Free and forced vibrations of damped and undamped, single and multi-degree-of-freedom systems. Modal analysis - eigenvalue problem. Time-domain vs. frequency-domain analysis methods. Classical approximate methods. Analysis and stability of structural components. Equivalent modeling of structural systems. Design of systems for deterministic and stochastic loads.
(a) EEGR 505: Advanced Engineering Mathematics or equivalent.
(b) IEGR 331: Probability & Statistics or equivalent
(c) Experience with a higher level programming language – C, FORTRAN, Pascal
OBJECTIVES OF THE COURSE
This course is intended to provide:
a) Introduction to dynamical systems – emphasis on structural systems
b) Analytical methods to solve the dynamic response of SDOF and MDOF structures under deterministic and random loads
Free and forced vibrations of undamped and damped, single-degree-of-freedom and multi-degree-of-freedom systems. Transient and steady-state vibrations. Vibration absorbers, isolators and measurement devices. Lagrange’s equations; Eigenvalue analysis for natural frequencies and normal modes; Analysis and stability of structural components (including beams, cables and large systems onshore, offshore, and in space). Time-domain vs. frequency-domain analysis; Classical approximate methods – Rayleigh method, Dunkerley’s equation, Rayleigh-Ritz Method, Myklestad’s Method for beams. Response to impulsive loading. Evaluation of structural property matrices. Numerical solution techniques.
Introduction to Random Vibrations – statistical description of stochastic loads and structural response. Response Spectrum approach to earthquake response of structures.
RELATIONSHIP TO CURRICULUM SEQUENCE
This course establishes the analytical groundwork for analysis of dynamic systems. Even though the majority of examples are taken from structural engineering, the analytical generality can easily be extended to stochastic electrical and mechanical systems. This course is recomended for students specializing in Structural Mechanics and Structural Engineering.
RELATIONSHIP TO SIMILAR OFFERINGS IN THE DEPARTMENT
This course can be viewed as a complement to structural analysis and design courses that focus on the static or pseudostatic behavior of structures subjected to constant loads. This course can be followed and complemented by the course on Wind and Earthquake Engineering
The course is focused on the analytical treatment of dynamic systems – in increasing degree of difficulty. The schedule of topics will be –
Week 1 Simple mass-spring system. 2nd order differential equation. Oscillatory motion. Damping. Damped harmonic oscillator. Free response of undamped and damped harmonic oscillators. Logarithmic Decrement to estimate damping.
Week 2 Forced response of single-degree-of-freedom systems – harmonic input – steady state solution. Resonance. Response to base excitation. Response to unbalanced (harmonically) rotating loads.
Week 3 Periodic excitation – use of Fourier Series. Nonharmonic excitation – impulse response, step response. Response to impulsive loading – use of shock spectra in design for blast loads.
Week 4 Analytical Dynamics - Lagrange’s Equations. Multi-degree-of-freedom systems. Eigenvalue problem. Extraction of natural frequencies and mode shapes. Coupling – dynamic and static.
Week 5 Multi-degree-of-freedom systems – orthogonal and orthonormal modes. Natural coordinates. Response of MDOF systems to initial conditions and harmonic excitation.
Week 6 Approximate methods – solution of the eigenvalue problem by matrix iteration. Rayleigh’s quotient.
Week 7 Continuous systems – exact solutions – axial vibration of rods, bending vibration of beams. Wave equation.
Week 8 Evaluation of structural property matrices – stiffness, mass and damping. Transformation of real structure into equivalent lumped model – consistent vs. lumped formulation.
Week 9 Inclusion of damping into analytical models – viscous, modal and Rayleigh damping. Complex stiffness approach to damping.
Week 10 Numerical Simulation of dynamical systems – step by step integration methods.
Week 11 Introduction to random vibrations – stationary and ergodic processes. Time and ensemble averages. Probability density functions. Autocorrelation functions. Fourier Transforms.
Week 12 Probability distribution of extreme values. Design considerations. Relation between input and output. Power spectral density. Response of SDOF systems to random excitation.
Week 13 Earthquake engineering – seismicity. Response Spectra. Use of response spectra in the seismic design of structures.
Week 14 Deterministic and stochastic response of structures subjected to earthquake loads.
The overall grade for the course will be based on 3 components – midterm examination, course project and a final. The grade distribution will be as follows:
Midterm Examination 30%
Final Examination 40%
Grades will be assigned according to the following chart:
Leonard Meirovitch. Elements of Vibration Analysis, 2nd ed. McGraw Hill, 1986
Clough & Penzien. Dynamics of Structures. 2nd ed. McGraw Hill, 1993.
Thomson & Dayleh. Theory of Vibration with Applications, 5th ed. Prentice Hall, 1998.