Mode Acceleration and Inertia Relief for a Semi-definite Structural Dynamics System

NASAsoundingrocket

Aircraft and launch vehicles behave as unconstrained systems in flight, with six rigid-body modes. These vehicles may be “trimmed” using aerodynamic control surfaces and thrust vector control to prevent rotation about the vehicle center-of-gravity (CG).

There is a need to calculate the vehicle’s displacement response to wind, gusts, buffeting and other external forces. This process requires separating the rigid-body response from the elastic response. The elastic response is the relative displacement referenced to the CG displacement. The stress and strain can then be calculated from the elastic displacement response. The method is carried out by inertia relief, where rigid-body motion is constrained by applying an inertial acceleration that is opposite to the acceleration resulting from the external forces.

Here is a paper, which is a work-in-progress inertia_relief.pdf

See also:  The Mode Acceleration Method  MA_method.pdf

– Tom Irvine

Modal Test Problem & Solution

Ideally, a modal test on a structure would be performed with completely free boundary conditions.  This configuration can be approximated by mounting the structure on compliant air cushions, or by suspending it with elastic cords, so that the mounted natural frequency is much smaller that the structure’s fundamental frequency.

Other choices would be to test the structure with one boundary fixed, or in its final installation configuration.

But there may be certain cases where a structure can only be tested at its “next higher level of assembly.”  NASA is facing this issue for a launch vehicle which can only be tested on its launch platform and tower assembly due to cost and schedule reasons.

The modal test results will thus be for the complete system rather than the vehicle by itself.  But the need is for the vehicle’s modal parameters, which can then be used to calibrate the stiffness in a finite element model.  This would be for the case immediately after liftoff when the vehicle boundary conditions are free-free.  The vehicle natural frequencies and mode shapes are needed to check control stability, structural stresses, etc.

Here is paper which offers a potential solution by extracting the subsystem stiffness matrix from system level modal test results with a known mass matrix.  A simple three-degree-of-freedom system is used.  The parameters are conceptual only and do not represent those of the launch vehicle and its platform.

Note the reduction method in this paper may be similar to System Equivalent Reduction Expansion Process (SEREP) which is used in the automotive industry.

See also:

Receptance Decoupling for Two Rigidly Connected Subsystems

Determination of the Fixed-Base Natural Frequencies for a Two-degree-of-freedom System via Modal Test Receptance

– Tom Irvine

Transverse Vibration of a Rotating Beam via the Finite Element Method

rotating_beam

A function for calculating the natural frequencies and mode shapes for an elastic beam undergoing rotation is given in:

Matlab script: Vibrationdata Signal Analysis Package

The function can be accessed via:

>> vibrationdata > Miscellaneous > Structural Dynamics > Beam Bending > Rotating Beam, FEA

An option is included for calculating the response to a uniform force/length.

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See also:  Beam_FEM_rotating.pdf

– Tom Irvine

Beam Supported by End Springs

beam_springs
A function for calculating the natural frequencies and mode shapes for an elastic beam supported by end springs is included in:

Matlab script: Vibrationdata Signal Analysis Package

The function can be accessed via:

>> vibrationdata > Miscellaneous > Structural Dynamics > Beam Bending > Beam with End Springs, FEA

Options for calculating the beam response to an applied force will be added soon.

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See also:  Beam Bending Finite Element Analysis

– Tom Irvine

SDOF System Response to Initial Velocity & Displacement

Mass_spring

I have added a function for calculating the free vibration response of an SDOF system to initial conditions to the Vibrationdata GUI package.

Matlab script: Vibrationdata Signal Analysis Package

>> vibrationdata  > Miscellaneous > Structural Dynamics > Spring-Mass Systems > SDOF Free Vibration Response to Initial Conditions

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Reference: free.pdf

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– Tom Irvine

Cylindrical Shell Natural Frequencies & Mode Shapes

A function for calculating cylindrical shell natural frequencies via the wave method is now included in the Vibrationdata GUI package.

Matlab script: Vibrationdata Signal Analysis Package

This function may be accessed via:
vibrationdata > Structural Dynamics > Rings & Cylinders > Cylinder, Wave Model

Reference Papers

Study on applicability of modal analysis of thin finite length
cylindrical shells using wave propagation approach:  Link

Natural Frequencies of a Finite, Thin-Walled Cylindrical Shell: cylindrical_shell.pdf

– Tom Irvine

Rectangular Plate Bending Frequencies and Mode Shapes, Finite Element Method, GUI version

mm1

I have added a feature for rectangular plate bending modes to the Matlab GUI package.

Matlab script: Vibrationdata Signal Analysis Package

The script allows for the addition of point masses and point constraints.  This is a work-in-progress.  The next step will be to add base excitation options.

The new feature can be accessed via:

vibrationdata > Miscellaneous > Structural Dynamics > Plates, Rectangular & Circular > Rectangular Plate, Finite Element Method

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Here is a reference paper: FEA_plate_bending.pdf

See also:  Dynamic Response to Enforced Motion

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– Tom Irvine

Two-degree-of-freedom System Subjected to an Applied Force

twodof

I have added a feature for a two-degree-of-freedom system subjected to an applied force to the Matlab GUI package. The applied force options are steady-state sine, arbitrary time history and PSD.

Matlab script: Vibrationdata Signal Analysis Package

The new feature can be accessed via:

Vibrationdata > Miscellaneous > Structural Dynamics > Spring-Mass Systems > Two-DOF System Applied Force

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Here are some related papers:

Calculating Transfer Functions from Normal Modes
An Introduction to Frequency Response Functions
Semidefinite System Force
General Coordinate Method

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See also:  Two-degree-of-freedom Systems

– Tom Irvine

Plate Vibration Response to Oblique Acoustic Pressure Field

oblique

The Steady-State Response of a Baffled Plate Simply-Supported on All Sides Subjected to Harmonic Pressure Wave Excitation at Oblique Incidence:  ss_plate_oblique_incidence.pdf

A related paper is:

The Steady-State Response of a Baffled Plate Simply-Supported on All Sides Subjected to Random Pressure Wave Excitation at Oblique Incidence:  ss_plate_plane_wave_random.pdf

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The scripts for performing this calculation are given at:
Vibrationdata Signal Analysis Package

>> vibrationdata > Acoustics & Vibroacoustics > Vibroacoustics

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See also:

Steady-State Response of a Rectangular Plate Simply-Supported on All Sides to a Uniform Pressure:  ss_plate_uniform_pressure.pdf

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– Tom Irvine

Webinar 30 – Rectangular Plate Shock & Vibration

PowerPoint Slides: Webinar_30_rectangular_plates.pptx

Audio Visual File:

NESC Academy Rectangular Plate Shock & Vibration – Recommend viewing in Firefox with Sliverlight Plugin

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References:

Leissa, Vibration of Plates

Plate Base Excitation: plate_base_excitation.pdf

The Natural Frequency of a Rectangular Plate Point-Supported at Each Corner: plate_point_corner.pdf

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Matlab script: Vibrationdata Signal Analysis Package

Data files included in the Package zip file:

srs1000G_accel.txt & navmat_spec.psd

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See also: Vibrationdata Webinars

Thank you,

Tom Irvine