Example Utilization

Example Utilization#

This section demonstrates the basic utilization of ForceFinder on an example beam system. The examples cover:

import sdynpy as sdpy
import numpy as np

No GPU Found, Geometry plotting will not work!

Creating the Example Beam System#

All the examples use a finite element model of the simple beam system that is shown in the image below. Each dot in the image represents a node in the beam model. It is connected to the ground via translating and torsion springs (note that the torsion springs aren’t shown in the image).

Note

The nodes are labeled 101-109 from left to right. The translating direction on the beam (vertical on the page) is Z+ and the rotating direction is RY+.

Example Beam System

The beam is modeled with standard functions in SDynPy using Euler-Bernoulli beam elements, where the axial degrees of freedom (DOFs) have been excluded. The beam properties are:

  • Beam length - 4 m

  • Beam width - 0.02 m

  • Beam height - 0.015 m

  • Material properties - aluminum

  • Translating spring stiffness to ground - 1e9 N/m

  • Torsion spring stiffness to ground - 1e2 Nm/radian

  • Beam damping ratio - 4% modal damping for the free-free beam

The resulting system matrices saved in a SDynPy System object, which includes the mass, damping, and stiffness matrices.

Tip

DOFs are defined in SDynPy as a CoordinateArray and will be referred to as “coordinates” in the code blocks. The numbers three and five correspond to the Z+ and RY+ directions.

youngs_modulus = 69.8e9 # pascals
density = 2700 # kg/m^3
poissons_ratio = 0.33
translating_spring = 1e9 #N/m
torsion_spring = 1e2 #Nm/rad
beam_damping_ratio = 0.04 # ratio
    
# Basic beam properties
beam_length = 4 # meters
beam_width = 0.02 # meters
beam_height = 0.015 # meters
number_nodes = 9

beam_coordinate = sdpy.coordinate_array(node=np.arange(number_nodes)+101, 
                                        direction=[3,5], force_broadcast=True)

beam_stiffness, beam_mass = sdpy.beam.beamkm_2d(beam_length, beam_width, 
                                                beam_height, number_nodes, 
                                                youngs_modulus, density, 
                                                poissons_ratio, axial=False)

beam_system = sdpy.System(beam_coordinate, beam_mass, beam_stiffness)
beam_system.assign_modal_damping(beam_damping_ratio)

# Translating spring to ground on left side
beam_system.stiffness[0,0] += translating_spring # N/m
# Torsion spring to ground on left side 
beam_system.stiffness[1,1] += torsion_spring # Nm/rad 

# Translating spring to ground on right side
beam_system.stiffness[-2,-2] += translating_spring # N/m
# Torsion spring to ground on right side 
beam_system.stiffness[-1,-1] += torsion_spring # Nm/rad

Beam Modal Properties#

The beam system modal properties are shown in the table below.

print(beam_system.eigensolution())

   Index,  Frequency,    Damping,     # DoFs
    (0,),     2.5358,    1.5364%,         18
    (1,),     9.0563,    2.4143%,         18
    (2,),    19.9007,    3.1257%,         18
    (3,),    35.1521,    3.3233%,         18
    (4,),    54.9795,    3.4809%,         18
    (5,),    79.6855,    3.5745%,         18
    (6,),   109.6482,    3.6710%,         18
    (7,),   154.1341,    3.6553%,         18
    (8,),   191.9346,    3.6277%,         18
    (9,),   244.8775,    3.6781%,         18
   (10,),   309.2425,    3.7002%,         18
   (11,),   387.0350,    3.7040%,         18
   (12,),   478.9956,    3.6912%,         18
   (13,),   579.7017,    3.6604%,         18
   (14,),   668.1026,    3.6172%,         18
   (15,),   704.5058,    3.7964%,         18
   (16,), 29435.8118,    0.0646%,         18
   (17,), 29436.8999,    0.0647%,         18

Beam System Geometry#

The beam geometry is saved into a SDynPy Geometry where the acronym “FEM” is used to refer to the geometry object.

beam_locations = np.array([[0,   0, 0], 
                           [0.5, 0, 0],
                           [1, 0, 0], 
                           [1.5, 0, 0],
                           [2,   0, 0], 
                           [2.5, 0, 0],
                           [3, 0, 0], 
                           [3.5, 0, 0],
                           [4,   0, 0]], dtype=float)

system_cs = sdpy.geometry.coordinate_system_array()

beam_nodes = sdpy.geometry.node_array(id=np.unique(beam_coordinate.node), 
                                      coordinate=beam_locations)
beam_fem = sdpy.Geometry(node=beam_nodes, coordinate_system=system_cs)

beam_fem.add_traceline(np.unique(beam_coordinate.node))

Saving the SDynPy Objects#

The SDynPy System and Geometry objects are saved into the documentation folder of the package, so they can be used in several examples.

Tip

The documentation and example content (the scripts and data) is not installed with a standard pip installation. This content must be acquired from the ForceFinder Github page.

beam_system.save('./example_system/example_system.npz')
beam_fem.save('./example_system/example_geometry.npz')
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