Explosion Protection of Buildings
|
|
- Georgiana Banks
- 5 years ago
- Views:
Transcription
1 Explosion Protection of Buildings Author: Miroslav Mynarz
2 Explosion Protection of Buildings Introduction to the Problems of Determination of Building Structure's Response
3 3 Classification of actions According to response of the structure: Static (no significant accelerations originate in the structure) Dynamic (significant accelerations originate in the structure)
4 4 Classification of actions According to its nature: Force (including moments) Deformation or inematic (forced by displacement or rotation)
5 5 Types of actions Deterministic: Periodical (repeating regularly) Non-periodical (repeating irregularly)
6 6 Types of actions Periodical: Harmonic (simple) y = A sinωt e. g. rotary motion of unbalanced movement of machine advance motion of piston in cylinder simple tone of musical instrument movement of a bell etc.
7 7 Types of actions Periodical : Compact of i-harmonic components F(t) = F st + F i sin(ω i t + φ i ) (Fourier transform) e. g. movement of a floor loaded by more machines movement of bridge dec loaded by more vehicles compound tones, noises etc.
8 8 Types of actions Non-periodical: Impuls (impuls of various shape) I = t F (τ) Δτ e. g. effects of explosions or blast waves push of drop hammers, ramming of piles acoustic stroes etc.
9 9 Types of actions Non-periodical: Non-periodical force F t Combination: Periodical impulses F t e. g. effects of flat wheels to bridge forging of repeated pushes of drop hammer
10 0 Types of actions Non-deterministic (random, stochastic): Stationary (steady, still) Non-stationary (unsteady, unstable)
11 Dynamic ~ time variable load Dynamic load load whose size, direction or place is changing in time. Response of building structure to dynamic load final deformations of the structure (deflections, displacements or rotations) and their derivation in time (velocity, acceleration) as well as internal forces (normal and shear forces or bending moments) and stresses are variable in time.
12 Dynamic ~ time variable load Determination of dynamic response analysis of the structure at dynamic load. Methods depend on character of the load: deterministic load, non-deterministic load.
13 3 Dynamic ~ time variable load Deterministic procedures: time course is nown; material and shape of the structure is nown; boundary conditions (for example founding of the structure) are defined.
14 4 Dynamic ~ time variable load Non-deterministic procedures: time course of load is not nown fully; character of load is variable; character of material properties of the structure is variable.
15 5 Analysis of the response of the building structure Determination of deformation quantities displacements, deflections; rotations; their time derivations. Following force quantities are derivated from deformation quantities internal forces; stress.
16 6 Building structure Number of degrees of freedom (number of components of possible displacements or rotations of structure s mass in particular points which are not precluded by sliding supports): real structures infinite number of degrees of freedom; solid body in 3-D space (point mass in space) 6 degrees of freedom; solid slab in plane (point mass in plane) 3 degrees of freedom. For each degree of freedom, equation of equilibrium (equation of motion) can be written.
17 7 Building structure Characteristics of the structure (material, structural arrangement and boundary conditions): stiffness; mass of inertia (weight); damping properties. Tuning of the structure spectrum of natural frequencies and corresponding free vibration mode shapes (function of stiffness and mass of the structures).
18 8 Building structure Mode shape (antinode versus node): natural mode shape; free vibration; forced vibration.
19 9 Quantities displacement (deflection) s(t)=s 0 f(t) rotation φ(t)=φ 0 f(t) velocity of vibration v(t)=ds/dt φ(t)=dφ/dt acceleration of vibration a(t)=d s/dt φ(t)=d φ /dt momentum of (point) mass H(t)=m v(t) momentum of body inertia force H(t)= V v(t) dm F(t)= m a(t)
20 0 Quantities in direction: of coordinate axes; related to the structure (e. g. in the middle of a span); related to trajectory (e. g. radial, tangential, centrifugal, centripetal). Equation of motion (summary condition of equilibrium of forces in direction of appropriate degree of freedom): displacement; rotational (spin, angular).
21 Quantities Newton's second law: F(t) = d/dt (m ds/dt) = m d s/dt = m a(t) F(t) force vector s(t) trajectory (displacement, rotation) ~ position vector p(t) = s(t) internal forces in the structure (elastic) m a(t) P(t) inertia forces in the structure (volume) external forces (load)
22 Quantities p(t) = s(t) m a(t) P(t) internal forces in the structure (elastic) inertia forces in the structure (volume) external forces (load) d Alembert's principle (motion of a point mass ~ a body) : P(t) - m d s/dt = 0
23 3 Equations of motion Structure under dynamic load: The course of load F(t) is variable in time whereas time change of load is usually in order of seconds or milliseconds. More precise definition of time change of load results from comparison of time change of load F(t) and natural periods of the structure T(i). When time changes of load are long-term, thus in order of hours, days, years, we usually do not tal about dynamic load but about variability of static load.
24 4 Equations of motion Equations of motion (control equations of the system): Equations of motion (control equations of the system) are derivated from equations of equilibrium in appropriate possible direction of motion (deflection or rotation) which is denoted as degrees of freedom: F I (t) + F D (t) + F E (t) = F(t)
25 5 Equations of motion Time change of displacement y(t) is considered as a solution of equation of equilibrium, inertia forces F I (t) = m y (t) thus product of mass m and acceleration of vibration y (t) (from Newton's second law); damping forces F D (t) = C y (t) are usually formulated as product of dumping constant C and velocity of vibration y (t); elastic forces F E (t) = y (t) product of stiffness of the structure and displacement of vibration y(t).
26 6 Equations of motion M y (t) + C y (t) + K y (t) = F where M, C and K are mass, damping and stiffness matrixes of the system; y, y, y are vectors of acceleration, velocity and displacement of vibration; F is a vector of the right sides (load of single parts points of the structure).
27 7 Equations of motion Members of a mass matrix after discretization of a system consist of masses of moving point masses and mass moments of inertia of rotating point masses. Particular elements m ij of a mass matrix can be determined as forces in a point i corresponding to a unit acceleration y in a point j.
28 8 Equations of motion Mass matrix: When using software, consistent mass matrix is usually used; it means a mass matrix with nonzero elements even at non-diagonal positions. More simply, generally for manual calculation or computation of small systems or systems with limited number of degrees of freedom, only lumped mass matrix is used it is diagonal matrix with zero elements at non-diagonal positions. It means that inertia forces in i-th point of the system generated by the load applied to this point do not induce inertia response of other points.
29 9 Equations of motion Let s divide bent beam to several particular elements where mass of single elements is focused into nodes of discrete system. If diagonal matrix is used for calculation then at each position of diagonal, mass of particular node is placed there and it performs only advanced motion. If we tae into account that masses in nodes can also rotate then mass matrix has nonzero elements even at non-diagonal positions it is considered as consistent mass matrix.
30 30 Equations of motion Damping matrix: For compilation of damping matrix C it is usually assumed that damping is directly proportional to velocity of motion (see formulation of damping forces of equation of motion) this damping is denoted as viscous damping. Then single elements of damping matrix correspond to force in node i at action of unit vector of velocity y in point j: F D = c y
31 3 Equations of motion In software, implementation of damping to equation of motion with help of coefficients a and b is used with advantage. In this case, damping matrix C is more simply considered to be linear combination of mass matrix and stiffness matrix C = a M + b K Simplification consists especially in determination of damping size with only two parameters a and b that are valid for all frequency components of vibration.
32 3 Equations of motion De facto, damping of vibrations is usually higher on higher frequencies it means that calculated dynamic response of the structure can be actually different.
33 33 Equations of motion Stiffness matrix: Particular elements of this matrix correspond to static stiffnesses of single nodal points of 3D discretizated system at unit displacements (and also rotations) of these relevant nodal points. As well as in the cases of mass and damping matrixes, single elements of stiffness matrix ij correspond to a force (moment) in a node i at action of unit displacement (rotation) y in point j: F E = y
34 34 Equations of motion Calculation decomposes in two cases: - equation without right side (unloaded structure): calculation of natural vibration (tuning of the structure); - equation with right side (loaded structure): calculation of forced vibration. There are many methods for solution of free and forced vibration. The user of particular software should now used method of solution. It is not advisable to use the software as a blac box through which the user gains the results and believes them absolutely. He or she should then now the possibilities of the method in so far to estimate its proximities bringing into the results of response calculation.
35 35 Equations of motion Development to the free vibration mode shapes: In this case, it highly depends on a number of mode shapes used for calculation. For small amount of mode shapes (up to 5 or 0 shapes), the calculation could involve only 50 to 60 % intensity of a real response. According to Eurocode 8, minimum number of considered global mode shapes is: 3 n or T 0. s n is a number of floors above the foundation or solid subsoil.
36 36 Equations of motion Damping of vibrations: A size of damping can affect even calculated quantities of the response especially in the case when the response of the structure is close to resonant frequency. It is usually assumed that damping is proportional to the velocity of vibration. This damping is called viscous. Most software generally enables the implementation of viscous damping to the calculation. Viscous damping is also suitable approximation to other assumptions about damping (e. g. hysteretic damping that is proportional to the deflection of vibrations).
37 37 Equations of motion Damping of vibrations: For determination of a level of viscous damping, it is suitable to result from parameters measured at particular structure. Damping is frequency-dependent. Generally, structural damping is lowest on lowest natural frequencies and damping is increasing with increasing frequency. Relation between the damping ratio D and the logarithmic decrement of damping δ follows: D δ f ω i i δ π
38 38 Equations of motion Damping of vibrations: - chosen values of the logarithmic decrement of damping δ and the damping ratio D (according to ČSN ) Material Constructive system, element D (%) Steel Masonry Timber Concrete engineering structures (building, halls) towers, smoestacs, high-rise structures single elements columns, beams springs (wound) vertical bric masonry (walls, partition walls) masonry vaults in steel beams stone masonry on lime mortar stone masonry on cement-lime mortar glued beams nailed beams roof trusses tower structures compression elements without cracs elements with tension (without cracs) elements with cracs slab and rib floors crane beams and columns high prefabricated houses ,
39 deflection y VŠB Technical University of Ostrava 39 Equations of motion Experimental methods for determination of damping: deflection in resonance frequency of excitation f
40 40 VŠB Technical University of Ostrava Experimental methods for determination of damping: Equations of motion the logarithmic decrement of damping m n n n n y y ln m y y ln δ the damping ratio from rezonant gain pea st y y D method of the width of the resonance curve f f f f D
41 4 Equations of motion Implementation of a size of damping to the calculation: In software, Rayleigh formulation of damping is generally implemented (through method of integration of equations of motion). It results from solution of matrix equation of motion whose damping matrix C is considered as a linear combination of mass matrix M and stiffness matrix K: C = α M + β K
42 4 Equations of motion Implementation of a size of damping to the calculation: For a system with one degree of freedom: α D ω β It could be derived even for a system with more degrees of freedom it is then necessary to now relevant number of natural frequencies. D ω
43 43 Equations of motion Implementation of a size of damping to the calculation: At calculation of systems with more dominant frequency components of excitation or at excitation with general time course of a load, it is necessary to decide which size of damping will be used for which frequency component. At other frequency components whose damping is not nown more precisely a size of response is burdened with bigger or smaller error.
44 44 Standards requirements for determination of dynamic response of the structure Standards usually do not specify process of calculation and chosen method. For example, standards (ČSN , Eurocode 8 or UBC 997 etc.) enable simplified (seismic this procedure can be applied even to the blast effects) calculation of simpler structures at use of cantilever model (the object is replaced by a bar model cantilever with masses focused in single floors).
45 45 Standards requirements for determination of dynamic response of the structure Simplification of geometry (numerical model) versus calculation method Cantilever model is suitable for high slender structures but not for structures with wide plan (e. g. industrial halls), underground structures, indented buildings or structures with dominant effect in vertical direction (bridges, huge cantilever length of floors in buildings). In such cases it is correct to use 3D or at least D model and to perform dynamic computation on these models. Simplification of input data: material characteristic and load.
46 46 Damping of system with one degree of freedom Natural undamped vibration: F in F re m d v dt d v m dt v t t t v m mass of a system t 0 - inertia force - restoring force of the spring v(t) vertical displacement of the spring stiffness (the spring constant)
47 47 Damping of system with one degree of freedom v Natural undamped vibration: t A sinω t B cosω t v sinω t A φ0 0 v0 cos φ0 B v sinφ v A B φ 0 B A 0 0 B φ arcg 0 A 0 0
48 48 Damping of system with one degree of freedom Initial conditions: t dv v t ω A cosω t B sinω t dt v0 ω0 cos ω0t φ0 v0 ω0 sin ω0t φ t π d v v t ω A sinω t B cosω t dt ω t φ v ω sinω t φ π v0 ω0 sin
49 49 Damping of system with one degree of freedom π T0 π f ω 0 0 m ω 0 m f 0 0 T 0 ω π π m Natural frequency f 0 = f ()
50 50 Damping of system with one degree of freedom v Natural damped vibration: d v m dt d v dt t t ω dv b dt b dv dt t t v ω t 0 t 0 For solution in shape of v(t)=e αt it has characteristic equation: with roots α α ω α ω b ω ω ω, b b 0 ωbt ωbt t e A sinω t B cosω t v e sinω t d d d d v 0 d φ 0
51 5 Damping of system with one degree of freedom Following three cases can occur: - critical damping, when ω b = ω 0, b r = ; characteristic equation has one real, double root: α, =-ω b, - overdamping, when ω b > ω 0, b r > ; characteristic equation has two different real roots: α ω ω ω, b b 0 - underdamping, when ω b < ω 0, b r < ; characteristic equation has two imaginary complex conjugate roots: α ω i ω ω ω iω, b 0 b b d ω d ω ω 0 b is natural angular (circular) frequency of damping vibration.
52 5 Damping of system with one degree of freedom υ ln v v ω f b ωbt d d πω ω d b υ πω ω d b b r π b r υ πb r the damping ratio b r = D
53 53 Damping of system with one degree of freedom Harmonic excitation undamped vibration: m d v dt t v v t v t sinωt F sinωt v F F ω0 v st mω m ω ω ω ω 0 0 v st δ
54 54 Damping of system with one degree of freedom Harmonic excitation undamped vibration: force force resonance deflection deflection
55 55 Damping of system with one degree of freedom Harmonic excitation undamped vibration: v m d v dt d v dt t t ω c b dv dt t t dv dt v ω 0 v t t Ft t F m t A sinωt B cosωt v sinωt φ v F A t B F sinωt tgφ B A
56 56 VŠB Technical University of Ostrava Damping of system with one degree of freedom Harmonic excitation undamped vibration: η b η η F ω ω ω ω ω ω m F A r b 0 4 b η η b η F ω ω ω ω ωω m F B r r b b 4 4 η π υ η η υ π υ η δ
57 57 Damping of system with one degree of freedom Harmonic excitation undamped vibration: π Under-resonant area for η= resonance δ υ over-resonant area. v v st F mω ω b m ω ω ω 0 ω ω 0 4ω ω b 0 v st δ F 4ω ωb
58 58 Damping of system with two degrees of freedom Simple beam undamped vibration: m m d v dt t v d v dt t t v t 0 v t v t 0 ω, m m 4 m m m m a v v 0 0 m ω m ω a v v 0 0 m ω m ω
59 59 Damping of system with two degrees of freedom Simple beam undamped vibration: I m d ζ dt t d u dt t a ζζ uu a ζ u m m t ut 0 ζu t ζ t 0 uζ I m a b
60 60 Damping of system with two degrees of freedom Simple beam undamped vibration: EI b b ζζ l l l 6EI b EI 3 l l uu l ζu ω, uu m ζζ I 4 uu m ζζ I uζ mi a a i i I m
61 6 VŠB Technical University of Ostrava Damping of system with two degrees of freedom 0 t v t v dt t v d m Simple beam undamped vibration: 0 t v t v dt t v d m 4 m m m m m m ω,
62 6 Damping of system with three degrees of freedom
63 63 Than you for your attention.
Introduction to structural dynamics
Introduction to structural dynamics p n m n u n p n-1 p 3... m n-1 m 3... u n-1 u 3 k 1 c 1 u 1 u 2 k 2 m p 1 1 c 2 m2 p 2 k n c n m n u n p n m 2 p 2 u 2 m 1 p 1 u 1 Static vs dynamic analysis Static
More informationDr.Vinod Hosur, Professor, Civil Engg.Dept., Gogte Institute of Technology, Belgaum
STRUCTURAL DYNAMICS Dr.Vinod Hosur, Professor, Civil Engg.Dept., Gogte Institute of Technology, Belgaum Overview of Structural Dynamics Structure Members, joints, strength, stiffness, ductility Structure
More informationDynamics of Machinery
Dynamics of Machinery Two Mark Questions & Answers Varun B Page 1 Force Analysis 1. Define inertia force. Inertia force is an imaginary force, which when acts upon a rigid body, brings it to an equilibrium
More informationDynamics of structures
Dynamics of structures 2.Vibrations: single degree of freedom system Arnaud Deraemaeker (aderaema@ulb.ac.be) 1 Outline of the chapter *One degree of freedom systems in real life Hypothesis Examples *Response
More informationStructural Dynamics. Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma).
Structural Dynamics Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma). We will now look at free vibrations. Considering the free
More informationCE 6701 Structural Dynamics and Earthquake Engineering Dr. P. Venkateswara Rao
CE 6701 Structural Dynamics and Earthquake Engineering Dr. P. Venkateswara Rao Associate Professor Dept. of Civil Engineering SVCE, Sriperumbudur Difference between static loading and dynamic loading Degree
More informationWEEKS 8-9 Dynamics of Machinery
WEEKS 8-9 Dynamics of Machinery References Theory of Machines and Mechanisms, J.J.Uicker, G.R.Pennock ve J.E. Shigley, 2011 Mechanical Vibrations, Singiresu S. Rao, 2010 Mechanical Vibrations: Theory and
More informationIntroduction to Vibration. Professor Mike Brennan
Introduction to Vibration Professor Mie Brennan Introduction to Vibration Nature of vibration of mechanical systems Free and forced vibrations Frequency response functions Fundamentals For free vibration
More informationCIVL 8/7117 Chapter 12 - Structural Dynamics 1/75. To discuss the dynamics of a single-degree-of freedom springmass
CIV 8/77 Chapter - /75 Introduction To discuss the dynamics of a single-degree-of freedom springmass system. To derive the finite element equations for the time-dependent stress analysis of the one-dimensional
More informationCOPYRIGHTED MATERIAL. Index
Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,
More informationSTRUCTURAL DYNAMICS BASICS:
BASICS: STRUCTURAL DYNAMICS Real-life structures are subjected to loads which vary with time Except self weight of the structure, all other loads vary with time In many cases, this variation of the load
More informationSection 3.7: Mechanical and Electrical Vibrations
Section 3.7: Mechanical and Electrical Vibrations Second order linear equations with constant coefficients serve as mathematical models for mechanical and electrical oscillations. For example, the motion
More informationModeling and Experimentation: Mass-Spring-Damper System Dynamics
Modeling and Experimentation: Mass-Spring-Damper System Dynamics Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin July 20, 2014 Overview 1 This lab is meant to
More informationVibrations Qualifying Exam Study Material
Vibrations Qualifying Exam Study Material The candidate is expected to have a thorough understanding of engineering vibrations topics. These topics are listed below for clarification. Not all instructors
More informationUNIT-I (FORCE ANALYSIS)
DHANALAKSHMI SRINIVASAN INSTITUTE OF RESEACH AND TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK ME2302 DYNAMICS OF MACHINERY III YEAR/ V SEMESTER UNIT-I (FORCE ANALYSIS) PART-A (2 marks)
More informationChapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation..14
Table of Contents Chapter 1: Research Objectives and Literature Review..1 1.1 Introduction...1 1.2 Literature Review......3 1.2.1 Describing Vibration......3 1.2.2 Vibration Isolation.....6 1.2.2.1 Overview.
More informationA Guide to linear dynamic analysis with Damping
A Guide to linear dynamic analysis with Damping This guide starts from the applications of linear dynamic response and its role in FEA simulation. Fundamental concepts and principles will be introduced
More informationOutline of parts 1 and 2
to Harmonic Loading http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March, 6 Outline of parts and of an Oscillator
More informationIntroduction to Vibration. Mike Brennan UNESP, Ilha Solteira São Paulo Brazil
Introduction to Vibration Mike Brennan UNESP, Ilha Solteira São Paulo Brazil Vibration Most vibrations are undesirable, but there are many instances where vibrations are useful Ultrasonic (very high
More informationCodal Provisions IS 1893 (Part 1) 2002
Abstract Codal Provisions IS 1893 (Part 1) 00 Paresh V. Patel Assistant Professor, Civil Engineering Department, Nirma Institute of Technology, Ahmedabad 38481 In this article codal provisions of IS 1893
More informationStructural Dynamics A Graduate Course in Aerospace Engineering
Structural Dynamics A Graduate Course in Aerospace Engineering By: H. Ahmadian ahmadian@iust.ac.ir The Science and Art of Structural Dynamics What do all the followings have in common? > A sport-utility
More informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.
More informationChapter 23: Principles of Passive Vibration Control: Design of absorber
Chapter 23: Principles of Passive Vibration Control: Design of absorber INTRODUCTION The term 'vibration absorber' is used for passive devices attached to the vibrating structure. Such devices are made
More informationFinal Exam Solution Dynamics :45 12:15. Problem 1 Bateau
Final Exam Solution Dynamics 2 191157140 31-01-2013 8:45 12:15 Problem 1 Bateau Bateau is a trapeze act by Cirque du Soleil in which artists perform aerial maneuvers on a boat shaped structure. The boat
More informationDynamics of Structures: Theory and Analysis
1. Free vibrations 2. Forced vibrations 3. Transient response 4. Damping mechanisms Dynamics of Structures: Theory and Analysis Steen Krenk Technical University of Denmark 5. Modal analysis I: Basic idea
More informationPreliminary Examination - Dynamics
Name: University of California, Berkeley Fall Semester, 2018 Problem 1 (30% weight) Preliminary Examination - Dynamics An undamped SDOF system with mass m and stiffness k is initially at rest and is then
More informationLaboratory notes. Torsional Vibration Absorber
Titurus, Marsico & Wagg Torsional Vibration Absorber UoB/1-11, v1. Laboratory notes Torsional Vibration Absorber Contents 1 Objectives... Apparatus... 3 Theory... 3 3.1 Background information... 3 3. Undamped
More informationon the figure. Someone has suggested that, in terms of the degrees of freedom x1 and M. Note that if you think the given 1.2
1) A two-story building frame is shown below. The mass of the frame is assumed to be lumped at the floor levels and the floor slabs are considered rigid. The floor masses and the story stiffnesses are
More informationChapter a. Spring constant, k : The change in the force per unit length change of the spring. b. Coefficient of subgrade reaction, k:
Principles of Soil Dynamics 3rd Edition Das SOLUTIONS MANUAL Full clear download (no formatting errors) at: https://testbankreal.com/download/principles-soil-dynamics-3rd-editiondas-solutions-manual/ Chapter
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationIntroduction to Mechanical Vibration
2103433 Introduction to Mechanical Vibration Nopdanai Ajavakom (NAV) 1 Course Topics Introduction to Vibration What is vibration? Basic concepts of vibration Modeling Linearization Single-Degree-of-Freedom
More informationT1 T e c h n i c a l S e c t i o n
1.5 Principles of Noise Reduction A good vibration isolation system is reducing vibration transmission through structures and thus, radiation of these vibration into air, thereby reducing noise. There
More informationLecture XXVI. Morris Swartz Dept. of Physics and Astronomy Johns Hopkins University November 5, 2003
Lecture XXVI Morris Swartz Dept. of Physics and Astronomy Johns Hopins University morris@jhu.edu November 5, 2003 Lecture XXVI: Oscillations Oscillations are periodic motions. There are many examples of
More informationAA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS 1 / 50 AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of n-dof Systems These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations:
More informationStructural Dynamics Lecture 2. Outline of Lecture 2. Single-Degree-of-Freedom Systems (cont.)
Outline of Single-Degree-of-Freedom Systems (cont.) Linear Viscous Damped Eigenvibrations. Logarithmic decrement. Response to Harmonic and Periodic Loads. 1 Single-Degreee-of-Freedom Systems (cont.). Linear
More informationSome Aspects of Structural Dynamics
Appendix B Some Aspects of Structural Dynamics This Appendix deals with some aspects of the dynamic behavior of SDOF and MDOF. It starts with the formulation of the equation of motion of SDOF systems.
More informationD : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each.
GTE 2016 Q. 1 Q. 9 carry one mark each. D : SOLID MECHNICS Q.1 single degree of freedom vibrating system has mass of 5 kg, stiffness of 500 N/m and damping coefficient of 100 N-s/m. To make the system
More informationMathematical Modeling and response analysis of mechanical systems are the subjects of this chapter.
Chapter 3 Mechanical Systems A. Bazoune 3.1 INRODUCION Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. 3. MECHANICAL ELEMENS Any mechanical system consists
More informationEVALUATING DYNAMIC STRESSES OF A PIPELINE
EVALUATING DYNAMIC STRESSES OF A PIPELINE by K.T. TRUONG Member ASME Mechanical & Piping Division THE ULTRAGEN GROUP LTD 2255 Rue De La Province Longueuil (Quebec) J4G 1G3 This document is provided to
More informationEN40: Dynamics and Vibrations. Final Examination Wed May : 2pm-5pm
EN40: Dynamics and Vibrations Final Examination Wed May 10 017: pm-5pm School of Engineering Brown University NAME: General Instructions No collaboration of any kind is permitted on this examination. You
More informationOscillatory Motion SHM
Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A
More informationEQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION
1 EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION The course on Mechanical Vibration is an important part of the Mechanical Engineering undergraduate curriculum. It is necessary for the development
More informationThis equation of motion may be solved either by differential equation method or by graphical method as discussed below:
2.15. Frequency of Under Damped Forced Vibrations Consider a system consisting of spring, mass and damper as shown in Fig. 22. Let the system is acted upon by an external periodic (i.e. simple harmonic)
More informationKEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY OSCILLATIONS AND WAVES PRACTICE EXAM
KEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY-10012 OSCILLATIONS AND WAVES PRACTICE EXAM Candidates should attempt ALL of PARTS A and B, and TWO questions from PART C. PARTS A and B should be answered
More informationStructural System, Machines and Load Cases
Machine-Induced Vibrations Machine-Induced Vibrations In the following example the dynamic excitation of two rotating machines is analyzed. A time history analysis in the add-on module RF-DYNAM Pro - Forced
More informationDynamics of structures
Dynamics of structures 2.Vibrations: single degree of freedom system Arnaud Deraemaeker (aderaema@ulb.ac.be) 1 One degree of freedom systems in real life 2 1 Reduction of a system to a one dof system Example
More informationLecture 27: Structural Dynamics - Beams.
Chapter #16: Structural Dynamics and Time Dependent Heat Transfer. Lectures #1-6 have discussed only steady systems. There has been no time dependence in any problems. We will investigate beam dynamics
More informationExam Question 6/8 (HL/OL): Circular and Simple Harmonic Motion. February 1, Applied Mathematics: Lecture 7. Brendan Williamson.
in a : Exam Question 6/8 (HL/OL): Circular and February 1, 2017 in a This lecture pertains to material relevant to question 6 of the paper, and question 8 of the Ordinary Level paper, commonly referred
More informationIdentification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016
Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural
More informationStep 1: Mathematical Modeling
083 Mechanical Vibrations Lesson Vibration Analysis Procedure The analysis of a vibrating system usually involves four steps: mathematical modeling derivation of the governing uations solution of the uations
More information4.9 Free Mechanical Vibrations
4.9 Free Mechanical Vibrations Spring-Mass Oscillator When the spring is not stretched and the mass m is at rest, the system is at equilibrium. Forces Acting in the System When the mass m is displaced
More informationThe... of a particle is defined as its change in position in some time interval.
Distance is the. of a path followed by a particle. Distance is a quantity. The... of a particle is defined as its change in position in some time interval. Displacement is a.. quantity. The... of a particle
More informationTable of Contents. Preface... 13
Table of Contents Preface... 13 Chapter 1. Vibrations of Continuous Elastic Solid Media... 17 1.1. Objective of the chapter... 17 1.2. Equations of motion and boundary conditions of continuous media...
More informationChapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.
Chapter 14 Oscillations Oscillations of a Spring Simple Harmonic Motion Energy in the Simple Harmonic Oscillator Simple Harmonic Motion Related to Uniform Circular Motion The Simple Pendulum The Physical
More informationInternational Journal of Emerging Technologies in Computational and Applied Sciences (IJETCAS)
International Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research) International Journal of Emerging Technologies in Computational
More informationChapter 15. Oscillatory Motion
Chapter 15 Oscillatory Motion Part 2 Oscillations and Mechanical Waves Periodic motion is the repeating motion of an object in which it continues to return to a given position after a fixed time interval.
More informationLANMARK UNIVERSITY OMU-ARAN, KWARA STATE DEPARTMENT OF MECHANICAL ENGINEERING COURSE: MECHANICS OF MACHINE (MCE 322). LECTURER: ENGR.
LANMARK UNIVERSITY OMU-ARAN, KWARA STATE DEPARTMENT OF MECHANICAL ENGINEERING COURSE: MECHANICS OF MACHINE (MCE 322). LECTURER: ENGR. IBIKUNLE ROTIMI ADEDAYO SIMPLE HARMONIC MOTION. Introduction Consider
More informationPhysics 8 Monday, December 4, 2017
Physics 8 Monday, December 4, 2017 HW12 due Friday. Grace will do a review session Dec 12 or 13. When? I will do a review session: afternoon Dec 17? Evening Dec 18? Wednesday, I will hand out the practice
More information202 Index. failure, 26 field equation, 122 force, 1
Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic
More informationEngineering Science OUTCOME 2 - TUTORIAL 3 FREE VIBRATIONS
Unit 2: Unit code: QCF Level: 4 Credit value: 5 Engineering Science L/60/404 OUTCOME 2 - TUTORIAL 3 FREE VIBRATIONS UNIT CONTENT OUTCOME 2 Be able to determine the behavioural characteristics of elements
More informationENG1001 Engineering Design 1
ENG1001 Engineering Design 1 Structure & Loads Determine forces that act on structures causing it to deform, bend, and stretch Forces push/pull on objects Structures are loaded by: > Dead loads permanent
More informationDynamic analysis of a reinforced concrete shear wall with strain rate effect. Synopsis. Introduction
Dynamic analysis of a reinforced concrete shear wall with strain rate effect Synopsis A simplified analysis method for a reinforced concrete shear wall structure considering strain rate effects is presented.
More informationMATH 251 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam
MATH 51 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam A collection of previous exams could be found at the coordinator s web: http://www.math.psu.edu/tseng/class/m51samples.html
More informationMeasurement Techniques for Engineers. Motion and Vibration Measurement
Measurement Techniques for Engineers Motion and Vibration Measurement Introduction Quantities that may need to be measured are velocity, acceleration and vibration amplitude Quantities useful in predicting
More informationChapter 14 Periodic Motion
Chapter 14 Periodic Motion 1 Describing Oscillation First, we want to describe the kinematical and dynamical quantities associated with Simple Harmonic Motion (SHM), for example, x, v x, a x, and F x.
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationANALYSIS OF HIGHRISE BUILDING STRUCTURE WITH SETBACK SUBJECT TO EARTHQUAKE GROUND MOTIONS
ANALYSIS OF HIGHRISE BUILDING SRUCURE WIH SEBACK SUBJEC O EARHQUAKE GROUND MOIONS 157 Xiaojun ZHANG 1 And John L MEEK SUMMARY he earthquake response behaviour of unframed highrise buildings with setbacks
More informationMaterials: engineering, science, processing and design, 2nd edition Copyright (c)2010 Michael Ashby, Hugh Shercliff, David Cebon.
Modes of Loading (1) tension (a) (2) compression (b) (3) bending (c) (4) torsion (d) and combinations of them (e) Figure 4.2 1 Standard Solution to Elastic Problems Three common modes of loading: (a) tie
More informationEngineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration
Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration Module 15 Lecture 38 Vibration of Rigid Bodies Part-1 Today,
More informationLecture 9: Harmonic Loads (Con t)
Lecture 9: Harmonic Loads (Con t) Reading materials: Sections 3.4, 3.5, 3.6 and 3.7 1. Resonance The dynamic load magnification factor (DLF) The peak dynamic magnification occurs near r=1 for small damping
More informationThe student will experimentally determine the parameters to represent the behavior of a damped oscillatory system of one degree of freedom.
Practice 3 NAME STUDENT ID LAB GROUP PROFESSOR INSTRUCTOR Vibrations of systems of one degree of freedom with damping QUIZ 10% PARTICIPATION & PRESENTATION 5% INVESTIGATION 10% DESIGN PROBLEM 15% CALCULATIONS
More informationJEPPIAAR ENGINEERING COLLEGE
JEPPIAAR ENGINEERING COLLEGE Jeppiaar Nagar, Rajiv Gandhi Salai 600 119 DEPARTMENT OFMECHANICAL ENGINEERING QUESTION BANK VI SEMESTER ME6603 FINITE ELEMENT ANALYSIS Regulation 013 SUBJECT YEAR /SEM: III
More informationTOPIC E: OSCILLATIONS EXAMPLES SPRING Q1. Find general solutions for the following differential equations:
TOPIC E: OSCILLATIONS EXAMPLES SPRING 2019 Mathematics of Oscillating Systems Q1. Find general solutions for the following differential equations: Undamped Free Vibration Q2. A 4 g mass is suspended by
More informationMECHANICS OF STRUCTURES SCI 1105 COURSE MATERIAL UNIT - I
MECHANICS OF STRUCTURES SCI 1105 COURSE MATERIAL UNIT - I Engineering Mechanics Branch of science which deals with the behavior of a body with the state of rest or motion, subjected to the action of forces.
More informationChapter 5 Design. D. J. Inman 1/51 Mechanical Engineering at Virginia Tech
Chapter 5 Design Acceptable vibration levels (ISO) Vibration isolation Vibration absorbers Effects of damping in absorbers Optimization Viscoelastic damping treatments Critical Speeds Design for vibration
More informationFLEXIBILITY METHOD FOR INDETERMINATE FRAMES
UNIT - I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 1. What is meant by indeterminate structures? Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 015 14. Oscillations Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License This wor
More informationFinal Exam. June 10, 2008, 1:00pm
PHYSICS 101: Fundamentals of Physics Final Exam Final Exam Name TA/ Section # June 10, 2008, 1:00pm Recitation Time You have 2 hour to complete the exam. Please answer all questions clearly and completely,
More informationStudy of coupling between bending and torsional vibration of cracked rotor system supported by radial active magnetic bearings
Applied and Computational Mechanics 1 (2007) 427-436 Study of coupling between bending and torsional vibration of cracked rotor system supported by radial active magnetic bearings P. Ferfecki a, * a Center
More informationRutgers University Department of Physics & Astronomy. 01:750:271 Honors Physics I Fall Lecture 19. Home Page. Title Page. Page 1 of 36.
Rutgers University Department of Physics & Astronomy 01:750:271 Honors Physics I Fall 2015 Lecture 19 Page 1 of 36 12. Equilibrium and Elasticity How do objects behave under applied external forces? Under
More informationCMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation
CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University September 26, 2005 1 Sinusoids Sinusoids
More informationCHAPTER 3 VIBRATION THEORY. Single Degree of Freedom Systems (SDOF) Stiffness, k Member Stiffness (Member Rigidity).
HPTER 3 VIRTION THEORY Single egree of Freedom Systems (SOF).. 3- Stiffness, k. 3- Member Stiffness (Member Rigidity). 3-3 Natural Period.. 3-3 Natural Frequency... 3-5 ngular Natural Frequency 3-5 Structural
More informationChapter 7 Vibration Measurement and Applications
Chapter 7 Vibration Measurement and Applications Dr. Tan Wei Hong School of Mechatronic Engineering Universiti Malaysia Perlis (UniMAP) Pauh Putra Campus ENT 346 Vibration Mechanics Chapter Outline 7.1
More informationSprings: Part I Modeling the Action The Mass/Spring System
17 Springs: Part I Second-order differential equations arise in a number of applications We saw one involving a falling object at the beginning of this text (the falling frozen duck example in section
More informationSeismic design of bridges
NAIONAL ECHNICAL UNIVERSIY OF AHENS LABORAORY FOR EARHQUAKE ENGINEERING Seismic design of bridges Lecture 4 Ioannis N. Psycharis Seismic isolation of bridges I. N. Psycharis Seismic design of bridges 2
More information2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity
2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity Energy 7 4 Kinematics Free fall Collisions 3 5 Dynamics
More informationTranslational Motion Rotational Motion Equations Sheet
PHYSICS 01 Translational Motion Rotational Motion Equations Sheet LINEAR ANGULAR Time t t Displacement x; (x = rθ) θ Velocity v = Δx/Δt; (v = rω) ω = Δθ/Δt Acceleration a = Δv/Δt; (a = rα) α = Δω/Δt (
More informationDynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras
Dynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 09 Characteristics of Single Degree - of -
More informationStudy Sheet for Exam #3
Physics 121 Spring 2003 Dr. Dragt Study Sheet for Exam #3 14. Physics knowledge, like all subjects having some substance, is cumulative. You are still responsible for all material on the Study Sheets for
More informationComputational Simulation of Dynamic Response of Vehicle Tatra T815 and the Ground
IOP Conference Series: Earth and Environmental Science PAPER OPEN ACCESS Computational Simulation of Dynamic Response of Vehicle Tatra T815 and the Ground To cite this article: Jozef Vlek and Veronika
More informationVibration Control Prof. Dr. S. P. Harsha Department of Mechanical & Industrial Engineering Indian Institute of Technology, Roorkee
Vibration Control Prof. Dr. S. P. Harsha Department of Mechanical & Industrial Engineering Indian Institute of Technology, Roorkee Module - 1 Review of Basics of Mechanical Vibrations Lecture - 2 Introduction
More informationProblem 1: Find the Equation of Motion from the static equilibrium position for the following systems: 1) Assumptions
Problem 1: Find the Equation of Motion from the static equilibrium position for the following systems: 1) Assumptions k 2 Wheels roll without friction k 1 Motion will not cause block to hit the supports
More informationDesign of Reinforced Concrete Structures (II)
Design of Reinforced Concrete Structures (II) Discussion Eng. Mohammed R. Kuheil Review The thickness of one-way ribbed slabs After finding the value of total load (Dead and live loads), the elements are
More information1859. Forced transverse vibration analysis of a Rayleigh double-beam system with a Pasternak middle layer subjected to compressive axial load
1859. Forced transverse vibration analysis of a Rayleigh double-beam system with a Pasternak middle layer subjected to compressive axial load Nader Mohammadi 1, Mehrdad Nasirshoaibi 2 Department of Mechanical
More informationFree Vibration of Single-Degree-of-Freedom (SDOF) Systems
Free Vibration of Single-Degree-of-Freedom (SDOF) Systems Procedure in solving structural dynamics problems 1. Abstraction/modeling Idealize the actual structure to a simplified version, depending on the
More informationEffects of Damping Ratio of Restoring force Device on Response of a Structure Resting on Sliding Supports with Restoring Force Device
Effects of Damping Ratio of Restoring force Device on Response of a Structure Resting on Sliding Supports with Restoring Force Device A. Krishnamoorthy Professor, Department of Civil Engineering Manipal
More informationImpact. m k. Natural Period of Vibration τ. Static load Gray area Impact load t > 3 τ. Absorbing energy. Carrying loads
Impact also called shock, sudden or impulsive loading driving a nail with a hammer, automobile collisions. dashpot a) Rapidly moving vehicles crossing a bridge To distinguish: b) Suddenly applied c) Direct
More informationStochastic Dynamics of SDOF Systems (cont.).
Outline of Stochastic Dynamics of SDOF Systems (cont.). Weakly Stationary Response Processes. Equivalent White Noise Approximations. Gaussian Response Processes as Conditional Normal Distributions. Stochastic
More informationCAEFEM v9.5 Information
CAEFEM v9.5 Information Concurrent Analysis Corporation, 50 Via Ricardo, Thousand Oaks, CA 91320 USA Tel. (805) 375 1060, Fax (805) 375 1061 email: info@caefem.com or support@caefem.com Web: http://www.caefem.com
More informationDynamic Loads CE 543. Examples. Harmonic Loads
CE 543 Structural Dynamics Introduction Dynamic Loads Dynamic loads are time-varying loads. (But time-varying loads may not require dynamic analysis.) Dynamics loads can be grouped in one of the following
More information