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 Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD 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 Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD 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. Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Importance of studying Vibration Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD 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. Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD 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). F(t) Excitations (input): Initial conditions of external force T Mechanical Vibrations U r(t) Responses (output) D Energy dissipation Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Modeling systems • All mechanical and structural systems can be modeled as mass-springdamper systems Real system Mechanical Model Mathematical Model ๐๐ฅแท + ๐ ๐ฅแถ + ๐๐ฅ = 0 Solution Analysis Mechanical Vibrations ๐๐ฅแท + ๐ ๐ฅแถ + ๐๐ฅ = 0 Respuesta análisis del sistema para que la respuesta sea coherente. Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Respuesta análisis del sistema para que la respuesta sea coherente. Prof. Carmen Muller-Karger, PhD Degrees of freedom The minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time Single DoF systems: Two DOF System Three DoF systems: Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD 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. Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Types of response • Undamped and damped Vibration • Linear of nonlinear Vibration ๐๐แท + ๐๐๐ ๐๐๐ = 0 • Deterministic ond Random Vibration Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD 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 Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Non-linear spring element: Spring Elements ๐น = ๐๐ฅ + ๐๐ฅ 3 ๐= ๐๐น แค ๐๐ฅ ๐ฅ ∗ The force related to a elongation or reduction in length opposes to the displacement of the end of the spring and is given by : ๐น = ๐๐ฅ The work done (U) in deforming a spring is stored as strain or potential energy in the spring, and it is given by ๐= 1 2 ๐๐ฅ 2 Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Example: Spring constant of a rod • Find the equivalent spring constant of a uniform rod of length l, crosssectional area A, Area Moment of Inertia I and Young’s modulus E 1. Subjected to an axial tensile (or compressive) force F 2. Cantilever bean subjected to a transversal load at the free end. ๐๐ 3 ๐ฟ= 3๐ธ๐ผ ๐ฟ ๐น๐ ๐ฟ = ๐ = ๐๐ = ๐ ๐ด๐ธ ๐น๐๐๐๐ ๐น ๐ด๐ธ ๐= = = ๐๐๐๐๐๐ฅ๐๐๐ ๐ฟ ๐ Mechanical Vibrations ๐= Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition ๐น๐๐๐๐ ๐ 3๐ธ๐ผ = = 3 ๐๐๐๐๐๐ฅ๐๐๐ ๐ฟ ๐ Prof. Carmen Muller-Karger, PhD Equivalent spring constants Original system Equivalent model Spring Constant of a Rod under axial load ๐= ๐ด๐ธ ๐ Cantilever beam with end force ๐= 3๐ธ๐ผ ๐3 Simple support beam with load in the middle ๐= Propeller Shaft subjected to a torsional moment Mechanical Vibrations ๐บ๐ฝ ๐= ๐ Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition 48๐ธ๐ผ ๐3 ๐(๐ท4 − ๐4 ) ๐ฝ= 32 Prof. Carmen Muller-Karger, PhD Combination of Springs Springs in Parallel Springs in Series ๐ฟ1 = ๐2 (๐ฟ2 −๐ฟ1 ) W= ๐1 ๐ฟ๐ ๐ก + ๐2 ๐ฟ๐ ๐ก W= (๐1 +๐2 )๐ฟ๐ ๐ก W= ๐1 ๐ฟ1 = ๐2 (๐ฟ๐ ๐ก −๐ฟ1 ) = ๐๐๐ ๐ฟ๐ ๐ก ๐ ๐ฟ1 = ๐1 W= (๐๐๐ )๐ฟ๐ ๐ก ๐๐๐ = (๐1 +๐2 ) Mechanical Vibrations ๐ฟ๐ ๐ก = ๐2 ๐ ๐ฟ๐ ๐ก − =๐ ๐1 ๐ ๐๐๐ Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition ๐ ๐ ๐ฟ๐ ๐ก = + ๐2 ๐1 1 1 1 = + ๐๐๐ ๐1 ๐2 Prof. Carmen Muller-Karger, PhD 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 Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Kinetic energy In general for one rigid body the kinetic energy can be calculated as ๐= For a system with several rigid bodies is the sum of the kinetic energy of each body: 1 1 ๐(๐ฃ๐าง )2 + ๐ผ๐๐ง๐ง (๐๐ง )2 +๐ฃ๐าง โ ๐ เดฅ๐ง × ๐๐๐บาง 2 2 ๐ฃาง = ๐ฃ๐าง + ๐ เดฅ๐ง × ๐าง dm y y´ ๐ผ ๐บ ๐ x´ ๐ ๐ x Mechanical Vibrations GENERAL MOTION Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Example 1: Translational masses by a rigid bar ๐= 1 1 1 ๐1 (๐ฅแถ 1 )2 + ๐2 (๐ฅแถ 2 )2 + ๐3 (๐ฅแถ 3 )2 2 2 2 ๐ฅแถ ๐๐ = ๐ฅแถ 1 • Let’s choose ๐ฅแถ 1 as the parameter that will describe the motion of the system and find the equivalent kinetic energy 1 1 ๐2 ๐ = ๐1 (๐ฅแถ 1 )2 + ๐2 ๐ฅแถ 2 2 ๐1 1 1 ๐2 ๐ = ๐1 + ๐2 2 ๐1 ๐ ๐ฅแถ 2 = ๐2 ๐ฅแถ 1 1 ๐ ๐ฅแถ 3 = ๐3 ๐ฅแถ 1 1 Mechanical Vibrations ๐๐๐ = ๐1 + ๐2 2 2 + ๐3 ๐2 ๐1 1 ๐3 + ๐3 ๐ฅแถ 2 ๐1 1 ๐3 ๐1 2 + ๐3 2 2 (๐ฅแถ 1 )2 ๐3 ๐1 2 Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Example 2: Translational and rotational masses coupled together Since the system is rotating without slipping and the gear is fix at its center ๐แถ = ๐ฅ/๐ แถ ๐= • If we choose as the parameter that will describe the motion of the system ๐ฅแถ ๐๐ = ๐แถ 1 1 แถ 2 ๐(๐ฅ) แถ 2 + ๐ฝ๐ (๐) 2 2 • If we choose as the parameter that will describe the motion of the system ๐ฅแถ ๐๐ = ๐ฅแถ 1 1 ๐ฅแถ ๐ = ๐(๐ฅ) แถ 2 + ๐ฝ๐ 2 2 ๐ ๐= 1 1 แถ 2 ๐(๐ฅ) แถ 2 + ๐ฝ๐ (๐) 2 2 ๐= 1 1 แถ 2 + ๐ฝ๐ ๐แถ 2 ๐(๐๐ ) 2 2 2 ๐= Mechanical Vibrations 1 แถ 2 ๐๐ 2 + ๐ฝ๐ (๐) 2 Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition ๐๐๐ = ๐๐ 2 + ๐ฝ๐ Prof. Carmen Muller-Karger, PhD 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 Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD In general viscous Dampers Non-linear damper element: ๐น = ๐๐ฃ Dampers in Series 1 1 1 = + ๐๐๐ ๐1 ๐2 Dampers in Parallel ๐๐๐ = (๐1 +๐2 ) Non-linear damper element: ๐= ๐๐น แค ๐๐ฅแถ ๐ฅ ∗ Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Damping constant of parallel plates separated by viscous fluid. • According to Newton’s laws of viscous flow: ๐=๐ ๐๐ข ๐๐ฆ • Gradient of the velocity: • ๐๐ข ๐ฃ = ๐๐ฆ โ If A is the surface area at the moving plate, and expressing the force in term of the damping constant: ๐๐ด๐ฃ ๐น = ๐๐ด = โ ๐น = ๐๐ฃ = ๐= ๐๐ด ๐ฃ โ ๐๐ด โ Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD 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: ๐๐ข ๐=๐ ๐๐ ๐๐ข ๐ฃ ๐๐ • Gradient of the velocity: = = ๐๐ ๐ ๐ If A is the surface area at the moving shaft (2๐๐ ๐), and expressing the torque T=F R in term of the damping constant: ๐ = (๐๐ด)๐ = ๐(๐๐ ) 2๐๐ ๐ ๐ ๐ ๐ = ๐๐ ๐2๐๐ 3 ๐ ๐= ๐ Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD 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 Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Harmonic Motion or Periodic Motion, Definitions and Terminology Harmonic motion: Motion is repeated after equal intervals of time Mechanical Vibrations 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. ๐ด Period of oscillation: The time taken to complete one cycle of motion, is denoted by τ. ๐= 2๐ ๐ Frequency of oscillation: The number of cycles per unit time ๐= 1 ๐ = ๐ 2๐ Circular frequency of oscillation: The number of cycles per unit time. ๐= 2๐ ๐ Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD 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. Displacement: ๐ฅ ๐ก = sin(๐๐ก) Velocity: ๐ฅแถ ๐ก = −ω๐๐๐ (๐๐ก) Acceleration: ๐ฅแท ๐ก = −๐2 sin(๐๐ก) P1 wt+f O wt P2 Vectorial representation Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Complex-number representation • Any vector in the xy-plane can be represented as a complex number: ๐เดค = a + ib = ๐ด cos ๐๐ก + ๐๐ ๐๐ ๐๐ก • The magnitude A = ๐2 + ๐ 2 Displacement: Velocity: • Using Euler form ๐เดค = A๐ ๐๐๐ก = A๐ ๐๐ = ๐ด cos ๐๐ก + ๐๐ ๐๐ ๐๐ก Acceleration: or ๐เดค = A๐ ๐๐๐ก เดค ๐(A๐ ๐๐๐ก ) ๐ ๐ ๐เดคแถ = = = ๐ωA๐ ๐๐๐ก ๐๐ก ๐๐ก ๐2 ๐เดค ๐2 (A๐ ๐๐๐ก ) แท ๐เดค = 2 = = −ω2 A๐ ๐๐๐ก 2 ๐๐ก ๐๐ก ๐เดค = A๐ −๐๐๐ก = A๐ −๐๐ = ๐ด cos ๐๐ก − ๐๐ ๐๐ ๐๐ก Mechanical Vibrations Expansion by series For very small angles: ๐2 ๐4 ๐6 cos ๐ = 1 − + − +โฏ 2! 4! 6! cos ๐ ≈ 1 ๐3 ๐5 ๐7 sin ๐ = ๐ − + − +โฏ 3! 5! 7! sin ๐ ≈ ๐ Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Adding harmonic motion • Recall trigonometry identities: sin ๐ + ๐ = sin ๐ ๐๐๐ ๐ + cos ๐ ๐ ๐๐ ๐ cos ๐ + ๐ = ๐๐๐ ๐ ๐๐๐ ๐ − sin ๐ ๐ ๐๐ ๐ • Two different ways of write a harmonic motion: ๐๐ (๐) = ๐จ๐ ๐๐จ๐ฌ ๐๐ +๐จ๐ ๐ฌ๐ข๐ง ๐๐ • Adding harmonic motions: or ๐ฅ1 (๐ก) = ๐ด1 cos ๐๐ก ๐๐ (๐) = ๐จ๐๐๐ ๐๐ − ๐ถ ๐ฅ2 (๐ก) = ๐ด2 ๐ ๐๐ ๐๐ก ๐จ= ๐ฅ๐ก (๐ก) = ๐ฅ1 ๐ก + ๐ฅ2 ๐ก = ๐ด1 cos ๐๐ก +๐ด2 sin ๐๐ก = ๐ด๐๐๐ ๐๐ก − ๐ผ ๐ถ = ๐๐๐−๐ ๐ฅ๐ก (๐ก) = ๐ด๐๐๐ ๐๐ก − ๐ผ = Acos(๐ผ) cos ๐๐ก + Asin(๐ผ) sin ๐๐ก ๐ด1 = Acos(๐ผ) ๐ด2 = Asin(๐ผ) ๐ด= ๐ด12 + ๐ด22 = ๐ผ = ๐ก๐๐−1 Mechanical Vibrations ๐จ๐๐ + ๐จ๐๐ ๐จ๐ ๐จ๐ (Acos(๐ผ))2 +(Asin(๐ผ))2 ๐ด2 ๐ด1 Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Adding harmonic motion • Recall trigonometry identities: cos A + cos B = 2 cos ๐ด+๐ต ๐ด−๐ต cos 2 2 cos A − cos B = −2 sin ๐ด+๐ต ๐ด−๐ต sin 2 2 sin A + sin B = 2 sin ๐ด+๐ต ๐ด−๐ต cos 2 2 Phenomenon Beats: occurs when adding two harmonic motion with frequencies close too one another the resultant motion ๐๐ (๐) = X๐๐๐ ๐ ๐ ๐๐ (๐) = X๐๐๐ ๐ + ๐น ๐ ๐(๐) = X๐๐๐ ๐ ๐+X๐๐๐ ๐ + ๐น ๐ ๐(๐) = ๐ ๐๐จ๐ฌ ๐น ๐ ๐ ๐๐๐ ๐ + ๐น ๐ ๐ ๐ด+๐ต ๐ด−๐ต sin A − sin B = 2 cos sin 2 2 Beats frequency is twice the frequency of the term ๐ ๐๐จ๐ฌ ๐น ๐ ๐ since two peaks pass in each cycle. Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Harmonic Analysis: Fourier Series Any periodic function of time can be represented by Fourier series as an infinite sum of sine and cosine terms ๐ฅ ๐ก = ๐0 + ๐1 ๐๐๐ ๐๐ก 2 + ๐2 ๐๐๐ 2๐๐ก + โฏ ๐1 ๐ ๐๐ ๐๐ก + ๐2 ๐ ๐๐ 2๐๐ก + โฏ ∞ ๐0 ๐ฅ ๐ก = + เท ๐๐ ๐๐๐ ๐๐๐ก + ๐๐ ๐ ๐๐ ๐๐๐ก 2 ๐=1 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. ๐ 2๐/๐ 2 ๐ ๐0 = เถฑ ๐ฅ ๐ก ๐๐ก = เถฑ ๐ฅ ๐ก ๐๐ก ๐ 0 ๐ 0 ๐ 2๐/๐ 2 ๐ ๐๐ = เถฑ ๐ฅ ๐ก ๐๐๐ ๐๐๐ก ๐๐ก = เถฑ ๐ฅ ๐ก ๐๐๐ ๐๐๐ก ๐๐ก ๐ 0 ๐ 0 ๐ 2๐/๐ 2 ๐ ๐๐ = เถฑ ๐ฅ ๐ก ๐ ๐๐ ๐๐๐ก ๐๐ก = เถฑ ๐ฅ ๐ก ๐ ๐๐ ๐๐๐ก ๐๐ก ๐ 0 ๐ 0 Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Harmonic Analysis: Fourier Series Fourier series can also be represented by the sum of sine terms only or cosine terms only. ๐ฅ ๐ก = ๐0 + ๐1 ๐๐๐ ๐๐ก − ๐1 + ๐2 ๐๐๐ 2๐๐ก − ๐1 + โฏ ∞ ๐ฅ ๐ก = ๐0 + เท ๐๐ ๐๐๐ ๐๐๐ก − ๐๐ ๐=1 Where: ๐0 = ๐0 2 ๐๐ = ๐๐2 + ๐๐2 ๐๐ = ๐ก๐๐−1 ๐๐ ๐๐ Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Fourier Series in complex numbers Since: ๐ ๐ ๐๐๐ก cos ๐๐ก = cos ๐๐ก + ๐๐ ๐๐ ๐๐ก −๐๐๐ก ๐ ๐๐๐ก +๐ −๐๐๐ก = 2 sin ๐๐ก = = cos ๐๐ก − ๐๐ ๐๐ ๐๐ก ๐ ๐๐๐ก −๐ −๐๐๐ก 2 The Fourier Series can be written as : ∞ ๐0 ๐ ๐๐๐๐ก + ๐ −๐๐๐๐ก ๐ ๐๐๐๐ก − ๐ −๐๐๐๐ก ๐ฅ ๐ก = + เท ๐๐ + ๐๐ 2 2 2 ∞ = ๐ ๐๐0 ๐=1 With: ๐๐ = ๐๐ − ๐๐๐ 2 ๐−๐ = ๐๐ + ๐๐๐ 2 ๐0 ๐0 ๐๐ − ๐๐๐ ๐๐ + ๐๐๐ −๐ + เท ๐ ๐๐๐๐ก + ๐ −๐๐๐ก 2 2 2 2 ๐=1 ๐0 = 0 The Fourier Series can be written in a very compact form : ∞ ๐ฅ ๐ก = เท ๐๐ ๐ ๐๐๐๐ก ๐=−∞ Mechanical Vibrations with ๐ 2๐/๐ 1 ๐ ๐๐ = เถฑ ๐ฅ ๐ก ๐๐๐ ๐๐๐ก − ๐ ๐๐ ๐๐๐ก ๐๐ก = เถฑ ๐ฅ ๐ก ๐ −๐๐๐๐ก ๐๐ก ๐ 0 ๐ 0 Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD Time and frequency domain representations • The Fourier series expansion permits the description of any periodic function using either a time-domain or a frequencydomain representation. • Note that the amplitudes ๐๐ and the phase angles ๐๐ corresponding to the frequencies ωn can be used in place of the amplitudes ๐๐ and ๐๐ for representation in the frequency domain. Mechanical Vibrations Figures and content adapted from Textbook: Singiresu S. Rao. Mechanical Vibration, Pearson sixth edition Prof. Carmen Muller-Karger, PhD
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