2. MATHEMATICAL FORMULATION OF LARGE DISPLACEMENTS-INDUCED
2.3 Deriving the Equations of Motion of the 3-rotor structural system
The equations of dynamic motion are derived based on D’Alembert’s principle of dynamic equations. D’Alembert’s principle, equation (13), is a basic classical law of motion. It describes that the summation of all externally applied loads, reactions, and inertia forces should be zero in the direction of interest. In other words, equation (13) explains the equilibrium relationship while each corresponding DOF is released.
( i i i) 0
i
m
F a (13) where Fi is the equivalent external force that has been applied in the direction of interest, mi is the mass of the ith displaced point; ai is the corresponding acceleration of the ith point that has been27
derived based on the position vector theory, and miaiis the inertia force of the mass in the direction of interest.
It should be mentioned that in the current research all the position, velocity, and acceleration vectors of each rotor are derived based on the sign convention mentioned earlier in figure 5. Therefore, in deriving the EOMs, all the inertia forces and moments of each direction will be considered in the opposite direction of the defined Cartesian coordinate. This assumption would provide a consistent and clearer procedure in deriving the EOMs of a complex dynamic system, such as the current MRWT system. The corresponding inertia forces and moments to each DOF are shown in figures 15 to 19 respectively.
Accordingly, by considering real-time translational displacement in the x-direction as the first DOF of the triangular building block, figure 15, the first equilibrium equation of motion is defined in equation (14),
1 1 2 2 3 3 damping of the tower in the x-direction; kx defines the translational stiffness of the tower in the x-direction; Fxe(t) is real-time equivalent of horizontal forces (thrusts) applied on the rotors of the top structure, and ax1(t) to ax3(t) are the time-dependent accelerations of rotors 1-3 in the x-direction. Acceleration values of ax1(t), ax2(t), and ax3(t) are defined based on the component i of equations (6), (9), and (12), respectively. ( )u t and u t( ) also defines real-time velocity and acceleration of the origin point O according to the displacement u(t) in the x-direction.
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Considering the values of the parameters described above, equation (15) defines the first EOM of the triangular building block,
rotation of the top structure about the y-axis, β t( ) and β t( ) are respectively the corresponding rotational velocity and rotational acceleration. Similarly, γ(t) is the rotation of the top structure about the z-axis, γ t( ) and γ t( ) are respectively the corresponding rotational velocity and rotational acceleration.Although the first EOM derivation phases and calculations have been described comprehensively, to decrease the complexity of fundamental calculations and provide a generic formulation, the symmetric building block model with 3 rotors will be considered. Assuming the mass of each rotor as m, the tower mass as mO, and the distances of rotors 1 and 2 to the origin point O equal to l, equation (16) presents the first EOM of the symmetric triangular building block.
2( ) 3 O ( ) cos ( ) ( ) sin ( ) x ( ) x ( ) xe( ) 0
u t mm β t mR β t β t mR β t c u t k u t F t (16)
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Figure 15: Deformation of the top structure in the x-direction, equivalent resultant external force, due to the wind-induced thrust on the turbines, inertia, and reaction forces
Similarly, by considering the translational displacement in the y-direction as the second DOF of the triangular building block, figure 16, the second equilibrium equation of motion is defined in equation (17),
1 1 2 2 3 3 damping of the tower in the y-direction. ky defines the translational stiffness of the tower in the y-direction; Fye(t) is real-time equivalent of horizontal forces applied on the top structure, and ay1(t) to ay3(t) are the time-dependent accelerations of rotors 1-3 in the y-direction. Acceleration values
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of ay1(t), ay2(t), and ay3(t) are defined based on the component j of equations (6), (9), and (12), respectively. v t( ) and v t( ) also defines real-time velocity and acceleration of the origin point O according to the displacement v(t) in the y-direction.
Considering the values of the parameters described above, equation (18) defines the second EOM of the triangular building block,
where l1, l2 and R are the distances of rotors 1-3 to the origin point O respectively; α(t) is real-time rotation of the top structure about the y-axis, α t( ) and α t( ) are respectively the time-dependent corresponding rotational velocity and rotational acceleration. Similarly, γ(t) is the rotation of the top structure about the z-axis, γ t( )and γ t( ) are respectively the corresponding rotational velocity and rotational acceleration.
Although the second EOM derivation phases and calculations have been described comprehensively, to decrease the complexity of fundamental calculations and provide a generic formulation, the symmetric building block model with 3 rotors will be considered. Assuming the mass of each rotor as m, the tower mass as mO, and the distances of rotors1 and 2 to the origin point O equal to l, equation (19) presents the second EOM of the symmetric triangular building block.
2( ) 3 O ( ) cos ( ) ( ) sin ( ) y ( ) y ( ) ye( ) 0
v t m m α t mR α t α t mR α t c v t k v t F t (19)
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Figure 16: Deformation of the top structure in the y-direction, equivalent resultant external force, inertia, and reaction forces
Similarly, by considering figure 17, which shows the roll forces and the rotation about the x-axis as the third DOF of the triangular building block, the third equilibrium equation of motion is defined in equation (20),
defines the rotational damping of the tower due to the rotation about the x-axis. kθx defines the rotational stiffness of the tower in the x-direction. Mxe(t) is real-time equivalent resultant external moment due to the wind-induced torque applied on the top structure, and ay1(t) to ay3(t) are the time-dependent accelerations of rotors 1-3 in the y-direction. Acceleration values of ay1(t), ay2(t), and ay3(t) are defined based on the component j of the equations (6), (9), and (12) respectively.
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α(t) is the rotation of the origin point O about the x-axis. α t( ) and α t( ) also define real-time rotational velocity and acceleration, respectively. dy1 to dy3 are respectively, the relative distance of rotors 1-3 in the y-direction to the origin point O. Similarly, az1(t) to az3(t) are the time-dependent accelerations of rotors 1-3 in the z-direction. Acceleration values of az1(t), az2(t) and az3(t) are defined based on the component k of equations (6), (9), and (12), respectively. dz1 to dz3
are subsequently, the relative distances of rotors 1-3 in the z-direction to the origin point O. The distances mentioned are described by geometric relationships in figure 17.
Figure 17: Rotation of top structure about the x-axis, equivalent roll forces due to the wind-induced torque on the turbines, and inertia forces
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Furthermore, the moment of inertia, IO, for a structure consisted of lumped masses is defined by equation IO
m di i2 where mi is the mass of rotor i, and di is the distance of rotor i to the axis of rotation O. As mentioned earlier, according to one of the basic assumptions of the current research, the structural system point of rotation is considered to be at the origin point O, and the substructure dynamic characteristics are also applied at point O. Therefore, the distance between the point of rotation and the substructure mass is zero, which results in omitting the termO ( )
I α t that describes the inertia torque resultant of the substructure.
Considering the values of the described parameters above, equation (21) defines the general formulation of the third EOM.
results in equation (22).34
Finally, equation (23) is the simplified format of the third EOM of the triangular building block,
rotational velocity and rotational acceleration. Similarly, γ(t) is real-time rotation of the top structure about the z-axis, γ t( ) and γ t( ) are respectively the corresponding rotational velocity and rotational acceleration. Furthermore, α(t) is real-time rotation of the top structure about the x-axis, α t( ) and α t( ) are respectively the corresponding rotational velocity and rotational acceleration. v t( ) and v t( ) also respectively define real-time velocity and acceleration of the origin point O according to the displacement as v(t) in the y-direction.35
Although the third EOM derivation phases and calculations have been described comprehensively, to decrease the complexity of fundamental calculations and provide a generic formulation, the symmetric building block model with 3 rotors will be considered. Assuming the mass of each rotor as m, and the distances of rotors 1 and 2 to the origin point O equal to l, equation (24) presents the third EOM of the symmetric triangular building block.
Similarly, by considering figure 18 that shows the pitch force and the rotation about the y-axis as the fourth DOF of the triangular building block, the fourth equilibrium equation of motion is defined in equation (25),
stiffness of the tower in the y-direction. Mye(t) is real-time equivalent moment of pitch forces applied on the top structure, and ax3(t) is the time-dependent acceleration of rotor 3 in the x-direction that is defined based on the component i of equation (12). β(t) is the rotation of the origin point O about the y-axis. β t( ) and β t( ) also show real-time rotational velocity and acceleration,36
respectively. dx3 is the relative distance of rotor 3 in the x-direction to the origin point O, and it is equal to Rsin ( )β t . Similarly, az3(t) is the time-dependent acceleration of rotor 3 in the z-direction. Acceleration value of az3(t) is defined based on component k of equation (12), and dz3
is the relative distance of rotor 3 in the z-direction to the origin point O, and it is defined by cos ( )
R β t . The distances mentioned are described by geometric relationships in figure 16.
Furthermore, based on the assumptions and concepts explained previously, the moment of inertia of the substructure, IO, is zero, and term I β tO ( ) that described the resultant inertia moment of the substructure will be omitted from equation (25).
Figure 18: Rotation of top structure about the y-axis, equivalent pitch forces, inertia, and reaction forces
Considering the values of the parameters described above, equation (26) defines the general formulation of the fourth EOM.
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Finally, equation (27) is the simplified format of the fourth EOM of the triangular building block,
where R is the distance of rotor 3 to the origin point O. β(t) is real-time rotation of the top structure about the y-axis, β t( ) and β t( ) are respectively the corresponding rotational velocity and rotational acceleration. Furthermore, α(t) is real-time rotation of top structure about the x-axis,
( )
α t and ( )α t are respectively the corresponding rotational velocity and rotational acceleration.
( )
u t and u t( ) also respectively, define real-time velocity and acceleration of the origin point O according to the displacement u(t) in the x-direction.
Although the fourth EOM derivation phases and calculations have been described comprehensively, to decrease the complexity of fundamental calculations and provide a generic formulation, the symmetric building block model with 3 rotors will be considered. Assuming the mass of rotor 3 as the constant value m, and the distance of rotor 3 to the origin point O equal to R, equation (28) presents the fourth EOM of the symmetric triangular building block.
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Similarly, by considering the rotation about the z-axis as the fifth DOF of the triangular building block, figure 19, the fifth equilibrium equation of motion is defined in equation (29),
cθz is the rotational damping of the tower due to the rotation about the z-axis. kθz defines the rotational stiffness of the tower in the z-direction; Mze(t) is real-time equivalent moment of yaw forces applied on the top structure, and ay1(t) and ay2(t) are the time-dependent accelerations of rotors 1 and 2 in the y-direction. Acceleration values of ay1(t) and ay2(t) are defined based on the component j of equations (6) and (9) respectively. γ t( ) also defines real-time rotational acceleration of the origin point O according to the rotation of γ(t) about z-axis. dy1 and dy2 are respectively the relative distances of rotors 1 and 2 in the y-direction to the origin point O.
Similarly, ax1(t) and ax2(t) are the time-dependent accelerations of rotor 1 and rotor 2 in the x-direction. Acceleration values of ax1(t) and ax2(t) are defined based on the component i of equations (6) and (9) respectively.
In addition, dx1 and dx2 are respectively the relative distances of rotors 1 and 2 in the x-direction to the origin point O. The distances mentioned are described by geometric relationships in figure 17.
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Figure 19: Rotation of top structure about the z-axis, equivalent yaw forces, inertia, and reaction forces
Moreover, based on the assumptions and concepts explained previously, the moment of inertia of the substructure, IO, is zero, and term I γ tO ( ) that described the resultant inertia moment of the substructure will be omitted from equation (29).
Considering the values of the parameters described above, equation (30) defines the general formulation of the fifth EOM.
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Finally, equation (31) is the simplified format of the fifth EOM of the triangular building block,
where l1 and l2 are the distances of rotors 1 and 2 to the origin point O respectively. γ(t) is real-time rotation of the top structure about the z-axis, γ t( ) and γ t( ) are respectively the corresponding rotational velocity and rotational acceleration. Similarly, α(t) is real-time rotation of the top structure about the x-axis, α t( ) and α t( ) are the corresponding rotational velocity and rotational acceleration respectively. ( )v t and v t( ) also respectively define real-time velocity and acceleration of the origin point O according to the displacement as v(t) in the y-direction. In addition, u t( ) and ( )u t also define real-time velocity and acceleration of the origin point O according to the displacement as u(t) in the x-direction.
Although the fifth EOM derivation phases and calculations have been described comprehensively, to decrease the complexity of fundamental calculations and provide a generic formulation, the symmetric building block model with 3 rotors will be considered. Assuming the mass of each rotor as m, and the distances of rotors 1 and 2 to the origin point O equal to l, equation (32) presents the fifth EOM of the symmetric triangular building block.
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In summary, equations of motion (15) to (32) present the dynamic characteristics of the triangular building block structural system and its symmetric configuration. Consequently, the general EOMs and symmetric EOMs previously derived are summarized in tables 1 and 2.
Table 1: General formulation of EOMs of the 3-rotor triangular building block
EOM Formulation
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Table 2: EOMs of the symmetric configuration of 3-rotor triangular building block
EOM Formulation In-text
equation
44 Table 2 Continued.
EOM Formulation In-text
equation
Comparing table 1 and table 2 clearly demonstrates the differences between two types of derived EOMs. As it was clarified in sections 2.2 and 2.3, some rotational DOFs will cause additional translational displacements in the x, y and z directions as well. The induced displacements and corresponding velocities and accelerations would be partly omitted from the EOMs by considering geometric and material symmetry. For example, by applying the geometric symmetry to the position vectors of each pair of horizontal rotors, relative to the origin point O, the translational displacements and rotations of rotors 1 and 2 turn out to be the same in magnitude and opposite in direction. Therefore, for some of the DOFs, rotational or translational-coupled components in the basic derived EOMs will be canceled out.
As an example to clarify the comparison above, equations (15) and (16) are explained. Top structure rotation about the z-axis, induces translational displacements in the x-direction to the aligned rotors of 1 and 2. By applying the geometric symmetry in the formulation of equation (15), these described translational displacements turn out to be the same in magnitude and opposite in the direction. Therefore, the total resultant rotational component about the z-axis will be zero, and the simplified formulation is shown in equation (16).
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