# Beam Bending, Finite Element Analysis

Here is a paper which gives a derivation of the mass and stiffness matrices for beam bending: beam_FEM.pdf

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Here is a Matlab script for the modal analysis of a straight beam. beam.zip

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Here is Matlab script for the modal analysis of a rotating beam such as a helicopter blade: beam_rotating.zip

Here is a paper: Beam_FEM_rotating.pdf

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Here is a Matlab script for the beam bending response to base excitation: beam_base_accel_fea.zip

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Dynamic Response to Enforced Motion

Beam Bending Natural Frequencies & Mode Shapes

– Tom Irvine

# Stress-Velocity Relationship

Shock and vibration environments produce dynamic stresses which can cause material failure in structures. The potential failure modes include fatigue, yielding, and ultimate stress limit. F.V. Hunt wrote a seminal paper on this subject, titled “Stress and Stress Limits on the Attainable Velocity in Mechanical Vibration,” published in 1960. This paper gave the relationship between stress and velocity for a number of sample structures.

H. Gaberson continued research on stress and modal velocity with a series of papers and presentations.

The purpose of the paper sv_velocity.pdf  is to explore the work of Hunt, Gaberson, and others.  Derivations are given relating stress and velocity for a number of structures. Some of these examples overlap the work of previous sources. Other examples are original. In addition, this paper presents some unique data samples for shock events, with the corresponding spectra plotted in tripartite format.

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Shock Severity Limit

An empirical rule-of-thumb in MIL-STD-810E states that a shock response spectrum is considered severe only if one of its components exceeds the level

Threshold = [ 0.8 (G/Hz) * Natural Frequency (Hz) ]

For example, the severity threshold at 100 Hz would be 80 G.

This rule is effectively a velocity criterion.

MIL-STD-810E states that it is based on unpublished observations that military-quality equipment does not tend to exhibit shock failures below a shock response spectrum velocity of 100 inches/sec (254 cm/sec).

The above actually corresponds to 50 inches/sec.  It thus has a built-in 6 dB margin of conservatism.

Note that this rule was not included in MIL-STD-810F or G, however.

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SMC-TR-06-11 AEROSPACE REPORT NO. TR-2004(8583)-1 REV. A, Test Requirements for Launch, Upper-Stage, and Space Vehicles, Section 10.2.6 Threshold Response Spectrum for Shock Significance

A response velocity to the shock less than 50 inches/second is judged to be non-damaging.

This is the case if the shock response spectrum value in G is less than 0.8 times the frequency in Hz.

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The 100 ips threshold is defined in part by the observation that the severe velocities which cause yield point stresses in mild steel beams turn out to be about 130 ips.

Mild Steel Yield stress is: 36 ksi

Speed of sound in Steel is: c = 200,000 ips

rho = 0.00075122 lbf sec^2/in^4
khat = sqrt(3) for rectangular cross-section

Vmax = (yield stress)/(khat * rho * c)
= 36,000 lbf/in^2 /( sqrt(3) * 0.00075122 lbf sec^2/in^4 * 200,000 ips)
= 130 ips for beam (rounded slightly downward)

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Stress-velocity examples for beam bending are given in:  stress_velocity_examples.pdf

The base acceleration input used in the paper is:  avs.txt

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Great Amplitude Format Debate

Gaberson Papers

Shock and Vibration Severity Thresholds for Structures and Equipment

– Tom Irvine

# Modal Transient Analysis of a Beam with Enforced Motion via a Ramp Invariant Digital Recursive Filtering Relationship

These Matlab scripts calculate the natural frequencies of a beam via the finite element method:  beam_base_accel_fea.zip

They also have options for base excitation:

1. frequency response function
2. half-sine base input
3. arbitrary base input

A ramp invariant digital recursive relationship is used for the case of arbitrary base excitation.

The main function is:  beam_base_accel_fea.m

The remaining functions are supporting functions.

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The reference papers are:

modal_enforced_motion_beam_transient.pdf

beam.pdf

modal_enforced_motion_ramp_invariant.pdf

gencoord.pdf

beam_FEM.pdf

The numerical engine is taken from the applied force model because the temporal response is needed in its modal absolute form.  See: Modal Transient Analysis of a System Subjected to an Applied Force via a Ramp Invariant Digital Recursive Filtering Relationship: force_ramp_invariant.pdf
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Beam Bending Natural Frequencies and Modeshapes

Modal Transient Numerical Engine

– Tom Irvine

# Beam Bending Natural Frequencies & Mode Shapes

A rocket vehicle behaves as a free-free beam during flight. The vehicle’s body bending modes can be excited by wind gusts, aerodynamic buffeting, thrust offset, maneuvers, etc.

Note that the mass and stiffness properties vary along the vehicle’s length. Furthermore, the mass properties change with time as propellant is expelled.

The image shows the IRVE 2 launch from Wallops Island. The vehicle is a Black Brant 9, which has a Terrier Mark 70 first stage.

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Here is a Newsletter which identifies the fundamental body-bending mode in a Terrier-Black Brant from flight accelerometer data: August2010_NL.pdf

Here is the lowpass filtered accelerometer data for the lateral X-axis: terrier_black_brant_2KHz_lpf_x.zip (Use right mouse click – save as)

Matlab format:  terrier_black_brant_2KHz_lpf_x.mat

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The following paper gives derivations for natural frequencies, mode shapes, participation factors and effective modal mass values for beams with  common boundary conditions:
Bending Frequencies of Beams, Rods, and Pipes & Alternate Link

Beam on an Elastic Foundation

Multispan Beams

Effective Modal Mass

Static Bending Analysis of a Free-Pinned-Pinned-Free Beam

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Here is a Matlab GUI script for the beam bending frequency calculation:  beam_bending.zip

It also has options for calculating the response to base excitation.

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Here is a Matlab GUI script for a beam with rotary inertia and shear effects both included: beam_bending_inertia_shear.zip

It is appropriate for a short, stubby beam.

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Additional tutorial papers and corresponding Matlab scripts are given at:  Vibrationdata Beam Page

The papers give formulas for the responses of beams to base excitation and applied forces and pressures.

Topics include:

Transverse Vibration of a Beam Simply-Supported at Each End with Bending, Shear, and Rotary Inertia (Timoshenko Beam)
Natural Frequencies of Multispan Beams
Bending Frequencies of a Beam Supported on an Elastic Foundation
Natural Frequencies of Beams Subjected to a Uniform Axial Load
Transverse Vibration of a Beam via the Finite Element Method

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– Tom Irvine

Contact Form:

# Matlab Tips

This script demonstrates a Matlab anonymous function with variable coefficients. It also performs integration using the quad function, which uses adaptive Simpson quadrature. This example is from an actual engineering problem for beam vibration.

L=27.5;
beta=4.73004/L;
%
sigma1=( sinh(beta*L) +sin(beta*L))/(cosh(beta*L) -cos(beta*L));
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core=@(x) ((cosh(beta*x)-cos(beta*x))-sigma1*(sinh(beta*x) -sin(beta*x))).^2;
%