Mechanical Vibration CH 1 Introduction and basic concepts - for VIDEO (1).pdf
- 1. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Mechanical Vibrations FUNDAMENTALS OF VIBRATION Prof. Carmen Muller-Karger, PhD Florida International University Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition. Chapter 1: Fundamentals of Vibration
- 2. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Learning Objectives • Recognize the importance of studying Vibration • Describe a brief the history of vibration • Understand the definition of Vibration • State the process of modeling systems • Determine the Degrees of Freedom (DOF) of a system • Identify the different types of Mechanical Vibrations • Compute equivalent values for Spring elements, Mass elements and Damping elements • Characterize harmonic motion and the different possible representation • Add and subtract harmonic motions • Conduct Fourier series expansion of given periodic functions
- 3. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Importance of studying Vibration • All systems that have mass and any type of flexible components are vibrating system. • Examples are many: • We hear because our eardrums vibrate • Human speech requires the oscillatory motion of larynges • In machines, vibration can loosen fasteners such as nuts. • In balance in machine can cause problem to the machine itself or surrounding machines or environment. • Periodic forces bring dynamic responses that can cause fatigue in materials • The phenomenon known as Resonance leads to excessive deflections and failure. • The vibration and noise generated by engines causes annoyance to people and, sometimes, damage to property.
- 4. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Importance of studying Vibration
- 5. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Brief history • People became interested in vibration when they created the first musical instruments ( as long as 4000 B.C.). • Pythagoras ( 582 – 507 B.C) is considered the fisrt person to investigate musical sounds. • Galileo Galilei (1564-1642) is considered to be the founder of modern experimental science, he conduct experiments on the simple pendulum, describing the dependence of the frequency of vibration and the length. • Robert Hooke (1635–1703) also conducted experiments to find a relation between the pitch and frequency of vibration of a string. • Joseph Sauveur (1653–1716) coined the word “acoustics” for the science of sound. • Sir Isaac Newton (1642–1727) his law of motion is routinely used to derive the equations of motion of a vibrating body. • Brook Taylor (1685–1731), obtained the natural frequency of vibration observed by Galilei and Mersenne. • Daniel Bernoulli (1700–1782), Jean D’Alembert (1717–1783), and Leonard Euler (1707–1783)., introduced partial derivatives in the equations of motion. • J. B. J. Fourier (1768–1830) contributed on the development of the theory of vibrations and led to the possibility of expressing any arbitrary function using the principle of superposition. • Joseph Lagrange (1736–1813) presented the analytical solution of the vibrating string. • Charles Coulomb did both theoretical and experimental studies in 1784 on the torsional oscillations of a metal cylinder suspended by a wire. He also contributed in the modeling of dry friction. • E. F. F. Chladni (1756–1824) developed the method of placing sand on a vibrating plate to find its mode shapes. • Simeon Poisson (1781–1840) study vibration of a rectangular flexible membrane. • Lord Baron Rayleigh (1842 – 1919) Among the many contributions, he develop the method of finding the fundamental frequency of vibration of a conservative system by making use of the principle of conservation of energy.
- 6. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Definition of Vibration • Any motion that repeats itself after an interval of time. • A vibratory system, in general, includes a means for storing potential energy (spring or elasticity), a means for storing kinetic energy (mass or inertia), and a means by which energy is gradually lost (damper). Excitations (input): Initial conditions of external force Responses (output) T U D Energy dissipation F(t) r(t)
- 7. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Modeling systems • All mechanical and structural systems can be modeled as mass-spring- damper systems Real system Mechanical Model Mathematical Model Solution Analysis 𝑚 ሷ 𝑥 + 𝑐 ሶ 𝑥 + 𝑘𝑥 = 0 Respuesta análisis del sistema para que la respuesta sea coherente. 𝑚 ሷ 𝑥 + 𝑐 ሶ 𝑥 + 𝑘𝑥 = 0 Respuesta análisis del sistema para que la respuesta sea coherente.
- 8. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Degrees of freedom Single DoF systems: Two DOF System The minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time Three DoF systems:
- 9. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Classification of Vibration • Free Vibration: When a system, after an initial disturbance, is left to vibrate on its own. No external force acts on the system. The system oscillates at its natural frequency. Example: a pendulum. • Forced Vibration: When a system is subjected to an external force (often, a repeating type of force). The oscillation that arises in machines such as diesel engines is an example of forced vibration.
- 10. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Types of response • Undamped and damped Vibration • Linear of nonlinear Vibration • Deterministic ond Random Vibration 𝑙 ሷ 𝜃 + 𝑚𝑔𝑠𝑖𝑛𝜃 = 0
- 11. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Mechanical Vibrations SPRING, MASS and DAMPING ELEMENTS Prof. Carmen Muller-Karger, PhD Florida International University Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition. Chapter 1: Fundamentals of Vibration
- 12. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Spring Elements The force related to a elongation or reduction in length opposes to the displacement of the end of the spring and is given by : 𝐹 = 𝑘𝑥 The work done (U) in deforming a spring is stored as strain or potential energy in the spring, and it is given by 𝑈 = 1 2 𝑘𝑥2 𝐹 = 𝑎𝑥 + 𝑏𝑥3 Non-linear spring element: ቤ 𝑘 = 𝑑𝐹 𝑑𝑥 𝑥∗
- 13. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Example: Spring constant of a rod • Find the equivalent spring constant of a uniform rod of length l, cross- sectional area A, Area Moment of Inertia I and Young’s modulus E 𝛿 = 𝛿 𝑙 𝑙 = 𝜀𝑙 = 𝐹𝑙 𝐴𝐸 𝑘 = 𝐹𝑜𝑟𝑐𝑒 𝑑𝑒𝑓𝑙𝑒𝑥𝑖𝑜𝑛 = 𝑊 𝛿 = 3𝐸𝐼 𝑙3 𝛿 = 𝑊𝑙3 3𝐸𝐼 𝑘 = 𝐹𝑜𝑟𝑐𝑒 𝑑𝑒𝑓𝑙𝑒𝑥𝑖𝑜𝑛 = 𝐹 𝛿 = 𝐴𝐸 𝑙 1. Subjected to an axial tensile (or compressive) force F 2. Cantilever bean subjected to a transversal load at the free end.
- 14. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Original system Equivalent model Spring Constant of a Rod under axial load Cantilever beam with end force Simple support beam with load in the middle Propeller Shaft subjected to a torsional moment Equivalent spring constants 𝑘 = 𝐴𝐸 𝑙 𝑘 = 3𝐸𝐼 𝑙3 𝑘 = 𝐺𝐽 𝑙 𝐽 = 𝜋(𝐷4 − 𝑑4 ) 32 𝑘 = 48𝐸𝐼 𝑙3
- 15. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Combination of Springs Springs in Parallel Springs in Series W= 𝑘1𝛿𝑠𝑡 + 𝑘2𝛿𝑠𝑡 W= (𝑘1+𝑘2)𝛿𝑠𝑡 W= (𝑘𝑒𝑞)𝛿𝑠𝑡 𝑘𝑒𝑞 = (𝑘1+𝑘2) W= 𝑘1𝛿1 = 𝑘2(𝛿𝑠𝑡−𝛿1) = 𝑘𝑒𝑞𝛿𝑠𝑡 𝛿1 = 𝑊 𝑘1 1 𝑘𝑒𝑞 = 1 𝑘1 + 1 𝑘2 𝛿1 = 𝑘2(𝛿2−𝛿1) 𝛿𝑠𝑡 = 𝑊 𝑘𝑒𝑞 𝑘2 𝛿𝑠𝑡 − 𝑊 𝑘1 = 𝑊 𝛿𝑠𝑡 = 𝑊 𝑘2 + 𝑊 𝑘1
- 16. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Mass or Inertia Elements • The mass or inertia element is assumed to be a rigid body; it can gain or lose kinetic energy whenever the velocity of the body changes. • For single DoF system for a simple analysis, we can replace several masses by a single equivalent mass. • The key step is to choose properly a parameter that will describe the motion of the system and express all other parameters in term of the chosen one. • Calculate the kinetic energy of the system and make it equal to the kinetic energy of the equivalent system
- 17. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Kinetic energy ҧ 𝑣 = ҧ 𝑣𝑝 + ഥ 𝜔𝑧 × ҧ 𝑟 x´ y´ 𝑝 𝐺 GENERAL MOTION 𝜔 𝛼 dm x y 𝑜 In general for one rigid body the kinetic energy can be calculated as 𝑇 = 1 2 𝑚( ҧ 𝑣𝑝)2 + 1 2 𝐼𝑝𝑧𝑧(𝜔𝑧)2 + ҧ 𝑣𝑝 ∙ ഥ 𝜔𝑧 × 𝑚 ҧ 𝑟𝐺 For a system with several rigid bodies is the sum of the kinetic energy of each body:
- 18. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Example 1: Translational masses by a rigid bar 𝑇 = 1 2 𝑚1( ሶ 𝑥1)2 + 1 2 𝑚2( ሶ 𝑥2)2 + 1 2 𝑚3( ሶ 𝑥3)2 ሶ 𝑥𝑒𝑞 = ሶ 𝑥1 ሶ 𝑥3 = 𝑙3 𝑙1 ሶ 𝑥1 ሶ 𝑥2 = 𝑙2 𝑙1 ሶ 𝑥1 𝑇 = 1 2 𝑚1( ሶ 𝑥1)2 + 1 2 𝑚2 𝑙2 𝑙1 ሶ 𝑥1 2 + 1 2 𝑚3 𝑙3 𝑙1 ሶ 𝑥1 2 𝑇 = 1 2 𝑚1 + 𝑚2 𝑙2 𝑙1 2 + 𝑚3 𝑙3 𝑙1 2 ( ሶ 𝑥1)2 𝑚𝑒𝑞 = 𝑚1 + 𝑚2 𝑙2 𝑙1 2 + 𝑚3 𝑙3 𝑙1 2 • Let’s choose ሶ 𝑥1 as the parameter that will describe the motion of the system and find the equivalent kinetic energy
- 19. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Example 2: Translational and rotational masses coupled together Since the system is rotating without slipping and the gear is fix at its center ሶ 𝜃 = ሶ 𝑥/𝑅 𝑇 = 1 2 𝑚( ሶ 𝑥)2 + 1 2 𝐽𝑜( ሶ 𝜃)2 • If we choose as the parameter that will describe the motion of the system ሶ 𝑥𝑒𝑞 = ሶ 𝑥 𝑇 = 1 2 𝑚( ሶ 𝑥)2 + 1 2 𝐽𝑜 ሶ 𝑥 𝑅 2 • If we choose as the parameter that will describe the motion of the system ሶ 𝑥𝑒𝑞 = ሶ 𝜃 𝑇 = 1 2 𝑚( ሶ 𝑥)2 + 1 2 𝐽𝑜( ሶ 𝜃)2 𝑇 = 1 2 𝑚( ሶ 𝜃𝑅)2 + 1 2 𝐽𝑜 ሶ 𝜃 2 𝑚𝑒𝑞 = 𝑚𝑅2 + 𝐽𝑜 𝑇 = 1 2 𝑚𝑅2 + 𝐽𝑜 ( ሶ 𝜃)2
- 20. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Damping element • In many practical systems, the vibrational energy is gradually converted to heat or sound. Due to the reduction in the energy, the response, such as the displacement of the system, gradually decreases. Damping force exists only if there is relative velocity between the two ends of the damper. • We will consider three types of damping: • Viscous Damping: the damping force is proportional to the velocity of the vibrating body. • Coulomb or Dry-Friction Damping: damping force is constant in magnitude but opposite in direction to that of the motion • Material or Solid or Hysteretic Damping: due to friction between the internal planes, which slip or slide as the deformations take place
- 21. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations In general viscous Dampers Dampers in Parallel Dampers in Series 𝑐𝑒𝑞 = (𝑐1+𝑐2) 1 𝑐𝑒𝑞 = 1 𝑐1 + 1 𝑐2 Non-linear damper element: ቤ 𝑐 = 𝑑𝐹 𝑑 ሶ 𝑥 𝑥∗ Non-linear damper element: 𝐹 = 𝑐𝑣
- 22. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Damping constant of parallel plates separated by viscous fluid. • According to Newton’s laws of viscous flow: 𝜏 = 𝜇 𝑑𝑢 𝑑𝑦 𝐹 = 𝜏𝐴 = 𝜇𝐴𝑣 ℎ 𝑑𝑢 𝑑𝑦 = 𝑣 ℎ 𝐹 = 𝑐𝑣 = 𝜇𝐴 ℎ 𝑣 𝑐 = 𝜇𝐴 ℎ • If A is the surface area at the moving plate, and expressing the force in term of the damping constant: • Gradient of the velocity:
- 23. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Damping constant of a journal bearing • According to Newton’s laws of viscous flow, u is the radial velocity, and v the tangential velocity of the shaft: 𝜏 = 𝜇 𝑑𝑢 𝑑𝑟 𝑇 = (𝜏𝐴)𝑅 = 𝜇(𝜔𝑅) 2𝜋𝑅𝑙 𝑅 𝑑 𝑇 = 𝑐𝜔 𝑐 = 𝜇2𝜋𝑅3 𝑙 𝑑 𝑑𝑢 𝑑𝑟 = 𝑣 𝑑 = 𝜔𝑅 𝑑 If A is the surface area at the moving shaft (2𝜋𝑅𝑙), and expressing the torque T=F R in term of the damping constant: • Gradient of the velocity:
- 24. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Mechanical Vibrations Harmonic Motion Prof. Carmen Muller-Karger, PhD Florida International University Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition. Chapter 1: Fundamentals of Vibration
- 25. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Harmonic Motion or Periodic Motion, Definitions and Terminology Harmonic motion: Motion is repeated after equal intervals of time Cycle: The movement from one position, going to the other direction and returning to the same position Amplitude: The maximum displacement of a vibrating body from its equilibrium position. Period of oscillation: The time taken to complete one cycle of motion, is denoted by τ. Frequency of oscillation: The number of cycles per unit time Circular frequency of oscillation: The number of cycles per unit time. 𝜏 = 2𝜋 𝜔 𝑓 = 1 𝜏 = 𝜔 2𝜋 𝜔 = 2𝜋 𝜏 𝐴
- 26. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Harmonic Motion or Periodic Motion, Definitions and Terminology Synchronous motion : have the same frequency or angular velocity, ω. Need not have the same amplitude, and they need not attain their maximum values at the same time. Phase angle: means that the maximum of the second vector would occur f radians earlier than that of the first vector. wt wt+f O P1 P2 Vectorial representation 𝑥 𝑡 = sin(𝜔𝑡) ሶ 𝑥 𝑡 = −ω𝑐𝑜𝑠(𝜔𝑡) Displacement: Velocity: Acceleration: ሷ 𝑥 𝑡 = −𝜔2 sin(𝜔𝑡)
- 27. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Complex-number representation • Any vector in the xy-plane can be represented as a complex number: ത 𝑋 = a + ib = 𝐴 cos 𝜔𝑡 + 𝑖𝑠𝑖𝑛 𝜔𝑡 • The magnitude • Using Euler form ത 𝑋 = A𝑒𝑖𝜔𝑡 = A𝑒𝑖𝜃 = 𝐴 cos 𝜔𝑡 + 𝑖𝑠𝑖𝑛 𝜔𝑡 A = 𝑎2 + 𝑏2 ത 𝑋 = A𝑒−𝑖𝜔𝑡 = A𝑒−𝑖𝜃 = 𝐴 cos 𝜔𝑡 − 𝑖𝑠𝑖𝑛 𝜔𝑡 ത 𝑋 = A𝑒𝑖𝜔𝑡 or ሶ ത 𝑋 = 𝑑 ത 𝑋 𝑑𝑡 = 𝑑(A𝑒𝑖𝜔𝑡 ) 𝑑𝑡 = 𝑖ωA𝑒𝑖𝜔𝑡 ሷ ത 𝑋 = 𝑑2 ത 𝑋 𝑑𝑡2 = 𝑑2 (A𝑒𝑖𝜔𝑡 ) 𝑑𝑡2 = −ω2 A𝑒𝑖𝜔𝑡 Displacement: Velocity: Acceleration: Expansion by series cos 𝜃 = 1 − 𝜃2 2! + 𝜃4 4! − 𝜃6 6! + ⋯ sin 𝜃 = 𝜃 − 𝜃3 3! + 𝜃5 5! − 𝜃7 7! + ⋯ For very small angles: cos 𝜃 ≈ 1 sin 𝜃 ≈ 𝜃
- 28. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Adding harmonic motion • Recall trigonometry identities: 𝑥1(𝑡) = 𝐴1 cos 𝜔𝑡 𝑥2 (𝑡) = 𝐴2 𝑠𝑖𝑛 𝜔𝑡 𝑥𝑡(𝑡) = 𝑥1 𝑡 + 𝑥2 𝑡 = 𝐴1 cos 𝜔𝑡 +𝐴2 sin 𝜔𝑡 = 𝐴𝑐𝑜𝑠 𝜔𝑡 − 𝛼 sin 𝑎 + 𝑏 = sin 𝑎 𝑐𝑜𝑠 𝑏 + cos 𝑎 𝑠𝑖𝑛 𝑏 cos 𝑎 + 𝑏 = 𝑐𝑜𝑠 𝑎 𝑐𝑜𝑠 𝑏 − sin 𝑎 𝑠𝑖𝑛 𝑏 • Adding harmonic motions: 𝑥𝑡(𝑡) = 𝐴𝑐𝑜𝑠 𝜔𝑡 − 𝛼 = Acos(𝛼) cos 𝜔𝑡 + Asin(𝛼) sin 𝜔𝑡 𝐴1 = Acos(𝛼) 𝐴2 = Asin(𝛼) 𝐴 = 𝐴1 2 + 𝐴2 2 = (Acos(𝛼))2+(Asin(𝛼))2 𝛼 = 𝑡𝑎𝑛−1 𝐴2 𝐴1 • Two different ways of write a harmonic motion: 𝒙𝒕(𝒕) = 𝑨𝟏 𝐜𝐨𝐬 𝝎𝒕 +𝑨𝟐 𝐬𝐢𝐧 𝝎𝒕 𝒙𝒕(𝒕) = 𝑨𝒄𝒐𝒔 𝝎𝒕 − 𝜶 or 𝑨 = 𝑨𝟏 𝟐 + 𝑨𝟐 𝟐 𝜶 = 𝒕𝒂𝒏−𝟏 𝑨𝟐 𝑨𝟏
- 29. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Adding harmonic motion • Recall trigonometry identities: cos A + cos B = 2 cos 𝐴 + 𝐵 2 cos 𝐴 − 𝐵 2 Phenomenon Beats: occurs when adding two harmonic motion with frequencies close too one another the resultant motion 𝒙𝟏(𝒕) = X𝒄𝒐𝒔 𝝎 𝒕 cos A − cos B = −2 sin 𝐴 + 𝐵 2 sin 𝐴 − 𝐵 2 sin A − sin B = 2 cos 𝐴 + 𝐵 2 sin 𝐴 − 𝐵 2 sin A + sin B = 2 sin 𝐴 + 𝐵 2 cos 𝐴 − 𝐵 2 𝒙𝟐(𝒕) = X𝒄𝒐𝒔 𝝎 + 𝜹 𝒕 𝒙(𝒕) = X𝒄𝒐𝒔 𝝎 𝒕+X𝒄𝒐𝒔 𝝎 + 𝜹 𝒕 𝒙(𝒕) = 𝐗 𝐜𝐨𝐬 𝜹 𝟐 𝒕 𝒄𝒐𝒔 𝝎 + 𝜹 𝟐 𝒕 Beats frequency is twice the frequency of the term 𝐗 𝐜𝐨𝐬 𝜹 𝟐 𝒕 since two peaks pass in each cycle.
- 30. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Harmonic Analysis: Fourier Series where ω=2π/τ is the fundamental frequency. To determine the coefficients , we multiply by cos(nωt) and sin(nωt), respectively, and integrate over one period τ=2π/ω—for example, from 0 to 2π/ω. 𝑥 𝑡 = 𝑎0 2 + 𝑎1 𝑐𝑜𝑠 𝜔𝑡 + 𝑎2 𝑐𝑜𝑠 2𝜔𝑡 + ⋯ 𝑏1 𝑠𝑖𝑛 𝜔𝑡 + 𝑏2 𝑠𝑖𝑛 2𝜔𝑡 + ⋯ 𝑥 𝑡 = 𝑎0 2 + 𝑛=1 ∞ 𝑎𝑛 𝑐𝑜𝑠 𝑛𝜔𝑡 + 𝑏𝑛 𝑠𝑖𝑛 𝑛𝜔𝑡 Any periodic function of time can be represented by Fourier series as an infinite sum of sine and cosine terms 𝑎0 = 𝜔 𝜋 න 0 2𝜋/𝜔 𝑥 𝑡 𝑑𝑡 = 2 𝜏 න 0 𝜏 𝑥 𝑡 𝑑𝑡 𝑎𝑛 = 𝜔 𝜋 න 0 2𝜋/𝜔 𝑥 𝑡 𝑐𝑜𝑠 𝑛𝜔𝑡 𝑑𝑡 = 2 𝜏 න 0 𝜏 𝑥 𝑡 𝑐𝑜𝑠 𝑛𝜔𝑡 𝑑𝑡 𝑏𝑛 = 𝜔 𝜋 න 0 2𝜋/𝜔 𝑥 𝑡 𝑠𝑖𝑛 𝑛𝜔𝑡 𝑑𝑡 = 2 𝜏 න 0 𝜏 𝑥 𝑡 𝑠𝑖𝑛 𝑛𝜔𝑡 𝑑𝑡
- 31. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Harmonic Analysis: Fourier Series Where: 𝑥 𝑡 = 𝑑0 + 𝑑1 𝑐𝑜𝑠 𝜔𝑡 − 𝜑1 + 𝑑2 𝑐𝑜𝑠 2𝜔𝑡 − 𝜑1 + ⋯ 𝑥 𝑡 = 𝑑0 + 𝑛=1 ∞ 𝑑𝑛 𝑐𝑜𝑠 𝑛𝜔𝑡 − 𝜑𝑛 Fourier series can also be represented by the sum of sine terms only or cosine terms only. 𝑑0 = 𝑎0 2 𝑑𝑛 = 𝑎𝑛 2 + 𝑏𝑛 2 𝜑𝑛 = 𝑡𝑎𝑛−1 𝑏𝑛 𝑎𝑛
- 32. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Fourier Series in complex numbers 𝑥 𝑡 = 𝑎0 2 + 𝑛=1 ∞ 𝑎𝑛 𝑒𝑖𝑛𝜔𝑡 + 𝑒−𝑖𝑛𝜔𝑡 2 + 𝑏𝑛 𝑒𝑖𝑛𝜔𝑡 − 𝑒−𝑖𝑛𝜔𝑡 2 = 𝑒𝑖𝜔0 𝑎0 2 − 𝑖 𝑏0 2 + 𝑛=1 ∞ 𝑒𝑖𝑛𝜔𝑡 𝑎𝑛 − 𝑖𝑏𝑛 2 + 𝑒−𝑖𝜔𝑡 𝑎𝑛 + 𝑖𝑏𝑛 2 𝑐−𝑛 = 𝑎𝑛 + 𝑖𝑏𝑛 2 𝑒𝑖𝜔𝑡 = cos 𝜔𝑡 + 𝑖𝑠𝑖𝑛 𝜔𝑡 𝑒−𝑖𝜔𝑡 = cos 𝜔𝑡 − 𝑖𝑠𝑖𝑛 𝜔𝑡 cos 𝜔𝑡 = 𝑒𝑖𝜔𝑡+𝑒−𝑖𝜔𝑡 2 sin 𝜔𝑡 = 𝑒𝑖𝜔𝑡−𝑒−𝑖𝜔𝑡 2 Since: The Fourier Series can be written as : With: 𝑐𝑛 = 𝑎𝑛 − 𝑖𝑏𝑛 2 𝑏0 = 0 The Fourier Series can be written in a very compact form : 𝑥 𝑡 = 𝑛=−∞ ∞ 𝑐𝑛𝑒𝑖𝑛𝜔𝑡 𝑐𝑛 = 𝜔 𝜋 න 0 2𝜋/𝜔 𝑥 𝑡 𝑐𝑜𝑠 𝑛𝜔𝑡 − 𝑠𝑖𝑛 𝑛𝜔𝑡 𝑑𝑡 = 1 𝜏 න 0 𝜏 𝑥 𝑡 𝑒−𝑖𝑛𝜔𝑡 𝑑𝑡 with
- 33. Prof. Carmen Muller-Karger, PhD Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Mechanical Vibrations Time and frequency domain representations • The Fourier series expansion permits the description of any periodic function using either a time-domain or a frequency- domain representation. • Note that the amplitudes 𝑑𝑛 and the phase angles 𝜑𝑛 corresponding to the frequencies ωn can be used in place of the amplitudes 𝑎𝑛 and 𝑏𝑛 for representation in the frequency domain.