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

Size: px
Start display at page:

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

Transcription

1 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 relate the vector of system coordinates x and its time derivatives ẋ and ẍ to the forces acting on the system nodes, f S, f D and f I, respectively. In the end, we will see again the solution of a MDOF problem by superposition, and in general today we will revisit many of the subjects of our previous class, but you know that a bit of reiteration is really good for developing minds. We already met the mass and the stiffness matrix, M and K, and tangentially we introduced also the dampig matrix C. We have seen that these matrices express the linear relation that holds between the vector of system coordinates x and its time derivatives ẋ and ẍ to the forces acting on the system nodes, f S, f D and f I, elastic, damping and inertial force vectors. M ẍ + C ẋ + K x = pt) f I + f D + f S = pt) Also, we know that M and K are symmetric and definite positive, and that it is possible to uncouple the equation of motion expressing the system coordinates in terms of the eigenvectors, xt) = q i ψ i, where the q i are the modal coordinates and the eigenvectors ψ i are the non-trivial solutions to the equation of free vibrations, K ω M ) ψ = 0 Additional Orthogonality Relationships

2 Free Vibrations Additional From the homogeneous, undamped problem M ẍ + K x = 0 introducing separation of variables xt) = ψ A sin ωt + B cos ωt) we wrote the homogeneous linear system K ω M ) ψ = 0 Additional Orthogonality Relationships From K ψ s = ωs M ψ s premultiplying by ψ T r KM 1 we have ψ T r KM 1 K ψ s = ωs ψ T r K ψ s = δ rs ωr 4 M r, premultiplying the first equation by ψ T r KM 1 KM 1 Additional Orthogonality Relationships whose non-trivial solutions ψ i for ωi such that K ω i M = 0 are the eigenvectors. It was demonstrated that, for each pair of distint eigenvalues ωr and ωs, the corresponding eigenvectors obey the ortogonality condition, ψ T r KM 1 KM 1 K ψ s = ωs ψ T r KM 1 K ψ s = δ rs ωr 6 M r and, generalizing, ψ T r KM 1 ) b K ψs = δ rs ω r ) b+1 Mr. ψ T s M ψ r = δ rs M r, ψ T s K ψ r = δ rs ω r M r. Additional Relationships, Additional Relationships, 3 From M ψ s = ωs K ψ s premultiplying by ψ T r MK 1 we have ψ T r MK 1 M ψ s = ωs ψ T M s r M ψ s = δ rs ωs Additional Orthogonality Relationships Defining X rs k) = ψ T r M M 1 K ) k ψs we have X rs 0) = ψ T r Mψ s = δ rs ω s ) 0 Ms X rs 1) = ψ T ) r Kψ s = δ rs ω 1 s Ms X rs ) = ψ T r KM 1 ) 1 ) Kψs = δ rs ω s Ms X rs n) = ψ T r KM 1 ) n 1 ) Kψs = δ rs ω n s Ms Additional Orthogonality Relationships premultiplying the first eq. by ψ T r MK 1 ) we have Observing that M 1 K ) 1 = K 1 M ) 1 ψ T r MK 1 ) M ψs = ωs ψ T r MK 1 M s M ψ s = δ rs ωs 4 and, generalizing, X rs 1) = ψ T r X rs n) = ψ T r MK 1 ) 1 M ψs = δ rs ω s ) 1 Ms MK 1 ) n M ψs = δ rs ω s ) n Ms ψ T r MK 1 ) b M ψs = δ rs M s ω s b finally X rs k) = δ rs ω k s M s for k =,...,.

3 Flexibility Given a system whose state is determined by the generalized displacements x j of a set of nodes, we define the flexibility f jk as the deflection, in direction of x j, due to the application of a unit force in correspondance of the displacement x k. The matrix F = [ f jk ] is the flexibility matrix. In our context, the degrees of freedom are associated with external loads and/or inertial forces. Given a load vector p = { p k } of course the load components act in correspondence of the degrees of freedom), the individual displacement x j is or, in vector notation, x j = f jk p k Consistent m, J a b x x 1 The dynamical system The degrees of freedom 1 f 11 1 f 1 f 1 Displacements due to p 1 = 1 and due to p = 1. f Consistent x = F p Elastic Forces Each node shall be in equilibrium under the action of the external forces and the elastic forces, hence taking into accounts all the nodes, all the external forces and all the elastic forces it is possible to write the vector equation of equilibrium p = f S and, substituting in the previos vector expression of the displacements x = F f S Pre=multiplying by F 1, F 1 x = F 1 F f S = f S. Consistent The stiffness matrix K can be simply defined as the inverse of the flexibility matrix F, K = F 1. Alternatively the single coefficient k ij can be defined as the external force equal and opposite to the corresponding elastic force) applied to the DOF number i that gives place to a displacement vector x j) = { x n } = { δnj }, where all the components are equal to zero, except for x j) j = 1. f in k nj = Fk j = δ ij n Consistent where k s is the vector containing the coefficients k rs.

4 Collecting all the x j) in a matrix X, it is X = I and we have, writing all the equations at once, Finally, X = I = F [ k ij ], [ kij ] = K = F 1. p = f S = K x. Consistent The elastic strain energy V can be written in terms of displacements and external forces, p V = 1 pt x = 1 T F p, }{{} x x }{{} T K x. p T Because the elastic strain energy of a stable system is always greater than zero, K is a positive definite matrix. On the other hand, for an unstable system, think of a compressed beam, there are displacement patterns that are associated to zero strain energy. Consistent, When two sets of loads, p A and p B, are applied one after the other to an elastic system; the work done is V AB = 1 pat x A + p AT x B + 1 pb T x B. If we revert the order of application the work is Expressing the displacements in terms of F, p AT F p B = p B T Fp A, both terms are scalars so we can write p AT F p B = p B T Fp A) T = p A T F T p B. V BA = 1 pb T x B + p B T x A + 1 pat x A. Consistent Because this equation holds for every p, we conclude that Consistent The total work being independent of the order of loading, p AT x B = p B T x A. F = F T, and, as the inverse of a symmetric matrix is symmetric, K = K T.

5 A practical consideration FEM For the kind of structures we mostly deal with in our examples, problems, exercises and assignments, that is simple structures, it is usually convenient to compute the flexibility matrix applying the Principle of Virtual Displacements we have seen an example last week) and inverting the flexibilty to obtain the stiffness matrix, K = F 1. For general structures, large and/or complex, the PVD approach cannot work in practice, as the number of degrees of freedom necessary to model the structural behaviour exceed our ability to do pencil and paper computations... Different methods are required to construct the stiffness matrix for such large, complex structures. Consistent The most common procedure to construct the matrices that describe the behaviour of a complex system is the Finite Element Method, or FEM. The procedure can be sketched in the following terms: the structure is subdivided in non-overlapping portions, the finite elements, bounded by nodes, connected by the same nodes, the displacements, strains, stresses in the fe are described in terms of a linear combination of shape functions, weighted in according to the nodal displacements, element matrices are computed accordingly the element stiffness matrix, K el establishes a linear relation between an element nodal displacements and forces, the state of the structure can be described in terms of a vector x of generalized nodal displacements, there is a mapping between element and structure DOF s, i el r, for each FE, all local k ij s are contributed to the global stiffness k rs s, with i r and j s, taking in due consideration differences between local and global systems of reference. Note that in the r-th global equation of equilibrium we have internal forces caused by the nodal displacements of the FE that have nodes i el such that i el r, thus implying that global K is a sparse matrix. Consistent, Consider a -D inextensible beam element, that has 4 DOF, namely two transverse end displacements x 1, x and two end rotations, x 3, x 4. The element stiffness is computed using 4 shape functions φ i, the transverse displacement being vs) = i φ is)x i, the different φ i are such all end displacements or rotation are zero, except the one corresponding to index i. The shape functions for a beam are s ) s ) 3, s ) s ) 3, φ 1 s) = φ s) = 3 φ 3 s) = s 1 ) s s s ), φ 4 s) = s ) ) ). Consistent The element stiffness coefficients can be computed using, what else, the PVD: we compute the external virtual work done by a variation δ x i by the force due to a unit displacement x j, that is k ij, δ W ext = δ x i k ij, the virtual internal work is the work done by the variation of the curvature, δ x i φ i s) by the bending moment associated with a unit x j, φ j s)ejs), δ W int = 0 δ x i φ i s)φ j s)ejs) ds. Consistent

6 , 3 Blackboard Time! The equilibrium condition is the equivalence of the internal and external virtual works, so that simplifying δ x i we have For EJ = const, k ij = f S = EJ 3 φ 0 i s)φ j s)ejs) ds x Consistent EJ 4EJ EJ x x 3 x 1 Consistent The mass matrix maps the nodal accelerations to nodal inertial forces, and the most common assumption is to concentrate all masses in nodal point masses, without rotational inertia, computed lumping a fraction of each element mass or a fraction of the supported mass) on all its bounding nodes. This procedure leads to a so called lumped mass matrix, a diagonal matrix with diagonal elements greater than zero for all the translational degrees of freedom, and diagonal elements equal to zero for angular degrees of freedom. Consistent The mass matrix is definite positive only if all the structure DOF s are translational degrees of freedom, otherwise M is semi-definite positive and the eigenvalue procedure is not directly applicable. This problem can be overcome either by using a consistent mass matrix or using the static condensation procedure. Consistent

7 Consistent Consistent, A consistent mass matrix is built using the rigorous FEM procedure, computing the nodal reactions that equilibrate the distributed inertial forces that develop in the element due to a linear combination of inertial forces. Using our beam example as a reference, consider the inertial forces associated with a single nodal acceleration ẍ j, f I,j s) = ms)φ j s)ẍ j and denote with m ij ẍ j the reaction associated with the i-nth degree of freedom of the element, by the PVD δ x i m ij ẍ j = δ x i φ i s)ms)φ j s) ds ẍ j simplifying m ij = ms)φ i s)φ j s) ds. For ms) = m = const f I = m ẍ Consistent Pro some convergence theorem of FEM theory holds only if the mass matrix is consistent, sligtly more accurate results, no need for static condensation. Contra M is no more diagonal, heavy computational aggravation, static condensation is computationally beneficial, inasmuch it reduces the global number of degrees of freedom. Consistent For each element c ij = cs)φ i s)φ j s) ds and the damping matrix C can be assembled from element contributions. However, using the FEM C = Ψ T C Ψ is not diagonal and hence the modal equations are no more uncoupled! The alternative is to write directly the global damping matrix, in terms of the underdetermined coefficients c b and the infinite sequence of orthogonal matrices we described previously: C = c b M M 1 K ) b. b Consistent With our definition of C, C = b c b M M 1 K ) b, assuming normalized eigenvectors, we can write the individual component of C = Ψ T C Ψ cij = ψ T i C ψ j = δ ij c b ωj b due to the additional orthogonality relations, we recognize that now C is a diagonal matrix. Introducing the modal damping C j we have b Consistent C j = ψ T j C ψ j = b c b ω b j = ζ j ω j and we can write a system of linear equations in the c b.

8 We want a fixed, 5% damping ratio for the first three modes, taking note that the modal equation of motion is Using we have q i + ζ i ω i q i + ω i q i = p i C = c 0 M + c 1 K + c KM 1 K 0.05 ω 1 ω ω 3 1 ω = 1 ω ω ω 4 1 ω3 ω3 4 c 0 c 1 c Consistent Computing the coefficients c 0, c 1 and c to have a 5% damping at frequencies ω 1 =, ω = 5 and ω 3 = 8 we have c 0 = 100/9100, c 1 = 159/9100 and c = 1/9100. Writing ζω) = 1 c0 ω + c 1ω + c ω 3) we can plot the above function, along with its two term equivalent c 0 = 10/70, c 1 = 1/70). Damping ratio Two and three terms solutions three terms two terms Consistent Solving for the c s and substituting above, the resulting damping matrix is orthogonal to every eigenvector of the system, for the first three modes, leads to a modal damping ratio that is equal to 5% Circular frequency Negative damping? No, thank you: use only an even number of terms. s A common assumption is based on a linear approximation, for a beam element f G = N x N f 1 f N x 1 x f = f 1 f 1 = N x x 1 ) Following the same line of reasoning that we applied to find nodal inertial forces, by the PVD and the use of shape functions we have p i t) = ps, t)φ i s) ds. It is possible to compute the geometrical stiffness matrix using FEM, shape functions and PVD, k G,ij = Ns)φ is)φ js) ds, for constant N K G = N Consistent For a constant, uniform load ps, t) = p = const, applied on a beam element, p = p { } T 1 Consistent

9 Simplified Approach Some structural parameter is approximated, only translational DOF s are retained in dynamic analysis. Consistent Approach All structural parameters are computed according to the FEM, and all DOF s are retained in dynamic analysis. If we choose a simplified approach, we must use a procedure to remove unneeded structural DOF s from the model that we use for the dynamic analysis. Enter the Method. We have, from a FEM analysis, a stiffnes matrix that uses all nodal DOF s, and from the lumped mass procedure a mass matrix were only translational and maybe a few rotational) DOF s are blessed with a non zero diagonal term. In this case, we can always rearrange and partition the displacement vector x in two subvectors: x A all the DOF s that are associated with inertial forces and x B all the remaining DOF s not associated with inertial forces. { } xa x = x B,, 3 After rearranging the DOF s, we must rearrange also the rows equations) and the columns force contributions) in the structural matrices, and eventually partition the matrices so that { } [ ] } fi MAA M = AB {ẍa 0 M BA M BB ẍ B [ ] { } KAA K f S = AB xa K BA K BB x B... the equation of dynamic equilibrium, p A = M AA ẍ A + M AB ẍ B + K AA x A + K AB x B p B = M BA ẍ A + M BB ẍ B + K BA x A + K BB x B The highlighted terms are zero vectors, so we can simplify M AA ẍ A + K AA x A + K AB x B = p A K BA x A + K BB x B = p B with solving for x B in the nd equation and substituting M BA = M T AB = 0, M BB = 0, K BA = K T AB Finally we rearrange the loadings vector and write... x B = K 1 BB p B K 1 BB K BAx A p A K AB K 1 BB p B = M AA ẍ A + K AA K AB K 1 BB K BA) xa

10 , 4 Going back to the homogeneous problem, with obvious positions we can write K ω M ) ψ A = 0 but the ψ A are only part of the structural eigenvectors, because in essentially every application we must consider also the other DOF s, so we write { } ψa,i ψ i =, with ψ B,i = K 1 BB K BAψ A,i ψ B,i x x 3 4EJ EJ K BB = 4EJ EJ x 1 [ ] 3 1, K BB = K = EJ 3 3EJ [ ] 3 1, 1 3 K AB = 6EJ [ ] 1 1, KAB K 1 BB KT AB = 6EJ 6EJ 3EJ 4 = 9 EJ 3 The matrix K is K = K AA K AB K 1 BB KT AB = 4 9 )EJ 3 = 39 EJ 3

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

Multi Degrees of Freedom Systems

Multi Degrees of Freedom Systems Multi Degrees of Freedom Systems MDOF s http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 9, 07 Outline, a System

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

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

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

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

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

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

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

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

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

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

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

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

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. 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). 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 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

Codal Provisions IS 1893 (Part 1) 2002

Codal 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 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

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

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix

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

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

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

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras (Refer Slide Time: 00:25) Module - 01 Lecture - 13 In the last class, we have seen how

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

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

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

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods

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

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

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

A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS

A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A. Kroker, W. Becker TU Darmstadt, Department of Mechanical Engineering, Chair of Structural Mechanics Hochschulstr. 1, D-64289 Darmstadt, Germany kroker@mechanik.tu-darmstadt.de,

More information

Chapter 5 Structural Elements: The truss & beam elements

Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations

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

Dispersion relation for transverse waves in a linear chain of particles

Dispersion relation for transverse waves in a linear chain of particles Dispersion relation for transverse waves in a linear chain of particles V. I. Repchenkov* It is difficult to overestimate the importance that have for the development of science the simplest physical and

More information

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix

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

JEPPIAAR ENGINEERING COLLEGE

JEPPIAAR 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 information

A consistent dynamic finite element formulation for a pipe using Euler parameters

A consistent dynamic finite element formulation for a pipe using Euler parameters 111 A consistent dynamic finite element formulation for a pipe using Euler parameters Ara Arabyan and Yaqun Jiang Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721,

More information

ANALYSIS OF HIGHRISE BUILDING STRUCTURE WITH SETBACK SUBJECT TO EARTHQUAKE GROUND MOTIONS

ANALYSIS 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 information

Modal Analysis: What it is and is not Gerrit Visser

Modal Analysis: What it is and is not Gerrit Visser Modal Analysis: What it is and is not Gerrit Visser What is a Modal Analysis? What answers do we get out of it? How is it useful? What does it not tell us? In this article, we ll discuss where a modal

More information

General elastic beam with an elastic foundation

General elastic beam with an elastic foundation General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation

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

SPECIAL DYNAMIC SOIL- STRUCTURE ANALYSIS PROCEDURES DEMONSTATED FOR TWO TOWER-LIKE STRUCTURES

SPECIAL DYNAMIC SOIL- STRUCTURE ANALYSIS PROCEDURES DEMONSTATED FOR TWO TOWER-LIKE STRUCTURES 2010/2 PAGES 1 8 RECEIVED 21. 9. 2009 ACCEPTED 20. 1. 2010 Y. KOLEKOVÁ, M. PETRONIJEVIĆ, G. SCHMID SPECIAL DYNAMIC SOIL- STRUCTURE ANALYSIS PROCEDURES DEMONSTATED FOR TWO TOWER-LIKE STRUCTURES ABSTRACT

More information

Introduction to Vibration. Professor Mike Brennan

Introduction 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 information

. D CR Nomenclature D 1

. D CR Nomenclature D 1 . D CR Nomenclature D 1 Appendix D: CR NOMENCLATURE D 2 The notation used by different investigators working in CR formulations has not coalesced, since the topic is in flux. This Appendix identifies the

More information

Part D: Frames and Plates

Part D: Frames and Plates Part D: Frames and Plates Plane Frames and Thin Plates A Beam with General Boundary Conditions The Stiffness Method Thin Plates Initial Imperfections The Ritz and Finite Element Approaches A Beam with

More information

CIVL 8/7117 Chapter 12 - Structural Dynamics 1/75. To discuss the dynamics of a single-degree-of freedom springmass

CIVL 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 information

M.S Comprehensive Examination Analysis

M.S Comprehensive Examination Analysis UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... M.S Comprehensive

More information

Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati

Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 2 Simpul Rotors Lecture - 2 Jeffcott Rotor Model In the

More information

EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION

EQUIVALENT 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 information

Appendix C. Modal Analysis of a Uniform Cantilever with a Tip Mass. C.1 Transverse Vibrations. Boundary-Value Problem

Appendix C. Modal Analysis of a Uniform Cantilever with a Tip Mass. C.1 Transverse Vibrations. Boundary-Value Problem Appendix C Modal Analysis of a Uniform Cantilever with a Tip Mass C.1 Transverse Vibrations The following analytical modal analysis is given for the linear transverse vibrations of an undamped Euler Bernoulli

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

ABSTRACT Modal parameters obtained from modal testing (such as modal vectors, natural frequencies, and damping ratios) have been used extensively in s

ABSTRACT Modal parameters obtained from modal testing (such as modal vectors, natural frequencies, and damping ratios) have been used extensively in s ABSTRACT Modal parameters obtained from modal testing (such as modal vectors, natural frequencies, and damping ratios) have been used extensively in system identification, finite element model updating,

More information

Effect of Mass Matrix Formulation Schemes on Dynamics of Structures

Effect of Mass Matrix Formulation Schemes on Dynamics of Structures Effect of Mass Matrix Formulation Schemes on Dynamics of Structures Swapan Kumar Nandi Tata Consultancy Services GEDC, 185 LR, Chennai 600086, India Sudeep Bosu Tata Consultancy Services GEDC, 185 LR,

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

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

Deflections and Strains in Cracked Shafts due to Rotating Loads: A Numerical and Experimental Analysis

Deflections and Strains in Cracked Shafts due to Rotating Loads: A Numerical and Experimental Analysis Rotating Machinery, 10(4): 283 291, 2004 Copyright c Taylor & Francis Inc. ISSN: 1023-621X print / 1542-3034 online DOI: 10.1080/10236210490447728 Deflections and Strains in Cracked Shafts due to Rotating

More information

FLEXIBILITY METHOD FOR INDETERMINATE FRAMES

FLEXIBILITY 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 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

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

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

Response Analysis for Multi Support Earthquake Excitation

Response Analysis for Multi Support Earthquake Excitation Chapter 5 Response Analysis for Multi Support Earthquake Excitation 5.1 Introduction It is very important to perform the dynamic analysis for the structure subjected to random/dynamic loadings. The dynamic

More information

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 11 Last class, what we did is, we looked at a method called superposition

More information

Methods of Analysis. Force or Flexibility Method

Methods of Analysis. Force or Flexibility Method INTRODUCTION: The structural analysis is a mathematical process by which the response of a structure to specified loads is determined. This response is measured by determining the internal forces or stresses

More information

Discretization Methods Exercise # 5

Discretization Methods Exercise # 5 Discretization Methods Exercise # 5 Static calculation of a planar truss structure: a a F Six steps: 1. Discretization 2. Element matrices 3. Transformation 4. Assembly 5. Boundary conditions 6. Solution

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

AERSYS KNOWLEDGE UNIT

AERSYS KNOWLEDGE UNIT -7016 1. INTRODUCTION The scope of this document is to provide a clarification and a deeper understanding of the two different ways to move the mid plane of the element out of the nodal plane. Although

More information

Stress analysis of a stepped bar

Stress analysis of a stepped bar Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.

More information

CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES

CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES 14.1 GENERAL REMARKS In structures where dominant loading is usually static, the most common cause of the collapse is a buckling failure. Buckling may

More information

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2 Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1

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

Rayleigh s Classical Damping Revisited

Rayleigh s Classical Damping Revisited Rayleigh s Classical Damping Revisited S. Adhikari and A. Srikantha Phani Department of Aerospace Engineering, University of Bristol, Bristol, U.K. Email: S.Adhikari@bristol.ac.uk URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html

More information

Quintic beam closed form matrices (revised 2/21, 2/23/12) General elastic beam with an elastic foundation

Quintic beam closed form matrices (revised 2/21, 2/23/12) General elastic beam with an elastic foundation General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation

More information

Operating Deflection Shapes from Strain Measurement Data

Operating Deflection Shapes from Strain Measurement Data Operating Deflection Shapes from Strain Measurement Data Timothy G. Hunter, Ph.D., P.E. President Wolf Star Technologies, LLC 3321 N. Newhall St., Milwaukee, WI 53211 Abstract Strain gauges are often more

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

UNIT IV FLEXIBILTY AND STIFFNESS METHOD

UNIT IV FLEXIBILTY AND STIFFNESS METHOD SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : SA-II (13A01505) Year & Sem: III-B.Tech & I-Sem Course & Branch: B.Tech

More information

This procedure covers the determination of the moment of inertia about the neutral axis.

This procedure covers the determination of the moment of inertia about the neutral axis. 327 Sample Problems Problem 16.1 The moment of inertia about the neutral axis for the T-beam shown is most nearly (A) 36 in 4 (C) 236 in 4 (B) 136 in 4 (D) 736 in 4 This procedure covers the determination

More information

ME 475 Modal Analysis of a Tapered Beam

ME 475 Modal Analysis of a Tapered Beam ME 475 Modal Analysis of a Tapered Beam Objectives: 1. To find the natural frequencies and mode shapes of a tapered beam using FEA.. To compare the FE solution to analytical solutions of the vibratory

More information

MODELLING MIXED-MODE RATE-DEPENDENT DELAMINATION IN LAYERED STRUCTURES USING GEOMETRICALLY NONLINEAR BEAM FINITE ELEMENTS

MODELLING MIXED-MODE RATE-DEPENDENT DELAMINATION IN LAYERED STRUCTURES USING GEOMETRICALLY NONLINEAR BEAM FINITE ELEMENTS PROCEEDINGS Proceedings of the 25 th UKACM Conference on Computational Mechanics 12-13 April 217, University of Birmingham Birmingham, United Kingdom MODELLING MIXED-MODE RATE-DEPENDENT DELAMINATION IN

More information

Chapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation..14

Chapter 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 information

Numerical simulation of the coil spring and investigation the impact of tension and compression to the spring natural frequencies

Numerical simulation of the coil spring and investigation the impact of tension and compression to the spring natural frequencies Numerical simulation of the coil spring and investigation the impact of tension and compression to the spring natural frequencies F. D. Sorokin 1, Zhou Su 2 Bauman Moscow State Technical University, Moscow,

More information

Virtual distortions applied to structural modelling and sensitivity analysis. Damage identification testing example

Virtual distortions applied to structural modelling and sensitivity analysis. Damage identification testing example AMAS Workshop on Smart Materials and Structures SMART 03 (pp.313 324) Jadwisin, September 2-5, 2003 Virtual distortions applied to structural modelling and sensitivity analysis. Damage identification testing

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

EVALUATING DYNAMIC STRESSES OF A PIPELINE

EVALUATING 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 information

Shape Optimization of Revolute Single Link Flexible Robotic Manipulator for Vibration Suppression

Shape Optimization of Revolute Single Link Flexible Robotic Manipulator for Vibration Suppression 15 th National Conference on Machines and Mechanisms NaCoMM011-157 Shape Optimization of Revolute Single Link Flexible Robotic Manipulator for Vibration Suppression Sachindra Mahto Abstract In this work,

More information

k 21 k 22 k 23 k 24 k 31 k 32 k 33 k 34 k 41 k 42 k 43 k 44

k 21 k 22 k 23 k 24 k 31 k 32 k 33 k 34 k 41 k 42 k 43 k 44 CE 6 ab Beam Analysis by the Direct Stiffness Method Beam Element Stiffness Matrix in ocal Coordinates Consider an inclined bending member of moment of inertia I and modulus of elasticity E subjected shear

More information

Vibration of Thin Beams by PIM and RPIM methods. *B. Kanber¹, and O. M. Tufik 1

Vibration of Thin Beams by PIM and RPIM methods. *B. Kanber¹, and O. M. Tufik 1 APCOM & ISCM -4 th December, 23, Singapore Vibration of Thin Beams by PIM and RPIM methods *B. Kanber¹, and O. M. Tufik Mechanical Engineering Department, University of Gaziantep, Turkey. *Corresponding

More information

CIVL 7/8117 Chapter 4 - Development of Beam Equations - Part 2 1/34. Chapter 4b Development of Beam Equations. Learning Objectives

CIVL 7/8117 Chapter 4 - Development of Beam Equations - Part 2 1/34. Chapter 4b Development of Beam Equations. Learning Objectives CIV 7/87 Chapter 4 - Development of Beam Equations - Part /4 Chapter 4b Development of Beam Equations earning Objectives To introduce the work-equivalence method for replacing distributed loading by a

More information

CHAPTER 12 TIME DOMAIN: MODAL STATE SPACE FORM

CHAPTER 12 TIME DOMAIN: MODAL STATE SPACE FORM CHAPTER 1 TIME DOMAIN: MODAL STATE SPACE FORM 1.1 Introduction In Chapter 7 we derived the equations of motion in modal form for the system in Figure 1.1. In this chapter we will convert the modal form

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

Dynamics of Structures: Theory and Analysis

Dynamics 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 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

International Journal of Advanced Engineering Technology E-ISSN

International Journal of Advanced Engineering Technology E-ISSN Research Article INTEGRATED FORCE METHOD FOR FIBER REINFORCED COMPOSITE PLATE BENDING PROBLEMS Doiphode G. S., Patodi S. C.* Address for Correspondence Assistant Professor, Applied Mechanics Department,

More information

Preprocessor Geometry Properties )Nodes, Elements(, Material Properties Boundary Conditions(displacements, Forces )

Preprocessor Geometry Properties )Nodes, Elements(, Material Properties Boundary Conditions(displacements, Forces ) در برنامه يك تدوين براي بعدي دو يك سازه محيط MATLAB Preprocessor Geometry Properties )Nodes, Elements(, Material Properties Boundary Conditions(displacements, Forces ) Definition of Stiffness Matrices

More information

A BEAM FINITE ELEMENT MODEL INCLUDING WARPING

A BEAM FINITE ELEMENT MODEL INCLUDING WARPING A BEAM FINITE ELEMENT MODEL INCLUDING WARPING Application to the dynamic and static analysis of bridge decks Diego Lisi Department of Civil Engineering of Instituto Superior Técnico, October 2011 ABSTRACT

More information

Structural Analysis II Prof. P. Banerjee Department of Civil Engineering Indian Institute of Technology, Bombay Lecture 38

Structural Analysis II Prof. P. Banerjee Department of Civil Engineering Indian Institute of Technology, Bombay Lecture 38 Structural Analysis II Prof. P. Banerjee Department of Civil Engineering Indian Institute of Technology, Bombay Lecture 38 Good morning. We have been looking at influence lines for the last couple of lectures

More information

Mechanics of Inflatable Fabric Beams

Mechanics of Inflatable Fabric Beams Copyright c 2008 ICCES ICCES, vol.5, no.2, pp.93-98 Mechanics of Inflatable Fabric Beams C. Wielgosz 1,J.C.Thomas 1,A.LeVan 1 Summary In this paper we present a summary of the behaviour of inflatable fabric

More information

AEROELASTIC ANALYSIS OF SPHERICAL SHELLS

AEROELASTIC ANALYSIS OF SPHERICAL SHELLS 11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V) 6th European Conference on Computational Fluid Dynamics (ECFD VI) E. Oñate, J. Oliver

More information