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# Fundamentals of Vibrations 058 153

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This 18 page Class Notes was uploaded by Celine Senger on Friday October 23, 2015. The Class Notes belongs to 058 153 at University of Iowa taught by M. Bhatti in Fall. Since its upload, it has received 36 views. For similar materials see /class/228093/058-153-university-of-iowa in Mechanical Engineering at University of Iowa.

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Date Created: 10/23/15

53 58153 Lecture 1 Fundamental of Vibration Lecture 1 Introduction Reading materials Sections 11 16 1 Introduction 1 People became interested in Vibration when the first musical instruments probably whistles or drums were discovered ID Most human activities involve vibration in one form or other For example we hear because our eardrums vibrate and see because light waves undergo vibration i Any motion that repeats itself after an interval of time is called vibration or oscillation ID The general terminology of Vibration is used to describe oscillatory motion of mechanical and structural systems 10 The Vibration of a system involves the transfer of its potential energy to kinetic energy and kinetic energy to potential energy alternately 1 The terminology of Free Vibration is used for the study of natural vibration modes in the absence external loading ID The terminology of Forced Vibration is used for the study of motion as a result of loads that vary rapidly with time Loads that vary rapidly with time are called dynamic loads 49 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 however is called damped vibration C If the system is damped some energy is dissipated in each cycle of vibration and must be replaced by an external source if a state of steady vibration is to be maintained 53 58153 Lecture 1 Fundamental of Vibration 2 Branches of Mechanics an Rigid bodies Statics amp Dynamics Kinematics amp Dynamics of Mechanical Systems I Fluid mechanics D Deformable bodies Structural analysis assuming loads do not change over time or change very slowly Vibrations or Dynamic analysis considering more general case when loads vary with time Finite element analysis a powerful numerical method for both static and dynamic analysis 10 Vibration analysis procedure Step 1 Mathematical modeling Step 2 Derivation of governing equations Step 3 Solution of the governing equations Step 4 Interpretation of the results 3 Other Basic Concepts of Vibration 1D 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 ID The minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time defines the degree of freedom DOF of the system Examples 5358153 Lecture 1 Fundamental of Vibration A large number of practical systems can be described using a nite number of DOFs Systems with a nite number of DOFs are called discrete or lumped parameter systems 1 Some systems especially those involving continuous elastic members have an in nite number of DOFs Those systems are called continuous or distributed systems 4 Plane truss example Matrix method fl 1 15 4 10 5 o ic m 0 5 10 15 20 D Element equations 395 Global equations Boundary conditions a Stiffness matrix and unknown vaIiables 379 857 111803 7223 607 7111803 4 111803 559017 7111803 7559017 1394 7223 607 7111803 557 699 110485 us 7111803 7559017 110485 534789 1395 or Kd 5358153 Lecture 1 Fundamental ofVibraIion ft f0 Kd f 10 ft is a harmonic force ie ft 100 cos7 739 t 529039 0 00205121 0 a 0 529039 0 00205121 i 4 00205121 0 0170634 0 5 0 00205121 0 0170634 395 379 857 111803 7223 607 7111803 W 0 111803 559017 7111803 7559017 1394 100cosf7 f 7223 607 7111803 557 699 110485 us 0 7111803 7559017 110485 534789 1395 0 or M Kd f 0 Solutions Vernal mplacgmem a node 4 us 5358153 Lecture 1 Fundamental ofVibration Vm mnemmy m node 4 enxcax accelemnou u node 4 5358 153 Lecture 1 Fundammtal ofVibration Sixess cycles in elemem 1 I lbm1 5 PeIiod and Frequency O Teriod ofvibrat39on is the time that it takes to complete one cycle It is m asured in seconds I Trequen is the number of cycles per second It is measured in Hz 1 cyclesecond It could be also measured in mdianssecond Period ofvibmtion T Frequency ofvibration f 1139Hz or u 21t139radianss T2 my 1r 5358153 Lecture 1 Fundamental ofVibration 1 Example 1 1 A a A w 51 f u x u r 1 1 a u r 1 1 w us 11753 1 0 344681 0447453 056101 1 1 1 0116305 1 1 Cycle TLuJels 1 0116305 1 0101318 3 01139043 4 0102711 5 0113562 6 Dynamic loads on structures C Wind loads OBlast pressure Earthquakes Etc 53 58153 Lecture 1 Fundamental of Vibration 7 Importance of dynamic analysis I Load magnification amp Fatigue effects ID A static load is constant and is applied to the structure for a considerable part of its life For example the self weight of building Loads that are repeatedly exerted but are applied and removed very slowly are also considered static loads CD Fatigue phenomenon can be caused by repeated application of the load The number of cycles is usually low and hence this type of loading may cause what is known as lowcycle fatigue 10 Quasistatic loads are actually due to dynamic phenomena but remain constant for relatively long periods In Most mechanical and structural systems are subjected to loads that actually vary over time Each system has a characteristic time to determine whether the load can be considered static quasistatic or dynamic This characteristic time is the fundamental period of ee vibration of the system VD Dynamic Load Magnification factor DLF is the ratio of the maximum dynamic force experienced by the system and the maximum applied load I The small period of vibration results in a small DLF ID Fatigue phenomenon can be caused by repeated application of the load The number of cycles and the stress range are important factors in determining the fatigue life 5358153 Lecture 1 Fundamental ofVibration 8 Types of Vibratory Motion 0 Oscillaton motion may repeat itself regularly as in the case ofa simp e endulum or it may display considerable irregularity as in the case ofground motion during an earthquake 0 If the motion is repeated after equal intervals of time it is called periodic motion The simplest type of periodic motion is harmonic motion 0 Harmonic motion It is described by sine or cosine functions xi Asinw t A is the amplitude while a is the frequency radianssec a Acosiw t 7602A sinw t 76024 FPanodil Plot ofxm 1 sim39l m 5358153 Lecture 1 Fundamental ofVibra on harmonic motions having the same period andor amplitude could have different phase angle Plor of two haimonic funcrions in2 m and 2 sinfl r r zl O A harmonic motion can be Written in terms of exponential inctions zones cosa r f 2quot coswri isiuur A harmonic motion could be Written as A1 I 6 quot 0 Alternative forms for harmonic motion Generally a harmonic motion can be expressed as a combination of sine and cosine Waves vr A cos or B 3111 m vrl139 sintmr 6 10 535815 Lecture 1 Fundamental ofVibration Y 7 A2 32 6 mirlm B or 139I A cos a 7 B sinwr c rr 7139 5mm 7 6 Y costar 7 6 1 Periodic motion The motion repeats itself exactly x 5358153 Lecture 7 Fundamental ofVibration Lecture 7 Systems involving zero or repeat Frequencies Reading materials Sections 24 and 25 1 Systems involving zero frequency Some possible mode shapes may not involve any deformation They are called rigid body modes The corresponding frequencies are zero Example an unrestrained three springmass system 53 58153 Lecture 7 Fundamental of Vibration 5358 153 Lecture 7 Fundamental of Vibration OExact solution 5358 153 Lecture 7 Fundamental ofVibnmon anModal superposition solution Some dynarnic systenns exhibit rigidbody modes that are characterized by zero natural frequencies The bearn is not properly restrained Its rst rnode is a n39gid body rnode in which the bearn pivots around its le support Generally the uncoupled rnodal equations are lrtiFm il2l iq since the frequency is zero the conesponding uncoupled rnodal equation is i F m q 1 t0ot 1010 For free Vibration 5358153 Lecture 7 Fundamental ofVibration Example Y1 Y2 y Y3 k k 0N NW m m mg ml 50 m 100 7713 150 Iquot 1000 kg 500 50 0 0 1000 1000 0 0 m 0 100 0 Aquot 1000 1500 500 f 0 0 0 150 0 500 500 0 0 0 u 0 0 0 r 100 Eigenvalue Frequency rads Mode shape 1 992523 x10quot 0 0057735 007735 0051735 2 249597 uo49m9 o 3 555599 O10699 005441 4300 015quot lz 111 K f k g 0 F f 0 0 0 210o 2100 i1l0188675 z392 622935 z3 210 0 210 7301245 239 521055230 25Ogt0 i31053494s 5358153 Lecture 7 Fundamental of Vibration 21m 188675r 21m 144731 still19597 T 2quot 0i944137 su1566599 n yllll 01666671 0104567 dull49597 f 0101013 si111566599 r yin 0166667 1 0071995 Si11249597 fl 7 00611303 sint566599 t 3731 0166667 1 00828523 sinll49597 I 0i007032615im5665991 2 Systems involving repeated frequency In a special case all frequencies ofa dynamic system are not unique Example 1 0 0 44 74 0 m 0 12 0 39 k 714 34 0 O 0 l 0 O 5 5358 153 Lecture 7 Fundammtal OfVlbraLlOn 3 2 351115 hmsm rr 13 3 13V 12 quotz lsm r y1t7stir 13 3 135 asmHs r Vz07 J quot1 J z 39 I x W l 10 15 1p 5 10 1B 10 J V v y A a r n a r t L0 H 6 H H H m OVSHHHHwa HHAHMM H V W V v V I y t 5531PJ111 H 1h w x 0 v x U A U

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