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$           RIGID FORMAT No. 9, Transient Analysis - Direct Formulation
$          Transient Analysis of a Fluid-Filled Elastic Cylinder (9-3-1)
$ 
$ A. Description
$ 
$ The fluid-filled shell used for analysis of the third harmonic in
$ Demonstration Problem No. 7-2-1 is subjected to a step change in external
$ pressure at t = 0 of the form
$ 
$                 pi z
$    p  =  p  sin ---- cos n phi
$           o      l
$ 
$ The fluid is assumed incompressible in order to obtain an analytical solution
$ with reasonable effort. The harmonic used is n = 3.
$ 
$ In addition to the cards of Demonstration Problem No. 7-2-1, DAREA, PRESPT,
$ TLOAD2, and TSTEP cards are also used. Selected displacements and pressures
$ are plotted against time.
$ 
$ B. Input
$ 
$    Parameters used are:
$ 
$      B  =  infinity                   (Bulk modulus of fluid - incompressible)
$ 
$                    -2       2   4
$      p  =  1.8 x 10   lb-sec /in      (Fluid mass density)
$       f
$ 
$                    -2       2   4
$      p  =  6.0 x 10   lb-sec /in      (Structure mass density)
$       s
$                    5      2
$      E  =  1.6 x 10  lb/in            (Young's modulus for structure)
$ 
$                    4      2
$      G  =  6.0 x 10  lb/in            (Shear modulus for structure)
$ 
$      a  =  10.0 inch                  (Radius of cylinder)
$ 
$      l  =  10.0 inch                  (Length of cylinder)
$ 
$      h  =  0.01 inch                  (Thickness of cylinder wall)
$ 
$      p  =  2.0                        (Pressure load coefficient)
$       o
$ 
$ C. Theory
$ 
$ The theory was derived with the aid of Reference 16 as in Demonstration
$ Problem No. 7-2-1. Since the fluid is incompressible, it acts on the structure
$ like a pure mass. Neglecting the bending stiffness, the equation of force on
$ the structure is:
$ 
$                             2
$                    ..   1  a F
$    p   =  (m + m ) w  + -  ---                                             (1)
$     s           f       a    2
$                            aZ
$ 
$ where:
$ 
$    p  is the loading pressure on the structure (positive outward).
$     s
$ 
$    m = p h is the mass per area of the structure.
$         s
$ 
$    m  is the apparent mass of the fluid.
$     f
$ 
$    w is the normal displacement (positive outward).
$ 
$ The function F is defined by the equation,
$ 
$                        2
$            4      Eh  a w
$    gradient F  =  --  ---                                                  (2)
$                   a     2
$                       az
$ 
$ The spatial functions of pressure, displacement, and function F may be written
$ in the form:
$ 
$                  pi z
$    p   =  p  sin ----  cos n phi
$     s      o      l
$ 
$                 pi z
$    w  =  w  sin ----  cos n phi                                            (3)
$           o      l
$ 
$                 pi z
$    F  =  F  sin ----  cos n phi
$           o      l
$ 
$ where p , w , and F  are variables with respect to time only.
$        o   o       o
$ 
$ Substituting Equations 3 into Equation 2 we obtain:
$ 
$                               w
$             Eh       2         o
$    F   =  - -- (l/pi)  -----------------                                   (4)
$     o       a                        2 2
$                        [1 + (nl/pi a) ]
$ 
$ Substituting Equations 3 and 4 into Equation 1 we obtain:
$ 
$                   ..          Eh
$    p   = (m + m ) w  + -------------------  w                              (5)
$     o          f   o    2              2 2   o
$                        a [1 + (nl/pi a) ]
$ 
$ The incompressible fluid is described by the differential equation:
$ 
$            2
$    gradient  p  =  0                                                       (6)
$ 
$ Applying the appropriate boundary conditions to Equation 6 results in the
$ pressure distribution:
$ 
$                 pi z                  pi r
$    p  =  p  sin ----  cos (n phi) I  (----)                                (7)
$           r      l                 n   l
$ 
$ where I  is the modified Bessel function of the first kind and is an
$        n
$ undetermined variable. The balance of pressure and flow at the boundary of the
$ fluid, with no structural effects, is described by the equations:
$ 
$                    pi a
$    p   =  - p  I  (----)                                                   (8)
$     o        r  n   l
$ 
$      ..     ap |
$    p w  = - -- |                                                           (9)
$     f       ar |r=a
$ 
$ Substituting Equations 3 and 7 into Equation 9 results in:
$ 
$      ..     pi     pi a
$    p w   =  -- I' (----) p                                                (10)
$     f o     l   n   l     r
$ 
$ Eliminating P  with Equations 8 and 10 gives the expression for apparent mass,
$              f
$ m :
$  f
$ 
$                          ..
$                       p  w
$               pi a     f  o             ..
$    p   =  I  (----) ------------  =  m  w                                 (11)
$     o      n   l    pi     pi a       f  o
$                     -- I' (----)
$                     l   n   l
$ 
$ Substituting the expression for mf from Equation 11 into Equation 5 results in
$ a simple single degree of freedom system.  When the applied loading pressure
$ is a step function at t = 0,
$ 
$          p
$           o                     pi z
$    w  =  -- (1 - cos n phi) sin ---- cos n phi                            (12)
$          k                       l
$ 
$ where
$ 
$    w  =  sqrt(k/m )
$                  T
$ 
$ and
$ 
$                 Eh
$    k  =  -------------------
$           2              2 2
$          a [1 + (nl/pi a) ]
$ 
$ and
$ 
$                                 I (pi a/l)
$                              l   n
$    m  =  m + m   =  p h + p  -- ----------
$     T         f      s     f pi I'(pi a/l)
$                                  n
$ 
$ D. Results
$ 
$ A transient analysis was performed for the case n = 3 on the model and various
$ displacements and pressures were output versus time up to one second. The
$ theoretical frequency is calculated to be 1.580 Hertz and the period is 0.633
$ seconds.  The displacements at two points on the structure (Point 91 is located
$ at phi = 0, z = 5.0; Point 94 is located at phi = 18 degrees, z = 5.0) were
$ plotted versus time.
$ 
$ The maximum error for the first full cycle occurs at the end of the cycle.
$ The ratio of the error to maximum displacement is 4.75%. Changes in the time
$ step used in the transient integration algorithm did not affect the accuracy
$ to any great extent. The most probable causes for error were the mesh size of
$ the model and the method used to apply the distributed load. The applied load
$ was calculated by multiplying the pressure value at the point by an associated
$ area. The "consistent method" of assuming a cubic polynomial displacement and
$ integrating would eliminate the extraneous response of higher modes. The
$ method chosen in this problem, however, is typical of actual applications.
$ 
$ APPLICABLE REFERENCES
$ 
$ 16. J. G. Berry and E. Reissner, "The Effect of an Internal Compressible Fluid
$     Column on the Breathing Vibrations of a Thin Pressurized Cylindrical
$     Shell", Journal of the Aeronautical Sciences, Vol. 25, No. 5, pp 288-294,
$     May 1958.
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