# Dynamics of Structures

## Course Description:

Formulation and solution of equation of motion of one-degree of freedom systems for any external load. Systems of many degrees of freedom of motion. Free and forced vibrations of multi-degree-of-freedom systems. Damped of multi-degree-of-freedom systems. Dynamic analysis of multi-story building.

### Prerequisite Knowledge

Students is recommended to have basic knowledge of Mechanics of the rigid body, Mechanics οf Deformable Solids, Dynamics of the rigid body, Matrix Structural Analysis – 1D Finite Elements.

### Course Units

# Title Description Hours
1 Introduction Introduction. Differences in static, dynamic behavior of structures. Dynamic loads. Dynamic equilibrium. Degrees of freedom of a structure. Dynamic model and equation of motion. Formulation of equation of motion of one-degree of freedom with the method of direct equilibrium and with the principle of virtual work. 1Χ4=4
2 Study of single-degree-of-freedom systems vibrations Systems with one degree of freedom of motion. Free undamped and damped vibrations of single-degree-of-freedom systems. Forced vibrations of single-degree-of-freedom systems. Study of forced undamped and damped vibrations of single-degree-of-freedom systems subjected to harmonic and periodic forces. Resonance. Forced undamped and damped vibrations for any external load. Duhamel integral. Calculation of the Duhamel integral. Applications of the Duhamel integral. Response to step and harmonic loads. Study of forced vibrations of single-degree-of-freedom systems subject to ground motion. Response spectra. Influence of gravity on forced vibrations of single-degree-of-freedom system. 3Χ4=12
3 Numerical calculation of dynamic response Numerical calculation of dynamic response. Central Difference Method. Acceleration Method (Newmark). Numerical calculation of the Duhamel integral. Demonstration of the dynamic behavior of a single-degree-of-freedom system on PC. 1Χ4=4
4 Generalized single-degree-of-freedom systems Generalized single-degree-of-freedom systems. Shape functions. Calculation of elastic, kinetic energy, virtual work of non-conservative forces. 1Χ4=4
5 Formulation of equation of motion of multi-degree-of-freedom systems Systems with many degrees of freedom of motion. Elastic, inertial and damping forces of a structure. Formulation of stiffness matrix element with constant cross section. Formulation of stiffness matrix of a structure. Formulation of mass matrix of multi-degree-of-freedom systems with lumped and distributed mass. Geometric stiffness matrix structure. Formulation of stiffness matrix element with variable cross section. Static condensation of degrees-of-freedom. 2Χ4=8
6 Dynamic analysis of singe- and multi-story buildings Dynamic analysis of multi-story buildings. Eccentricity matrix. Transformation matrix. Stiffness matrix of a building. Mass matrix of a building. 1Χ4=4
7 Dynamic analysis of of multi-degree-of-freedom systems Free vibration of multi-degree-of-freedom systems. Frequency equation of multi-degree-of-freedom systems. Eigenvalues, mode shapes, natural mode shapes of vibration of multi-degree-of-freedom systems. Orthogonality conditions of modes shapes. Properties of the eigenfrequencies and modes shapes of free undamped of multi-degree-of-freedom systems. Forced vibrations of undamped of multi-degree-of-freedom systems. Generalized mass, stiffness, external force of multi-degree-of-freedom systems. Damped of multi-degree-of-freedom systems. Uncoupled damped equations of motion. Evaluation of damping matrix of multi-degree-of-freedom systems. Dynamic response of damped multi-degree-of-freedom systems. 3Χ4=12
8 Participation of the modes shape in the mode superposition method Participation of the modes shapes in the mode superposition method. Modal contribution. Modal contribution factor. Truncation error of higher modes. Base shear, Overturn moment of multi-degree-of-freedom building. 1Χ4=4

### Learning Objectives

After successful completion of the course, students will be able to:

1. study the dynamic behavior of single and multi-degree-of-freedom systems,
2. form the motion equations governing the behavior of these structures,
3. solve equations of motion with analytical and / or modern computational methods.

### Teaching Methods

 Teaching methods Lectures in class. Solve examples and problems in the classroom Presentations in the Table. Slides Power Point. Demonstrate an application to dynamic behavior of single degree-of-freedom system on PC with the help of free software on the internet. Students solve the problems with the help of teachers using EXCEL and MATLAB on PC. Students are examined in the solution of Dynamic analysis of multi-story building.

### Student Assessment

• Final written exam: 70%
• Mid-term exam: 20%
• Assignments (projects, reports): 10%

### Textbooks - Bibliography

1. Κατσικαδέλης Ι. Θ., Δυναμική Ανάλυση των Κατασκευών, Συμμετρία, 2012.
2. Chopra A. K., Δυναμική των Κατασκευών - Θεωρία και Εφαρμογές στη Σεισμική Μηχανική, εκδ. Γκιούρδας, 2007
3. Κολιόπουλος Π. K. - Μανώλης Γ. Δ., Δυναμική των Κατασκευών με Εφαρμογές στην Αντισεισμική Μηχανική, εκδ. Γκιούρδας, 2005
4. Clough R.W. and Penzien J., Dynamics of Structures, McGraw-Hill, New York, 1993

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