*heading
LONGITUDINAL VIBRATION OF "CLAMPED-FREE" (CANTILEVER) BEAM	
WITH AND WITHOUT ADDED MASS
*******************************************************************
***		RESULTS/ANALYTICAL SOLUTION
***
*** Analytical solution:
***	f_n = (n + 0.5) * (1 / 2L) * sqrt (E/(rho + rho_w))
*** where
***	f	is the frequency in cycles/time
***	n	is the number of the mode
***     r	is the radius of circular cross-section
***     L	is the length of the structure
***     E	is the Young's modulus of the material (of the structure)
***     rho	is the density of material
***     rho_w   is the density of fluid (or water)
***
***	F R E Q U E N C Y    T A B L E
*** Mode Analytical ABAQUS Analytical ABAQUS
***	   No Added Mass        Added Mass
***  1    4.2554   4.2554   3.9959   3.9958
***  2    12.766   12.765   11.988   11.986
***  3    21.277   21.272   19.979   19.974
***  4    29.788   29.773   27.971   27.957
***  5    38.299   38.267   35.963   35.933
*******************************************************************
**************MODEL********
*node
1,   0.0,    0.0
1001, 1000.0, 0.0
*ngen, nset=all_nodes
1,1001,10
*ngen, nset=nodes_bend
11,991,10
*nset,nset=loaded_nodes
1,991
*element, type=b21
1, 1,11
*elgen, elset=b21_elements
1, 100, 10, 10
*************AQUA**********
*aqua
-100.0, 100.0, 32.2, 2.0
0., 0., 0., 0.
*************MATERIAL/SECTION*******
*beam section, elset=b21_elements, section=circ, material=mat1
3.0, 
*material, name=mat1
*elastic
4.32e9, 
*density
14.91, 
**************BOUNDARY***********
*boundary
all_nodes,2,2,0.0
1   ,1,1,0.0
1   ,6,6,0.0
**************HISTORY/LOADING*******
*restart,write,f=100
*****
*step,perturbation
Without added mass
*frequency, eigensolver=lanczos, SIM=NO
5, , 
*endstep
*****
*step,perturbation
With added mass (FI1 and FI2 simulating lumped mass)
*frequency, eigensolver=lanczos, SIM=NO
5, , 
*d added mass
b21_elements, fi1, 1.0, 141.37
b21_elements, fi2, 1.0, 141.37
*endstep
*****
*step,perturbation
With added mass (TSI simulating lumped mass)
*frequency, eigensolver=lanczos, SIM=NO
5, , 
*c added mass
loaded_nodes,        tsi, 1.0, 141.37
1.0, 0.0, 0.0
nodes_bend, tsi, 1.0, 282.74
1.0, 0.0, 0.0
*endstep