General Information Mechanical Vibrations Lesson 1 Grade Breakdown: Midterm Exam 45% Final Exam 55% Homework and Quiz 5% (Extra)

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1 General Information Instructor: Name: Withit Chatlatanagulchai Office: 9/, 9th floor of Engineering Building Office Phone: ext 858 Mobile Phone: Website: Lecture: Section : Monday Room E33 Section 4: Friday Room E34 Section 45: Tuesday Room 63 Downloadable Class Notes: Class notes for all lessons can be downloaded from my website above. Required Textbook: [] Mechanical Vibrations, by Singiresu S. Rao, Prentice Hall, 4 Recommended Textbooks: [] Theory of Vibration with Applications, by William T. Thomson and Marie Dillon Dahleh, Prentice Hall, 998 [3] Mechanical Vibration, by William J. Palm III, Wiley, 7 References: [4] Fundamentals of Vibrations, by Leonard Meirovitch, Mc Graw Hill, [5] Principles and Techniques of Vibrations, by Leonard Meirovitch, Prentice Hall, 997 [6] Vibration with Control, by Daniel J. Inman, Wiley, 6 [7] Vibration Problems in Engineering, by William Weaver Jr, Stephen P. Timoshenko, Donovan H. Young, Wiley, 99 [8] Dynamics and Control of Structures, by Leonard Meirovitch, Wiley, 99 [9] Mechanical Vibration: Analysis, Uncertainties, and Control, by Haym Benaroya, Marcel Dekker, 4 83 Mechanical Vibrations Grade Breakdown: Midterm Exam 45% Final Exam 55% Homework and Quiz 5% (Extra) Evaluation: + A 8% 75 B < B< C < C < 55 4 D < 45 3 D< 4 F < 3 Office Hours: You are welcome to come into my office at any time. Class Structure: Three hours are divided into three sections of one hour. In each hour, the class structure is as follows: 5 minutes: Study new materials (or discuss homework due that day) minutes: Break Rules: ) Close notes and books during exams. Open own notes and books during quizzes. ) During exams and quizzes, only non-programmable calculator is allowed (Casio fx38 or compatible.) 3) Homework problems are from the required textbook []. Each homework is due at next class. Homework is graded based on completion not correctness. Homework is an individual effort. You may consult your friends or me, but copying is strictly prohibited. 4) I do not accept late homework. 5) All in-class quizzes are random. 6) Grade sheet, which includes your score, ranking, and estimated grade, is posted on my website weekly. 7) Be punctual. If you are late, wait until next break to get in. 8) I do not accept students who transfer from other sections. Copyright 7 by Withit Chatlatanagulchai

2 Course Schedule: Monday Tuesday Wednesday Thursday Friday Saturday Sunday 4 June 5 June 6 June 7 June 8 June 9 June June () (45) (4) June () Lesson June 3 June (45) Lesson 4 June 5 June (4) Lesson 6 June 7 June 8 June () Lesson 3 5 June () Lesson 4 July () Lesson 5 9 July () Lesson 6 6 July () Lesson 7 3 July () Lesson 8 3 July Midterm Week 6 August () Lesson 9 3 August Queen s Birthday Subst. August () 7 August () 3 September () September () 3 7 September () 4 4 September October Add/Drop (KU 3) 9 June June June June 3 June 4 June (45) Lesson 3 (4) Lesson 3 6 June 7 June 8 June 9 June 3 June July (45) Lesson 4 (4) Lesson 4 3 July 4 July 5 July 6 July 7 July 8 July (45) Lesson 5 (4) Lesson 5 Drop (No Grade W) July July July 3 July 4 July 5 July (45) Lesson 6 (4) Lesson 6 7 July 8 July 9 July July July July (45) Lesson 7 (4) Lesson 7 4 July 5 July 6 July 7 July 8 July 9 July Commencemenmenmenment Commence- Commence- Commence- Midterm Week Midterm Week 3 July Midterm Week August Midterm Week August Midterm Exam.-5.PM 3 August Midterm Week 4 August 5 August 7 August 8 August (45) Lesson 8 9 August August (4) Lesson 8 August August 4 August 5 August 6 August 7 August 8 August 9 August (45) Lesson 9 (4) Lesson 9 August August (45) 8 August 9 August (45) Drop 3 (Grade W) 4 September 5 September (45) September September (45) 3 8 September 9 September (45) 4 5 September October 6 September 3 October 3 August 4 August (4) 3 August 3 August (4) 6 September 7 September (4) 3 September 4 September (4) 3 September September (4) 4 7 September 4 October Final Exam 3.-6.PM 8 September 5 October 5 August 6 August September September 8 September 9 September 5 September 6 September September 3 September 9 September 3 September 6 October 7 October 83 Mechanical Vibrations Course Content: Contents and homework problems are based on the required textbook []. Lesson Contents Introduction to Vibrations Degrees of Freedom Categories of Vibrations Harmonic Motion Exponential Form Periodic Motion Fourier Series Vibration Terminology Vibration Analysis Procedure Vibration Model Mechanical Elements One Degree of Freedom, Free Vibrations Systems with Mass and Spring Natural Frequency 3 Systems with Mass, Spring, and Damper Underdamped Case Overdamped Case Critically Damped Case Logarithmic Decrement Coulomb Damping 4 EOM : Newton s Method EOM : Energy Method 5 EOM 3: Equivalent System Method (Rayleigh s Method) EOM 4: Virtual Work Method (D Alembert s Method) 6 One Degree of Freedom, Harmonically Excited Vibrations Forced Harmonic Vibration Frequency Response Curves Rotating Unbalance Support Motion 7 Vibration Isolation Sharpness of Resonance Energy Dissipated by Damping Vibration-Measuring Instruments Copyright 7 by Withit Chatlatanagulchai

3 Midterm Exam ( - 7) 83 Mechanical Vibrations 8 One Degree of Freedom, Nonperiodically Excited Vibrations Laplace Transformation Laplace Transform of Functions Laplace Transform of Derivatives Shifting Theorem Transformation of Ordinary Differential Equations Inverse Laplace Transform 9 Impulse Excitation Arbitrary Excitation Convolution Integral Laplace Transform Formulation Pulse Excitation Shock Response Spectrum Shock Isolation Numerical Methods Finite Difference Method Runge-Kutta Method Two Degrees of Freedom, Free Vibrations EOM : Newton s Method Normal Mode Analysis Initial Conditions Two Degrees of Freedom, Harmonically Excited Vibrations Forced Harmonic Vibration Vibration Absorber 3 Multidegrees of Freedom Coordinate Coupling Orthogonality of Eigenvectors Modal Matrix Decoupling Forced Vibration Equations 4 EOM 5: Lagrange s Method Generalized Coordinates Lagrange s Equation Final Exam (Lesson 8-4) 3 Copyright 7 by Withit Chatlatanagulchai

4 Introduction to Vibrations Any motion that repeats itself after an interval of time is called vibration or oscillation. Examples of vibration are we hear because our eardrums vibrate we see because light waves undergo vibration breathing is vibration of lungs walking involves oscillatory motion of legs and hands we speak due to the oscillation of tongue destruction of Tacoma Narrows bridge 83 Mechanical Vibrations A vibratory system includes mass (to store kinetic energy), spring (to store potential energy), and damper (means by which energy is gradually lost.). Degrees of Freedom Degrees of Freedom is the minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time. For example, a free particle undergoing general motion in space has three degrees of freedom (x,y,z) a rigid body in space has six degrees of freedom (three components for position and three components for orientation) a continuous elastic body has infinite number of degrees of freedom The coordinates necessary to describe the motion of a system constitute a set of generalized coordinates. Example : [3] Determine number of degrees of freedom and generalized coordinates of the following systems. Figure : Destruction of Tacoma Narrows bridge []. a) explosion of space shuttle Challenger Figure : Explosion of Challenger []. Solution DOF, θ. 4 Copyright 7 by Withit Chatlatanagulchai

5 b) e) 83 Mechanical Vibrations θ, θ, θ. Solution 3 DOF, 3 c) Solution 6 DOF, x, y, z, θ, θ, θ. 3 f) Solution infinite degrees of freedom d) Solution DOF, x, θ. x, y, z, θ, θ, θ, θ. Solution 7 DOF, r l 5 Copyright 7 by Withit Chatlatanagulchai

6 . Categories of Vibrations Discrete versus Continuous Systems Systems with a finite number of degrees of freedom are called discrete systems or lumped parameter systems. Systems with an infinite number of degrees of freedom are called continuous systems or distributed systems. Most of the time, continuous systems are approximated as discrete systems, and solutions are obtained in a simpler manner. Free versus Forced Vibrations Free vibration is vibration of a system, after an initial disturbance, is left to vibrate on its own without external force acting on the system. Forced vibration is vibration of a system that is subjected to an external force. The system under free vibration will vibrate at one or more of its natural frequencies. When the system is excited (forced), the system is forced to vibrate at the excitation frequency. Undamped versus Damped Vibrations If no energy is lost or dissipated in friction or other resistance during oscillation, the vibration is known as undamped vibration. If any energy is lost in this way, it is called damped vibration. Linear versus Nonlinear Vibrations A linear system obeys principle of superposition shown in Figure 3. If the system does not obey principle of superposition, the system is said to be nonlinear system and hence having nonlinear vibration. au u y Vibrating System u y Vibrating System + bu ay + by Vibrating System Figure 3: Principle of superposition of a linear system. 83 Mechanical Vibrations Deterministic versus Random Vibrations Vibration that results from excitation (force or motion) that can be predicted at any given time is called deterministic vibration. If the excitation at a given time cannot be predicted, the vibration is called nondeterministic or random vibrations. Harmonic Motion Figure 4: Deterministic and random excitations Harmonic motion is motion of an oscillating system that vibrates at one fixed frequency. Consider Figure 5, the harmonic motion can be represented by a vector OP of magnitude A rotating at a constant angular velocity ω. The projection of the tip of the vector OP on the vertical axis is given by y = Asin ωt, and the projection on the horizontal axis is given by x = Acos ωt. If we replace the x-y plane with a complex plane where x-axis represents real axis and y-axis represents imaginary axis, we can write the vector OP in complex format as Z = x+ iy = Acosωt+ i Asin ωt, where i =. 6 Copyright 7 by Withit Chatlatanagulchai

7 . Exponential Form Using Euler s equation 83 Mechanical Vibrations we can write the vector OP in exponential form as Z = Acosωt+ i Asinωt = Ae iωt. e iθ = cosθ + isin θ, ()... Algebra of the Exponential Form i t Let Z = Ae ω i t and Z = Ae ω exponential form, we have ( Z ) Z + Z = Ae + Ae Z Z Ae Ae iωt iωt iωt iωt = i( ω+ ω) t ZZ = AAe, Z A i e Z = A n n inω t = Ae be two different vectors in ( ω ω ). t,,, Figure 5: Harmonic motion as the projection of the end of a rotating vector. 7 Copyright 7 by Withit Chatlatanagulchai

8 Example : [3] Find the sum of the two harmonic motions ( ) = ωt and x ( t) = ( ωt+ ) x t cos 5cos. 83 Mechanical Vibrations 8 Copyright 7 by Withit Chatlatanagulchai

9 Solution Let the sum of the two harmonic motions be Then, ( ) = cos( ω + α) = ( ωt)( A α) ( ωt)( A α) xt A t cos cos sin sin. ( ) = ( ) + ( ) = cosωt+ 5cos( ωt+ ) = cosωt+ 5( cosωt cos sinωt sin ) = ( ωt)( + ) ( ωt)( ) xt x t x t Comparing () with (3), we have and cos 5cos sin 5sin. () (3) A cosα = + 5cos, (4) Asinα = 5sin. (5) 83 Mechanical Vibrations... Velocity and Acceleration of Harmonic Motion Consider a rotating vector Z = Ae iωt, time are The projection of whose derivatives are given as d x dz iωt = iωae = iωz, dt dz iωt = ω Ae = ω Z. dt i t Z = Ae ω its derivatives with respect to on the horizontal axis is dx = ωasinωt = ωacos ωt+, dt dt ( ) = ω Acosωt = ω Acos ωt+. (6) x = Acosωt We can see from both (6) and (7) that the acceleration vector leads the velocity vector by 9 degrees, and the velocity vector leads the displacement vector by 9 degrees as shown in Figure 6. (7) From (4) and (5), we have ( 5cos ) ( 5sin ) A = + + = 4.48, + 5cos α = cos 4.48 =.3 rad. Therefore, we have the sum as ( ) = ( ωt+ ) xt 4.48cos.3. Figure 6: Displacement, velocity, and acceleration vectors as rotating vectors. 9 Copyright 7 by Withit Chatlatanagulchai

10 3 Periodic Motion Periodic motion is motion of an oscillating system that vibrates at several different frequencies simultaneously. Examples of periodic motions are the vibration of a violin string is composed of the fundamental frequency f and all of its harmonics, f,3 f, and so on free vibration of a multidegree-of-freedom system, to which the vibrations at each natural frequency contribute. 83 Mechanical Vibrations Example 3: [5] Find the Fourier series of the following periodic function f ( x) k, < x< = k,< x< f with f ( x+ ) = f ( x) ( x) k 3. Fourier Series A French mathematician J. Fourier (768-83) showed that any periodic signal can be approximated by a series of sines and cosines that are harmonically related. Any periodic signal with period Fourier series τ = L can be approximated by the nx nx f ( x) = a + ancos + bnsin. n= L L (8) a,, a n and b n are called Fourier coefficients with following equations x k L a = f ( x) dx, L L L n x an = f ( x) cos dx, L L L L n x bn = f ( x) sin dx. L L L (9) Copyright 7 by Withit Chatlatanagulchai

11 83 Mechanical Vibrations S( x) S ( x) S3 ( x) Copyright 7 by Withit Chatlatanagulchai

12 Solution In (8), the period is From (9), we have L. Since the signal has period, we have L =, L =. a = f ( x) dx = kdx+ =, kdx 3... Complex Form 83 Mechanical Vibrations The Fourier series can be written in complex form using the Euler s equation () as follows = n in x / L, n= L cn = L in x / L f ( x) e dx ( ) f x c e L 3... Gibbs Phenomenon Figure 7 describes Gibbs phenomenon. As the number of terms of the Fourier series increases, the approximation can be seen to improve everywhere except near the discontinuity (point P.). n x an = f ( x) cos dx =, n x bn = f ( x) sin dx = k sin nx dx k sin nx dx + k k = cos nx cos nx n n k k k k = cos( n) cos n + n n n n k = ( cos n + ). n Figure 7: Gibbs phenomenon. Copyright 7 by Withit Chatlatanagulchai

13 3..3. Frequency Spectrum Frequency spectrum or spectral diagram is plot of amplitude of each term in the Fourier series (8) with its corresponding circular frequency ω. If we expand (8), using ω = / τ, we have 83 Mechanical Vibrations ( ) = + cosω + f x a a x b sinωx + acos ωx+ bsin ωx + a cos3ωx+ b sin 3ωx+ 3 3 Figure 8 shows an example of the frequency spectrum (using n and instead of and respectively.) a n b n d φ n Figure 8: Frequency spectrum of a typical periodic function of time. The following plots show two examples of the frequency spectrum. 3 Copyright 7 by Withit Chatlatanagulchai

14 4 Vibration Terminology Certain terminologies used in vibration analysis are as follows. Peak Value generally means the maximum value of a vibrating body. Amplitude is the maximum displacement of a vibrating body from its equilibrium position. Average Value or Mean Value indicates a steady or static value. It can be found by the time integral x = lim xt () dt. T T If the signal x( t ) is periodic, the formula above reduces to T 83 Mechanical Vibrations Example 4: [4] Find the mean value and the mean square value of the following signals: a) x t) = Asin t ( b) a rectified sine wave x( t) Asin t t where τ is the period of the signal x( t ). τ x = xt () dt, τ () Mean Square Value is the average of the squared values, integrated over some time interval T T = T T () x lim x t dt. If the signal x( t ) is periodic, the formula above reduces to τ x x t dt, where τ is the period of the signal x( t ). () = τ () 4 Copyright 7 by Withit Chatlatanagulchai

15 Solution a) Since the signal ( ) sin x t = A t has period, 83 Mechanical Vibrations using (), we get Using (), we have x = Asint dt =. () x = x t dt = A sin A cost = dt A = t sin t 4 A = A =. ( ) t dt b) Since the rectified sine wave has period, using (), we get x = Asin t dt A A = ( cos t ) =. 5 Copyright 7 by Withit Chatlatanagulchai

16 Using (), we get x = x t dt = A = A () sin t dt cos t dt A = t sin t 4 A = A =. Root Mean Square Value is the square root of the mean square value Decibel is a unit of measurement that is frequently used in vibration measurements. It is defined in terms of a power ratio p db = log. p Since power is proportional to the square of the amplitude or voltage, we have x x x x db = log = log. 83 Mechanical Vibrations Thus an amplifier with a voltage gain of 5 has a decibel gain of ( ) log 5 = + 4. Octave is any frequency span with the maximum value twice the minimum value. For example, each of the ranges 8-6 Hz, - Hz, 3-6 Hz can be called an octave. Cycle describes the movement of a vibrating body from its undisturbed or equilibrium position to its extreme position in one direction, then to the equilibrium position, then to its extreme position in the other direction, and back to equilibrium position. Period of Oscillation is the time taken to complete one cycle of motion τ =, ω when ω is called the circular frequency. Frequency of Oscillation is the number of cycles per unit time ω f = =. τ Phase Difference generally means the phase difference between two signals. Consider x = Asin ωt, x = A sin t+, ( ω φ ) the phase difference is φ that means the maximum of the second vector would occur φ radians earlier than that of the first vector. Natural Frequency is the frequency of any system left to vibrate on its own without external force. A vibratory system having n degrees of freedom has n distinct natural frequencies of vibration. Beating Phenomenon describes motion resulting from adding two harmonic motions with frequencies close to each other. For example, if we add the two signals 6 Copyright 7 by Withit Chatlatanagulchai

17 ( ) = cos ω, () = ( ω + δ ) x t A t x t Acos t, where δ is small value, using the trigonometry identity A + B A B A B cos + cos = cos cos, we have References 83 Mechanical Vibrations [] [] [3] Mechanical Vibrations, by Singiresu S. Rao, Prentice Hall, 4 [4] Theory of Vibration with Applications, by William T. Thomson and Marie Dillon Dahleh, Prentice Hall, 998 [5] Advanced Engineering Mathematics, by Erwin Kreyszig, Wiley, 6 ( ) = ( ) + ( ) xt x t x t The plot of x( t ) is given in Figure 9. δt δ = Acos cos ω + t. Figure 9: Beating phenomenon. Homework Problems ,.5,.5,.58,.66,.7 Homework problems are from the required textbook (Mechanical Vibrations, by Singiresu S. Rao, Prentice Hall, 4) 7 Copyright 7 by Withit Chatlatanagulchai