5-Seakeeping - Ship Motions in Regular Waves (2012)

Seakeeping – Motions in Regular Waves SEAKEEPING – SHIP MOTIONS IN REGULAR WAVES MODULE Note: These notes are drawn from those issued by Dr Jonathan Duffy to students of JEE329 Seakeeping & Manoeuvring at the Australian Maritime College Dr Duffy has used edited extracts from the main reference books for the subject and which are listed at the end of the module General A ship in waves is almost always in oscillatory motion However, only heaving, pitching and rolling are purely oscillatory motions, as these motions are subject to a restoring force or moment when the ship is disturbed from its equilibrium position Therefore, heave, pitch and roll motions are particularly important as they possess natural response periods and the potential for resonance For surge, sway and yaw motions the ship does not return to its original equilibrium position after being disturbed unless the exciting forces or moments act alternately from opposite directions Figure The x, y and z axes of a ship (Bhattacharyya 1978) In Figure the ship motions are defined as follows:  Translation along the x axis is surging  Translation along the y axis is swaying  Translation along the z axis is heaving  Rotation around the x axis is rolling  Rotation around the y axis is pitching  Rotation around the z axis is yawing Seakeeping – Motions in Regular Waves Although a ship experiences all types of motion simultaneously, each motion will be dealt with separately here It is important to note that any one kind of motion is not independent of the others; however, to simplify the problem the coupling between the motions will be neglected in this course Spring Mass System The heave, pitch and roll motions of a ship can be compared to the oscillatory motion of a mass on a spring, as these motions are subject to a restoring force or moment when the ship is disturbed from its equilibrium position If a mass attached to the end of a spring is disturbed from its equilibrium condition and the motion is considered to be undamped, the oscillation around the equilibrium position is a sine wave and is an example of simple harmonic motion The force exerted on the stretched spring is: 𝐹 = −𝑐𝑥 (1) Where c is the spring constant and x is the distance moved by the mass Applying Newton’s second law of motion (F= 𝑚𝑎) we obtain: 𝑑2𝑥 −𝑐𝑥 = 𝑚𝑎 = 𝑚 𝑑𝑡 (2) 𝑑2𝑥 ∴ 𝑚 𝑑 𝑡 + 𝑐𝑥 = (3) Equation is the equation for simple harmonic motion The term cx represents the spring restoring force 2.1 Damped Spring Mass System In addition to the two terms in the equation for simple harmonic motion there are two other factors that apply to a ship in waves, i.e damping and a forcing function Damping of a ship’s motion can be caused by many factors, including friction and wave energy dissipation The forcing function is due to the waves Therefore, the oscillatory behaviour of a ship in waves is fundamentally similar to the response of the classical damped spring mass system acted on by an oscillating force Therefore an understanding of the characteristics of such a system is a good basis for the study of ship motions Note that for a ship the motion will generally be in degrees of freedom, however, to explain the concept it is easiest to consider one degree of freedom only Consider the damped spring mass system in Figure 1.In this case the mass is denoted by m, b denotes the damping coefficient, c denotes the spring stiffness and F is an oscillatory force acting on the structure The forces acting in the damped spring mass system are: 𝑑2𝑥  The inertial force, 𝑚 𝑑 𝑡  The damping force, 𝑏 𝑑𝑡  The spring restoring force, 𝑐𝑥 𝑑𝑥 Seakeeping – Motions in Regular Waves  The time dependent oscillatory force on the structure, F The forces must be in equilibrium, thus the differential equation of motion of the system is: 𝐹 − 𝑚𝑥 − 𝑏𝑥 − 𝑐𝑥 = (4) Or 𝑚𝑥 + 𝑏𝑥 + 𝑐𝑥 = 𝐹 (5) Note that a dot above symbols denotes differentiation with respect to time In the following sections this fundamental differential equation will be applied to the oscillatory motions experienced by a ship and solutions will be provided to give the motion of the ship Figure Damped spring mass system with oscillating exciting force Heave Let us assume that a ship is forced down deeper into the water from its equilibrium position and suddenly released An oscillatory motion will occur, known as heaving This oscillatory motion occurs as when the ship is forced down deeper into the water the buoyancy force is greater than the weight of the ship, causing a restoring force The ship accelerates upward towards the equilibrium position due to the restoring force As the ship approaches the equilibrium position, the restoring force decreases and the acceleration toward the equilibrium position diminishes When the ship reaches its equilibrium position the restoring force and acceleration vanish, but by this time the ship has reached its maximum velocity and it will continue to rise due to momentum As the ship rises above its equilibrium position the ship weight is greater than the buoyancy force and hence the upward velocity will be reduced Eventually the ship will reach its extreme position where the velocity of the ship is zero At this point the weight of the ship exceeds the magnitude of the buoyancy force causing a restoring force moving the ship downward until it reaches an extreme position below the equilibrium position If there is no damping force the oscillatory motion will continue indefinitely This is free, undamped oscillation and is known as free oscillation This movement of the ship is a simple harmonic motion If damping is introduced we have a free, damped oscillation (sometimes termed damped oscillation) Seakeeping – Motions in Regular Waves If we now assume that the ship is being oscillated vertically up and down by a fluctuating force that is periodic in nature the motion will be irregular and is known as transient oscillation Due to the damping the irregularities disappear and a steady-state oscillation occurs This is a forced, damped oscillation (sometimes termed a forced oscillation), in which the amplitude and frequency of the motion are dependent on the amplitude and frequency of the exciting force The damping will also affect the amplitude of the forced oscillation 3.1 General Form of the Heave Equation Applying the equation for a damped spring mass system to a ship heaving in a seaway we obtain: 𝑎𝑧 + 𝑏𝑧 + 𝑐𝑧 = 𝐹0 𝑐𝑜𝑠𝜔𝑒 𝑡 (6) Where  o o 𝑎𝑧 is the inertial force, which is present when the ship is in oscillatory motion 𝑎 is the virtual mass of the ship, which is the ship mass plus the added mass 𝑧 = 𝑑 𝑧 𝑑𝑡 is the vertical acceleration  o o 𝑏𝑧 is the damping force, which resists the motion 𝑏 is the heave damping coefficient 𝑧 = 𝑑𝑧 𝑑𝑡 is the vertical velocity  𝑐𝑧 is the restoring force (additional buoyancy force due to an increase in draught), which tends to bring the ship back to its equilibrium position 𝑐 is the restoring or spring constant 𝑧 is the displacement of the centre of gravity of the ship o o  o o o 𝐹0 𝑐𝑜𝑠𝜔𝑒 𝑡 is the exciting (or encountering) force due to the waves, which acts on the mass of the ship 𝐹0 is the amplitude of the exciting (or encountering) force 𝜔𝑒 is the circular frequency of the encountering force 𝑡 is time 3.2 Heave in Calm Water Now we will look at two simplified cases before moving onto the case with all forces acting, as follows:   Case – free, undamped heaving motion (𝐹0 = 0, 𝑏 = 0) Case – free, damped heaving motion (𝐹0 = 0) Case Free, undamped heaving motion (𝑭𝟎 = 𝟎, 𝒃 = 𝟎) Let us assume that a ship is forced down deeper into the water from its equilibrium position and suddenly released If damping is assumed to be negligible the resulting motion is a free, undamped oscillation (also termed free oscillation) In the case of free oscillation the distance above the equilibrium position is the same as the distance from the equilibrium position to the maximum downward distance travelled by the ship Seakeeping – Motions in Regular Waves The magnitude of motion on either side of the equilibrium position is termed the amplitude of the heaving motion The heaving period is defined as the time required for one complete cycle Since the free heaving motion is a simple harmonic motion, the period of oscillation is independent of the amplitude and is thus termed the natural period Since both 𝐹0 and 𝑏 =0 for this case the equation for the condition of equilibrium is: 𝑎𝑧 + 𝑐𝑧 = (7) Where 𝑎 = virtual mass of the ship 𝑐 = heave restoring force coefficient The solution to this differential equation is: 𝑧 = 𝐴𝑠𝑖𝑛𝜔𝑧 𝑡 + 𝐵𝑐𝑜𝑠𝜔𝑧 𝑡 (8) or 𝑧 = 𝐴𝑠𝑖𝑛(𝜔𝑧 𝑡 − 𝛽) (9) Where 𝐴 is a constant that can be found from initial conditions 𝐵 is a constant that can be found from initial conditions 𝛽 is the phase angle 𝜔𝑧 is the natural frequency of the heaving motion 𝜔𝑧 = 2𝜋 𝑇 = 𝑐 𝑎 𝑧 𝑇𝑧 is the heaving period, which may be considered to be constant for small and moderate motions and not dependent upon the amplitude of the motion Therefore the natural frequency of the heaving motion is: 𝜔𝑧 = 𝜌𝑔 𝐴 𝑤 𝑝 𝑚+𝑎 𝑧 (10) Case Free, damped heaving motion (𝑭𝟎 = 𝟎) This is sometimes termed damped oscillation Since 𝐹0 is zero for this case the equation for the condition of equilibrium is: 𝑎𝑧 + 𝑏𝑧 + 𝑐𝑧 = (11) Provided that 𝑏 < 2𝑎𝑐, the solution to the equation is: 𝑧 = 𝑒 −𝜈𝑡 (𝐶1 𝑐𝑜𝑠𝜔𝑑 𝑡 + 𝐶2 𝑠𝑖𝑛𝜔𝑑 𝑡) (12) or 𝑧 = 𝑒 −𝜈𝑡 𝐴𝑠𝑖𝑛 𝜔𝑑 𝑡 − 𝛿 (13) Where Seakeeping – Motions in Regular Waves 𝜈 is the decaying constant (𝜈 = 𝑏 2𝑎) 𝜔𝑑 is the circular frequency of the damped oscillation (𝜔𝑑 = 𝜔𝑧2 − 𝜈 ) 𝜔𝑧 is the natural circular frequency of the free, undamped oscillation 𝐶1 , 𝐶2 , 𝐴 𝑎𝑛𝑑 𝛿 are constants that can be determined from initial conditions The damped heaving period is: 𝑇𝑑 = 2𝜋 𝜔𝑑 (14) The damped heaving period is greater than the natural heaving period since 𝜔𝑑 < 𝜔𝑧 An example of free damped oscillation is given in Figure Figure Free damped oscillation (Bhattacharyya 1978) Note that if 𝜈 > 𝜔𝑧 (damping is very large) the motion is no longer oscillatory and is known as aperiodic Such cases are not of interest to us as the magnitude of 𝜈 is always very small compared to the magnitude of 𝜔𝑧 for a ship in waves Damping acts in the opposite direction to the motion, which slows the motion The amplitude of the heaving motion therefore gradually decreases until the ship finally comes to rest at the equilibrium position The period of oscillation for the case with damping will be slightly larger than for a case without damping 3.3 Forced Damped Heave (in Regular Waves) This case is analogous to a ship heaving in waves as it includes an oscillatory exciting force due to waves Since all forces act for this case the equation for the condition of equilibrium is: 𝑎𝑧 + 𝑏𝑧 + 𝑐𝑧 = 𝐹0 𝑐𝑜𝑠𝜔𝑒 𝑡 (15) The solution to this equation is: 𝑧 = 𝑒 −𝜈𝑡 𝐶1 𝑐𝑜𝑠𝜔𝑑 𝑡 + 𝐶2 𝑠𝑖𝑛𝜔𝑑 𝑡 + 𝑧𝑎 cos⁡ (𝜔𝑒 𝑡 − 𝜀2 ) (16) or 𝑧 = 𝐴𝑒 −𝜈𝑡 sin⁡ (𝜔𝑑 𝑡 − 𝛽) + 𝑧𝑎 cos⁡ (𝜔𝑒 𝑡 − 𝜀2 ) (17) Seakeeping – Motions in Regular Waves Where 𝑧𝑎 is the amplitude of the forced motion 𝜀2 is the phase angle of the forced motion in relation to the exciting force Equation 17 describes the sum of two oscillations It can be seen that the first term is the same as the solution for free, damped heaving oscillation The second term describes a sinusoidal oscillation with constant amplitude with the same circular frequency as that of the exciting force for heaving and a phase angle of 𝜀2 to the exciting force At time t=0 the motion is said to be transient as both oscillations exist As t increases the first term diminishes (at a rate depending on the value of 𝜈) and the motion reaches a steady state oscillation governed by the second term Therefore the solution for the steady-state condition (when the first term dies out with time) is: 𝑧 = 𝑧𝑎 cos(𝜔𝑒 𝑡 − 𝜀2 ) (18) Figure shows an example of the forced response, natural response and the total response Figure Forced response, natural response and total response An example of just the total response is shown in Figure The damped forced vibration corresponds to the transient oscillation, whilst the steady forced vibration corresponds to the steady state forced response Figure Example response for forced damped heave (steady amplitude = 𝑧𝑎 ) Seakeeping – Motions in Regular Waves Let us consider the steady forced oscillation If it is assumed that a wave of a certain length and amplitude passes along rather slowly, so that the ship is in position to balance itself statically on the wave at every instant of its passage, the ship will then rise and fall slowly with the encountering frequency so as to keep balance between weight and buoyancy, and a static amplitude, zst, will result If the wave is now considered to move at its correct velocity, a dynamic amplitude, za, will be produced The ratio of the amplitude in the dynamic case to that in the static case is called the magnification factor and is denoted by 𝜇𝑧 : 𝜇𝑧 = 𝑧𝑎 𝑧𝑠𝑡 (19) 𝐹 The static heaving amplitude 𝑧𝑠𝑡 can be given by 𝑐 The quantity 𝑧𝑠𝑡 represents the heaving amplitude of the ship due to the maximum heave force under static conditions The amplitude of the steady forced heave response (𝑧𝑎 ) can then be predicted from: 𝑧𝑎 = 𝑧𝑠𝑡 𝜇𝑧 (20) The magnification factor may be expressed in terms of the tuning factor Λ: 𝜇𝑧 = (21) (1−Λ )2 +4k Λ Where Λ= 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑒𝑛𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝜔𝑒 = 𝑁𝑎𝑡𝑢𝑟𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝜔𝑧 𝜐 k is a non-dimensional damping factor (k = 𝜔 ) 𝑧 𝜈 is the decaying constant (𝜈 = 𝑏 2𝑎) 𝜔𝑧 is the natural circular frequency of the free, undamped oscillation (𝜔𝑧 = 𝑐 𝑎) 2kΛ 𝜀2 is the phase angle of the forced motion in relation to the exciting force 𝜀2 = 𝑡𝑎𝑛−1 1−Λ It is important to know the phase angle when studying ship motions as this affects such things as slamming and deck wetness It can be seen that the phase angle depends on damping For an undamped system (k = 0) the phase angle is for frequencies less than the natural frequency and 180 degrees for frequencies greater than the natural frequency The magnification factor can be plotted as a function of tuning factor, which yields the figure for dynamic response, as illustrated in Figure From Figure 5, for low damping, if the encountering frequency of the exciting force is close to the natural frequency of heaving for the ship the amplitude of the steady-state oscillation may be much larger than the static amplitude and the magnification factor, 𝜇𝑧 , is much greater than one The maximum response is obtained by differentiating Equation 21 with respect to Λ and equating to zero From this it can be shown that the magnitude of the magnification factor is maximum when Λ = − 2k Seakeeping – Motions in Regular Waves Hence, for very small values of k (low damping) the maximum magnification factor occurs when the tuning factor is very close to Therefore, when designing a ship for a seaway when damping is low Λ = should be avoided to minimize motions It can be seen that for cases with larger damping the maximum value of magnification factor shifts towards the left Figure Magnification factor as a function of tuning factor (Bhattacharyya 1978) From Figure it can be seen that if the frequency of the exciting force is very low then the amplitude of oscillation in the steady state condition will be equal to the static amplitude, i.e the ship will rise and fall slowly with the encountering frequency with a balance maintained between weight and buoyancy, hence 𝑧𝑎 = 𝑧𝑠𝑡 and 𝜇𝑧 = It can be seen that the natural frequency of a ship is an important parameter when designing for a given seaway, as this influences the tuning factor and this is important in finding the region of resonance In order to predict the amplitude of the steady forced heave motion the terms a, b, c and 𝐹0 need to be predicted The terms are discussed in the following sections along with methods to predict them 3.4 Inertial Heave Force The following description of the prediction of heave inertial force is taken from Bhattacharyya (1978) A body having an accelerated motion in a continuous medium of fluid experiences a force that is greater than the mass of the body times the acceleration Since this increment of force can be Seakeeping – Motions in Regular Waves defined as the product of the body acceleration and a quantity having the same dimension as the mass, it is termed added mass This concept is needed to discuss the inertial force of a ship The inertial force is the body accelerating force (ship mass x acceleration) plus a liquid accelerating force: 𝑑2𝑧 𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 = 𝑎𝑧 = 𝑎 𝑑 𝑡 𝑑2𝑧 (22) 𝑑2𝑧 𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 = 𝑚 𝑑𝑡 + 𝑎𝑧 𝑑𝑡 (23) 𝑑2𝑧 𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 = (𝑚 + 𝑎𝑧 ) 𝑑𝑡 (24) Where 𝑚 is the ship mass 𝑎𝑧 is the added mass in heave 𝑑2𝑧 𝑚 𝑑 𝑡 is the body accelerating force 𝑑2𝑧 𝑎𝑧 𝑑𝑡 is the liquid accelerating force One should remember that the concept of added mass is introduced into fluid mechanics for convenience of evaluation and does not have any physical significance For example, one should not imagine that a body accelerating in an ideal fluid in a certain direction drags with it a certain amount of fluid mass A method to predict the added mass of a ship section follows According to Lewis (1929), an inertial coefficient C is defined as: 𝐶= 𝐴𝑑𝑑𝑒𝑑 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡𝑕, 𝑏𝑒𝑎𝑚 𝐵𝑛 , 𝑎𝑛𝑑 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑑𝑟𝑎𝑢𝑔𝑕𝑡 𝑇𝑛 𝑡𝑜 𝑏𝑜𝑡𝑡𝑜𝑚 𝑜𝑓 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝐻𝑎𝑙𝑓 𝑜𝑓 𝑡𝑕𝑒 𝑎𝑑𝑑𝑒𝑑 𝑚𝑎𝑠𝑠 𝑓𝑜𝑟 𝑎 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡𝑕 𝑎𝑛𝑑 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝐵𝑛 Now, half of the added mass for a circular section segment of unit length and diameter 𝐵𝑛 is: 𝜌𝜋𝑟 = 𝜌𝜋 𝐵𝑛 (Lewis 1929) (25) Where breadth 𝐵𝑛 = 2𝑟 For shapes other than semi-circular ones, the added mass of a ship section, an, is found to be: 𝑎𝑛 = 𝐶 𝜌𝜋 𝐵𝑛 (Lewis 1929) (26) The coefficient C for Lewis-form sections for two dimensional floating bodies is obtained from Figure as a function of:    the draught/beam ratio 𝐵𝑛 T the sectional area coefficient 𝛽𝑛 = 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 the circular frequency of oscillation 𝐵𝑛 ∗ 𝑇 10 Seakeeping – Motions in Regular Waves or 𝐼𝑥𝑥 = 𝑚𝑘𝑥𝑥 (89) Where m is the total mass of the ship 5.2 General Form of the Equation for Roll The equation of motion for roll has the same form as that for pitch There are four moments which act in roll in a seaway: Inertial moment Damping moment Restoring moment Exciting moment The equation of motion of roll in a seaway can be written as: 𝑎∅ + 𝑏∅ + 𝑐∅ = 𝑀0 𝑐𝑜𝑠𝜔𝑒 𝑡 (90) Where  𝑎∅ is the inertial moment, which is present when the ship is in oscillatory motion o 𝑎 is the virtual mass moment of inertia of the ship in roll, which is the ship mass moment of inertia plus the added mass moment of inertia ′ 𝑎 = 𝐼𝑥𝑥 = 𝐼𝑥𝑥 + 𝛿𝐼𝑥𝑥 o ∅ = 𝑑 ∅ 𝑑𝑡 is the angular roll acceleration  o o 𝑏∅ is the roll damping moment, which resists the motion 𝑏 is the damping moment coefficient ∅ = 𝑑∅ 𝑑𝑡 is the angular velocity  𝑐∅ is the roll restoring moment, which tends to bring the ship back to its equilibrium position o 𝑐 is the restoring moment coefficient o ∅ is the angular displacement in roll  𝑀0 𝑐𝑜𝑠𝜔𝑒 𝑡is the exciting (or encountering)moment due to the waves, which acts on the mass of the ship o 𝑀0 is the amplitude of the exciting (or encountering) moment o 𝜔𝑒 is the circular frequency of the encountering moment o 𝑡 is time 5.3 Roll in Calm Water As with heave and pitch we will look at the case of roll in calm water to determine the natural frequency for roll 25 ... damping will be slightly larger than for a case without damping 3.3 Forced Damped Heave (in Regular Waves) This case is analogous to a ship heaving in waves as it includes an oscillatory exciting... spring restoring force 2.1 Damped Spring Mass System In addition to the two terms in the equation for simple harmonic motion there are two other factors that apply to a ship in waves, i.e damping... 10 Seakeeping – Motions in Regular Waves Figure Added mass coefficients for two-dimensional floating bodies in heaving motion (Bhattacharyya