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Chapter 10 Elasticity & Oscillations Elasticity and Oscillations • Elastic Deformations • Hooke’s Law • Stress and Strain • Shear Deformations • Volume Deformations • Simple Harmonic Motion • The Pendulum • Damped Oscillations, Forced Oscillations, and Resonance MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 2 Elastic Deformation of Solids A deformation is the change in size or shape of an object. An elastic object is one that returns to its original size and shape after contact forces have been removed. If the forces acting on the object are too large, the object can be permanently distorted. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 3 Hooke’s Law F F Apply a force to both ends of a long wire. These forces will stretch the wire from length L to L+L. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 4 Stress and Strain Define: L strain L The fractional change in length F stress A Force per unit crosssectional area Stretching ==> Tensile Stress Squeezing ==> Compressive Stress MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 5 Hooke’s Law Hooke’s Law (Fx) can be written in terms of stress and strain (stress strain). F L Y A L The spring constant k is now YA k L Y is called Young’s modulus and is a measure of an object’s stiffness. Hooke’s Law holds for an object to a point called the proportional limit. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 6 A steel beam is placed vertically in the basement of a building to keep the floor above from sagging. The load on the beam is 5.8104 N and the length of the beam is 2.5 m, and the cross-sectional area of the beam is 7.5103 m2. Find the vertical compression of the beam. Force of ceiling on beam Force of floor on beam F L Y A L F L L A Y For steel Y = 200109 Pa. 4 2.5 m F L 5.8 10 N 4 L 1 . 0 10 m 3 2 9 2 A Y 7.5 10 m 200 10 N/m MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 7 Example (text problem 10.7): A 0.50 m long guitar string, of cross-sectional area 1.0106 m2, has a Young’s modulus of 2.0109 Pa. By how much must you stretch a guitar string to obtain a tension of 20 N? F L Y A L 0.5 m F L 20.0 N L 6 2 9 2 A Y 1.0 10 m 2.0 10 N/m 5.0 10 3 m 5.0 mm MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 8 Beyond Hooke’s Law Elastic Limit If the stress on an object exceeds the elastic limit, then the object will not return to its original length. Breaking Point An object will fracture if the stress exceeds the breaking point. The ratio of maximum load to the original cross-sectional area is called tensile strength. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 9 Ultimate Strength The ultimate strength of a material is the maximum stress that it can withstand before breaking. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 10 An acrobat of mass 55 kg is going to hang by her teeth from a steel wire and she does not want the wire to stretch beyond its elastic limit. The elastic limit for the wire is 2.5108 Pa. What is the minimum diameter the wire should have to support her? F stress elastic limit A Want F mg A elastic limit elastic limit 2 mg D elastic limit 2 D MFMcGraw 4mg 1.7 10 3 m 1.7 mm elastic limit Chap 10c - Elas & Vibrations - Revised 4-3-10 11 Shear Deformations A shear deformation occurs when two forces are applied on opposite surfaces of an object. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 12 Stress and Strain Define: Shear Force F Shear Stress Surface Area A displaceme nt of surfaces x Shear Strain separation of surfaces L Hooke’s law (stressstrain) for shear deformations is F x S A L MFMcGraw where S is the shear modulus Chap 10c - Elas & Vibrations - Revised 4-3-10 13 Example (text problem 10.25): The upper surface of a cube of gelatin, 5.0 cm on a side, is displaced by 0.64 cm by a tangential force. The shear modulus of the gelatin is 940 Pa. What is the magnitude of the tangential force? F F x S A L F From Hooke’s Law: x F SA L 0.64 cm 940 N/m 0.0025 m 0.30 N 5.0 cm MFMcGraw 2 2 Chap 10c - Elas & Vibrations - Revised 4-3-10 14 Volume Deformations An object completely submerged in a fluid will be squeezed on all sides. F volume stress pressure A The result is a volume strain; MFMcGraw V volume strain V Chap 10c - Elas & Vibrations - Revised 4-3-10 15 For a volume deformation, Hooke’s Law is (stressstrain): V P B V where B is called the bulk modulus. The bulk modulus is a measure of how easy a material is to compress. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 16 An anchor, made of cast iron of bulk modulus 60.0109 Pa and a volume of 0.230 m3, is lowered over the side of a ship to the bottom of the harbor where the pressure is greater than sea level pressure by 1.75106 Pa. Find the change in the volume of the anchor. V P B V VP 0.230 m 3 1.75 106 Pa V B 60.0 109 Pa 6.7110 6 m 3 MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 17 Deformations Summary Table Tensile or compressive Shear Volume Stress Force per unit cross-sectional area Shear force divided by the area of the surface on which it acts Pressure Strain Fractional change in length Ratio of the relative displacement to the separation of the two parallel surfaces Fractional change in volume Constant of Young’s modulus Shear modulus (S) proportionality (Y) MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 Bulk Modulus (B) 18 Simple Harmonic Motion Simple harmonic motion (SHM) occurs when the restoring force (the force directed toward a stable equilibrium point) is proportional to the displacement from equilibrium. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 19 The Spring The motion of a mass on a spring is an example of SHM. Equilibrium position y x x The restoring force is F = kx. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 20 Equation of Motion & Energy Assuming the table is frictionless: F Classic form for SHM Also, MFMcGraw x kx max k a x t xt m 1 2 1 2 E t K t U t mv t kx t 2 2 Chap 10c - Elas & Vibrations - Revised 4-3-10 21 Spring Potential Energy MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 22 Simple Harmonic Motion At the equilibrium point x = 0 so a = 0 too. When the stretch is a maximum, a will be a maximum too. The velocity at the end points will be zero, and it is a maximum at the equilibrium point. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 23 MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 24 What About Gravity? When a mass-spring system is oriented vertically, it will exhibit SHM with the same period and frequency as a horizontally placed system. The effect of gravity is cancelled out. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 25 Spring Compensates for Gravity MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 26 Representing Simple Harmonic Motion MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 27 A simple harmonic oscillator can be described mathematically by: xt A cos t x vt A sin t t v at A 2 cos t t Or by: xt A sin t x vt A cos t t v at A 2 sin t t MFMcGraw where A is the amplitude of the motion, the maximum displacement from equilibrium, A = vmax, and A2 = amax. Chap 10c - Elas & Vibrations - Revised 4-3-10 28 Linear Motion - Circular Functions MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 29 Projection of Circular Motion MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 30 The Period and the Angular Frequency The period of oscillation is where is the angular frequency of the oscillations, k is the spring constant and m is the mass of the block. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 T 2 . k m 31 The period of oscillation of an object in an ideal mass-spring system is 0.50 sec and the amplitude is 5.0 cm. What is the speed at the equilibrium point? At equilibrium x = 0: 1 2 1 2 1 2 E K U mv kx mv 2 2 2 Since E = constant, at equilibrium (x = 0) the KE must be a maximum. Here v = vmax = A. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 32 Example continued: The amplitude A is given, but is not. 2 2 12.6 rads/sec T 0.50 s and v Aω 5.0 cm 12.6 rads/sec 62.8 cm/sec MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 33 The diaphragm of a speaker has a mass of 50.0 g and responds to a signal of 2.0 kHz by moving back and forth with an amplitude of 1.8104 m at that frequency. (a) What is the maximum force acting on the diaphragm? 2 2 F F ma m A mA 2 f 4 mAf max max 2 2 The value is Fmax=1400 N. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 34 Example continued: (b) What is the mechanical energy of the diaphragm? Since mechanical energy is conserved, E = Kmax = Umax. U max K max 1 2 kA 2 1 2 mvmax 2 The value of k is unknown so use Kmax. K max 1 2 1 1 2 2 mvmax m A mA2 2f 2 2 2 The value is Kmax= 0.13 J. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 35 Example (text problem 10.47): The displacement of an object in SHM is given by: yt 8.00 cmsin 1.57 rads/sec t What is the frequency of the oscillations? Comparing to y(t) = A sint gives A = 8.00 cm and = 1.57 rads/sec. The frequency is: 1.57 rads/sec f 0.250 Hz 2 2 MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 36 Example continued: Other quantities can also be determined: The period of the motion is 2 2 T 4.00 sec 1.57 rads/sec xmax A 8.00 cm vmax A 8.00 cm 1.57 rads/sec 12.6 cm/sec amax A 2 8.00 cm 1.57 rads/sec 19.7 cm/sec 2 2 MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 37 The Pendulum A simple pendulum is constructed by attaching a mass to a thin rod or a light string. We will also assume that the amplitude of the oscillations is small. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 38 The pendulum is best described using polar coordinates. The origin is at the pivot point. The coordinates are (r, φ). The r-coordinate points from the origin along the rod. The φcoordinate is perpendicualr to the rod and is positive in the counterclock wise direction. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 39 2nd Apply Newton’s Law to the pendulum bob. F mg sin ma 2 v Fr T mg cos m r 0 If we assume that φ <<1 rad, then sin φ φ and cos φ 1, the angular frequency of oscillations is then: F mg sin ma mL mg sin mL ( g / L)sin ( g / L) The period of oscillations is MFMcGraw T Chap 10c - Elas & Vibrations - Revised 4-3-10 g L 2 2 L g 40 Example (text problem 10.60): A clock has a pendulum that performs one full swing every 1.0 sec. The object at the end of the string weighs 10.0 N. What is the length of the pendulum? L T 2 g Solving for L: MFMcGraw gT 2 9.8 m/s 2 1.0 s L 0.25 m 2 2 4 4 Chap 10c - Elas & Vibrations - Revised 4-3-10 2 41 The gravitational potential energy of a pendulum is U = mgy. Taking y = 0 at the lowest point of the swing, show that y = L(1-cos). Lcos L L y L(1 cos ) y=0 MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 42 The Physical Pendulum A physical pendulum is any rigid object that is free to oscillate about some fixed axis. The period of oscillation of a physical pendulum is not necessarily the same as that of a simple pendulum. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 43 The Physical Pendulum http://hyperphysics.phy-astr.gsu.edu/HBASE/pendp.html MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 44 Damped Oscillations When dissipative forces such as friction are not negligible, the amplitude of oscillations will decrease with time. The oscillations are damped. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 45 Graphical representations of damped oscillations: MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 46 Damped Oscillations Overdamped: The system returns to equilibrium without oscillating. Larger values of the damping the return to equilibrium slower. Critically damped : The system returns to equilibrium as quickly as possible without oscillating. This is often desired for the damping of systems such as doors. Underdamped : The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. Source: Damping @ Wikipedia MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 47 Damped Oscillations The larger the damping the more difficult it is to assign a frequency to the oscillation. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 48 Forced Oscillations and Resonance A force can be applied periodically to a damped oscillator (a forced oscillation). When the force is applied at the natural frequency of the system, the amplitude of the oscillations will be a maximum. This condition is called resonance. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 49 Tacoma Narrows Bridge Nov. 7, 1940 MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 50 Tacoma Narrows Bridge Nov. 7, 1940 MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 51 Tacoma Narrows Bridge The first Tacoma Narrows Bridge opened to traffic on July 1, 1940. It collapsed four months later on November 7, 1940, at 11:00 AM (Pacific time) due to a physical phenomenon known as aeroelastic flutter caused by a 67 kilometres per hour (42 mph) wind. The bridge collapse had lasting effects on science and engineering. In many undergraduate physics texts the event is presented as an example of elementary forced resonance with the wind providing an external periodic frequency that matched the natural structural frequency (even though the real cause of the bridge's failure was aeroelastic flutter[1]). Its failure also boosted research in the field of bridge aerodynamics/ aeroelastics which have themselves influenced the designs of all the world's great long-span bridges built since 1940. - Wikipedia http://www.youtube.com/watch?v=3mclp9QmCGs MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 52 Chapter 10: Elasticity & Oscillations • • • • • • • • • • MFMcGraw Elastic deformations of solids Hooke's law for tensile and compressive forces Beyond Hooke's law Shear and volume deformations Simple harmonic motion The period and frequency for SHM Graphical analysis of SHM The pendulum Damped oscillations Forced oscillations and resonance Chap 10c - Elas & Vibrations - Revised 4-3-10 53 Extra MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 54 Aeroelasticity Aeroelasticity is the science which studies the interactions among inertial, elastic, and aerodynamic forces. It was defined by Arthur Collar in 1947 as "the study of the mutual interaction that takes place within the triangle of the inertial, elastic, and aerodynamic forces acting on structural members exposed to an airstream, and the influence of this study on design." MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 55 Aeroelastic Flutter Flutter Flutter is a self-feeding and potentially destructive vibration where aerodynamic forces on an object couple with a structure's natural mode of vibration to produce rapid periodic motion. Flutter can occur in any object within a strong fluid flow, under the conditions that a positive feedback occurs between the structure's natural vibration and the aerodynamic forces. That is, that the vibrational movement of the object increases an aerodynamic load which in turn drives the object to move further. If the energy during the period of aerodynamic excitation is larger than the natural damping of the system, the level of vibration will increase, resulting in self-exciting oscillation. The vibration levels can thus build up and are only limited when the aerodynamic or mechanical damping of the object match the energy input, this often results in large amplitudes and can lead to rapid failure. Because of this, structures exposed to aerodynamic forces - including wings, aerofoils, but also chimneys and bridges - are designed carefully within known parameters to avoid flutter. It is however not always a destructive force; recent progress has been made in small scale (table top) wind generators for underserved communities in developing countries, designed specifically to take advantage of this effect. In complex structures where both the aerodynamics and the mechanical properties of the structure are not fully understood flutter can only be discounted through detailed testing. Even changing the mass distribution of an aircraft or the stiffness of one component can induce flutter in an apparently unrelated aerodynamic component. At its mildest this can appear as a "buzz" in the aircraft structure, but at its most violent it can develop uncontrollably with great speed and cause serious damage to or the destruction of the aircraft. In some cases, automatic control systems have been demonstrated to help prevent or limit flutter related structural vibration. Flutter can also occur on structures other than aircraft. One famous example of flutter phenomena is the collapse of the original Tacoma Narrows Bridge. MFMcGraw Chap 10c - Elas & Vibrations - Revised 4-3-10 56