Enviar búsqueda
Cargar
Base excitation of dynamic systems
•
Descargar como PPT, PDF
•
19 recomendaciones
•
27,480 vistas
University of Glasgow
Seguir
Base excitation of SDOF systems, dynamics due to rotating unbalance
Leer menos
Leer más
Educación
Denunciar
Compartir
Denunciar
Compartir
1 de 27
Descargar ahora
Recomendados
Finite Element Analysis - UNIT-4
Finite Element Analysis - UNIT-4
propaul
Mechanical Vibration
Mechanical Vibration
Ankur Shukla
Approximate Methods
Approximate Methods
Teja Ande
Finite Element Analysis - UNIT-3
Finite Element Analysis - UNIT-3
propaul
Energy methods for damped systems
Energy methods for damped systems
University of Glasgow
L5 determination of natural frequency & mode shape
L5 determination of natural frequency & mode shape
Sam Alalimi
Mechanical vibration note
Mechanical vibration note
Mohammed Imran
Mechanical Vibrations all slides
Mechanical Vibrations all slides
Ebrahim Hanash
Recomendados
Finite Element Analysis - UNIT-4
Finite Element Analysis - UNIT-4
propaul
Mechanical Vibration
Mechanical Vibration
Ankur Shukla
Approximate Methods
Approximate Methods
Teja Ande
Finite Element Analysis - UNIT-3
Finite Element Analysis - UNIT-3
propaul
Energy methods for damped systems
Energy methods for damped systems
University of Glasgow
L5 determination of natural frequency & mode shape
L5 determination of natural frequency & mode shape
Sam Alalimi
Mechanical vibration note
Mechanical vibration note
Mohammed Imran
Mechanical Vibrations all slides
Mechanical Vibrations all slides
Ebrahim Hanash
Unit 4 Forced Vibration
Unit 4 Forced Vibration
Parrthipan B K
Constant strain triangular
Constant strain triangular
rahul183
Finite Element Analysis - UNIT-1
Finite Element Analysis - UNIT-1
propaul
Vibration Isolation and Base Excitation
Vibration Isolation and Base Excitation
Himanshi Gupta
single degree of freedom systems forced vibrations
single degree of freedom systems forced vibrations
KESHAV
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ASHOK KUMAR RAJENDRAN
Dynamics of Machinery Unit IV
Dynamics of Machinery Unit IV
Vaidyanathan Ramakrishnan
Mechanical Vibration- An introduction
Mechanical Vibration- An introduction
Hareesha N Gowda, Dayananda Sagar College of Engg, Bangalore
Multiple Degree of Freedom (MDOF) Systems
Multiple Degree of Freedom (MDOF) Systems
Mohammad Tawfik
Sdof
Sdof
Teja Ande
Finite Element Analysis - UNIT-2
Finite Element Analysis - UNIT-2
propaul
Chapter 1 introduction to mechanical vibration
Chapter 1 introduction to mechanical vibration
Bahr Alyafei
Vibration and damping
Vibration and damping
Divya Lattoo
Force Damped Vibrations
Force Damped Vibrations
Manthan Kanani
Undamped Free Vibration
Undamped Free Vibration
Urvish Patel
Unsymmetrical bending.ppt
Unsymmetrical bending.ppt
Venkatesh Ca
ME6603 - FINITE ELEMENT ANALYSIS UNIT - II NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - II NOTES AND QUESTION BANK
ASHOK KUMAR RAJENDRAN
Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5
propaul
Chapter 2 lecture 1 mechanical vibration
Chapter 2 lecture 1 mechanical vibration
Bahr Alyafei
ME6503 - DESIGN OF MACHINE ELEMENTS UNIT - I NOTES
ME6503 - DESIGN OF MACHINE ELEMENTS UNIT - I NOTES
ASHOK KUMAR RAJENDRAN
Dynamic response to harmonic excitation
Dynamic response to harmonic excitation
University of Glasgow
Dynamic response of oscillators to general excitations
Dynamic response of oscillators to general excitations
University of Glasgow
Más contenido relacionado
La actualidad más candente
Unit 4 Forced Vibration
Unit 4 Forced Vibration
Parrthipan B K
Constant strain triangular
Constant strain triangular
rahul183
Finite Element Analysis - UNIT-1
Finite Element Analysis - UNIT-1
propaul
Vibration Isolation and Base Excitation
Vibration Isolation and Base Excitation
Himanshi Gupta
single degree of freedom systems forced vibrations
single degree of freedom systems forced vibrations
KESHAV
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ASHOK KUMAR RAJENDRAN
Dynamics of Machinery Unit IV
Dynamics of Machinery Unit IV
Vaidyanathan Ramakrishnan
Mechanical Vibration- An introduction
Mechanical Vibration- An introduction
Hareesha N Gowda, Dayananda Sagar College of Engg, Bangalore
Multiple Degree of Freedom (MDOF) Systems
Multiple Degree of Freedom (MDOF) Systems
Mohammad Tawfik
Sdof
Sdof
Teja Ande
Finite Element Analysis - UNIT-2
Finite Element Analysis - UNIT-2
propaul
Chapter 1 introduction to mechanical vibration
Chapter 1 introduction to mechanical vibration
Bahr Alyafei
Vibration and damping
Vibration and damping
Divya Lattoo
Force Damped Vibrations
Force Damped Vibrations
Manthan Kanani
Undamped Free Vibration
Undamped Free Vibration
Urvish Patel
Unsymmetrical bending.ppt
Unsymmetrical bending.ppt
Venkatesh Ca
ME6603 - FINITE ELEMENT ANALYSIS UNIT - II NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - II NOTES AND QUESTION BANK
ASHOK KUMAR RAJENDRAN
Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5
propaul
Chapter 2 lecture 1 mechanical vibration
Chapter 2 lecture 1 mechanical vibration
Bahr Alyafei
ME6503 - DESIGN OF MACHINE ELEMENTS UNIT - I NOTES
ME6503 - DESIGN OF MACHINE ELEMENTS UNIT - I NOTES
ASHOK KUMAR RAJENDRAN
La actualidad más candente
(20)
Unit 4 Forced Vibration
Unit 4 Forced Vibration
Constant strain triangular
Constant strain triangular
Finite Element Analysis - UNIT-1
Finite Element Analysis - UNIT-1
Vibration Isolation and Base Excitation
Vibration Isolation and Base Excitation
single degree of freedom systems forced vibrations
single degree of freedom systems forced vibrations
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
Dynamics of Machinery Unit IV
Dynamics of Machinery Unit IV
Mechanical Vibration- An introduction
Mechanical Vibration- An introduction
Multiple Degree of Freedom (MDOF) Systems
Multiple Degree of Freedom (MDOF) Systems
Sdof
Sdof
Finite Element Analysis - UNIT-2
Finite Element Analysis - UNIT-2
Chapter 1 introduction to mechanical vibration
Chapter 1 introduction to mechanical vibration
Vibration and damping
Vibration and damping
Force Damped Vibrations
Force Damped Vibrations
Undamped Free Vibration
Undamped Free Vibration
Unsymmetrical bending.ppt
Unsymmetrical bending.ppt
ME6603 - FINITE ELEMENT ANALYSIS UNIT - II NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - II NOTES AND QUESTION BANK
Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5
Chapter 2 lecture 1 mechanical vibration
Chapter 2 lecture 1 mechanical vibration
ME6503 - DESIGN OF MACHINE ELEMENTS UNIT - I NOTES
ME6503 - DESIGN OF MACHINE ELEMENTS UNIT - I NOTES
Similar a Base excitation of dynamic systems
Dynamic response to harmonic excitation
Dynamic response to harmonic excitation
University of Glasgow
Dynamic response of oscillators to general excitations
Dynamic response of oscillators to general excitations
University of Glasgow
Capítulo 11 (5th edition)rewrweerww
Capítulo 11 (5th edition)rewrweerww
Frank Diego Quispe Vigo
solucionario mecanica vectorial para ingenieros - beer & johnston (dinamica...
solucionario mecanica vectorial para ingenieros - beer & johnston (dinamica...
Sohar Carr
Principles of soil dynamics 3rd edition das solutions manual
Principles of soil dynamics 3rd edition das solutions manual
Human2379
Absorber
Absorber
Massimo Cavacece
dynamical analysis of soil and structures
dynamical analysis of soil and structures
HaHoangJR
Solutions Manual for Foundations Of MEMS 2nd Edition by Chang Liu
Solutions Manual for Foundations Of MEMS 2nd Edition by Chang Liu
HildaLa
Fox And Mcdonald's Introduction To Fluid Mechanics 8th Edition Pritchard Solu...
Fox And Mcdonald's Introduction To Fluid Mechanics 8th Edition Pritchard Solu...
qedobose
25 johnarry tonye ngoji 250-263
25 johnarry tonye ngoji 250-263
Alexander Decker
'Documents.mx dynamics solucionario-riley.pdf'
'Documents.mx dynamics solucionario-riley.pdf'
jhameschiqui
sdof-1211798306003307-8.pptx
sdof-1211798306003307-8.pptx
SahilDhanvijay2
11.vibrational characteristics of visco elstic plate with varying thickness
11.vibrational characteristics of visco elstic plate with varying thickness
Alexander Decker
Chapter 14 solutions_to_exercises(engineering circuit analysis 7th)
Chapter 14 solutions_to_exercises(engineering circuit analysis 7th)
Maamoun Hennache
Platoon Control of Nonholonomic Robots using Quintic Bezier Splines
Platoon Control of Nonholonomic Robots using Quintic Bezier Splines
Kaustav Mondal
Similar a Base excitation of dynamic systems
(15)
Dynamic response to harmonic excitation
Dynamic response to harmonic excitation
Dynamic response of oscillators to general excitations
Dynamic response of oscillators to general excitations
Capítulo 11 (5th edition)rewrweerww
Capítulo 11 (5th edition)rewrweerww
solucionario mecanica vectorial para ingenieros - beer & johnston (dinamica...
solucionario mecanica vectorial para ingenieros - beer & johnston (dinamica...
Principles of soil dynamics 3rd edition das solutions manual
Principles of soil dynamics 3rd edition das solutions manual
Absorber
Absorber
dynamical analysis of soil and structures
dynamical analysis of soil and structures
Solutions Manual for Foundations Of MEMS 2nd Edition by Chang Liu
Solutions Manual for Foundations Of MEMS 2nd Edition by Chang Liu
Fox And Mcdonald's Introduction To Fluid Mechanics 8th Edition Pritchard Solu...
Fox And Mcdonald's Introduction To Fluid Mechanics 8th Edition Pritchard Solu...
25 johnarry tonye ngoji 250-263
25 johnarry tonye ngoji 250-263
'Documents.mx dynamics solucionario-riley.pdf'
'Documents.mx dynamics solucionario-riley.pdf'
sdof-1211798306003307-8.pptx
sdof-1211798306003307-8.pptx
11.vibrational characteristics of visco elstic plate with varying thickness
11.vibrational characteristics of visco elstic plate with varying thickness
Chapter 14 solutions_to_exercises(engineering circuit analysis 7th)
Chapter 14 solutions_to_exercises(engineering circuit analysis 7th)
Platoon Control of Nonholonomic Robots using Quintic Bezier Splines
Platoon Control of Nonholonomic Robots using Quintic Bezier Splines
Más de University of Glasgow
A hollow future for engineering structures
A hollow future for engineering structures
University of Glasgow
Homogeneous dynamic characteristics of damped 2 d elastic lattices
Homogeneous dynamic characteristics of damped 2 d elastic lattices
University of Glasgow
Projection methods for stochastic structural dynamics
Projection methods for stochastic structural dynamics
University of Glasgow
Eh5 Design case studies and future scopes
Eh5 Design case studies and future scopes
University of Glasgow
Eh4 energy harvesting due to random excitations and optimal design
Eh4 energy harvesting due to random excitations and optimal design
University of Glasgow
Eh3 analysis of nonlinear energy harvesters
Eh3 analysis of nonlinear energy harvesters
University of Glasgow
Eh2 piezoelectric energy harvesting due to harmonic excitations
Eh2 piezoelectric energy harvesting due to harmonic excitations
University of Glasgow
EH1 - Reduced-order modelling for vibration energy harvesting
EH1 - Reduced-order modelling for vibration energy harvesting
University of Glasgow
Dynamic Homogenisation of randomly irregular viscoelastic metamaterials
Dynamic Homogenisation of randomly irregular viscoelastic metamaterials
University of Glasgow
Dynamic response of structures with uncertain properties
Dynamic response of structures with uncertain properties
University of Glasgow
Computational methods for nanoscale bio sensors
Computational methods for nanoscale bio sensors
University of Glasgow
Random vibration energy harvesting
Random vibration energy harvesting
University of Glasgow
Free vibration analysis of composite plates with uncertain properties
Free vibration analysis of composite plates with uncertain properties
University of Glasgow
Dynamics of structures with uncertainties
Dynamics of structures with uncertainties
University of Glasgow
Transient response of delaminated composite shell subjected to low velocity o...
Transient response of delaminated composite shell subjected to low velocity o...
University of Glasgow
Dynamic stiffness and eigenvalues of nonlocal nano beams
Dynamic stiffness and eigenvalues of nonlocal nano beams
University of Glasgow
Dynamics of nonlocal structures
Dynamics of nonlocal structures
University of Glasgow
Dynamics of multiple degree of freedom linear systems
Dynamics of multiple degree of freedom linear systems
University of Glasgow
Dynamics of wind & marine turbines
Dynamics of wind & marine turbines
University of Glasgow
Multiscale methods for graphene based nanocomposites
Multiscale methods for graphene based nanocomposites
University of Glasgow
Más de University of Glasgow
(20)
A hollow future for engineering structures
A hollow future for engineering structures
Homogeneous dynamic characteristics of damped 2 d elastic lattices
Homogeneous dynamic characteristics of damped 2 d elastic lattices
Projection methods for stochastic structural dynamics
Projection methods for stochastic structural dynamics
Eh5 Design case studies and future scopes
Eh5 Design case studies and future scopes
Eh4 energy harvesting due to random excitations and optimal design
Eh4 energy harvesting due to random excitations and optimal design
Eh3 analysis of nonlinear energy harvesters
Eh3 analysis of nonlinear energy harvesters
Eh2 piezoelectric energy harvesting due to harmonic excitations
Eh2 piezoelectric energy harvesting due to harmonic excitations
EH1 - Reduced-order modelling for vibration energy harvesting
EH1 - Reduced-order modelling for vibration energy harvesting
Dynamic Homogenisation of randomly irregular viscoelastic metamaterials
Dynamic Homogenisation of randomly irregular viscoelastic metamaterials
Dynamic response of structures with uncertain properties
Dynamic response of structures with uncertain properties
Computational methods for nanoscale bio sensors
Computational methods for nanoscale bio sensors
Random vibration energy harvesting
Random vibration energy harvesting
Free vibration analysis of composite plates with uncertain properties
Free vibration analysis of composite plates with uncertain properties
Dynamics of structures with uncertainties
Dynamics of structures with uncertainties
Transient response of delaminated composite shell subjected to low velocity o...
Transient response of delaminated composite shell subjected to low velocity o...
Dynamic stiffness and eigenvalues of nonlocal nano beams
Dynamic stiffness and eigenvalues of nonlocal nano beams
Dynamics of nonlocal structures
Dynamics of nonlocal structures
Dynamics of multiple degree of freedom linear systems
Dynamics of multiple degree of freedom linear systems
Dynamics of wind & marine turbines
Dynamics of wind & marine turbines
Multiscale methods for graphene based nanocomposites
Multiscale methods for graphene based nanocomposites
Último
Keynote by Prof. Wurzer at Nordex about IP-design
Keynote by Prof. Wurzer at Nordex about IP-design
MIPLM
Proudly South Africa powerpoint Thorisha.pptx
Proudly South Africa powerpoint Thorisha.pptx
thorishapillay1
Full Stack Web Development Course for Beginners
Full Stack Web Development Course for Beginners
Sabitha Banu
Model Call Girl in Tilak Nagar Delhi reach out to us at 🔝9953056974🔝
Model Call Girl in Tilak Nagar Delhi reach out to us at 🔝9953056974🔝
9953056974 Low Rate Call Girls In Saket, Delhi NCR
USPS® Forced Meter Migration - How to Know if Your Postage Meter Will Soon be...
USPS® Forced Meter Migration - How to Know if Your Postage Meter Will Soon be...
Postal Advocate Inc.
Difference Between Search & Browse Methods in Odoo 17
Difference Between Search & Browse Methods in Odoo 17
Celine George
Roles & Responsibilities in Pharmacovigilance
Roles & Responsibilities in Pharmacovigilance
SamikshaHamane
DATA STRUCTURE AND ALGORITHM for beginners
DATA STRUCTURE AND ALGORITHM for beginners
Sabitha Banu
How to do quick user assign in kanban in Odoo 17 ERP
How to do quick user assign in kanban in Odoo 17 ERP
Celine George
FINALS_OF_LEFT_ON_C'N_EL_DORADO_2024.pptx
FINALS_OF_LEFT_ON_C'N_EL_DORADO_2024.pptx
Conquiztadors- the Quiz Society of Sri Venkateswara College
Choosing the Right CBSE School A Comprehensive Guide for Parents
Choosing the Right CBSE School A Comprehensive Guide for Parents
navabharathschool99
Science 7 Quarter 4 Module 2: Natural Resources.pptx
Science 7 Quarter 4 Module 2: Natural Resources.pptx
MaryGraceBautista27
Raw materials used in Herbal Cosmetics.pptx
Raw materials used in Herbal Cosmetics.pptx
Ashokrao Mane college of Pharmacy Peth-Vadgaon
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
Humphrey A Beña
Gas measurement O2,Co2,& ph) 04/2024.pptx
Gas measurement O2,Co2,& ph) 04/2024.pptx
Dr.Ibrahim Hassaan
Field Attribute Index Feature in Odoo 17
Field Attribute Index Feature in Odoo 17
Celine George
How to Add Barcode on PDF Report in Odoo 17
How to Add Barcode on PDF Report in Odoo 17
Celine George
What is Model Inheritance in Odoo 17 ERP
What is Model Inheritance in Odoo 17 ERP
Celine George
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
Anupkumar Sharma
call girls in Kamla Market (DELHI) 🔝 >༒9953330565🔝 genuine Escort Service 🔝✔️✔️
call girls in Kamla Market (DELHI) 🔝 >༒9953330565🔝 genuine Escort Service 🔝✔️✔️
9953056974 Low Rate Call Girls In Saket, Delhi NCR
Último
(20)
Keynote by Prof. Wurzer at Nordex about IP-design
Keynote by Prof. Wurzer at Nordex about IP-design
Proudly South Africa powerpoint Thorisha.pptx
Proudly South Africa powerpoint Thorisha.pptx
Full Stack Web Development Course for Beginners
Full Stack Web Development Course for Beginners
Model Call Girl in Tilak Nagar Delhi reach out to us at 🔝9953056974🔝
Model Call Girl in Tilak Nagar Delhi reach out to us at 🔝9953056974🔝
USPS® Forced Meter Migration - How to Know if Your Postage Meter Will Soon be...
USPS® Forced Meter Migration - How to Know if Your Postage Meter Will Soon be...
Difference Between Search & Browse Methods in Odoo 17
Difference Between Search & Browse Methods in Odoo 17
Roles & Responsibilities in Pharmacovigilance
Roles & Responsibilities in Pharmacovigilance
DATA STRUCTURE AND ALGORITHM for beginners
DATA STRUCTURE AND ALGORITHM for beginners
How to do quick user assign in kanban in Odoo 17 ERP
How to do quick user assign in kanban in Odoo 17 ERP
FINALS_OF_LEFT_ON_C'N_EL_DORADO_2024.pptx
FINALS_OF_LEFT_ON_C'N_EL_DORADO_2024.pptx
Choosing the Right CBSE School A Comprehensive Guide for Parents
Choosing the Right CBSE School A Comprehensive Guide for Parents
Science 7 Quarter 4 Module 2: Natural Resources.pptx
Science 7 Quarter 4 Module 2: Natural Resources.pptx
Raw materials used in Herbal Cosmetics.pptx
Raw materials used in Herbal Cosmetics.pptx
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
Gas measurement O2,Co2,& ph) 04/2024.pptx
Gas measurement O2,Co2,& ph) 04/2024.pptx
Field Attribute Index Feature in Odoo 17
Field Attribute Index Feature in Odoo 17
How to Add Barcode on PDF Report in Odoo 17
How to Add Barcode on PDF Report in Odoo 17
What is Model Inheritance in Odoo 17 ERP
What is Model Inheritance in Odoo 17 ERP
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
call girls in Kamla Market (DELHI) 🔝 >༒9953330565🔝 genuine Escort Service 🔝✔️✔️
call girls in Kamla Market (DELHI) 🔝 >༒9953330565🔝 genuine Escort Service 🔝✔️✔️
Base excitation of dynamic systems
1.
2.4 Base Excitation •
Important class of vibration analysis – Preventing excitations from passing from a vibrating base through its mount into a structure • Vibration isolation – Vibrations in your car – Satellite operation – Building under earthquake – Disk drives, etc. © Eng. Vib, 3rd Ed. 1/27 @ProfAdhikari, #EG260
2.
FBD of SDOF
Base Excitation System Sketch x(t) m k y(t) System FBD m c k ( x − y ) c(x − y) base x ∑ F =-k(x-y)-c(x-y)=m m x + kx = cy + ky x+c © Eng. Vib, 3rd Ed. 2/27 College of Engineering (2.61)
3.
SDOF Base Excitation
(cont) Assume: y(t) = Y sin(ω t) and plug into Equation(2.61) m x + kx = cω + kY sin(ω t) (2.63) x+c Y cos(ω t) harmonic forcing functions For a car, ω= 2π 2πV = τ λ The steady-state solution is just the superposition of the two individual particular solutions (system is linear). f0 s 2 2 ζω n x + ω n x = 2ζω nω Y cos(ω t) + ω n Y sin(ω t) x+2 f0 c © Eng. Vib, 3rd Ed. 3/27 College of Engineering (2.64)
4.
Particular Solution (sine
term) With a sine for the forcing function, 2 ζω n x + ω n x =f0 s sin ω t x+2 x ps = As cos ω t + Bs sin ω t = X s sin(ω t − φ s ) where As = Bs = © Eng. Vib, 3rd Ed. 4/27 −2ζω nω f0 s (ω − ω ) + ( 2ζω nω ) 2 n 2 2 2 (ω − ω ) f0 s 2 n 2 2 2 (ω − ω ) + ( 2ζω nω ) 2 n 2 College of Engineering Use rectangular form to make it easier to add the cos term
5.
Particular Solution (cos
term) With a cosine for the forcing function, we showed 2 ζω n x + ω n x =f0c cos ω t x+2 x pc = Ac cosω t + Bc sin ω t = Xc cos(ω t − φ c ) where Ac = Bc = © Eng. Vib, 3rd Ed. 5/27 (ω − ω ) f0c 2 n 2 2 2 (ω − ω ) + ( 2ζω nω ) 2 n 2 2ζω nω f0c (ω − ω ) + ( 2ζω nω ) 2 n 2 2 2 College of Engineering
6.
Magnitude X/Y Now add
the sin and cos terms to get the magnitude of the full particular solution X = f 02c + f 02s (ω − ω ) + ( 2ζω nω ) 2 n 2 2 2 = ω nY (2ζω ) 2 + ω n2 (ω − ω ) + ( 2ζω nω ) 2 n 2 2 2 where f 0 c = 2 ζω nωY and f 0 s = ω n2Y if we define r = ω ωn this becomes X =Y (1 − r ) + ( 2ζ r ) X 1 + (2ζ r ) 2 = 2 2 2 Y (1 − r ) + ( 2ζ r ) © Eng. Vib, 3rd Ed. 6/27 1 + (2ζ r ) 2 College of Engineering 2 2 (2.71) 2 (2.70)
7.
The relative magnitude
plot of X/Y versus frequency ratio: Called the Displacement Transmissibility 40 ζ =0.01 ζ =0.1 ζ =0.3 ζ =0.7 30 X/Y (dB) 20 10 0 -10 -20 0 © Eng. Vib, 3rd Ed. 7/27 0.5 Figure 2.13 1 1.5 2 Frequency ratio r College of Engineering 2.5 3
8.
From the plot
of relative Displacement Transmissibility observe that: • X/Y is called Displacement Transmissibility Ratio • Potentially severe amplification at resonance • Attenuation for r > sqrt(2) Isolation Zone • If r < sqrt(2) transmissibility decreases with damping ratio Amplification Zone • If r >> 1 then transmissibility increases with damping ratio Xp~2Yξ /r © Eng. Vib, 3rd Ed. 8/27 College of Engineering
9.
Next examine the
Force Transmitted to the mass as a function of the frequency ratio FT = −k(x − y) − c( x − y ) = m x From FBD At steady state, x(t) = X cos(ω t − φ ), ω 2 X cos(ω t − φ ) so x=x(t) FT = mω X = kr X 2 m 2 FT k c y(t) © Eng. Vib, 3rd Ed. 9/27 College of Engineering base
10.
Plot of Force
Transmissibility (in dB) versus frequency ratio 40 ζ =0.01 ζ =0.1 F/kY (dB) 30 ζ =0.3 ζ =0.7 20 10 0 -10 -20 0 0.5 Figure 2.14 © Eng. Vib, 3rd Ed. 10/27 1 1.5 2 Frequency ratio r College of Engineering 2.5 3
11.
Figure 2.15 Comparison
between force and displacement transmissibility Force Transmissibility Displacement Transmissibility © Eng. Vib, 3rd Ed. 11/27 College of Engineering
12.
Example 2.4.1: Effect
of speed on the amplitude of car vibration © Eng. Vib, 3rd Ed. 12/27 College of Engineering
13.
Model the road
as a sinusoidal input to base motion of the car model Approximation of road surface: y(t) = (0.01 m)sin ω bt 1 hour 2π rad ω b = v(km/hr) = 0.2909v rad/s 0.006 km 3600 s cycle ω b (20km/hr) = 5.818 rad/s From the data given, determine the frequency and damping ratio of the car suspension: ωn = k = m c ζ= = 2 km 2 © Eng. Vib, 3rd Ed. 13/27 4 × 10 4 N/m = 6.303 rad/s ( ≈ 1 Hz) 1007 kg 2000 Ns/m (4 × 10 4 ) N/m (1007 kg ) College of Engineering = 0.158
14.
From the input
frequency, input amplitude, natural frequency and damping ratio use equation (2.70) to compute the amplitude of the response: r= ω b 5.818 = ω 6.303 1 + (2ζ r)2 X =Y (1 − r 2 )2 + (2ζ r)2 = (0.01 m ) 1 + [2(0.158)(0.923)] 2 (1 − (0.923) ) + (2 (0.158)(0.923)) 2 2 2 = 0.0319 m What happens as the car goes faster? See Table 2.1. © Eng. Vib, 3rd Ed. 14/27 College of Engineering
15.
Example 2.4.2: Compute
the force transmitted to a machine through base motion at resonance From (2.77) at r =1: 1/2 FT 1 + (2ζ ) = kY (2ζ )2 2 ⇒ FT = kY 1 + 4ζ 2 2ζ From given m, c, and k: ζ = c 900 = ≅ 0.04 2 km 2 40, 000g 3000 From measured excitation Y = 0.001 m: kY (40, 000 N/m)(0.001 m) 2 FT = 1 + 4ζ = 1 + 4(0.04)2 = 501.6 N 2ζ 2(0.04) © Eng. Vib, 3rd Ed. 15/27 College of Engineering
16.
2.5 Rotating Unbalance • • • • Gyros Cryo-coolers Tires Washing machines m0 e Machine
of total mass m i.e. m0 ω rt included in m e = eccentricity mo = mass unbalance ω r t =rotation frequency © Eng. Vib, 3rd Ed. 16/27 College of Engineering k c
17.
Rotating Unbalance (cont) Rx ω
rt What force is imparted on the structure? Note it rotates with x component: m0 e θ Ry xr = e sin ω r t ⇒ a x = r = −eω r2 sin ω r t x From the dynamics, R 0 = es = es r = x m i −ω t a ω m iω n θ n x m− 2 o r 2 o r 2 o r 2 o r R 0 = ec = ec r = y m o −ωω a ωθm ot s s y m− © Eng. Vib, 3rd Ed. 17/27 College of Engineering
18.
Rotating Unbalance (cont) The
problem is now just like any other SDOF system with a harmonic excitation m0 eω r2 sin(ω r t) + cx + kx = mo eω r2 sin ω r t mx x(t) m k c © Eng. Vib, 3rd Ed. 18/27 (2.82) mo 2 or + 2ζω n x + ω x = x eω r sin ω r t m 2 n Note the influences on the forcing function (we are assuming that the mass m is held in place in the y direction as indicated in Figure 2.18) College of Engineering
19.
Rotating Unbalance (cont) •
Just another SDOF oscillator with a harmonic forcing function • Expressed in terms of frequency ratio r x p (t ) = X sin(ω r t − φ ) (2.83) moe r2 X= m (1 − r 2 )2 + (2ζ r )2 (2.84) 2ζ r φ = tan 2 1− r (2.85) −1 © Eng. Vib, 3rd Ed. 19/27 College of Engineering
20.
Figure 2.20: Displacement
magnitude vs frequency caused by rotating unbalance © Eng. Vib, 3rd Ed. 20/27 College of Engineering
21.
Example 2.5.1:Given the
deflection at resonance (0.1m), ζ = 0.05 and a 10% out of balance, compute e and the amount of added mass needed to reduce the maximum amplitude to 0.01 m. At resonance r = 1 and mX 1 1 0.1 m 1 = = ⇒ 10 = = 10 ⇒ e = 0.1 m m0 e 2ζ 2(0.05) e 2ζ Now to compute the added mass, again at resonance; m X = 10 m0 0.1 m Use this to find Δm so that X is 0.01: m + ∆m 0.01 m m + ∆m = 100 ⇒ ∆m = 9m = 10 ⇒ m0 0.1 m (0.1)m Here m0 is 10%m or 0.1m © Eng. Vib, 3rd Ed. 21/27 College of Engineering
22.
Example 2.5.2 Helicopter
rotor unbalance Given Fig 2.21 k = 1 × 10 5 N/m mtail = 60 kg mrot = 20 kg m0 = 0.5 kg ζ = 0.01 Fig 2.22 Compute the deflection at 1500 rpm and find the rotor speed at which the deflection is maximum © Eng. Vib, 3rd Ed. 22/27 College of Engineering
23.
Example 2.5.2 Solution The
rotating mass is 20 + 0.5 or 20.5. The stiffness is provided by the Tail section and the corresponding mass is that determined in the example of a heavy beam. So the system natural frequency is k ωn = = 33 m+ mtail 140 The frequency of rotation is 10 5 N/m = 53.72 rad/s 33 20.5 + 60 kg 140 rev min 2π rad ω r = 1500 rpm = 1500 = 157 rad/s min 60 s rev 157 rad/s ⇒ r= = 2.92 53.96 rad/s © Eng. Vib, 3rd Ed. 23/27 College of Engineering
24.
Now compute the
deflection at r = 2.91 and ζ =0.01 using eq (2.84) m0 e r2 X= m (1 − r 2 )2 + (2ζ r)2 (0.5 kg )(0.15 m ) = 34.64 kg (2.92 )2 2 2 (1 − (2.92) ) − (2(0.01)(2.92))2 = 0.002 m At around r = 1, the max deflection occurs: rad rev 60 s r = 1 ⇒ ω r = 53.72 rad/s = 53.72 = 515.1 rpm s 2π rad min At r = 1: m0 e 1 (0.5 kg )(0.15 m ) 1 X= = = 0.108 m or 10.8 cm meq 2ζ 34.34 kg 2(0.01) © Eng. Vib, 3rd Ed. 24/27 College of Engineering
25.
2.6 Measurement Devices •
A basic transducer used in vibration measurement is the accelerometer. x • This device can be ∑ F = - k(x-y) - c(x-y) = m modeled using the ⇒ m = -c( x − y) - k(x − y) x base equations (2.86) and (2.61) developed in the Here, y(t) is the measured previous section response of the structure © Eng. Vib, 3rd Ed. 25/27 College of Engineering
26.
Base motion applied
to measurement devices Let z(t) = x(t) − y(t) (2.87) : ⇒ 2 m + cz(t) + kz(t) = mω b Y cosω bt (2.88) z Z r2 ⇒ = Y (1− r 2 )2 + (2ζ r)2 (2.90) Accelerometer and 2ζ r θ = tan −1 2÷ 1− r (2.91) These equations should be familiar from base motion. Here they describe measurement! © Eng. Vib, 3rd Ed. 26/27 Strain Gauge College of Engineering
27.
Magnitude and sensitivity plots
for accelerometers. Effect of damping on proportionality constant Fig 2.27 Fig 2.26 Magnitude plot showing Regions of measurement In the accel region, output voltage is nearly proportional to displacement © Eng. Vib, 3rd Ed. 27/27 College of Engineering
Notas del editor
ASK CLASS TO CHECK NUMBERS
The equivalent mass is m+(33/140)ms = 34.34
Descargar ahora