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$       RIGID FORMAT No. 10, Complex Eigenvalue Analysis - Modal Formulation
$                   Rocket Guidance and Control Problem (10-1-1)
$ 
$ A. Description
$ 
$ This problem, although a simplified model, contains all of the elements used
$ in a linear control system analysis. The flexible structure consists of three
$ sections: two sections are constructed of structural finite elements; the
$ third section is formulated in terms of its modal coordinates. A sensor is
$ located at an arbitrary point on the structure and connected to a structural
$ point with multipoint constraints. The measured attitude and position of the
$ sensor point is used to generate a control voltage for the gimbal angle of the
$ thrust nozzle. The nozzle control is in itself a servomechanism consisting of
$ an amplifier, a motor, and a position and velocity feedback control. The
$ nozzle produces a force on the structure due to its mass and the angle of
$ thrust. The motion of any point on the structure is dependent on the elastic
$ motions, free-body notions, and large angle effects due to free-body rotation.
$ 
$ The definitions for the variables and coefficients along with values for the
$ coefficients are given in Table 1. The use of the Transfer Function data card
$ (TF) allows the direct definition of the various relations.
$ 
$ B. Theory
$ 
$ A section of the structure is defined by its modaI coordinates by using a
$ modification of the method given in the NASTRAN Theoretical Manual. The
$ algorithm is given as follows:
$ 
$ Define  xi , i = 1, n  - modal deflections scalar points
$           i
$                   u    - grid point components used as nonredundant
$                    r     supports for modal test. These may or may not be
$                          connected to the rest of the structure.
$ 
$                   u    - grid point components to be connected to the
$                    c     remaining structure (not u  points)
$                                                    r
$ 
$         x   i = 1, n   - rigid body component degrees of freedom for the
$          i               nonzero nodes
$ 
$ The relations between these variables are defined by using multipoint
$ constraints with the following relationships:
$ 
$    {u }  =  [phi  ]{xi } + [D  ]{u }                                       (1)
$      c          ci    i      cr   r
$ 
$ where phi   is the angular deflection of point u  for mode i. D   is the
$          ci                                     c              cr
$ deflection of point u  when the structure is rigid and point u  is given a
$                      c                                        r
$ unit deflection.
$ 
$                 -1   T
$    {x }  =  [K ]  [H] {u }  =  [G]{u }                                     (2)
$      i        i         r           r
$ 
$ where [K ] is a diagonal matrix. Each term K , the modal stiffness, is defined
$         i                                   i
$ as:
$ 
$              2
$    K   =  m w                      (w  not equal 0)                        (3)
$     i      i i                       i
$ 
$ where m  is the modal mass and w  is the natural frequency in radians per
$        i                        i
$ second. [H] is determined by the forces on the support points due to each
$ nonzero eigenvector:
$ 
$    P   =  - sum from i H   xi      (w  not equal 0)                        (4)
$     r                   ri   i       i
$ 
$ The structure to be added in this problem consists of a simply supported
$ uniform bed. The support points, u , are y   and y  . The additional degree of
$                                   r       16      19
$ freedom to be connected is u  = theta  . Four modes are used in the test
$                             c        16
$ problem. The following data is used to define and connect the modal
$ coordinates of this substructure. The mode shapes are
$ 
$                    n pi x
$    phi (x)  =  sin ------                                                  (5)
$       n              l
$ 
$ The modal frequencies, masses, and stiffness in terms of normal beam
$ terminology are
$ 
$            2  2
$           n pi   EI
$    w   =  -----  --                (n = 1, 2, 3, 4)                        (6)
$     n       2    pA
$            l
$ 
$           pAl
$    m   =  ---                                                              (7)
$     n      2
$ 
$            4  4
$           n pi EI
$    K   =  -------                                                          (8)
$     n         3
$             2l
$ 
$ The forces of support for each mode are
$ 
$                         3
$                     EIpi    3
$    P (16)  =  sum - -----  n                                               (9)
$     y                 3
$                      l
$ 
$                            3
$                      n EIpi    3
$    P (19) =  sum (-1)  -----  n                                           (10)
$     y                    3
$                         l
$ 
$ The motion theta   is defined by multipoint constraints:
$                 16
$ 
$              1                          n pi
$    theta   = - (y   - y  ) + sum from n ----  xi                          (11)
$         16   l   19    16                l      n
$ 
$ The free-body components of the modes are defined, using multipoint constraints,
$ as:
$ 
$    +     +                          +            +
$    |  x  |                          | 1      1   |
$    |   1 |                          |            |
$    |     |                          |            |  +     +
$    |  x  |           3         3    | 1/2   -1/2 |  | y   |
$    |   2 |         2l      EIpi     |            |  |  16 |
$    |     |   =  - (-----) (-----)   |            |  |     |               (12)
$    |  x  |           4       3      | 1/3    1/3 |  | y   |
$    |   3 |         pi EI    l       |            |  |  19 |
$    |     |                          |            |  +     +
$    |  x  |                          | 1/4   -1/4 |
$    |   4 |                          |            |
$    +     +                          +            +
$ 
$ The mass of the nozzle would normally be included with the structural
$ modeling. However, to demonstrate the flexibility of the Transfer Function
$ data, it is modeled as part of the guidance system. Defining the angle of
$ thrust, gamma, to be measured relative to the deformed structure, the forces
$ which result are
$ 
$                  2     ..      ..           ..
$    T  =  (I   + x m )(gamma + theta ) - m x y                             (13)
$            no    n i               1     n n 1
$ 
$                        ..      ..
$    F   = m y  - x m  (gamma + theta ) - F  gamma                          (14)
$     y     n 1    n n               1     n
$ 
$ Using the thrust force, F , as a constant, the transfer functions are
$                          n
$ 
$       2                2               2
$    I s  gamma - T + I s  theta  - x m s y   =  0                          (15)
$     n                n        1    n n   1
$ 
$       2           2                    2
$    m s y  - (x m s  + F ) gamma - x m s  theta   =  0                     (16)
$     n   1     n n      n           n n        1
$ 
$    (0) theta  + T  =  0                                                   (17)
$             1
$ 
$ where
$ 
$                  2
$    I   =  I   + x  m   =  500                                             (18)
$     n      no    n  n
$ 
$ The large angle motion must be included in the analysis since it contributes
$ to the linear terms. The equations of motion of the structure are formed
$ relative to a coordinate system parallel to the body. The accelerations are
$ coupled when the body rotates.
$                               ..
$ Since the axial acceleration, x, is constant throughout the body, the
$ vertical acceleration at any point, to the first order, is
$ 
$    ..       ..     ..           ..     ..
$    y     =  y    + x theta   =  y    + y                                  (19)
$     abs      rel          1      rel    theta
$ 
$ An extra degree of freedom y      is added to the problem and coupled by
$                             theta
$ the equations:
$ 
$     ..
$    my       =  F  theta                                                   (20)
$      theta      n      1
$ 
$    y     =  y    + y                                                      (21)
$     abs      rel    theta
$ 
$ The center of gravity (point 101) and the sensor location (point 100) are
$ rigidly connected to the nearest structural point with multipoint constraints.
$ For instance the sensor point is located a distance of 4.91 from point 8.
$ 
$ It is desired to leave point 101 as an independent variable point. Therefore
$ point 8 is defined in terms of point 101 by the equations:
$ 
$    y   =  y    + 4.91 theta                                               (22)
$     8      101             101
$ 
$    theta   =  theta                                                       (23)
$         8          101
$ 
$ C. Results
$ 
$ A comparison of the NASTRAN complex roots and those derived by a conventional
$ analysis described below are given in Table 2. The resulting eigenvectors were
$ substituted into the equations of motion to check their validity. The
$ equations of motion for a polynomial solution may be written in terms of the
$ rigid body motions of the center of gravity plus the modal displacements. The
$ equations of motion using Laplace transforms are
$ 
$      2
$    ms  y    =  F (theta  + gamma)                                         (24)
$         cg      n      1
$ 
$      2
$    Is  theta    =  - F  x  gamma                                          (25)
$             cg        n  1
$ 
$ The inertia forces of the nozzle on the structure may be ignored.
$ 
$ The motion of the nozzle, as explained above, is
$ 
$       2                                                         2
$      s                       ~                                 s
$    (---- + tau s = 1) gamma  =  (a + bs)y  + (c + ds)theta  - ---- +
$     beta                                 s                x   beta
$ 
$     2
$    s m x
$       n n
$    -------  y                                                             (26)
$    beta I    1
$          n
$ 
$ where gamma is defined as the relative angle between the nozzle and the
$ structure.
$ 
$ The flexible motions at the sensor point, y  and theta , may be defined in
$                                            s          s
$ terms of the modal coefficients and the rigid motions of the center of
$ gravity:
$ 
$    y   =  y   + x  theta   + sum from i phi      xi                       (27)
$     s      cg    2      cg                 100,i   i
$ 
$    theta   =  theta   + sum from i phi      xi                            (28)
$         s          cg                 100,i   i
$ 
$ The motions of the nozzle point, in terms of the modal and center of gravity
$ motions are
$ 
$                                       '
$    theta   =  theta   + sum from i phi    xi                              (29)
$         1          cg                 1,i   i
$ 
$    y   =  y   - x  theta   + sum from i phi    xi                         (30)
$     1      cg    1      cg                 1,i   i
$ 
$ The modal displacements are due primarily to the vertical component of the
$ nozzle force. Their equation of motion is
$ 
$        2    2
$    m (s  + w )xi   =  F gamma                                             (31)
$     i       i   i      n
$ 
$ where
$ 
$    phi    is the deflection of point j for mode i
$       j,i
$ 
$       '
$    phi    is the rotation of point j for mode i
$       j,i
$ 
$    m      is the nodal mass of mode i
$     i
$ 
$    w      is the natural frequency of mode i
$     i
$ 
$    xi     is the modal displacement of mode i
$      i
$ 
$ The determinant of the matrix forms a polynomial of order 10. The roots of
$ this polynomial were obtained by a standard computer library routine and are
$ presented in Table 2 as the analytical results. The rigid body solution is
$ also presented.
$ 
$ The differences between the two sets of answers is due to the differences in
$ models. The NASTRAN model produces errors due to the finite difference
$ approximation and the number of modes chosen to model the third stage. The
$ polynomial solution produces errors due to the approximations used in the
$ equations of motion as applied to control system problems.
$ 
$ As a further check the first eigenvalue (lambda = -1.41) was substituted into
$ the matrix and the matrix was normalized by dividing each row by its diagonal
$ value. The NASTRAN eigenvector was multiplied by the matrix, resulting in an
$ error vector which theoretically should be zero. Dividing each term in the
$ error vector by its corresponding term in the eigenvector resulted in very
$ small error ratios.
$ 
$                         Table 1. Variables and Parameters
$ 
$ EXTRA
$ POINT
$ NUMBER   SYMBOL       DESCRIPTION
$ 
$ 1010     e            Voltage describing y
$           y
$ 1011     e            Voltage describing theta
$           theta
$ 1020     E            Control voltage for y (Input)
$           yc
$ 1021     E            Control voltage for theta (Input)
$           thetac
$ 1030     E            Attitude error function
$           gamma
$ 1040     epsilon      Nozzle position error
$                 gamma
$ 1050     E            Voltage for nozzle servo
$           m
$ 1060     T            Torque for nozzle servo
$ 
$ 1070     tau          Nozzle thrust angle relative to structure
$ 
$ 1080     y            Position increment due to attitude
$           theta
$ 
$ 
$ PARAMETER   VALUE     DESCRIPTION
$ 
$ K           1.0       Servo amplifier gain
$  s
$ 
$ K           500       Servo gain
$  m
$ 
$ tau         .1414     Nozzle angular velocity feedback
$ 
$ x           3.0       Distance from nozzle c.g. to gimbal axis
$  n
$ 
$ I           500.0     Inertia of nozzle about gimbal axis
$  n
$                    6
$ F           4.25x10   Thrust of nozzle
$  n
$ 
$ m           50        Nozzle mass
$  n
$ 
$ beta        100.0     Overall voltage-to-angle ratio
$     theta
$ 
$ beta        1.0       Overall voltage-to-position ratio
$     y
$ 
$ a           .16       Position feedback coefficient
$ 
$ b           .28       Velocity feedback coefficient
$ 
$ c           15.0      Angle feedback coefficient
$ 
$ d           7.0       Angular velocity feedback coefficient
$                   4
$ m           8.5x10    Mass of structure
$ 
$           Table 2. Comparison of Complex Roots for NASTRAN Modeling vs.
$                          Simplified Polynomial Expansion
$       --------------------------------------------------------------------
$            Rigid Body Model                   Flexible Modes Model
$       --------------------------------------------------------------------
$       NASTRAN          POLYNOMIAL         NASTRAN          POLYNOMIAL
$       --------------------------------------------------------------------
$       -.540 +/- .821i  -.522 +/- .802i    -.507 +/- .819i  -.494 +/- .801i
$ 
$       -1.68 +/- 0i     -1.74 +/- 0i       -1.41 +/- 0i     -1.46 +/- 0i
$ 
$       +.751 +/- 5.96i  +.774 +/- 5.98i    +.520 +/- 3.82i  +.522 +/- 3.83i
$       --------------------------------------------------------------------
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