STRUCTURAL DYNAMICS (ENCE 365)

1 Introduction (6 hours)

1.1 Structural dynamics (Dynamic loading, structural behavior and response)

1.2 Essential characteristics of a dynamic problem

1.3 Equation of motion (Newton’s second law of motion and D’Alembert’s principle) and dynamic degrees of freedom

1.4 Types of vibrations (Free, forced, damped and undamped)

1.5 Types of dynamic loading (Simple harmonic, periodic, transient, impulsive, and random)

1.6 Damping: Definition, characteristics, and types (Viscous, coulomb, structural, aerodynamic, magnetic, viscoelastic, and active)

1.7 Dynamic response of structures

1.8 Time-domain and frequency-domain analysis

2 Single-degree-of-freedom System (20 hours)

2.1 Equations of motion: Modeling of single-degree-of-freedom system, response parameters, effect of gravity, effect of support excitation, combination of stiffness (Parallel and series)

2.2 Free vibration response: Undamped system; Damped system (Under critically, critically, and over critically)

2.3 Logarithmic decrement of response

2.4 Forced vibration response to harmonic loading: Undamped system; Damped systems (Transient response, steady-state response, general response, and resonance response)

2.5 Energy dissipated by damping (Viscous damping)

2.6 Forced vibration response to periodic loading: Undamped system (Transformation of loading and response using Fourier series)

2.7 Forced vibration response to impulsive load (Half-sine wave pulse, rectangular pulse, ramp pulse, triangular pulse)

2.8 Forced vibration response to general dynamic loading: Unit-impulse and Dirac-delta function; Unit-impulse response function; Duhamel integral and Convolution integral; Numerical evaluation of Duhamel integral

2.9 Vibration measuring instruments (Displacement meter and accelerometer)

2.10 Vibration isolation (Vertical oscillating force, harmonic motion of the base, force transmissibility, unbalanced mass amplitude of rotating unbalanced system)

2.11 Computer simulation of dynamic responses of single-degree-of-freedom system: Free; Forced; Undamped damped (Under critical, critical, and over critical); Resonance effect

3 Multi-degrees-of-freedom System (8 hours)

3.1 Simple system (Basic concept, uses and limitations)

3.2 Generalized coordinate and reduction of degrees of freedom (Kinematic constraints and static condensation)

3.3 Dynamic equilibrium: Mathematical modeling and dynamic equilibrium equation; influence coefficients (Stiffness influence coefficients, damping influence coefficients, and mass influence coefficients)

3.4 Free vibration analysis of undamped system: Eigen value problem (natural frequencies and mode shapes)

3.5 Free vibration response of undamped system: Modal expansion; Orthogonality conditions; Normalization; Normal coordinates; Uncoupled equations of motion; Mode superposition method

3.6 Modal response analysis of damped systems

3.7 Forced vibration response of damped and undamped system

3.8 Dynamic analysis of linear multi-degrees-of-freedom system: Response spectrum analysis; Element forces; Modal contribution factors and minimum number of modes; Base shear; Modal mass

3.9 Evaluation of natural frequencies and mode shapes (Iterative methods): Rayleigh’s method; Stodola’s method; Holzer’s method

3.10 Support excitation (Influence vector, synchronous support motion of a planar system, and structure with multiple support motions)

4 Continuous Systems (4 hours)

4.1 Partial differential equations of motion (Transverse vibration of a string and beam; axial vibration of a bar)

4.2 Evaluation of natural frequencies and mode shapes (Transverse vibration of a string and beam)

5 Application of Finite Element Method in Structural Dynamics (7 hours)

5.1 Basic concept of finite element method: Node and element; types and characteristics of finite element, local and global coordinate system; Discretization and meshing; shape functions; Nodal forces, Nodal displacements, Elemental stiffness matrix and structural (Global) stiffness matrix

5.2 Formulation of shape function and stiffness matrix (Bar and beam elements)

5.3 Convergence and compatibility

1. Evaluation of equivalent stiffness and fundamental frequency for structural systems with lateral-load-resisting elements in parallel and series combinations

2. Evaluation and plotting of vibration responses for an single-degree-of-freedom system (Undamped free vibration, undamped forced vibration, damped free vibration damped forced vibration, periodic and impulse loading)

3. Determination of the vibration response due to general dynamic loading using Duhamel’s integral by applying both the analytical approach and numerical integration methods

4. Evaluation of the modal frequencies and mode shapes using the Eigen value method including sketches of the mode shapes

5. Determination of displacements and velocities of the free and forced vibration with damped and undamped system for the given initial conditions

6. Determination of displacements and velocities of the undamped and damped free vibration system for given initial conditions

7. Determination of several parameters of damped free vibration system of realistic problems (Such as elevated water tank, portal frame, and others): Damping ratio, natural period, equivalent stiffness, weight, damped natural frequency, damping coefficient, number of cycles required for the targeted reduced displacement amplitude, logarithmic decrement, and others

8. Determination of force transmissibility ratio and force due to rotation unbalance of machine

9. Determination of responses due to several types of periodic and impulsive loading (Half-sine wave, triangular, step or rectangular, ramp, and blast)

10. Evaluation of responses due to general dynamic loading: (General integration and numerical evaluation approach)

11. Determination of modal frequencies and mode shape vector and plotting the mode shapes of Eigen value problem of MDOF system

12. Evaluation and plotting of vibration responses for multi-degrees-of-freedom system using modal superposition method

13. Determination of modal frequencies of multi-degrees-of-freedom system using iterative methods (Rayleigh’s method, Stodola’s method, and Holzer’s method)

14. Evaluation of the natural frequencies and plotting the mode shapes of vibrations (Transverse or axial) for continuous structural systems (String, bar and beam)

15. Evaluation of the nodal displacements of 1D bar problem and simple truss under point load and UDL

1. Free vibration of a simple cantilever beam

2. Mass-spring system: Determination of natural frequency

3. Damping in a simple pendulum

4. Dynamic response of a multi-story frame model

5. Modal analysis of MDOF system using FEM-based software

1. Clough, R. W., Penzien, J. (2015). Dynamics of structures. CBS Publishers & Distributors.

2. Paultre, P. (2013). Dynamics of structures. John Wiley & Sons.

3. Jain, A.K. (2016). Dynamics of structures with MATLAB applications. Pearson.

4. Chopra, A.K. (2017). Dynamics of structures: Theory and applications to earthquake engineering. Pearson.

5. Paz, M., Leigh, W. (2010). Dynamics of structures: Theory and computation. Springer.

6. Thompson, W.T., Dahleh, M. D. (1998). Theory of vibration with applications (Latest Edition). Prentice Hall.

7. Cook, R.D. (1981). Finite element analysis (Latest Edition). John Wiley & Sons.

8. Belegundu, A. D., Chandrupatla, T. R. (2011). Introduction to finite elements in engineering. Pearson.