Lecture Notes for OE3035 - Motion of Ships and Floating Systems

Author
Affiliation

Abhilash Somayajula

Marine Autonomous Vehicles (MAV) Laboratory, Department of Ocean Engineering, IIT Madras

Published

January 1, 2026

Course Overview

Welcome to OE3035: Motions of Ships and Floating Systems course. This set of course notes will act as the single document that the students need to refer to while taking this course.

Learning Outcomes

By the end of this course, students will be able to:

  • Explain the importance of undamped and damped natural frequency, damping ratio, homogeneous and particular response to sinusoidal forcing of a single degree of freedom system
  • Explain how Fourier transform can be used to reduce the seakeeping in irregular waves into multiple regular wave problems, and compute the response of a linear system to an arbitrary forcing using its impulse response function
  • Describe the position and orientation of a ship using Euler angles and rotation matrices, and derive its kinematic equations
  • Derive the rigid body equations of motion of a marine vehicle in the body coordinate system
  • Explain the origin of the hydrostatic restoring forces, the added mass and the radiation damping of a floating body, and assemble the equations of motion of a floating body in calm water
  • Explain the difference between incident, scattering and radiation forces and how to calculate them
  • Explain the effect of forward speed on the frequency at which a ship meets the waves
  • Interpret response amplitude operators (RAOs) for ships and offshore structures
  • Explain why the roll motion of a ship depends on viscous damping that potential flow theory cannot predict
  • Explain the statistical nature of the ocean and its characterization by sea states
  • Explain the importance of spectrum and autocorrelation function
  • Calculate the response of a ship to an irregular sea state, and estimate the short-term statistics of the response
  • Use the hydrodynamic coefficients of a ship from a panel method to estimate its motions in a sea state and the motion sickness incidence on board

Course Content

The chapters of these notes and the topics covered in each of them are listed below.

  • 1  Introduction
  • 2  Single Degree of Freedom Systems
    • Free vibration of spring-mass and spring-mass-damper systems
    • Complex representation of the response
    • Forced vibration, resonance and the response amplitude operator
    • Principle of superposition
    • Heave motion of a spar platform
  • 3  Fourier Series and Fourier Transform
    • Fourier series in the real and complex forms
    • Fourier transform and its properties
    • Dirac delta function
    • Frequency response and impulse response function of a single degree of freedom system
    • Causality and the convolution integral
  • 4  Rigid Body Kinematics
    • Global and body coordinate systems
    • Pose of a rigid body
    • Euler angles, axis-angle representation and rotation matrix
    • Order of rotations and gimbal lock
    • Kinematic equations and their linearization
  • 5  Rigid Body Dynamics
    • Translational and rotational equations of motion
    • Inertia tensor and parallel axes theorem
    • Rigid body mass matrix and Coriolis matrix
    • Linearized equations of motion
  • 6  Forces and Moments on a Floating Object in Calm Waters
    • Hydrostatic restoring forces and moments
    • Viscous and pressure forces, and the potential flow assumption
    • Setup of boundary value problem and its linearization
    • Radiation forces and moments: added mass and radiation damping
    • Cummins equations and the retardation function
  • 7  Forces and Moments on a Floating Object in Waves
    • Regular incident waves and the dispersion relation
    • Encounter frequency
    • Froude Krylov forces and moments
    • Scattering/diffraction forces and moments
  • 8  Response Amplitude Operator
    • Six degree of freedom coupled equations of motion in the time and frequency domains
    • Response amplitude operator, its magnitude and phase
    • Long and short wave limits, and the effect of forward speed
    • Viscous roll damping
    • Impulse response function from the RAO
    • Local and relative motions, green water, slamming and parametric rolling (exercises)
  • 9  Irregular Waves in the Ocean
    • Random processes and their statistical properties
    • Stationarity, ergodicity and Gaussian processes
    • Autocorrelation function and power spectral density
    • Distribution of upcrossings and peaks, and significant wave height
    • Wave spectra - Pierson Moskowitz and JONSWAP spectra
    • Directional spectra
    • Response of a single degree of freedom system to a random force
    • Response spectra and encounter spectrum
    • Short-term statistics of the response of a ship in an irregular sea
  • 10  Summary of the Course

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