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: (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

See also:

Beam on an Elastic Foundation

Multispan Beams

Beams with Axial Load

Effective Modal Mass

Visual C++ Beam Program

Beam Bending, Finite Element Analysis

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:

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:

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:

8 thoughts on “Beam Bending Natural Frequencies & Mode Shapes

  1. Pingback: Plate Bending Frequency « Vibrationdata

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  4. Dear Tom Irvine,

    I read your paper called “Effective Modal Mass” and I’m in doubt about the physical meaning between generalized and effective modal mass.

    As example, the generalized modal mass of a simply supported beam is known as (0.5 * total sctructural mass). According to the paper, the effective modal mass for the same case is ( (8/pi²) * p L) ). [page 19].

    In case of a tuned mass damper (TMD – absorber) implementation (such as on a pedestrian bridge, a gym floor), it’s very important to know the structural modal mass to design one carefully.

    Should I choose the generalized or effective modal mass in this case? Could you give me some directions about this subject?

    Thank you.


    Cássio Gaspar.

    • Cássio Gaspar.

      The participation factor is used for certain response calculation cases. Examples include base excitation or uniform distributed force.

      The effective modal mass is not directly used in these response calculations. Rather the effective modal mass is used to judge the model accuracy and to determine whether enough modes are included.

      See for example:

      But there are many other types of dynamic analyses…

      So the use of generalized or effective modal mass would depend on the context and any simplifying assumptions.

      Please contact me via Email if you have a specific analysis problem.

      Thank you,
      Tom Irvine

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