A rotating structure is usually modeled under the assumption that the coordinate system used in the description of the structure will rotate at a constant rate about a fixed axis. Displacements of the structure and forces applied to the structure are measured in this rotating system. Because the rotating system is accelerating relative to a stationary inertial system, the inertial or mass dependent impedance cannot be directly calculated in the rotating system. The inertial impedance in the rotating system must first be determined in the stationary reference frame, and then transformed to the rotating system.
The impedance due to stiffness and damping is unaffected by the choice of reference frames.
The stiffness and damping impedances in a rotating reference system are identical to those in a stationary reference system.
To determine the inertial impedance in a rotating system requires the development of general transformation techniques between stationary and rotating systems. The following section describe the general transformation of a vector between stationary and rotating system
coordinates. The second section discusses inertial forces defined in a rotating coordinate system.
The third section applies these results to develop the impedance of a rotating structure. The fourth section describes the DMAP procedure to add the gyroscopic terms to the structural matrices for analysis in the rotating system.
Vector Transformation from Stationary to Rotating Coordinates
It is important to keep in mind that the essential features of a vector are magnitude and
direction, but not location. A vector defined in one coordinate system can just as easily be defined in a different coordinate system through the use of a coordinate transformation matrix. Also keep in mind that vectors may be time dependent in both their magnitude and direction and that coordinate systems may rotate with respect to each other resulting in a time-dependent transformation matrix.
The remainder of this discussion concerns transforming a vector defined in stationary system coordinates to rotating system coordinates. The choice of which system is rotating and which system is stationary is completely arbitrary at this point. No physical significance (such as position, velocity, etc.) is assigned to the vector; the only important qualities are its magnitude and direction.
The general transformation of a time-dependent vector from a stationary coordinate system to a rotating coordinate system at the same origin may be written as
Equation 3-42.
where:
{n(t)r} = rotating coordinate system definition of time-dependent vector.
[A(t)] = time-dependent transformation matrix from rotating to stationary system.
{n(t)s} = stationary coordinate system definition of time-dependent vector.
The above transformation is valid for any vector, real or complex.
The fundamentals of this transformation are demonstrated for the two systems shown below.
Figure 3-1. Stationary and Rotating Coordinate Systems
Assume the rotating system spins about the z-axis at a constant rate, Ω, relative to the stationary system. x(t)s, y(t)s, and z(t)sare the time-dependent components of the vector in the stationary system. x(t)r, y(t)r, and z(t)rare the time-dependent components of the vector in the rotating system.
For this example the transformation of the coordinates from the stationary to rotating system is
Equation 3-43.
Advanced Dynamic Analysis User’s Guide 3-11
where q = Ωt. EquationEq. 3-43is simply an expanded form ofEq. 3-42.
Using complex identities for cosq and sinq,
Equation 3-44.
Equation 3-45.
EquationEq. 3-43is written as
Equation 3-46.
or
Equation 3-47.
where:
Equation 3-48.
and the conjugate matrix is
Equation 3-49.
Equation 3-50.
Additionally, we can write the vectors in terms of complex components
Equation 3-51.
where:
are time-dependent magnitudes and Ψx(t), Ψy(t), and Ψz(t) are time-dependent phase functions.
Substituting the above definition intoEq. 3-47produces the complex transformation
Equation 3-52.
Advanced Dynamic Analysis User’s Guide 3-13
Inspection of the above equation shows that a vector defined in stationary system coordinates can be defined as three vectors in rotating system coordinates. One vector will have the phasing of its motion advanced by Ωt with respect to the stationary system definition, the second will have its phasing delayed by Ωt, and the third will have its phase unaffected.
Progressive and Regressive Vectors
Equation 3-53.
where
Equation 3-54.
Equation 3-55.
Equation 3-56.
It is convenient to specify the vector defined byEq. 3-54as the progressive vector in rotating system coordinates, the vector defined byEq. 3-55as the regressive vector in rotating system coordinates, and the vector defined byEq. 3-56 as the collective vector in rotating system coordinates. Care must be taken when discussing progressive and regressive vectors to establish whether the definition is in terms of the rotating or nonrotating system. By reformulatingEq.
3-43as a transformation from rotating to nonrotating coordinates, one can define progressive and regressive vectors in terms of the nonrotating system.
Equation 3-57.
In this discussion, progressive and regressive are defined relative to the rotating system.
Inertial Forces in a Rotating Coordinate System
The calculation of inertial forces in a rotating coordinate system requires the introduction of an inertial reference system. An inertial reference system is defined as a coordinate system in which the motion of a particle with respect to these coordinates adhere to Newton’s laws of motion. Implied in these laws is that acceleration of a particle is measured in a nonaccelerating and nonrotating coordinate system. In the above discussions the stationary coordinate system can be considered an inertial coordinate system.
Inertial force is defined as the resistance of a particle to a change in its state of rest or straight-line motion. It is expressed as Newton’s well-known equation
Equation 3-58.
or
Equation 3-59.
where:
F = applied force m = mass of the particle
a = acceleration of the particle
The term −ma is referred to as the inertial force. Inertial force represents the resistance of a particle to a change in its velocity vector. It is equal in magnitude to the applied force but acts in the opposite direction.
To determine the inertial forces in a rotating coordinate system, it is necessary to transform the vector representing the acceleration of a particle in the inertial coordinate system to the rotating coordinate system. This requires: 1) determining the acceleration vector in the inertial coordinate system, then 2) transforming the acceleration vector from the inertial system coordinates to the rotating system coordinates. This results in the inertial coordinate system acceleration vector being defined in terms of rotating system vectors.
Eq. 3-57defines a vector transformation from rotating system coordinates to stationary system coordinates. Let a vector rs be defined as the position vector for a particle in the stationary system coordinates, and rr be defined as the position vector in the rotating system coordinates.
Advanced Dynamic Analysis User’s Guide 3-15
Equation 3-60.
Taking the second derivative with respect to time produces
Equation 3-61.
The equation above defines the acceleration vector in the stationary or inertial coordinate system.
Eq. 3-47represents the vector transformation from stationary system coordinates to rotating system coordinates. Applying this transformation to the stationary system acceleration vector given byEq. 3-61 results in the inertial system acceleration defined in rotating system coordinates, as follows:
Equation 3-62.
Using the identities
Equation 3-63.
Equation 3-64.
Equation 3-65.
Equation 3-66.
Equation 3-67.
Equation 3-68.
Equation 3-69.
Eq. 3-62reduces to
Equation 3-70.
The term 4ΩRe(i[T1]) is commonly referred to as the Coriolis acceleration and the term
−2Ω2Re(i[T1]) is referred to as the centripetal acceleration. This equation provides a means of representing inertial or stationary system acceleration in terms of rotating system coordinates.
UsingEq. 3-70, the inertial force on a particle whose position is measured relative to a rotating coordinate system is written as
Equation 3-71.
This equation can be written as
Advanced Dynamic Analysis User’s Guide 3-17
Equation 3-72.
where (for rotation about the z-axis)
Equation 3-73.
Equation 3-74.
Equation 3-75.
Structural Impedance of a Rotating Structure
InEq. 3-72, the inertial forces are dependent on the time-varying position of a mass particle. This position vector can be written in terms of an initial component and a time-varying component.
Equation 3-76.
Eq. 3-72can be rewritten using the above equation as
Equation 3-77.
Inertial forces due to the mass are resisted by structural stiffness and damping.
Equation 3-78.
SubstitutingEq. 3-77in the above equation and collecting all time-dependent terms on the left-hand side results in
Equation 3-79.
The above equation is the equation of motion for a particle in a rotating elastic structure with no externally applied forces. The loading on the right-hand side ofEq. 3-79 is often called the centripetal force. The centripetal force is always present on a rotating structure and produces an additional stiffness-like term. This additional term can be determined from consideration of the standard static equilibrium equation for a single element.
Equation 3-80.
where:
K is the element stiffness in global coordinates (coordinates used to define motion).
u is the displacement in global coordinates.
P is the load in global coordinates.
The deformation, u, produces element reaction forces, F, which balance the applied load, P.
The stiffness, K, can be viewed as the change in the element reaction forces with respect to a change in displacement.
Advanced Dynamic Analysis User’s Guide 3-19
Equation 3-81.
Equation 3-82.
Equation 3-83.
where:
F is the internal element force in global coordinates.
T is a transformation from element to global coordinates.
Feis the internal element force in element coordinates.
ueis a displacement in element coordinates.
For a linear material and a loading which is not dependent on displacement, the term dFe/dueis constant, the element forces are linearly related to the element displacements. The terms T and due/du are dependent only on the orientation of the element coordinate system with respect to the global coordinate system. For small displacement problems, T and due/du are assumed constant and are calculated from the undeformed geometry. The three terms (T, due/du, and dFe/due) are used to calculated the linear element stiffness matrix in global coordinates. The element stiffness matrices are assembled to create the complete linear stiffness matrix for the structure.
The second part of the stiffness calculation, Fe(dT/du), is usually ignored in standard linear analysis. It arises from small changes in the element orientation with respect to the applied load. For example, consider a bar under axial and transverse loading.
Figure 3-2. Cantilevered Bar With Axial and Transverse Loading In a standard linear analysis, the y-directed load is reacted by the bar bending and shear forces. The x-directed load is reacted by the bar axial forces. The calculated y displacement is independent of the x-directed load. Likewise, the calculated x displacement is independent of the y-directed load. On closer inspection, one would find that as the free end of the bar rotates with
respect to the applied loads, a component of the bar axial force reacts against the y-directed load.
Likewise, a component of the bar bending and shear react to the x-directed load.
Figure 3-3. Bar Forces Due to Axial Loading
This results in a stiffness coupling which is not taken into account by standard linear analysis.
This effect is due to the Fe(dT/du) term inEq. 3-83and is referred to as the differential stiffness term. Its calculation requires knowledge of the element force Feand the transformation matrix T. The total stiffness matrix, often referred to as the tangent stiffness matrix, is the sum of the linear and differential stiffnesses.
Eq. 3-79shows that a centripetal load is always present in a rotating structure. This constant loading results in a differential stiffness term which must be included for accurate analysis of rotating structures. The differential stiffness is added to the stiffness terms on the left-hand side of Eq. 3-79. The complete equation of motion for a rotating structure, neglecting the centripetal loading is
Equation 3-84.
The above equation can be used to determine the motion of a rotating structure about the deformation due to the centripetal loading.