Multi Degrees of Freedom Systems

Size: px
Start display at page:

Download "Multi Degrees of Freedom Systems"

Transcription

1 Multi Degrees of Freedom Systems MDOF s Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 9, 07 Outline, a System of Linear Differential Equations are are are a base EoM in Modal Coordinates Consider an undamped system with two masses and two degrees of freedom. p (t) p (t) m m k k k 3 x x are

2 We can separate the two masses, single out the spring forces and, using the D Alembert Principle, the inertial forces and, finally. write an equation of dynamic equilibrium for each mass. k x p m ẍ k (x x ) m ẍ + (k + k )x k x = p (t) are k (x x ) p m ẍ k 3 x m ẍ k x + (k + k 3 )x = p (t) equation of motion of a DOF system With some little rearrangement we have a system of two linear differential equations in two variables, x (t) and x (t): { m ẍ + (k + k )x k x = p (t), m ẍ k x + (k + k 3 )x = p (t). are equation of motion of a DOF system Introducing the loading vector p, the vector of inertial forces f I and the vector of elastic forces f S, { } { } { } p (t) fi, fs, p =, f p (t) I =, f S = f I, we can write a vectorial equation of equilibrium: f I + f S = p(t). f S, are

3 f S = K x It is possible to write the linear relationship between f S and the vector of displacements x = { } T x x in terms of a matrix product, introducing the so called stiffness matrix K. In our example it is [ ] k + k f S = k x = K x k k + k 3 stiffness matrix K has a number of rows equal to the number of elastic forces, i.e., one force for each DOF and a number of columns equal to the number of the DOF. stiffness matrix K is hence a square matrix K ndof ndof are f I = M ẍ Analogously, introducing the mass matrix M that, for our example, is [ ] m 0 M = 0 m we can write f I = M ẍ. Also the mass matrix M is a square matrix, with number of rows and columns equal to the number of DOF s. are Matrix Equation Finally it is possible to write the equation of motion in matrix format: M ẍ + K x = p(t). Of course it is possible to take into consideration also the damping forces, taking into account the velocity vector ẋ and introducing a damping matrix C too, so that we can eventually write M ẍ + C ẋ + K x = p(t). But today we are focused on undamped systems... are

4 K K is symmetrical. elastic force exerted on mass i due to an unit displacement of mass j, f S,i = k ij is equal to the force k ji exerted on mass j due to an unit diplacement of mass i, in virtue of Betti s theorem (also known as Maxwell-Betti reciprocal work theorem). K is a positive definite matrix. strain energy V for a discrete system is V = xt f S, and expressing f S in terms of K and x we have are V = xt K x, and because the strain energy is positive for x 0 it follows that K is definite positive. M Restricting our discussion to systems whose degrees of freedom are the displacements of a set of discrete masses, we have that the mass matrix is a diagonal matrix, with all its diagonal elements greater than zero. Such a matrix is symmetrical and definite positive. Both the mass and the stiffness matrix are symmetrical and definite positive. are Note that the kinetic energy for a discrete system can be written T = ẋt M ẋ. Generalisation of previous results findings in the previous two slides can be generalised to the structural matrices of generic structural systems, with two main exceptions.. For a general structural system, in which not all DOFs are related to a mass, M could be semi-definite positive, that is for some particular displacement vector the kinetic energy is zero.. For a general structural system subjected to axial loads, due to the presence of geometrical stiffness it is possible that for some particular displacement vector the strain energy is zero and K is semi-definite positive. are

5 problem k Graphical statement of the problem p(t) m k m x x k = k, k = k; m = m, m = m; p(t) = p 0 sin ωt. equations of motion m ẍ + k x + k (x x ) = p 0 sin ωt, m ẍ + k (x x ) = but we prefer the matrix notation... are steady state solution We prefer the matrix notation because we can find the steady-state response of a SDOF system exactly as we found the s-s solution for a SDOF system. Substituting x(t) = ξ sin ωt in the equation of motion and simplifying sin ωt, [ ] [ ] { } 3 k ξ mω 0 ξ = p dividing by k, with ω 0 = k/m, β = ω /ω 0 and st = p 0 /k the above equation can be written ([ ] [ ]) [ ] { } 3 β 0 3 β ξ = 0 β ξ = st. 0 are steady state solution determinant of the matrix of coefficients is Det = β 4 5β + but we want to write the polynomial in β in terms of its roots Det = (β /) (β ). Solving for ξ/ st in terms of the inverse of the coefficient matrix gives [ ] { } ξ β = st (β )(β ) 3 β 0 { } β = (β. )(β ) are

6 solution, graphically Normalized displacement 0 steady-state response for a dof system, harmonic load ξ /Δ st ξ /Δ st are β =ω /ω o Comment to the Steady State Solution steady state solution is x s-s = st (β )(β ) { } β sin ωt. As it s apparent in the previous slide, we have two different values of the excitation frequency for which the dynamic amplification factor goes to infinity. For an undamped SDOF system, we had a single frequency of excitation that excites a resonant response, now for a two degrees of freedom system we have two different excitation frequencies that excite a resonant response. are We know how to compute a particular integral for a MDOF system (at least for a harmonic loading), what do we miss to be able to determine the integral of motion? equation of motion To understand the behaviour of a MDOF system, we have to study the homogeneous solution. Let s start writing the homogeneous equation of motion, M ẍ + K x = 0. solution, in analogy with the SDOF case, can be written in terms of a harmonic function of unknown frequency and, using the concept of separation of variables, of a constant vector, the so called shape vector ψ: are x(t) = ψ(a sin ωt + B cos ωt). Substituting in the equation of motion, we have ( K ω M ) ψ(a sin ωt + B cos ωt) = 0

7 Eigenvalues previous equation must hold for every value of t, so it can be simplified removing the time dependency: ( K ω M ) ψ = 0. This is a homogeneous linear equation, with unknowns ψ i and the coefficients that depends on the parameter ω. Speaking of homogeneous systems, we know that there is always a trivial solution, ψ = 0, and non-trivial solutions are possible if the determinant of the matrix of coefficients is equal to zero, det ( K ω M ) = 0 are eigenvalues of the MDOF system are the values of ω for which the above equation (the equation of frequencies) is verified or, in other words, the frequencies of vibration associated with the shapes for which Kψ sin ωt = ω Mψ sin ωt. Eigenvalues, cont. For a system with N degrees of freedom the expansion of det ( K ω M ) is an algebraic polynomial of degree N in ω. A polynomial of degree N has exactly N roots, either real or complex conjugate. In Dynamics of Structures those roots ω i, i =,..., N are all real because the structural matrices are symmetric matrices. Moreover, if both K and M are positive definite matrices (a condition that is always satisfied by stable structural systems) all the roots, all the eigenvalues, are strictly positive: are ω i 0, for i =,..., N. Substituting one of the N roots ω i in the characteristic equation, ( K ω i M ) ψ i = 0 the resulting system of N linearly independent equations can be solved (except for a scale factor) for ψ i, the eigenvector corresponding to the eigenvalue ω i. are

8 scale factor being arbitrary, you have to choose (arbitrarily) the value of one of the components and compute the values of all the other N components using the N linearly indipendent equations. It is common to impose to each eigenvector a normalisation with respect to the mass matrix, so that ψ T i M ψ i =. are Please consider that, substituting different eigenvalues in the equation of free vibrations, you have different linear systems, leading to different eigenvectors. most general expression (the general integral) for the displacement of a homogeneous system is x(t) = ψ i (A i sin ω i t + B i cos ω i t). i= In the general integral there are N unknown constants of integration, that must be determined in terms of the initial conditions. are Usually the initial conditions are expressed in terms of initial displacements and initial velocities x 0 and ẋ 0, so we start deriving the expression of displacement with respect to time to obtain ẋ(t) = ψ i ω i (A i cos ω i t B i sin ω i t) i= and evaluating the displacement and velocity for t = 0 it is x(0) = ψ i B i = x 0, ẋ(0) = ψ i ω i A i = ẋ 0. i= i= are above equations are vector equations, each one corresponding to a system of N equations, so we can compute the N constants of integration solving the N equations ψ ji B i = x 0,j, i= ψ ji ω i A i = ẋ 0,j, j =,..., N. i=

9 ity - Take into consideration two distinct eigenvalues, ω r and ω s, and write the characteristic equation for each eigenvalue: K ψ r = ω rm ψ r K ψ s = ω sm ψ s premultiply each equation member by the transpose of the other eigenvector are ψ T s K ψ r = ω rψ T s M ψ r ψ T r K ψ s = ω sψ T r M ψ s ity - term ψ T s K ψ r is a scalar, hence ψ T s K ψ r = ( ψ T ) T s K ψ r = ψ T r K T ψ s but K is symmetrical, K T = K and we have ψ T s K ψ r = ψ T r K ψ s. By a similar derivation are ψ T s M ψ r = ψ T r M ψ s. ity - 3 Substituting our last identities in the previous equations, we have ψ T r K ψ s = ω rψ T r M ψ s ψ T r K ψ s = ω sψ T r M ψ s subtracting member by member we find that (ω r ω s) ψ T r M ψ s = 0 We started with the hypothesis that ω r ω s, so for every r s we have that the corresponding eigenvectors are orthogonal with respect to the mass matrix are ψ T r M ψ s = 0, for r s.

10 ity - 4 eigenvectors are orthogonal also with respect to the stiffness matrix: ψ T s K ψ r = ω rψ T s M ψ r = 0, for r s. By definition and consequently M i = ψ T i M ψ i ψ T i K ψ i = ω i M i. are M i is the modal mass associated with mode no. i while K i ω i M i is the respective modal stiffness. are a base eigenvectors are linearly independent, so for every vector x we can write x = ψ j q j. j= coefficients are readily given by premultiplication of x by ψ T i M, because N ψ T i M x = ψ T i M ψ jq j = ψ T i M ψ iq i = M i q i j= in virtue of the ortogonality of the eigenvectors with respect to the mass matrix, and the above relationship gives are a base EoM in Modal Coordinates q j = ψt j M x M j. are a base Generalising our results for the displacement vector to the acceleration vector and expliciting the time dependency, it is x(t) = x i (t) = ψ j q j (t), ẍ(t) = j= Ψ ij q j (t), ẍ i (t) = j= ψ j q j (t), j= ψ ij q j (t). j= are a base EoM in Modal Coordinates Introducing q(t), the vector of modal coordinates and Ψ, the eigenvector matrix, whose columns are the eigenvectors, we can write x(t) = Ψ q(t), ẍ(t) = Ψ q(t).

11 EoM in Modal Coordinates... Substituting the last two equations in the equation of motion, premultiplying by Ψ T M Ψ q + K Ψ q = p(t) Ψ T M Ψ q + Ψ T K Ψ q = Ψ T p(t) introducing the so called starred matrices, with p (t) = Ψ T p(t), we can finally write are a base EoM in Modal Coordinates M q + K q = p (t) vector equation above corresponds to the set of scalar equations p i = m ij q j + k ij q j, i =,..., N.... are N independent equations! We must examine the structure of the starred symbols. generic element, with indexes i and j, of the starred matrices can be expressed in terms of single eigenvectors, m ij = ψt i M ψ j = δ ij M i, k ij = ψt i K ψ j = ω i δ ijm i. where δ ij is the Kroneker symbol, are a base EoM in Modal Coordinates { i = j δ ij = 0 i j Substituting in the equation of motion, with p i = ψt i p(t) we have a set of uncoupled equations M i q i + ω i M iq i = p i (t), i =,..., N Revisited initial displacements can be written in modal coordinates, x 0 = Ψ q 0 and premultiplying both members by Ψ T M we have the following relationship: Ψ T M x 0 = Ψ T M Ψ q 0 = M q 0. Premultiplying by the inverse of M and taking into account that M is diagonal, are a base EoM in Modal Coordinates q 0 = (M ) Ψ T M x 0 q i0 = ψt i M x 0 M i and, analogously, q i0 = ψ i T M ẋ 0 M i

12 k m k m p(t) x x k = k, k = k; m = m, m = m; p(t) = p 0 sin ωt. { } { } x 0 x =, p(t) = sin ωt, x p 0 [ ] [ ] 0 3 M = m, K = k. 0 frequencies equation of frequencies is K ω M 3k ω m k = k k ω m = 0. Developing the determinant (m )ω 4 (7mk)ω + (k )ω 0 = 0 Solving the algebraic equation in ω ω = k 7 33 m 4 ω = k m ω = k m 4 ω = k m Substituting ω for ω in the first of the characteristic equations gives the ratio between the components of the first eigenvector, while substituting ω gives k ( )ψ kψ = 0 k ( )ψ kψ = 0. Solving with the arbitrary assignment ψ = ψ = gives the unnormalized eigenvectors, { } ψ =, ψ = { }

13 Normalization We compute first M and M, M = ψ T M ψ = { , } [ ] { } m m = {.6864m, m } { } =.453m M =.70346m the adimensional normalisation factors are α =.453, α = Applying the normalisation factors to the respective unnormalised eigenvectors and collecting them in a matrix, we have the matrix of normalized eigenvectors [ ] Ψ = Modal Loadings modal loading is p (t) = Ψ T p(t) [ ] { } = p 0 sin ωt { } = p 0 sin ωt Modal EoM Substituting its modal expansion for x into the equation of motion and premultiplying by Ψ T we have the uncoupled modal equation of motion { m q k q = p 0 sin ωt m q k q = p 0 sin ωt Note that all the terms are dimensionally correct. Dividing by m both equations, we have q + ω q = p 0 sin ωt m q + ω q = p 0 sin ωt m

14 Particular Integral We set ξ = C sin ωt, and substitute in the first modal EoM: solving for C C = p m with ω = K /m m = K /ω : ξ = ω C sin ωt ( C ω ω ) sin ωt = p sin ωt m ω ω C = p ω = K ω () st ω β with () st = p K =.047 p 0 k and β = ω ω of course C = () st β with () st = p K = p 0 k and β = ω ω Integrals integrals, for our loading, are thus q (t) = A sin ω t + B cos ω t + () sin ωt st β q (t) = A sin ω t + B cos ω t + () sin ωt st β for a system initially at rest q (t) = () st (sin ωt β β sin ω t) q (t) = () st (sin ωt β β sin ω t) we are interested in structural degrees of freedom, too... disregarding transient ( ) () st () ( st.096 x (t) = ψ + ψ β sin ωt = ) p0 sin ωt β β β k ( ) () st () ( st.3575 x (t) = ψ + ψ β sin ωt = ) p0 sin ωt β β β k response in modal coordinates To have a feeling of the response in modal coordinates, let s say that the frequency of the load is ω = ω 0, hence β = = and β = = q i /Δ st q (α)/δ s t α = ω o t q (α)/δ s t In the graph above, the responses are plotted against an adimensional time coordinate α with α = ω 0 t, while the ordinates are adimensionalised with respect to st = p0 k

15 response in structural coordinates Using the same normalisation factors, here are the response functions in terms of x = ψ q + ψ q and x = ψ q + ψ q : x i /Δ st x (α)/δ s t α = ω o t x (α)/δ s t

Structural Matrices in MDOF Systems

Structural Matrices in MDOF Systems in MDOF Systems http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 9, 2016 Outline Additional Static Condensation

More information

Outline. Structural Matrices. Giacomo Boffi. Introductory Remarks. Structural Matrices. Evaluation of Structural Matrices

Outline. Structural Matrices. Giacomo Boffi. Introductory Remarks. Structural Matrices. Evaluation of Structural Matrices Outline in MDOF Systems Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano May 8, 014 Additional Today we will study the properties of structural matrices, that is the operators that

More information

Matrix Iteration. Giacomo Boffi.

Matrix Iteration. Giacomo Boffi. http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 12, 2016 Outline Second -Ritz Method Dynamic analysis of MDOF

More information

Dr.Vinod Hosur, Professor, Civil Engg.Dept., Gogte Institute of Technology, Belgaum

Dr.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 information

Truncation Errors Numerical Integration Multiple Support Excitation

Truncation Errors Numerical Integration Multiple Support Excitation Errors Numerical Integration Multiple Support Excitation http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 10,

More information

Giacomo Boffi. April 7, 2016

Giacomo Boffi. April 7, 2016 http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 7, 2016 Outline Analysis The process of estimating the vibration

More information

Outline of parts 1 and 2

Outline 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 information

Generalized Single Degree of Freedom Systems

Generalized Single Degree of Freedom Systems Single Degree of Freedom http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March, 8 Outline Until now our were described

More information

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.

Structural 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 information

3. Mathematical Properties of MDOF Systems

3. Mathematical Properties of MDOF Systems 3. Mathematical Properties of MDOF Systems 3.1 The Generalized Eigenvalue Problem Recall that the natural frequencies ω and modes a are found from [ - ω 2 M + K ] a = 0 or K a = ω 2 M a Where M and K are

More information

1. Multiple Degree-of-Freedom (MDOF) Systems: Introduction

1. Multiple Degree-of-Freedom (MDOF) Systems: Introduction 1. Multiple Degree-of-Freedom (MDOF) Systems: Introduction Lesson Objectives: 1) List examples of MDOF structural systems and state assumptions of the idealizations. 2) Formulate the equation of motion

More information

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016

Identification 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 information

AA242B: MECHANICAL VIBRATIONS

AA242B: 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 information

Generalized Single Degree of Freedom Systems

Generalized Single Degree of Freedom Systems Single Degree of Freedom http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April, 16 Outline Until now our were described

More information

FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS

FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS Lecture Notes: STRUCTURAL DYNAMICS / FALL 2011 / Page: 1 FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS : : 0, 0 As demonstrated previously, the above Equation of Motion (free-vibration equation) has a solution

More information

Giacomo Boffi. Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano

Giacomo Boffi.  Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 21, 2017 Outline of Structural Members Elastic-plastic Idealization

More information

Reduction in number of dofs

Reduction in number of dofs Reduction in number of dofs Reduction in the number of dof to represent a structure reduces the size of matrices and, hence, computational cost. Because a subset of the original dof represent the whole

More information

Advanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One

Advanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian ahmadian@iust.ac.ir Elements of Analytical Dynamics Newton's laws were formulated for a single particle Can be extended to

More information

Some Aspects of Structural Dynamics

Some 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 information

TWO-STAGE ISOLATION FOR HARMONIC BASE EXCITATION Revision A. By Tom Irvine February 25, 2008

TWO-STAGE ISOLATION FOR HARMONIC BASE EXCITATION Revision A. By Tom Irvine   February 25, 2008 TWO-STAGE ISOLATION FOR HARMONIC BASE EXCITATION Revision A By Tom Irvine Email: tomirvine@aol.com February 5, 008 Introduction Consider a base plate mass m and an avionics mass m modeled as two-degree-of-freedom.

More information

Stochastic Dynamics of SDOF Systems (cont.).

Stochastic 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 information

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.

Introduction 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 information

Damping Matrix. Donkey2Ft

Damping Matrix. Donkey2Ft 1 Damping Matrix DonkeyFt Damping in a single-degree-of-freedom (SDOF) system is well studied. Whether the system is under-damped, over-damped, or critically damped is well known. For an under-damped system,

More information

SHOCK RESPONSE OF MULTI-DEGREE-OF-FREEDOM SYSTEMS Revision F By Tom Irvine May 24, 2010

SHOCK RESPONSE OF MULTI-DEGREE-OF-FREEDOM SYSTEMS Revision F By Tom Irvine   May 24, 2010 SHOCK RESPONSE OF MULTI-DEGREE-OF-FREEDOM SYSTEMS Revision F By Tom Irvine Email: tomirvine@aol.com May 4, 010 Introduction The primary purpose of this tutorial is to present the Modal Transient method

More information

Structural Dynamics A Graduate Course in Aerospace Engineering

Structural 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 information

Structural 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 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 information

Final Exam Solution Dynamics :45 12:15. Problem 1 Bateau

Final 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 information

Multiple Degree of Freedom Systems. The Millennium bridge required many degrees of freedom to model and design with.

Multiple Degree of Freedom Systems. The Millennium bridge required many degrees of freedom to model and design with. Multiple Degree of Freedom Systems The Millennium bridge required many degrees of freedom to model and design with. The first step in analyzing multiple degrees of freedom (DOF) is to look at DOF DOF:

More information

Lab #2 - Two Degrees-of-Freedom Oscillator

Lab #2 - Two Degrees-of-Freedom Oscillator Lab #2 - Two Degrees-of-Freedom Oscillator Last Updated: March 0, 2007 INTRODUCTION The system illustrated in Figure has two degrees-of-freedom. This means that two is the minimum number of coordinates

More information

D && 9.0 DYNAMIC ANALYSIS

D && 9.0 DYNAMIC ANALYSIS 9.0 DYNAMIC ANALYSIS Introduction When a structure has a loading which varies with time, it is reasonable to assume its response will also vary with time. In such cases, a dynamic analysis may have to

More information

STRUCTURAL DYNAMICS BASICS:

STRUCTURAL 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 information

Dynamics of Structures

Dynamics of Structures Dynamics of Structures Elements of structural dynamics Roberto Tomasi 11.05.2017 Roberto Tomasi Dynamics of Structures 11.05.2017 1 / 22 Overview 1 SDOF system SDOF system Equation of motion Response spectrum

More information

Linear Algebra & Geometry why is linear algebra useful in computer vision?

Linear Algebra & Geometry why is linear algebra useful in computer vision? Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia

More information

Introduction to structural dynamics

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 information

Dynamic Loads CE 543. Examples. Harmonic Loads

Dynamic 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

Vibrations Qualifying Exam Study Material

Vibrations 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 information

Design of Structures for Earthquake Resistance

Design of Structures for Earthquake Resistance NATIONAL TECHNICAL UNIVERSITY OF ATHENS Design of Structures for Earthquake Resistance Basic principles Ioannis N. Psycharis Lecture 3 MDOF systems Equation of motion M u + C u + K u = M r x g(t) where:

More information

COPYRIGHTED MATERIAL. Index

COPYRIGHTED 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 information

Math 1553, Introduction to Linear Algebra

Math 1553, Introduction to Linear Algebra Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level

More information

ME 563 HOMEWORK # 7 SOLUTIONS Fall 2010

ME 563 HOMEWORK # 7 SOLUTIONS Fall 2010 ME 563 HOMEWORK # 7 SOLUTIONS Fall 2010 PROBLEM 1: Given the mass matrix and two undamped natural frequencies for a general two degree-of-freedom system with a symmetric stiffness matrix, find the stiffness

More information

Structural Dynamics Lecture 7. Outline of Lecture 7. Multi-Degree-of-Freedom Systems (cont.) System Reduction. Vibration due to Movable Supports.

Structural Dynamics Lecture 7. Outline of Lecture 7. Multi-Degree-of-Freedom Systems (cont.) System Reduction. Vibration due to Movable Supports. Outline of Multi-Degree-of-Freedom Systems (cont.) System Reduction. Truncated Modal Expansion with Quasi-Static Correction. Guyan Reduction. Vibration due to Movable Supports. Earthquake Excitations.

More information

22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes

22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one

More information

Dynamics of structures

Dynamics 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 information

Response to Periodic and Non-periodic Loadings. Giacomo Boffi.

Response to Periodic and Non-periodic Loadings. Giacomo Boffi. Periodic and Non-periodic http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 7, 218 Outline Undamped SDOF systems

More information

Appendix A Equations of Motion in the Configuration and State Spaces

Appendix A Equations of Motion in the Configuration and State Spaces Appendix A Equations of Motion in the Configuration and State Spaces A.1 Discrete Linear Systems A.1.1 Configuration Space Consider a system with a single degree of freedom and assume that the equation

More information

Introduction to Geotechnical Earthquake Engineering

Introduction to Geotechnical Earthquake Engineering Module 1 Introduction to Geotechnical Earthquake Engineering by Dr. Deepankar Choudhury Professor Department of Civil Engineering IIT Bombay, Powai, Mumbai 400 076, India. Email: dc@civil.iitb.ac.in URL:

More information

Seminar 6: COUPLED HARMONIC OSCILLATORS

Seminar 6: COUPLED HARMONIC OSCILLATORS Seminar 6: COUPLED HARMONIC OSCILLATORS 1. Lagrangian Equations of Motion Let consider a system consisting of two harmonic oscillators that are coupled together. As a model, we will use two particles attached

More information

Outline. Introduction

Outline. Introduction Outline Periodic and Non-periodic Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano March 5, 4 A periodic loading is characterized by the identity p(t) = p(t + T ) where T is the period

More information

The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems: Non-Linear Dynamics Part I

The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems: Non-Linear Dynamics Part I The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems: Non-Linear Dynamics Part I Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos Agathos Lecture 5/Part A - 23 November,

More information

Repeated Eigenvalues and Symmetric Matrices

Repeated Eigenvalues and Symmetric Matrices Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one

More information

! 4 4! o! +! h 4 o=0! ±= ± p i And back-substituting into the linear equations gave us the ratios of the amplitudes of oscillation:.»» = A p e i! +t»»

! 4 4! o! +! h 4 o=0! ±= ± p i And back-substituting into the linear equations gave us the ratios of the amplitudes of oscillation:.»» = A p e i! +t»» Topic 6: Coupled Oscillators and Normal Modes Reading assignment: Hand and Finch Chapter 9 We are going to be considering the general case of a system with N degrees of freedome close to one of its stable

More information

Section 4.9; Section 5.6. June 30, Free Mechanical Vibrations/Couple Mass-Spring System

Section 4.9; Section 5.6. June 30, Free Mechanical Vibrations/Couple Mass-Spring System Section 4.9; Section 5.6 Free Mechanical Vibrations/Couple Mass-Spring System June 30, 2009 Today s Session Today s Session A Summary of This Session: Today s Session A Summary of This Session: (1) Free

More information

Introduction to Mechanical Vibration

Introduction 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 information

Response to Periodic and Non-periodic Loadings. Giacomo Boffi. March 25, 2014

Response to Periodic and Non-periodic Loadings. Giacomo Boffi. March 25, 2014 Periodic and Non-periodic Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano March 25, 2014 Outline Introduction Fourier Series Representation Fourier Series of the Response Introduction

More information

Assignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get

Assignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get Assignment 6 Goldstein 6.4 Obtain the normal modes of vibration for the double pendulum shown in Figure.4, assuming equal lengths, but not equal masses. Show that when the lower mass is small compared

More information

AA 242B / ME 242B: Mechanical Vibrations (Spring 2016)

AA 242B / ME 242B: Mechanical Vibrations (Spring 2016) AA 242B / ME 242B: Mechanical Vibrations (Spring 206) Solution of Homework #3 Control Tab Figure : Schematic for the control tab. Inadequacy of a static-test A static-test for measuring θ would ideally

More information

ME scope Application Note 28

ME scope Application Note 28 App Note 8 www.vibetech.com 3/7/17 ME scope Application Note 8 Mathematics of a Mass-Spring-Damper System INTRODUCTION In this note, the capabilities of ME scope will be used to build a model of the mass-spring-damper

More information

Small oscillations and normal modes

Small oscillations and normal modes Chapter 4 Small oscillations and normal modes 4.1 Linear oscillations Discuss a generalization of the harmonic oscillator problem: oscillations of a system of several degrees of freedom near the position

More information

DYNAMICS OF MACHINERY 41514

DYNAMICS OF MACHINERY 41514 DYNAMICS OF MACHINERY 454 PROJECT : Theoretical and Experimental Modal Analysis and Validation of Mathematical Models in Multibody Dynamics Holistic Overview of the Project Steps & Their Conceptual Links

More information

Chapter 23: Principles of Passive Vibration Control: Design of absorber

Chapter 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 information

2C9 Design for seismic and climate changes. Jiří Máca

2C9 Design for seismic and climate changes. Jiří Máca 2C9 Design for seismic and climate changes Jiří Máca List of lectures 1. Elements of seismology and seismicity I 2. Elements of seismology and seismicity II 3. Dynamic analysis of single-degree-of-freedom

More information

PROJECT 1 DYNAMICS OF MACHINES 41514

PROJECT 1 DYNAMICS OF MACHINES 41514 PROJECT DYNAMICS OF MACHINES 454 Theoretical and Experimental Modal Analysis and Validation of Mathematical Models in Multibody Dynamics Ilmar Ferreira Santos, Professor Dr.-Ing., Dr.Techn., Livre-Docente

More information

L = 1 2 a(q) q2 V (q).

L = 1 2 a(q) q2 V (q). Physics 3550, Fall 2011 Motion near equilibrium - Small Oscillations Relevant Sections in Text: 5.1 5.6 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion

More information

Session 2: MDOF systems

Session 2: MDOF systems BRUFACE Vibrations and Acoustics MA Academic year 7-8 Cédric Dumoulin (cedumoul@ulb.ac.be) Arnaud Deraemaeker (aderaema@ulb.ac.be) Session : MDOF systems Exercise : Multiple DOFs System Consider the following

More information

a) Find the equation of motion of the system and write it in matrix form.

a) Find the equation of motion of the system and write it in matrix form. .003 Engineering Dynamics Problem Set Problem : Torsional Oscillator Two disks of radius r and r and mass m and m are mounted in series with steel shafts. The shaft between the base and m has length L

More information

Differential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm

Differential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm Differential Equations 228 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 25 at 2:5pm Instructions: This in-class exam is 5 minutes. No calculators, notes, tables or books. No answer check is

More information

TOPIC E: OSCILLATIONS SPRING 2019

TOPIC E: OSCILLATIONS SPRING 2019 TOPIC E: OSCILLATIONS SPRING 2019 1. Introduction 1.1 Overview 1.2 Degrees of freedom 1.3 Simple harmonic motion 2. Undamped free oscillation 2.1 Generalised mass-spring system: simple harmonic motion

More information

Module 4: Dynamic Vibration Absorbers and Vibration Isolator Lecture 19: Active DVA. The Lecture Contains: Development of an Active DVA

Module 4: Dynamic Vibration Absorbers and Vibration Isolator Lecture 19: Active DVA. The Lecture Contains: Development of an Active DVA The Lecture Contains: Development of an Active DVA Proof Mass Actutor Application of Active DVA file:///d /chitra/vibration_upload/lecture19/19_1.htm[6/25/2012 12:35:51 PM] In this section, we will consider

More information

Structural Dynamics Lecture 2. Outline of Lecture 2. Single-Degree-of-Freedom Systems (cont.)

Structural 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 information

Stability of Mass-Point Systems

Stability of Mass-Point Systems Simulation in Computer Graphics Stability of Mass-Point Systems Matthias Teschner Computer Science Department University of Freiburg Demos surface tension vs. volume preservation distance preservation

More information

NORMAL MODES, WAVE MOTION AND THE WAVE EQUATION. Professor G.G.Ross. Oxford University Hilary Term 2009

NORMAL MODES, WAVE MOTION AND THE WAVE EQUATION. Professor G.G.Ross. Oxford University Hilary Term 2009 NORMAL MODES, WAVE MOTION AND THE WAVE EQUATION Professor G.G.Ross Oxford University Hilary Term 009 This course of twelve lectures covers material for the paper CP4: Differential Equations, Waves and

More information

Tuning TMDs to Fix Floors in MDOF Shear Buildings

Tuning TMDs to Fix Floors in MDOF Shear Buildings Tuning TMDs to Fix Floors in MDOF Shear Buildings This is a paper I wrote in my first year of graduate school at Duke University. It applied the TMD tuning methodology I developed in my undergraduate research

More information

22.2. Applications of Eigenvalues and Eigenvectors. Introduction. Prerequisites. Learning Outcomes

22.2. Applications of Eigenvalues and Eigenvectors. Introduction. Prerequisites. Learning Outcomes Applications of Eigenvalues and Eigenvectors 22.2 Introduction Many applications of matrices in both engineering and science utilize eigenvalues and, sometimes, eigenvectors. Control theory, vibration

More information

Keble College - Hilary 2014 CP3&4: Mathematical methods I&II Tutorial 5 - Waves and normal modes II

Keble College - Hilary 2014 CP3&4: Mathematical methods I&II Tutorial 5 - Waves and normal modes II Tomi Johnson 1 Keble College - Hilary 2014 CP3&4: Mathematical methods I&II Tutorial 5 - Waves and normal modes II Prepare full solutions to the problems with a self assessment of your progress on a cover

More information

EFFECTIVE MODAL MASS & MODAL PARTICIPATION FACTORS Revision F

EFFECTIVE MODAL MASS & MODAL PARTICIPATION FACTORS Revision F EFFECTIVE MODA MASS & MODA PARTICIPATION FACTORS Revision F By Tom Irvine Email: tomirvine@aol.com March 9, 1 Introduction The effective modal mass provides a method for judging the significance of a vibration

More information

On the comparison of symmetric and unsymmetric formulations for experimental vibro-acoustic modal analysis

On the comparison of symmetric and unsymmetric formulations for experimental vibro-acoustic modal analysis Acoustics 8 Paris On the comparison of symmetric and unsymmetric formulations for experimental vibro-acoustic modal analysis M. Ouisse a and E. Foltete b a FEMO-S UMR CNRS 24 chemin de l Epitaphe 25 Besançon

More information

A NEW METHOD FOR VIBRATION MODE ANALYSIS

A NEW METHOD FOR VIBRATION MODE ANALYSIS Proceedings of IDETC/CIE 25 25 ASME 25 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference Long Beach, California, USA, September 24-28, 25 DETC25-85138

More information

Modal Decomposition and the Time-Domain Response of Linear Systems 1

Modal Decomposition and the Time-Domain Response of Linear Systems 1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING.151 Advanced System Dynamics and Control Modal Decomposition and the Time-Domain Response of Linear Systems 1 In a previous handout

More information

Dynamics 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 Dynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras Lecture 25 Continuous System In the last class, in this, we will

More information

Chapter 4 Analysis of a cantilever

Chapter 4 Analysis of a cantilever Chapter 4 Analysis of a cantilever Before a complex structure is studied performing a seismic analysis, the behaviour of simpler ones should be fully understood. To achieve this knowledge we will start

More information

Final Exam December 11, 2017

Final Exam December 11, 2017 Final Exam Instructions: You have 120 minutes to complete this exam. This is a closed-book, closed-notes exam. You are NOT allowed to use a calculator with communication capabilities during the exam. Usage

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 11, 2016 1:00PM to 3:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of

More information

Table of Contents. Preface... 13

Table 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 information

Lecture 27: Structural Dynamics - Beams.

Lecture 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 information

Dynamics of structures

Dynamics 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 information

The quantum mechanics approach to uncertainty modeling in structural dynamics

The quantum mechanics approach to uncertainty modeling in structural dynamics p. 1/3 The quantum mechanics approach to uncertainty modeling in structural dynamics Andreas Kyprianou Department of Mechanical and Manufacturing Engineering, University of Cyprus Outline Introduction

More information

DAMPING MODELLING AND IDENTIFICATION USING GENERALIZED PROPORTIONAL DAMPING

DAMPING MODELLING AND IDENTIFICATION USING GENERALIZED PROPORTIONAL DAMPING DAMPING MODELLING AND IDENTIFICATION USING GENERALIZED PROPORTIONAL DAMPING S. Adhikari Department of Aerospace Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR (U.K.)

More information

Math Ordinary Differential Equations

Math Ordinary Differential Equations Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x

More information

Dynamic Response of Structures With Frequency Dependent Damping

Dynamic Response of Structures With Frequency Dependent Damping Dynamic Response of Structures With Frequency Dependent Damping Blanca Pascual & S Adhikari School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/

More information

The Phasor Analysis Method For Harmonically Forced Linear Systems

The Phasor Analysis Method For Harmonically Forced Linear Systems The Phasor Analysis Method For Harmonically Forced Linear Systems Daniel S. Stutts, Ph.D. April 4, 1999 Revised: 10-15-010, 9-1-011 1 Introduction One of the most common tasks in vibration analysis is

More information

LECTURE 14: DEVELOPING THE EQUATIONS OF MOTION FOR TWO-MASS VIBRATION EXAMPLES

LECTURE 14: DEVELOPING THE EQUATIONS OF MOTION FOR TWO-MASS VIBRATION EXAMPLES LECTURE 14: DEVELOPING THE EQUATIONS OF MOTION FOR TWO-MASS VIBRATION EXAMPLES Figure 3.47 a. Two-mass, linear vibration system with spring connections. b. Free-body diagrams. c. Alternative free-body

More information

Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche

Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio - DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second

More information

Multivariate Statistical Analysis

Multivariate Statistical Analysis Multivariate Statistical Analysis Fall 2011 C. L. Williams, Ph.D. Lecture 4 for Applied Multivariate Analysis Outline 1 Eigen values and eigen vectors Characteristic equation Some properties of eigendecompositions

More information

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections ) c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side

More information

VIBRATION PROBLEMS IN ENGINEERING

VIBRATION PROBLEMS IN ENGINEERING VIBRATION PROBLEMS IN ENGINEERING FIFTH EDITION W. WEAVER, JR. Professor Emeritus of Structural Engineering The Late S. P. TIMOSHENKO Professor Emeritus of Engineering Mechanics The Late D. H. YOUNG Professor

More information

Newtonian Mechanics. Chapter Classical space-time

Newtonian Mechanics. Chapter Classical space-time Chapter 1 Newtonian Mechanics In these notes classical mechanics will be viewed as a mathematical model for the description of physical systems consisting of a certain (generally finite) number of particles

More information

Chapter 7 Vibration Measurement and Applications

Chapter 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 information

Computational Stiffness Method

Computational Stiffness Method Computational Stiffness Method Hand calculations are central in the classical stiffness method. In that approach, the stiffness matrix is established column-by-column by setting the degrees of freedom

More information

Program System for Machine Dynamics. Abstract. Version 5.0 November 2017

Program System for Machine Dynamics. Abstract. Version 5.0 November 2017 Program System for Machine Dynamics Abstract Version 5.0 November 2017 Ingenieur-Büro Klement Lerchenweg 2 D 65428 Rüsselsheim Phone +49/6142/55951 hd.klement@t-online.de What is MADYN? The program system

More information

Background Mathematics (2/2) 1. David Barber

Background Mathematics (2/2) 1. David Barber Background Mathematics (2/2) 1 David Barber University College London Modified by Samson Cheung (sccheung@ieee.org) 1 These slides accompany the book Bayesian Reasoning and Machine Learning. The book and

More information