled as d F d c (13) c is the viscous daping coefficient. If is a displaceent then c has units of N s 1. In practice, c ay vary with velocity, but a useful analysis ay be conducted by assuing it is a constant, in which case the daping is tered linear. The equation of otion ( F = a ) for a daped ass-spring syste is d d c or d d c 0 (14) Dividing by, this can be written where d c d ( ) ω 0 ω / is the natural frequency of the undaped syste. The undaped syste (c = 0) has solutions of the for A sin( ωt ). In the presence of daping we epect the solution to decay in agnitude and, possibly, have a slightly different frequency. Hence we ight anticipate solutions of the for e λ t A sin(ω t ) (16) d where A and are arbitrary constants and λ and ω d are to be found. If you don t lie the analysis which follows, you can try siply substituting this into equation (15) and (after quite a lot of algebra) deriving the sae results as below. c (15) 3. Solution For Different Daping Levels Equation (15) is a hoogeneous, linear, second-order differential equation with constant coefficients. Hence, following the ethod taught in your aths course we see solutions of the for: pt Ae where p is a constant to be evaluated. Substituting in (15) we obtain the auiliary equation Mechanics Topic E (Oscillations) - 13 David Apsley
p with roots c p ω 0 c c ( ) 4ω c c p ω ( ) 1 (17) ω ω (For convenience) we define c ζ (daping ratio) (18) ω Then p ωζ ω ζ 1 (19) As you now fro your aths course, there are 3 possibilities, depending on the sign of the quantity under the square root (here, ζ 1) in equation (19): 1. two cople conjugate roots if ζ < 1;. two distinct negative real roots if ζ > 1; 3. two equal (negative, real) roots if ζ = 1. For cople conjugate roots ( p p r ip )the general solution can be written i prtipit prtipit prt Ae Be e ( C cos pit Dsin pit) where A and B (or C and D) are arbitrary constants. The last bracet also has an equivalent aplitude/phase-angle for. Case 1: ζ < 1 ( c < ω): under-daped syste The general solution ay be written e λ t A sin(ω ) (0) where c ωd ω 1 ζ, λ ωζ (1) There is oscillation with reduced aplitude (decaying eponentially as frequency, ω d. λt e ) and reduced The aplitude reduction factor over one cycle ( ω t π t π/ ω d or d πζ ep{ } () 1 ζ This ay be used to deterine the daping ratio ζ eperientally. Case 1 also includes the special case of no daping (ζ = 0), in which case λ = 0 and ω d = ω. ) is Mechanics Topic E (Oscillations) - 14 David Apsley
Case : ζ > 1 (c > ω): over-daped syste The general solution is of the for λ t t A 1 λ e Be (3) where λ ω(ζ ζ 1) (4) 1, There is no oscillation and, as both of the roots p are negative (i.e. λ positive), the displaceent decays to zero, ore slowly as the daping ratio ζ increases. Case 3: ζ = 1 (c = ω): critically-daped syste The general solution has the for: ωt ( A Bt)e (5) There is no oscillation and the aplitude decays rapidly to zero. Suary The daped ass-spring equation d d c 0 has solutions that depend on: undaped natural frequency: daping ratio: ω ζ c ω If ζ< 1 the syste is under-daped, oscillating, but with eponentially-decaying aplitude and reduced frequency ω d ω 1 ζ. If ζ > 1 the syste is over-daped and no oscillation occurs. The fastest return to equilibriu occurs when the syste is critically daped (ζ = 1). Mechanics Topic E (Oscillations) - 15 David Apsley
displaceent () Eaple. = 1 g, = 64 N 1 ω = 8 rad s 1 0 = 0.05, ( d /d t ) 0 0. 0.05 0.04 0.03 0.0 over-daped critically daped Cases in the graph: c = 1.6 N s 1 (ζ = 0.1) c = 16 N s 1 (ζ = 1) c = 64 N s 1 (ζ = 4) 0.01 0.00 0.0 0.5 1.0 1.5.0.5 3.0-0.01 tie (s) Eercise. Use Microsoft Ecel or any other coputer pacage to copute and plot the solution for various cobinations of,, c. -0.0-0.03-0.04-0.05 under-daped Eaple 14. Analyse the otion of the syste shown. What is the daping ratio? Does it oscillate? If so, what is the period? What value of c would be required for the syste to be critically daped? c = 60 N s/ = 700 N/ 40 g Eaple 15. (Ea, May 016) A carriage of ass 0 g is attached to a wall by a spring of stiffness 180 N 1. When required, a hydraulic daper can be attached to provide a resistive force with agnitude proportional to velocity; the constant of proportionality c = 40 N/( s 1 ). (a) If the carriage is displaced, write down its equation of otion, in ters of the displaceent, in the case when the hydraulic daper is in place. (b) (c) If there is no daping, find the period of oscillation. If the hydraulic daper is attached, find the period of oscillation and the fraction by which the aplitude is reduced on each cycle. c = 40 N/(/s) 0 g = 180 N/ Try this question fro first principles, rather than citing daping forulae. Mechanics Topic E (Oscillations) - 16 David Apsley
4. FORCED OSCILLATION Systes that oscillate about a position of equilibriu under restoring forces at their own preferred or natural frequency are said to undergo free oscillation. Systes that are perturbed by soe eternally-iposed periodic forcing are said to undergo forced oscillation. Eaples which ay be encountered in civil engineering are: concert halls and stadius; bridges (e.g. traffic- or wind-induced oscillations); earthquaes (the lateral oscillations of the foundations are equivalent to periodic forcing in a reference frae oving with the surface). The natural frequency and daping ratio are iportant paraeters for systes responding to eternally-iposed forces. Large-aplitude oscillations occur when the iposed frequency is close to the natural frequency (the phenoenon of resonance). In general, the syste s freeoscillation properties will affect both the aplitude and the phase of response to eternal forcing. 4.1 Matheatical Epression of the Proble c F sin t 0 The general for of the equation of otion with haronic forcing is d d c F0 sin Ωt (6) or d c d F0 ω sin Ωt (7) where the undaped natural frequency is given by ω Second-order differential equations of the for (7) are dealt with in ore detail in your atheatics courses. The general solution is the su of a copleentary function (containing two arbitrary constants and obtained by setting the RHS of (7) to 0) and a particular integral (which is any particular solution of the equation and is obtained by a trial function based on the nature of the RHS). The copleentary function is a free-oscillation solution and, if there is any daping at all, will decay eponentially with tie. The largetie behaviour of the syste, therefore, is deterined by the particular integral which, fro the for of the RHS, should be a cobination of sin Ωt and cos Ωt. Mechanics Topic E (Oscillations) - 17 David Apsley
4. Static Load A useful coparison is with the displaceent under a steady load of the sae aplitude F 0, rather than a variable load F0 sin Ωt. The steady-state displaceent s is given by the position of static equilibriu: s F 0 whence F0 F0 s (8) ω 4.3 Undaped Forced Oscillation In the special case of no frictional daping, the equation of otion is F 0sin t d F0 ω sin Ωt The for of the forcing function suggests a particular integral of the for Csin Ωt. By substituting this in the equation of otion to find C, one finds that F0 C s (9) ( ω Ω ) 1 Ω /ω where, as above, 0/ω s F is the displaceent under a static load of the sae aplitude. Hence the agnitude of forced oscillations is M s, where the aplitude ratio or agnification factor M is given by 1 M (30) 1 Ω / ω Key Points (1) The response of the syste to eternal forcing depends on the ratio of the forcing frequency Ω to the natural frequency ω. () There is resonance (M ) if the forcing frequency approaches the natural frequency (Ω ω). (3) If Ω < ω the oscillations are in phase with the forcing (C has the sae sign as F 0 ), because the syste can respond fast enough. (4) If Ω > ω the oscillations are 180 out of phase with the forcing (C has the opposite sign to F 0 ) because the iposed oscillations are too fast for the syste to follow. (5) If Ω >> ω (very fast oscillations) the syste will barely ove (C 0). Mechanics Topic E (Oscillations) - 18 David Apsley
4.4 Daped Forced Oscillation The coplete analysis of oscillations for a SDOF syste involves forced otion and daping. The equation of otion is d c d F0 ( ) ω sin Ωt Because of the d/ ter, a particular integral ust contain both sin Ωt and cos Ωt. Substituting a trial solution of the for Csin Ωt Dcos Ωt produces, after a lot of algebra, a solution (optional eercise) of the for 1 Ω /ω ζω/ω C s, D s (31) (1 Ω /ω ) (ζω/ω) (1 Ω /ω ) (ζω/ω) where, as in Section 3, the daping ratio is c ζ ω and the displaceent under static load is s F0 /ω This is ost conveniently written in the aplitude/phase-angle for Ms sin( Ωt ) (3) where the aplitude ratio M is given by C D 1 M (33) s (1 Ω /ω ) (ζω/ ω) and the phase lag by D ζω/ω tan C 1 Ω / ω (34) Key Points (1) The response of the syste to forcing depends on both the ratio of forcing to natural frequencies (Ω/ω) and the daping ratio ζ. () Daping prevents the coplete blow-up (M ) as Ω ω. (3) The iposed frequency at which the aiu aplitude oscillations (resonance) occur is slightly less than the undaped natural frequency ω. In fact Ω a ω 1 ζ (4) The phase lag varies fro 0 (as Ω 0) to π (as Ω ). However, the phase lag is always π/ when Ω = ω, irrespective of the level of daping. Mechanics Topic E (Oscillations) - 19 David Apsley
Eaple 16. Write down general solutions for the following differential equations: d (a) 4 1 d d (b) 4 5 15 130sin t Eaple 17. (Meria and Kraige, odified) The seisoeter shown is attached to a structure which has a horizontal haronic oscillation at 3 Hz. The instruent has a ass = 0.5 g, a spring stiffness = 150 N 1 and a viscous daping coefficient c = 3 N s 1. If the aiu recorded value of in its steady-state otion is 5, deterine the aplitude of the horizontal oveent B of the structure. (t) B c Mechanics Topic E (Oscillations) - 0 David Apsley
Nuerical Answers to Eaples in the Tet Full wored answers are given in a separate docuent online. Eaple 1..01 s Eaple. (a) ω = 3, A = 13, = tan 1 (1/5) = 1.18 radians = 67.4 (b) ω =, A = 5, = tan 1 ( 4/3) =.1 radians = 16.9 Eaple 3. (a) = 160 N 1 ; (b) 4 rad s 1 ; (c) 0.637 Hz; (d) 1.57 s; (e) 0.4 s 1 Eaple 4. (b) 0.679 s; 0.3 ; (c).78 s 1 ; 5.7 s ; (d).07 s 1 Eaple 5. (a) = 400 N 1 ; (b) 0.39 ; (c) 1.6 s Eaple 6. 0.164 ; no (the string becoes unstretched). Eaple 7. 3 π Eaple 8. d (a) 500( 1.1) ; (b) = 1.1 ; (c) (i) 0.81 s; (ii).4 s 1 ; 50 s Eaple 9. 1.74 s Eaple 10. (a).0 g ; (b) 1.70 s. Eaple 11. 1.69 s Eaple 1. 6 M Eaple 13. (a) PE MgL cos θ constant, KE ; (b) gl ; 1 ML θ Mechanics Topic E (Oscillations) - 1 David Apsley
(c) 1 1 ML θ MgLθ constant ; (d) π L g Eaple 14. It does oscillate; ζ = 0.179, period = 1.53 s; for critical daping, c = 335 N s 1 Eaple 15. d d (a) 9 0 ; (b).09 s (c). s; 0.109 Eaple 16. (a) C cos t Dsin t 3 t (b) e ( C cos t Dsin t) 316cos t sin t Eaple 17. 1.77 Mechanics Topic E (Oscillations) - David Apsley