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Added Mass - MIT OpenCourseWare

Hydrodynamics Reading #6. Hydrodynamics Prof. Techet Added mass For the case of unsteady motion of bodies underwater or unsteady flow around objects, we must consider the additional effect (force) resulting from the fluid acting on the structure when formulating the system equation of motion. This Added effect is Added mass . Most floating structures can be modeled, for small motions and linear behavior, by a system equation with the basic form similar to a typical mass -spring-dashpot system described by the following equation: mx + bx + kx = f (t ) ( ). where m is the system mass , b is the linear damping coefficient, k is the spring coefficient, f(t) is the force acting on the mass , and x is the displacement of the mass .

2.016 Hydrodynamics Reading #6 2.016 Hydrodynamics Prof. A.H. Techet Added Mass For the case of unsteady motion of bodies underwater or unsteady flow around objects, we

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Transcription of Added Mass - MIT OpenCourseWare

1 Hydrodynamics Reading #6. Hydrodynamics Prof. Techet Added mass For the case of unsteady motion of bodies underwater or unsteady flow around objects, we must consider the additional effect (force) resulting from the fluid acting on the structure when formulating the system equation of motion. This Added effect is Added mass . Most floating structures can be modeled, for small motions and linear behavior, by a system equation with the basic form similar to a typical mass -spring-dashpot system described by the following equation: mx + bx + kx = f (t ) ( ). where m is the system mass , b is the linear damping coefficient, k is the spring coefficient, f(t) is the force acting on the mass , and x is the displacement of the mass .

2 The natural frequency of the system is simply k = . ( ). m In a physical sense, this Added mass is the weight Added to a system due to the fact that an accelerating or decelerating body (ie. unsteady motion: dU dt 0 ) must move some volume of surrounding fluid with it as it moves. The Added mass force opposes the motion and can be factored into the system equation as follows: mx + bx + kx = f (t ) ma . x ( ). where ma is the Added mass . Reordering the terms the system equation becomes: ( m + ma ) x + bx + kx = f (t ) ( ). From here we can treat this again as a simple spring- mass -dashpot system with a new mass m = m + ma such that the natural frequency of the system is now k k = = ( ).

3 M m + ma It is important in ocean engineering to consider floating vessels or platforms motions in more than one direction. Added mass forces can arise in one direction due to motion in a different direction, and thus we can end up with a 6 x 6 matrix of Added mass coefficients. version updated 8/30/2005 -1- 2005 A. Techet Hydrodynamics Reading #6. Looking simply at a body in two-dimensions we can have linear motion in two directions and rotational motion in one direction. (Think of these coordinates as if you were looking down on a ship.). Two dimensional motion with axis (x,y) fixed on the body. 1: Surge, 2: Sway, 6: Yaw The unsteady forces on the body in the three directions are: du1 du du F1 = m11 + m12 2 + m16 6 ( ).

4 Dt dt dt du1 du du F2 = m21 + m22 2 + m26 6 ( ). dt dt dt du1 du du F6 = m61 + m62 2 + m66 6 ( ). dt dt dt Where F1, F2, and F6, are the surge (x-) force, sway (y-) force and yaw moments respectively. It is common practice in Ocean Engineering and Naval Architecture to write the moments for roll, pitch, and yaw as F4, F5, and F6 and the angular motions in these directions as X4, X5, and X6. This set of equations, ( )-( ), can be written in matrix form, F = [ M ]u , du1 .. m11 m12 m16 dt . F = m21 m26 2 . du m22 ( ). dt . m61 m62 m66 . du6 .. dt . version updated 8/30/2005 -2- 2005 A. Techet Hydrodynamics Reading #6. Considering all six degrees of freedom the Force Matrix is: m11 m12 m13 m14 m15 m16 u 1.

5 M . 21 m22 m23 m24 m25 m26 u 2 . m m32 m33 m34 m35 m36 u 3 . F = 31 ( ). m41 m42 m43 m44 m45 m46 u 4 . m51 m52 m53 m54 m55 m56 u 5 .. m61 m62 m63 m64 m65 m66 u 6 . We will often abbreviate how we write the Force matrix given in ( ) using tensor notation. The force vector is written as F = Fi , where i = 1N. , 2, 3 , 4N. , 5, 6 , ( ). Linear Moments Forces the acceleration vector as u i = [u1 , u2 , u3 , u4 , u5 , u6 ] , ( ). and the Added mass matrix [ma] as mij where i, j = 1, 2, 3, 4, 5, 6 . ( ). A good way to think of the Added mass components, mij , is to think of each term as mass associated with a force on the body in the i th direction due to a unit acceleration in the j th direction.

6 For symmetric geometries the Added mass tensor simplifies significantly. For example, figure 2 shows Added mass values for a circle, ellipse, and square. In the case of the circle and square, movement in the 1 and 2 directions yields similar geometry and identical Added mass coefficients ( m11 = m22 ). version updated 8/30/2005 -3- 2005 A. Techet Hydrodynamics Reading #6. Circle Ellipse Square m11 = m22 = d 2 m11 = b 2 m11 = m22 = a 2. m66 = 0 m22 = a 2 m66 = a 4. 2. m66 = a 2 b 2 . Two dimensional Added mass coefficients for a circle, ellipse, and square in 1: Surge, 2: Sway, 6: Yaw Using these coefficients and those tabulated in Newman's Marine Hydrodynamics on we can determine the Added mass forces quite simply.

7 In three-dimensions, for a sphere (by symmetry): 1. m11 = m22 = m33 = = mA ( ). 2. ALL OTHER mij TERMS ARE ZERO ( i j ). version updated 8/30/2005 -4- 2005 A. Techet Hydrodynamics Reading #6. General 6 DOF forces and moments on a Rigid body moving in a fluid: Velocities: G. Translation Velocity : U (t ) = (U1 , U 2 ,U 3 ) ( ). G. Rotational Velocity : (t ) = ( 1 , 2 , 3 ) (U 4 ,U 5 ,U 6 ) ( ). All rotation is taken with respect to Origin of the coordinate system (often placed at the center of gravity of the object for simplicity!). Forces: (force in the jth direction). ( i = 1, 2, 3, 4, 5, 6 and j, k , l = 1, 2, 3). Fj = U i mij jkl U i k mli ( ).

8 Moments: ( i = 1, 2, 3, 4, 5, 6 and j, k , l = 1, 2, 3). M j = U i m j +3, i jkl U i k ml +3, i jkl U kU i mli ( ). Einstein's summation notation applies! The alternating tensor jkl is simply 0; if any j , k , l are equal . jkl = 1; if j , k , l are in cyclic order ( ). 1; if j , k , l are in anti-cyclic order . The full form of the force in the x-direction (F1) is summed over all values of i: F = U 1 m11 U 2 m21 U 3 m31 U 4 m41 U 5 m51 U 6 m61. N1 N . j=1 i=1 i=2 i=3 i=4 i=5 i=6. 1kl U1 k ml1 1kl U 2 k ml 2 1kl U 3 k ml 3 1kl U 4 k ml 4 ( ).. i=1 i=2 i=3 i=4. 1kl U 5 k ml 5 1kl U 6 k ml 6.. i=5 i=6. for k , l = 1, 2, 3 . version updated 8/30/2005 -5- 2005 A.

9 Techet Hydrodynamics Reading #6. Next we can choose the index k to cycle through. It is helpful to note that the only terms where k plays a role, contain jkl . Following the definition for jkl given in ( ) and since j = 1, all terms will be zero for k = 1. Therefore k can only take the value of 2 or 3: F = U 1 m11 U 2 m21 U 3 m31 U 4 m41 U 5 m51 U 6 m61. N1 N . j =1 i =1 i =2 i =3 i =4 i =5 i =6. 12l U1 2 ml1 12l U 2 2 ml 2 12l U 3 2 ml 3 12l U 4 2 ml 4 12l U 5 2 ml 5 12l U 6 2 ml 6.. i =1 i =2 i =3 i=4 i =5 i =6. k =2. 13l U1 3 ml1 13l U 2 3 ml 2 13l U 3 3 ml 3 13l U 4 3 ml 4 13l U 5 3 ml5 13l U 6 3ml 6.. i =1 i =2 i =3 i=4 i =5 i =6.

10 K =3. ( ). Finally we cycle through the index l. Again it is helpful to note that the only terms where l plays a role, contain jkl . Following the definition for jkl given in ( ) and since j = 1, and k = 2 or 3, then all terms will be zero for l = 1 and some zero for the case l = 2 and others zero when l =3. Like before l can only take the value of 2 or 3 such that l k j : F = U 1 m11 U 2 m21 U 3 m31 U 4 m41 U 5 m51 U 6 m61. N1 N . j =1 i =1 i =2 i =3 i=4 i =5 i =6. 123 U1 2 m31 123 U 2 2 m32 123 U 3 2 m33 123 U 4 2 m34 123 U 5 2 m35 123 U 6 2 m36.. i =1 i=2 i =3 i=4 i =5 i =6. k = 2; l =3. 132 U1 3 m21 132 U 2 3 m22 132 U 3 3 m23 132 U 4 3 m24 132 U 5 3 m25 132 U 6 3m26.


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