TMR4220 Lecture

1
TMR4220 Lecture 2, 2004.04.11

• Content overview
• Review standard tests
• Stability forms
• Main dimensions and dynamic stability
• Axis system
• Nomenclature
• Linear equations of motion
• Characteristic equations for stability analysis

2
Learning objectives linear equations

• Show development of linear equations of motion
  for a surface vessel
• Apply linear equations to check if a vessel is
  dynamically stable
• Explain the meaning of stability arms
• Explain how the linear equations are used to
  investigate course and path stability
• Apply Rouths method to check course stability

3
Ship controllability

• Inherent controllability
• Open loop characteristic of a vessel
• Piloted controllability
• Human or autopilot controlled vessel, performing
  a manoeuvre such that deviations from a preset
  mission remain within acceptable limits

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Stability forms 1

• Dynamic stability
• Constant speed, external disturbance
• Locked rudder, no use of thrusters
• How will yaw speed change due to the disturbance?
• Also called Straight line stability

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Stability forms 2

• Course stability
• Constant speed, external disturbance
• Disturbance to be controlled by rudder
• How will yaw speed change due to initial
  disturbance and rudder control?
• Also called Directional stability

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Stability forms 3

• Path stability
• Constant speed, external disturbance
• Disturbance to be controlled by rudder
• How will course and lateral position change due
  to initial disturbance and rudder control?
• Also called Positional motion stability

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Buzz group no. 3

• What actions should be taken when
• Designing the aft body of a dynamically unstable
  vessel?
• Selecting control units for a dynamically
  unstable vessel
• Output of buzz work
• Adjust aft body lines to improve water flow to
  propeller and rudder
• Add area in aft body, skegs and fins (see PROBO
  example in course notes)
• Use high lift rudder
• Use asipod

8
Axis system and nomenclature

• International Towing Tank Conference (ITTC) and
  Society of Naval Architects and Marine Engineers
  (SNAME) definitions will be used
• Forces and moments
• X, Y, Z (surge, sway, heave)
• K, M, N (roll, pitch, yaw)
• Linear and angular speeds
• u, v, w
• p, q, r

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Linear equations for surge, sway and yaw

• Linear equations reduce to coupled sway and yaw
  motion
• Non-dimensional form
• Dynamic stability calculations
• Nomotos equation
• State space form

10
Linear equations

• Coupled sway and yaw equations
• LE can be used if the vessel is dynamically
  stable and the motion is small perturbations
  around a steady state condition defined as
  straight line motion at constant speed
• LE equations are used to check if a vessel is
  dynamically stable

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Stability arms

• The criteria for dynamic stability can be
  expressed by the arms of linear damping forces
  for sway and yaw respectively
• Rigid body terms depending on sway and yaw speed
  are included in the linear damping forces and
  moments
• If the yaw damping arm is greater then the sway
  damping arm, the vessel is dynamically stable

12
Pivot point

• The pivot point is defined as the point along the
  centre plane of the vessel where the local sway
  speed is zero
• v xr 0
• The pivot point is moving during a manoeuvre

13
Course stability

• Rudder control is included
• Rudder angle controlled based on deviation in yaw
  speed and course angle
• Remove sway speed/acceleration variables (as for
  dynamic stability)
• The characteristic equation will be of 3rd order
• A polynom solver can be used to calculate the
  roots of the 3rd order characteristic equation
• Rouths method can be used to check if the vessel
  is course stable

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Nomotos equation

• Remove sway speed and sway acceleration as
  variables in the linear equations of motion
• Express the resulting equation in terms of the
  time constant for the homogeneous part of the
  second order differential equation
• (AD2 BD C 0 )
• A non-linear version of the Nomoto equation
  usually includes a third order yaw speed term

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State space form

• Use sway and yaw speed as state variables (x)
• Use rudder angle as control vector variable (u)
• Introduce stiffness, damping and control matrices
• State space form
• xdot Ax Bu
• Euler integration used for time domain solution
  of the state space equations

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Summary lecture no. 2

• Linear equations of motions can be used for
  dynamically stable ships when small rudder angles
  are used
• Linear motion equations are used to determine if
  a ship has dynamic/course/track stability
• Pull-out manouvre can be used to check dynamic
  stability
• Most modern full bodied ship types are
  dynamically unstable
• High efficiency rudders may improve the
  manoeuvring characteristics of modern ships