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Structural Dynamics -  Beam Vibration

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BEAMS - BENDING VIBRATION - BERNOULLI-EULER BEAMS

Bending Frequencies of Beams, Rods, and Pipes: beam.pdf  
Bending Frequency of a Fixed-Fixed Beam via the Rayleigh Method: fixed_fixed_beam.pdf
Natural Frequencies of Composite Beams: compbeam.pdf  
Matlab script:  composite_beam.m
Bending Frequencies of a Beam Supported on an Elastic Foundation: beam_elastic_foundation.pdf
Free Vibration of a Cantilever Beam: free_cant.pdf

Vibration of a Simply Supported Beam with Concentrated Mass: ss_beam_cm.pdf
Matlab script:  ss_beam_mass.m
BEAMS - BENDING VIBRATION - BERNOULLI-EULER BEAMS - BASE EXCITATION
Steady-State Vibration Response of a Cantilever Beam Subjected to Base Excitation: steady_cant.pdf
Steady-State Vibration Response of a Simply-Supported Beam Subjected to Base Excitation: steady_ss_beam.pdf
Steady-State Vibration Response of a Fixed-Fixed Beam Subjected to Base Excitation: steady_fixed_fixed_beam.pdf

Modal Transient Vibration Response of a Cantilever Beam Subjected to Base Excitation:  cantilever_modal_transient_base.pdf

  __________________

Matlab Script with optional frequency response function, half-sine base excitation & arbitrary base excitation:
continuous_beam_base_accel.m

Supporting functions:

plot_PSD.m
plot_PSD_two.m

srs_coefficients avd.m
continuous_beam_base_PSD.m
psd_mult_trans.m
find_plot_limits.m
calculate_PSD_slopes.m

continuous_beam_base_arbitrary.m
continuous_beam_base_sine.m
continuous_beam_base_half_sine.m
continuous_beam_base_frf.m
enter_modal_damping.m
continuous_beam_bc.m
geometry_entry.m
materials.m
BEAMS - BENDING VIBRATION - BERNOULLI-EULER BEAMS -
APPLIED CONCENTRATED FORCE 

Vibration Response of a Cantilever Beam Subjected to a Concentrated Force at Its Free End: cant_beam_force.pdf

Matlab scripts:
cant_beam_force_frf.m
geometry_materials.m

The Response of a Cantilever Beam Subjected to a Concentrated Half-Sine Force Pulse: cant_beam_half_sine_force.pdf

Matlab scripts:  
cant_beam_half_sine_force.m
geometry_materials.m

frf_from_th.m
fix_size.m
progressbar.m

frf_median_filter.m

The Response of a Cantilever Beam Subjected to a Concentrated Sine Force

Matlab scripts:
cant_beam_sine_force.m
geometry_materials.m
fix_size.m
progressbar.m


The Response of a Cantilever Beam Subjected to an Applied Concentrated Arbitrary Force:  cant_beam_arbitrary_force.pdf

Matlab scripts:

cant_beam_arbit_force.m
enter_time_history.m
geometry_materials.m
progressbar.m

The Response of a Cantilever Beam Subjected to an Applied Concentrated Impulse Force:
cant_beam_impulse_force.pdf

BEAMS - BENDING VIBRATION - BERNOULLI-EULER BEAMS -
UNIFORMLY DISTRIBUTED APPLIED FORCE 

Vibration Response of a Fixed-Fixed Beam Subjected to an Applied Uniform Distributed Force:  steady_fixed_beam_force.pdf

Matlab script: fixed_fixed_beam_frf.m

Vibration Response of a Simply-Supported Beam Subjected to an Applied Uniform Distributed Force:  steady_ss_beam_force.pdf

Matlab Scripts:  ss_ss_beam_frf.m

Common Supporing Functions:

fixed_fixed_beam_modes.m
ss_beam_modes.m
beam_frf.m
fix_size.m

materials.m
damping_entry.m
geometry_entry.m
BEAMS - BENDING VIBRATION - BERNOULLI-EULER BEAMS -
- APPLIED FORCE WITH SPATIAL VARIATION 
Steady-State Vibration Response of a Simply-Supported Beam Subjected to an Applied Force with Spatial Variation:  steady_ss_beam_force_spatial.pdf

Matlab script: ss_ss_beam_spatial_frf.m

Steady-State Vibration Response of a Fixed-Fixed Beam Subjected to an Applied Force with Spatial Variation:  steady_fixed_fixed_beam_force_spatial.pdf

Matlab script: fixed_fixed_beam_spatial_frf.m

Supporing Functions:materials.m
damping_entry.m
geometry_entry.m
MULTISPAN BEAMS - BENDING VIBRATION 
Natural Frequencies of Multispan Beams: mspan.pdf
Matlab script for Clamped-Pinned-Free Beam:

clamped_pinned_free.m

Supporting Functions:
find_root_mbeam.m
cpf_C.m
cpf_C_coefficients.m

Matlab script for Clamped-Pinned-Clamped Beam:

clamped_pinned_clamped.m

Supporting Functions:
find_root_mbeam.m
cpc_C.m
cpc_C_coefficients.m
Matlab script (FEA implementation):   beam_multispan.m
Supporting Functions: 
beam_data_entry_ms.m
local_stiffness.m
local_mass.m
assembly_ms.m
plot_modes_ms.m
Generalized_Eigen.m

BEAMS WITH AXIAL LOAD - BENDING VIBRATION - BERNOULLI-EULER BEAMS

Natural Frequencies of Beams Subjected to a Uniform Axial Load: beam_axial_load.pdf

F. Shaker, Effect of Axial Load on Mode Shapes and Frequencies of Beams, NASA TN D-8109: NASA_TN_8109.pdf
BEAMS - BENDING & SHEAR VIBRATION - TIMOSHENKO BEAMS
Transverse Vibration of a Beam Simply-Supported at Each End with Bending, Shear, and Rotary Inertia: bending_shear _beam.pdf
Transverse Vibration of a Fixed-Free Timoshenko Beam: bending_Timoshenko_fixed_free.pdf
Matlab Script: beam_shear_inertia.m
Natural Frequencies of a Shear Beam: shear_beam.pdf
BEAM VIBRATION - FINITE ELEMENT METHOD - Natural Frequencies
Transverse Vibration of a Beam via the Finite Element Method: beam_FEM.pdf
Transverse Vibration of a Rotating Beam via the Finite Element Method:  Beam_FEM_rotating.pdf
Vibration of a Beam-Column via the Finite Element Method: beam_column_FEM.pdf
Transverse Vibration of a Cantilever Beam Subjected to a Constant Axial Load: beam_FEM_cantilever_axial.pdf
Transverse Vibration of a Simply-Supported Beam Subjected to a Constant Axial Load: beam_FEM_ss_axial.pdf
Transverse Vibration of a Fixed-Fixed Beam Subjected to a Constant Axial Load: beam_FEM_ff_axial.pdf
BEAM VIBRATION - FINITE ELEMENT METHOD - Enforced Motion
Frequency Response Function for Beam Bending with Enforced Acceleration via Finite Element Analysis:  modal_enforced_motion_beam_fea_frf.pdf

Modal Transient Analysis of a Beam with Enforced Motion via a Ramp Invariant Digital Recursive Filtering Relationship: modal_enforced_motion_beam_transient.pdf

 

 
  Matlab Scripts

Mass properties of a dynamic system using Guyan reduction. i.e center of gravity and moments of inertia.   The required inputs are the mass and stiffness matrices.

mass_properties.m

Generalized_Eigen.m

Mass properties of a dynamic system using geometric rigid-body modes.  The required inputs are a mass matrix and the nodal coordinates.

mass_properties_GRB.m
Natural Frequencies and Mode Shapes of a Beam via the Finite Element Method, with two degrees-of-freedom per node.

This program also calculates the participation factors and effective modal mass values.
It allows for the addition of a point mass.
 
beam.zip
This script calculates the natural frequencies of a beam via the finite element method. It also calculates the frequency response functions for base excitation. beam_base_accel_fea.zip
Natural Frequencies and Mode Shapes of a Rotating Beam (like a helicopter blade)  via the Finite Element Method with two degrees-of-freedom per node. 

This program also calculates the participation factors and effective modal mass values.
It allows for the addition of a point mass
beam_rotating.zip

Natural Frequencies and Mode Shapes of a Beam via the Finite Element Method with six degrees-of-freedom per node.

The input file is a Nastran *.bdf or *.nas file generated by Femap or some other preprocessor.

The beam elements may be in a three-dimensional frame.

beam_3D_nastran.m

Functions:

beam_3D_nastran_find.m
beam_3D_mass_condensation.m
Generalized_Eigen.m
beam_3D_modeplot.m

Natural Frequencies and Mode Shapes of a Beam via the Finite Element Method with six degrees-of-freedom per node.

It allows for the addition of a point mass.

The beam elements may be in a three-dimensional frame.

beam_3D.m

Functions:

beam_3D_mass_condensation.m
Generalized_Eigen.m
beam_3D_modeplot.m

User Guide

beam_3D_guide.pdf
Additional scripts are given at:  Matlab

 Software Description

 Executable

  Source
Natural frequencies of beam modes, both lateral and longitudinal.
 beam.exe
 _
Natural frequencies of rectangular plates.
 plate.exe

 _
Natural frequencies of circular plates.
 circular.exe

  _
Single-degree-of-freedom System Natural Frequency.
 sdof_fn.exe

 _
 Holzer method for torsional systems.
 Holzer.exe
  Holzer.cpp
Unit conversion  program.
units.exe

 

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