Free and forced vibration of single degree of freedom systems and small multi-DOF systems. Formulation of the equation of motion for simple systems. Eigenvalue and eigenvector calculations for MDOF systems. Vibration isolation. Rigid body motion - resilient mounting and floating body in waves, including coupled motions. Eigenfrequency-calculation of beams, wires, and rods using the differential equation energy method. Modelling of continuous systems using generalized co-ordinates. Calculation of forced response in time and frequency domain, modal superposition. Numerical time-domain integration methods. Irregular waves and wave spectra, short-term and long-term statistics of waves. Stochastic analysis of structural responses, including fatigue calculation and extreme response estimation.

## Learning outcomes

At the end of the course, the student will be able to:

- formulate the equations of motion for simple SDOF and MDOF systems
- explain how stiffness, damping and inertia forces will influence response for varying load frequency for SDOF systems, and provide examples of sources of stiffness, damping, inertia, and external loads in the marine environment
- apply frequency-domain methods, Laplace transforms, state-space formulation, Runge-Kuttas method, or constant average acceleration to find the dynamic response of SDOF systems subjected to different kinds of forcing
- explain the concepts of natural frequencies, mode shapes, and orthogonal eigenmodes for MDOF systems
- apply frequency-domain methods to find the response of 2DOF systems to harmonic forcing
- establish and solve the differential equations for free vibration of beams, tensioned strings, and rods in axial or torsional vibration
- transfer simple continuous systems to single degree of freedom systems by use of generalized coordinates, and understand the limitations in accuracy when using approximate mode shapes
- apply modal analysis to find the response of continuous systems
- account for that a rigid body is a model of stiff body whose motions are fully described by 6 DOFs that follow from a generalized version of Newtons second law
- understand the relation between reality and linear potential flow theory, and formulate the linearized boundary value problem (BVP) for a rigid body floating in waves
- derive the linear, coupled 6DOF equations of motion for a rigid body floating in waves, and explain the physical meaning of the different terms (added mass, damping, restoring and wave excitation)
- account for limitations and application areas of the Morison equation, both for the inertia term and the viscous term
- explain physically and mathematically how irregular waves are described using the wave spectrum for both long-crested and short-crested waves.
- explain the main properties of standard wave spectra (Pierson-Moskowitz type spectra, JONSWAP spectrum) including standard directional spectra
- define and explain the meaning of the spectral moments and wave parameters such as significant wave height, spectral peak period, mean zero-crossing period etc.
- explain the physical meaning of a stationary narrow- band Gaussian wave process and to apply the Rayleigh distribution to calculate short term statistics of waves including the statistics of the largest wave heights
- explain the meaning of long term statistics based on using the Rayleigh distribution together with joint frequency tables/scatter diagrams; also how scatter diagrams are used to determine e.g Hm0, Tp corresponding to a return period of 100 years.
- find transfer functions for simple systems and use these to describe stochastic response for fatigue analysis and extreme response estimation based on short term and long term statistics.

## Files/Documents

## ISCED Categories