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# Note for MATH 3321 at UH

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Date Created: 02/06/15

36 Vibrating Mechanical Systems Undamped Vibrations A spring of length lo units is suspended from a support When an object of mass m is attached to the spring the spring stretches to a length 1 units If the object is then pulled down or pushed up an additional yo units at time t 0 and then released what is the resulting motion of the object That is what is the position yt of the object at time t gt 0 Assume that time is measured in seconds We begin by analyzing the forces acting on the object at time t gt 0 First there is the weight of the object gravity F1 mg This is a downward force We choose our coordinate system so that the positive direction is down Next there is the restoring force of the spring By Hooke s Law this force is proportional to the total displacement 1 yt and acts in the direction opposite to the displacement F2 7km with k gt 0 The constant of proportionality k is called the spn39ng constant If we assume that the spring is frictionless and that there is no resistance due to the surrounding medium for example air resistance then these are the only forces acting on the object Under these conditions the total force is F F1 F2 mg klll 2405 mg kll 905 Before the object was displaced the system was in equilibrium so the force of gravity mg plus the force of the spring ikll must have been 0 mg 7 kll 0 Therefore the total force F reduces to F 7kyt By Newton s Second Law of Motion F ma force mass x acceleration we have k 7k 25 d 77 t ma yo an a myo Therefore at any time t we have ayquotlttgtiylttgt or y lttgt5ylttgto m m 39 When the acceleration is a constant negative multiple of the displacement the object is said to be in simple harmonic motion 106 Since km gt 07 we can set w wkm and write this equation as ya man 0 1 a second order linear homogeneous equation with constant coefficients The characteristic equation is 72 w2 0 and the characteristic roots are iwi The general solution of 1 is y 01 cos wt Cg sin wt ln Exercises 36 Problem 5 you are asked to show that the general solution can be written as y Asinwt o7 2 where A and 0 are constants with A gt 0 and 0 6 027139 For our purposes here7 this is the preferred form The motion is periodic with period T given by 727139 T 7 LA a complete oscillation takes 27rw seconds The reciprocal of the period gives the number of oscillations per second This is called the frequency denoted by f 1 727139 1 Since sin wt 0 oscillates between 71 and 17 yt Asin wt 0 oscillates between 7A and A The number A is called the amplitude of the motion The number 0 is called the phase constant or the phase shift The gure gives a typical graph of Figure 1 107 Example 1 Find an equation for the oscillatory motion of an object given that the period is 27r3 and at time t 0 y 1 y 3 SOLUTION In general the period is 27rw so that here 2 2 l l and therefore an 3 w 3 The equation of motion takes the form yt Asin 3w 0 Differentiating the equation of motion gives y t 3A cos3t 0 Applying the initial conditions we have y0 1 Asin 0 y 0 3 3A cos 0 and therefore Asin 01 Acos 01 Adding the squares of these equations we have 2 A2 sin2 0 A2 cos2 0 A2 Since A gt O A Finally to nd 0 note that 2sin 01 and 2cos 01 These equations imply that 0 7r4 Thus the equation of motion is W sin3t in l Damped Vibrations If the spring is not frictionless or if there the surrounding medium resists the motion of the object for example air resistance then the resistance tends to dampen the oscillations Experiments show that such a resistant force R is approximately proportional to the velocity v y and acts in a direction opposite to the motion R icy with c gt 0 Taking this force into account the force equation reads F 7km 7 cm 108 Newton s Second Law F ma my then gives my t 7W0 7 Clot which can be written as k C y 724 7y 0 m 37 k m all constant m This is the equation of motion in the presence of a damping factor The characteristic equation k rzir70 m m has roots 76 i V 02 7 4km r 2m There are three cases to consider 0274kmlt0 0274kmgt0 0274km0 Case 1 02 7 4km lt 0 In this case the characteristic equation has complex roots 0 4 c 4 x4km 7 02 r1 772w7 r2 777zw Wherew 2m 2m 2m The general solution is y eke27quot 01 cos wt Cg sin wt which can also be written as yt Amie27quot sin wt 0 4 Where7 as before7 A and 0 are constants7 A gt 07 0 6 027139 This is called the underdamped case The motion is similar to simple harmonic motion 702mt 5 The oscillations continue inde nitely with constant frequency f w27r but except that the damping factor causes yt 7 0 as t 7 oo diminishing amplitude Ae c2mt This motion is illustrated in Figure 2 I Y Figure 2 109 Case 2 02 7 4km gt 0 In this case the characteristic equation has two distinct real roots 76 V 02 7 4km 7c 7 V62 7 4km r r 1 2m 7 2 2m The general solution is yt y Clem Cgent 5 This is called the ouerdamped case The motion is nonoscillatory Since V6274kmlt V6 07 T1 and r2 are both negative and yt 7 0 as 257 00 I Case 3 02 7 4km 0 In this case the characteristic equation has only one real root 3 T1 7 2m7 and the general solution is 240 y 016702mt 02te7c2mt39 This is called the critically damped case Once again7 the motion is nonoscillatory and yt70 as 25700 I In both the overdamped and critically damped cases7 the object moves back to the equilibrium position 7 0 as t 7 00 The object may move through the equilibrium position once7 but only once Two typical examples of the motion are shown in Figure 3 Figure 3 Forced Vibrations The vibrations that we have considered thus far result from the interplay of three forces gravity7 the restoring force of the spring7 and the retarding force of friction or the surround ing medium Such vibrations are called free vibrations 110 The application of an external force to a freely vibrating system modi es the vibrations and produces what are called forced uibrations As an example we ll investigate the effect of a periodic external force F0 cos 39yt where F0 and 39y are positive constants In an undamped system the force equation is F 7km F0 cos 39yt and the equation of motion takes the form H k F0 y iyicos 39yt m m We set w km and write the equation of motion as F y Lazy 70 cos 39yt 7 m As we ll see the nature of the motion depends on the relation between the applied frequency 39y27r and the natural frequency of the system w27r Case 1 39y 7 Ad In this case the method of undetermined coefficients gives the particular solution Fom t 7 cos t Z wg 7 2 v and the general equation of motion is Fom y Asinwt 0 m cos 39yt 8 If wy is rational the vibrations are periodic lf wy is not rational then the vibrations are not periodic and can be highly irregular In either case the vibrations are bounded by Fom 1141 L724 Case 2 39y Ad In this case the method of undetermined coefficients gives F 2t t sin wt and the general solution has the form 1 F i yAs1nwt 0 tsin wt 9 The system is said to be in resonance The motion is oscillatory but because of the t factor in the second term it is not periodic As t a co the amplitude of the vibrations increases without bound 111 A typical illustration of the motion is given in Figure 4 I Figure 4 Exercises 36 1 An object is in simple harmonic motion Find an equation for the motion given that the period is in39 and7 at time t 07 y 17 y 0 What is the amplitude What is the frequency 2 An object is in simple harmonic motion Find an equation for the motion given that the frequency is 17r and7 at time t 07 y 07 y 72 What is the amplitude What is the period 3 An object is in simple harmonic motion with period T and amplitude A What is the velocity at the equilibrium point y 0 4 An object in simple harmonic motion passes through the equilibrium point y 0 at time t 0 and every three seconds thereafter Find the equation of motion given that y0 5 5 Show that simple harmonic motion yt 01 cos wt 02 sin wt can be written as a W ASinwt 0 b ylttgt Acosltwtwogt 6 What is the effect of an increase in the resistance constant c on the amplitude and frequency of the vibrations given by 4 7 Show that the motion given by 5 can pass through the equilibrium point at most once How many times can the motion change directions 8 Show that the motion given by 6 can pass through the equilibrium point at most once How many times can the motion change directions 9 Show that if 39y 7 w then the method of undetermined coefficients applied to 7 gives F 2 07177 cos ME 112 10 Show that if w7 is rational then the Vibrations given by 8 are periodic 11 Show that if y M then the method of undetermined coe icients applied to 7 gives F0 2 t sin wt 2wm 113

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