PNA Controlabilidade_3 Motion stability and linear equations_2
PNA Controlabilidade. Sessão 3, Motion Stability and Linear Equations
3.1 Definitions of Motion Stability
The concept of fast keeping is related to course stability or stability of direction. A body possesses stability if it returns to its initial equilibrium state after a disturbing force ceases. In the case of fast keeping, the most obvious source of disturbance is a wave or gust of wind. Whether the ship returns to its initial state without rudder action depends on the type of motion stability it possesses.
Figure 3 illustrates types of motion stability based on the attributes of the initial state that are maintained after the disturbance ceases. The BNA presents four cases of stability, focusing on which attributes of the initial state are retained in the final state after the perturbation.
Straight Line Stability: The ship, initially moving in a straight line, is disturbed. After the disturbance, it resumes moving in a straight line but in a different direction and position.
Direction Stability (with Oscillation): The ship, initially moving, is disturbed. After the disturbance, it maintains movement in a straight line and returns to the same direction, but oscillates.
Direction Stability (without Oscillation): Similar to case 2, but without oscillation after the disturbance.
Positional Motion Stability: After the disturbance, the ship returns to three attributes of the original movement: straight line, direction, and position.
These four cases form an ascending hierarchy. If a ship has positional motion stability, it necessarily has directional and straight-line stability. Straight line stability is the usual goal for manually governed ships.
If the controls are fixed (controls fixt),the maximum stability a ship can have is straight line stability, if it has any at all. Other cases require degrees of automatic control.
3.2 Course Stability with Controls Fixed and Working
The four types of stability make sense with the use of control surfaces either working, fixed at zero (controls fixt), or free to rotate. The term "stability" alone usually means controls fixt, straight line stability. However, the term can also have meaning with controls working, depending on the context.
The author analyzes stability in the vertical and horizontal planes, noting they are quite different.
Vertical Plane: A ship has positional motion stability because forces and hydrostatic moments induce this type of stability, due to gravity being balanced by buoyancy.
Horizontal Plane (Controls Fixed): A ship cannot have positional or directional stability because changes in buoyancy that stabilize the ship in the vertical plane do not exist in the horizontal plane.
Horizontal plane (Controls Working): A ship should have both positional and directional stability, in addition to any it has with controls fixed.
Each type of stability has an associated numerical index whose sign indicates stability or instability and whose magnitude indicates the degree of stability or instability.
3.3 Assumption of Linearity
In controllability, the force X is a function of v, \dot{v}, \ddot{v}, \dot{\psi}, and \ddot{\psi}. To calculate this force, there are complex hydrodynamic equations that are non-linear functions.
Consider a graph plotting x on the abscissa and any function of x on the ordinate. When x varies by a magnitude of \Delta x, the function of x varies by another magnitude. Although this function is non-linear, it can be linearized using Taylor expansion to simplify calculations.
Taylor expansion takes a small variation of x and studies how much the function of x varies. In a very small interval of variation of x, the curve approximates a straight line. However, for a large variation of x, the curve deviates from the straight line.
This mathematical tool works for very small variations of dependent and independent variables. Therefore, the assumption that \Delta x is small is compatible with the study of motion stability. It is acceptable to use Taylor expansion to linearize the study of Motion Stability. Motion stability determines if a small perturbation of equilibrium will increase or decrease over time.
3.4 Notation of Force and Moment Derivatives
A derivative is the variation of one variable divided by the variation of another. For example, consider the consumption of gasoline during a trip. In the first part of the trip, one liter of gasoline is consumed for every 10 kilometers traveled. The derivative of distance traveled in relation to gasoline spent is 10 kilometers per liter. In the second part, while driving up a hill, one liter of gasoline is consumed for every 6 kilometers. The derivative changes, and so does the slope of the curve.
In academia, the derivative of Y in relation to x is denoted as \frac{\Delta Y}{\Delta x}. Naval architects use the notation Yx, so Yv = \frac{\Delta Y}{\Delta v}. This simplifies writing mathematical equations.
u is the component of the velocity vector on the x-axis.
v is the component of the velocity vector on the y-axis.
r is the angular velocity on the z-axis.
When a dot is placed over a letter, it represents the first derivative of that variable with respect to time (e.g., \dot{u} is acceleration in X). Y_{vv} denotes the quantification of the derivative.
For example, if the derivative is 10, and 5 liters of gasoline are consumed, the distance traveled is 5 \times 10 = 50 kilometers. In exams, this is written as kml, where kml is the derivative (10) multiplied by l (5 liters).
Analogously, y_vv denotes the component on the y-axis of the hydrodynamic force Y that develops in the center of gravity due to a transversal velocity. The fact that the force needed to accelerate a body in fluid is always greater than the mass multiplied by the acceleration (Newton's law) leads to the concept of trainer mass or added mass.
However, this rational force should be interpreted as the hydrodynamic force that arises due to acceleration of the body in the fluid. The force needed to push a car is its mass times the acceleration. Pushing a boat transversally requires more force due to the water that must be moved along with it.
By including it in one of the complex equations of y as an addition to the mass term, the term y_{\dot{v}} is called the virtual mass coefficient. The three terms of this equation have the dimension of force. The term inside the parentheses has the dimension of mass, and the term outside (\dot{v}) is acceleration. Therefore, mass times acceleration gives force.
The derivative y{\dot{v}} is always negative and is an addition to the ship's mass. The term y{\dot{v}} acts in opposition to \dot{v}, i.e., the force is a reaction to the transversal acceleration. If a gust of wind makes the ship accelerate to port (negative), a hydrodynamic force arises to starboard (positive). Negative with positive gives negative. This derivative is always negative.
Analogously, the term N{\dot{r}}, which appears as an addition to Iz, is called the Virtual Moment of Inertia Coefficient. If a gust of wind causes a rotational acceleration to port, a hydrodynamic moment appears to starboard. The hydrodynamic moment is the difference between the force at the bow and the force at the stern. In this case, the force at the bow is greater than at the stern.
In the equation, Iz is the mass moment of inertia. N{\dot{r}} appears as an addition to the mass moment of inertia.
For example, consider a refrigerator door full of beverages. Applying force at the extreme end, as far from the hinge as possible, results in a large mass and a large distance, therefore a large moment of inertia. Removing all beverages and applying force in the middle means a smaller distance from the point of application to the hinge. Since mass and distance are small, the moment of inertia is small.
The same applies to a ship, considering the distance from the point of force application to the center of gravity and whether the ship is loaded or light. The derivatives y{\dot{r}} and N{\dot{v}} are called coupled virtual inertia coefficients.
Looking at y{\dot{r}}, if a gust of wind causes a rotational acceleration to port, and if the bow dominates, a force in y appears to starboard, and the derivative is negative. If the stern dominates, a force in y appears to port, and the derivative is positive. Similarly, examining N{\dot{v}}, if a gust of wind causes a transversal acceleration to port and the bow dominates, a moment to starboard appears, and the derivative is negative. If the stern dominates, a moment to port appears, and the derivative is positive.
These derivatives are called coupled virtual inertia coefficients because they couple the Y and Z axes.Y{\dot{r}} and N{\dot{v}} would be zero if the hulls and appendages were symmetrical on the YZ plane, i.e., if the bow were equal to the stern.
3.5 Control Forces and Moments
Until now, we have been discussing gusts of wind or any perturbation creating transverse velocity or acceleration, or rotational velocity or acceleration, and what hydrodynamic forces arise from these perturbations. In section 3.5, we study the hydrodynamic forces and moments that arise from a deflection of the rudder (\delta_r).
In Figure 5, \deltar is shown with the rudder deflected to port. Rudder to port is positive because it generates a hydrodynamic force that moves the stern to starboard. Y{\delta} \delta_r is the linearized component in Y of the force acting at the center of gravity. This force is created by the rudder deflection.
Analogously, N{\delta} \deltar is the linearized component of the moment in Z created by the rudder deflection. The ship follows a path but the lateral force of the deflected rudder creates a moment to turn the ship. This yaw causes the ship to assume an angle of attack in the water. In Figure 5, beta is the angle between the velocity vector and the ship's heading.
A well-designed ship acts as a hydrodynamic profile, with an angle of attack in the fluid, generating lateral forces that create a moment N{vv} which greatly increases the moment of the rudder. Thus, N{\delta} \deltar is the moment created by the rudder deflection, and N{vv} is the moment created by the hydrodynamic forces acting on the hull due to the angle of attack beta. This concludes section 3.