Course content

The theory of vibrations is one of the most interesting and useful topics in engineering. It allows for understanding the behavior and design of mechanical systems and structures, and has many applications.
In this course we focus on engineering systems that behave linearly and are relatively simple (single or only a few degrees of freedom). The loads that will be considered, on the other hand, arise from natural phenomena such as water waves, wind and earthquakes. These loads are irregular and partially random, and are therefore either given as irregular time series, or described by load spectra. For linear systems this leads to a beautiful theory: the input loading is decomposed into a number of frequency components, the system response is characterized by the reaction of the system to each of these single frequencies, and the total response is simply the sum of the responses to each single frequency component. The relationship between load and response with respect to frequency is known as a "transfer function", and this approach is also known as the "frequency domain method". From the response spectrum that one obtains one can calculate probabilities for the size or maximum of the response, the number of cycles, and the size of peaks.

Learning outcome

Knowledge
The candidate should have knowledge of:
- Dynamic equilibrium for mechanical systems.
- The equations of motion for single-degree-of-freedom mechanical systems.
- The concept of eigenfrequencies and assumed modes.
- Harmonic motion.
- Dynamic amplification and phase relationships in forced vibrations.
- Transfer functions and impulse-response function for dynamic systems.
- The concept of a random variable and its distribution.
- Stochastic processes.
- Gaussian, Rayleigh and Poisson statistical distributions and their properties.
- Definition and interpretation of covariance, autocorrelation, and the spectrum.
- Special classes of stochastic processes and their properties: Gaussian, narrow-banded, and white noise processes.
- The concept of level crossing rate.
- Peak formulas for narrow-banded and Gaussian processes.
- Standard functions in MATLAB for mathematics, statistics, integration of differential equations, and plotting.

Skills
The candidate should be able to:
- Establish the dynamic equilibrium equations for systems with a few degrees of freedom.
- Use complex numbers in the formulation of the equations of motions.
- Calculate the response of a system from a given deterministic load.
- Determine the mean and variance as well as the response spectrum of a system subjected to stochastic irregular loads.
- Create synthetic time series of environmental loads.
- Compute and interpret a spectrum for a given time series.
- Conduct statistical analysis of combined stochastic processes.
- Calculate up-crossing frequencies for the response process.

General competence
The candidate has:
- Basic understanding of the theory and practice of both deterministic and random vibrations.
- The ability to model mechanical or structural systems with differential equations, using the equations of dynamic equilibrium and the principle of virtual work.
- Detailed understanding of the dynamics associated with single degree of freedom mechanical systems.
- The ability to use MATLAB for analyzing and visualizing the behavior of mechanical systems.
- A sound basis for further studies in structural dynamics, stochastic dynamics, and response modelling.

Learning methods and activities

Lectures and exercises. The subject is taught in English. Students have to come prepared for the lectures (by reading specified parts of the study materials). Active participation and problem solving is important. Exercises will be given weekly and discussed in class after the deadline. Two larger problems will be solved in small student groups during class (problem-based learning). MATLAB will be used both for teaching (demonstrations) and exercises. A tutorial on the use of MATLAB will be given for first time users. Basic knowledge of MATLAB can be expected in the exam. If there is a re-sit examination, the form of the examination may be changed from written to oral.

Compulsory assignments

• Exercises

Recommended previous knowledge

A minimum of 30 credits in mathematics and statistics, as well as basic knowledge of mechanics equivalent to the first 2.5 years of the Civil and Environmental Engineering studyprogram at NTNU. Knowledge of calculus, ordinary differential equations, Fourier analysis and probability theory. Experience with MATLAB and programming. Both TBA4265 Marine Physical Environment and TKT4201 Structural Dynamics provide useful background for parts of the course.

Course materials

Lecture notes: Dynamic response to irregular loadings (available at Department of Civil and Transport Engineering).
Additional papers and materials distributed via itslearning.
Recommended study materials:
S.G. Kelly: Mechanical vibrations - Theory and applications.
L. Lutes & S. Sarkani: Random vibrations (available electronically).

Credit reductions