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 & Alternate Link

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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

Webinar 29 – Stress-Velocity Relationship

PowerPoint Slides:

Webinar_29_stress_velocity.pptx

Audio Visual Files:

Recommend viewing in Firefox with Sliverlight Plugin

NESC Academy Stress-Velocity Relationship Part I

NESC Academy Stress-Velocity Relationship Part II

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Acceleration Time History: Time (sec) & Accel (G) srs1000G_accel.txt

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References:

Stress-Velocity Relationship: sv_velocity.pdf

Bending Frequencies of Beams, Rods, and Pipes: beam.pdf

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

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Matlab script: Vibrationdata Signal Analysis Package

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See also:

Vibrationdata Webinars

Dr. Howard Gaberson’s Papers

Thank you,

Tom Irvine

Webinar 28 – Multi-degree-of-freedom SRS

PowerPoint Slides:  Webinar_28_mdof_SRS.pptx

Audio/Visual Files:

Recommend viewing in Firefox with Sliverlight Plugin

NESC Academy Multi-degree-of-freedom SRS Part I

NESC Academy Multi-degree-of-freedom SRS Part II

Three-dof System Example:   three_dof_srs_revC.pptx

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Acceleration Time History:  Time (sec) & Accel (G)    srs2000G_accel.txt

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References:

Two-Stage Isolation for Harmonic Base Excitation: two_stage_isolation.pdf

Shock Response of Multi-degree-of-freedom Systems: mdof_srs.pdf

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Matlab script: Vibrationdata Signal Analysis Package

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See also:

Vibrationdata Webinars

Thank you,

Tom Irvine

Tacoma Narrows Bridge

Tacoma Narrows Bridge Torsional Oscillation

Tacoma Narrows Bridge Torsional Oscillation

The original Tacoma Narrows Bridge was opened to traffic on July 1, 1940. It was located in Washington State, near Puget Sound.

Strong winds caused the bridge to collapse on November 7, 1940. Initially, 35 mile per hour winds excited the bridge’s transverse vibration mode, with an amplitude of 1.5 feet. This motion lasted 3 hours.

The wind then increased to 42 miles per hour. In addition, a support cable at mid-span snapped, resulting in an unbalanced loading condition. The bridge response thus changed to a 0.2 Hz torsional vibration mode, with an amplitude up to 28 feet. The bridge collapsed as a result.

Here is a paper that gives an explanation of the failure in terms of aerodynamic self-excitation: Tacoma Narrows Bridge Failure

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Here is a brief video clip that shows both the bending and torsional motion: video

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See also:

Volgograd Bridge Oscillation

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– Tom Irvine

Low Frequency Structural Loads for Secondary Structures & Components

Method 1

Secondary & Component loads can be specified in the form of a Mass-Acceleration curve as described in:

NASA-STD-5002, Paragraphs 4.2.1.2, 5.3.2, 5.4.

The Mass-Acceleration curve can be derived from measured flight or test data, or analytical loads. Care must be taken to avoid “double dipping,” if two or more sources are used.

This method effectively assumes that the component’s fundamental frequency is much higher than the excitation frequencies. The resulting analysis becomes a rigid-body analysis.

Method 2

Component loads can be expressed as a Vibration Response Spectrum (VRS).

This assumes that the component responds as a single-degree-of-freedom (SDOF) system subjected to base acceleration.

The Y-axis for this function is peak response acceleration. The X-axis is natural frequency (Hz). The amplification factor Q must also be noted. A family of curves for various Q factors can be included in a single plot.

The VRS curve(s) can be derived from measured flight or test data, or analytical loads. Again, care must be taken to avoid “double dipping,” if two or more sources are used.

The resulting VRS can then be used for design purposes by picking off the peak response acceleration value for a given natural frequency and Q.

A similar VRS can be derived for relative displacement. This is important for cases where clearance, sway space, alignment, or isolator deflection are concerns.

My colleagues and I used the VRS method at my previous workplace, Orbital Sciences Corporation, Chandler, AZ.

I have posted references for this method at:  VRS Link

Method 3

A measured or synthesized acceleration time history can be used to base drive a finite element model of the component, as a modal transient analysis.

The time history can be derived from measured flight or test data, or analytical loads. Again, care must be taken to avoid “double dipping,” if two or more sources are used.

I also occasionally used this method at Orbital Sciences Corporation.

Shaker Table Testing

Avionics components are typically tested over the frequency domain 20 to 2000 Hz for random vibration.

Components may also need to withstand transportation vibration below 20 Hz.

See SMC-TR-06-11, sections 3.24 – 3.26

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See also:

Launch Vehicle Coupled Loads Analysis (CLA) Upper Frequencies

Equivalent Static Loads for Random Vibration

Mass Acceleration Curves

Effective Modal Mass

JPL D-5882 Trubert

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– Tom Irvine

Launch Vehicle Coupled Loads Analysis (CLA) Upper Frequencies

NASA-STD-5002, Load Analyses of Spacecraft and Payloads, 1996

5.3.2 Transient load analysis (excerpt)

Statistical variation of parameters governing the forcing functions is accommodated by generating multiple cases of forcing functions for a single flight event. After all cases are analyzed, the maximum transient load is taken as the largest load from any of the cases. The frequency range of transient load analysis is limited by the accuracy of both the model and the forcing functions. For Shuttle liftoff and landing, transient analysis generally accounts only for frequencies from 0 to 35 Hz. For some expendable launch vehicles, significant axial loads are generated at higher frequencies during engine cutoff events, and transient analyses must be run to 60 Hz or higher. For other events such as docking, robotics berthing, plume impingement, and spacecraft landing, the modal content must be selected to adequately capture the dynamic response.

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NASA-HDBK-7005, Dynamic Environmental Criteria, 2001

5.1 Low Frequency Vibration and Transient Responses. Low frequency vibration and transient responses of payloads and spacecraft result in loads and motions that must be determined analytically to evaluate structural integrity and functionality. From the Sections referenced in Table 5.1, the primary sources of these low frequency loads are pre-launch events(ground winds and possible seismic loads), liftoff (engine/motor thrust buildup, ignition overpressure, and pad release), airloads (buffet, gust, and static-elastic), and liquid engine ignitions and shutdowns. These events have an upper frequency limit that is dependent on the launch vehicle and the stage of its operation, e.g., 35 Hz for Shuttle, 50-60 Hz in most cases for expendables. Major load events due to spacecraft operation are deployments and transients peculiar to the mission, such as docking and landing. Most cases involve linear system response, but nonlinear responses occur in certain cases. Examples are the account of trunnion sliding for Shuttle payloads, the liftoff release mechanism for Atlas, and the response of spacecraft deployment and docking mechanisms.

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SMC-TR-06-11,Test Requirements for Launch, Upper-Stage,
and Space Vehicles, AEROSPACE REPORT NO. TR-2004(8583)-1 REV. A

3.24 Maximum Predicted Acceleration

The maximum predicted acceleration, defined for structural loads analysis and test purposes, is the highest acceleration determined from the combined effects of quasi-steady acceleration, vibration and acoustics, and transient flight events (liftoff, engine ignitions and shutdowns, flight through transonic and maximum dynamic pressure, gust, and vehicle separation). The frequency range of concern is usually limited to below 70 Hz for structural loads resulting from the noted transient events, and to below 300 Hz for secondary structural loads resulting from the vibration and acoustic environments. Maximum accelerations are predicted for each of three mutually perpendicular axes in both positive and negative directions. When a statistical estimate is applicable, the maximum predicted acceleration is at least the acceleration that is not expected to be exceeded on 99 percent of flights, estimated with 90 percent confidence (P99/90) (10.2.1).

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European Cooperation for Spacecraft Standardization (ECSS), Spacecraft Mechanical Loads Analysis Handbook:
ECSS-E-HB-32-26A_19February2013

4.6.1 Spacecraft flight environments and dynamic loads

Launch consists of a series of events, each of which has several independent sources of load for the launch vehicle and payload. The flight environments that generate static and dynamic loads on spaceflight hardware are normally categorized as follows (e.g. [1] [2]):

• The static acceleration, generated by constant external forces or which change slowly with time so that the dynamic response of the structure is not significant (also called quasi-static acceleration associated to a quasi-static event).
• The low-frequency dynamic response, typically from 0 Hz to 100 Hz, of the launch vehicle/payload system to transient flight events. However for some small launch vehicles the range of low-frequency dynamic response can be up to 150 Hz.
• The high-frequency random vibration environment, which typically has significant energy in the frequency range from 20 Hz to 2000 Hz, transmitted from the launch vehicle to the payload at the launch vehicle/payload interfaces.
• The high frequency acoustic pressure environment, typically from 20 Hz to 8000 Hz, inside the payload compartment. The payload compartment acoustic pressure environment generates dynamic loads on components in two ways: (1) by direct impingement on the surfaces of exposed components, and (2) by the acoustic pressure impingement upon the component mounting structures, which induces random vibrations that are mechanically transmitted to the components.
• Shock events. The energy spectrum is usually concentrated at or above 500 Hz and is measured in a frequency range of 100 Hz to 10 KHz [15].

4.6.6 Spacecraft-launcher coupled loads analysis

The structural response of the spacecraft to transient flight events (low frequency mechanical environment) is simulated by spacecraft-launcher coupled dynamic analysis, which is a key task of the loads cycle process. The coupled loads analysis is a transient (or harmonic) analysis performed by using the mathematical models of the spacecraft and launcher, merged together, and by applying the forcing functions for the different launch events.
The main objective of the CLA is to calculate the loads on the spacecraft, where the term “loads” refers to the set of internal forces, displacements and accelerations that characterise the structural response to the applied forces. The loads of the spacecraft derived from the analysis are taken as a basis to verify the dimensioning of the spacecraft itself.

The low frequency domain typically ranges from 0 to up 100 Hz and corresponds to the frequency content of the forcing functions used in the CLA. The excitation may be of aerodynamic origin (wind, gust, buffeting at transonic velocity) or may be induced by the propulsion system (e.g. thrust build up or tail-off transient, acoustic loads in the combustion chambers). Of primary interest are the spacecraft interface accelerations and interface forces. The interface accelerations can be used to derive an equivalent sine spectrum at the spacecraft interface. The interface forces can be employed to calculate the “equivalent accelerations” at the spacecraft centre of gravity (quasi-static accelerations). Of large interest is also the recovery of the internal responses which are used to verify the structural integrity of the spacecraft and its components. The computed responses and their deduced minimum and maximum levels can be employed within the design, verification and test phases of the spacecraft. For example, secondary structures and flexible components such as solar arrays, booms, instruments and propellant tanks are also designed (and test verified) to withstand the dynamic environment induced at the base of the spacecraft. The dynamic loads (e.g. accelerations, forces, stresses) on these components can be verified directly by means of the CLA (apart from acoustic loads under the fairing which are analysed separately).

In the test verification phase of the spacecraft, the equivalent sine spectrum computed by means of the CLA is often used to assess and justify the reduction, at specific resonant frequencies, of the spectrum specified by the launcher authority. This might be required to avoid possible damage to the spacecraft structure itself or its components (e.g. solar arrays, booms).

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See also:

Low Frequency Structural Loads for Secondary Structures & Components

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– Tom Irvine

Webinar 17 – SDOF Response to Applied Force

PowerPoint File:

webinar_17_SDOF_force.pptx

Audio/Visual File:

NESC Academy SDOF – Response to Applied Force – Recommend viewing in Firefox with Sliverlight plugin

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Reference Papers:

T. Irvine, Machine Mounting for Vibration Attenuation, Rev B, Vibrationdata, 2000 link

Bruel & Kjaer Booklets:
Mobility Measurement link
Modal Testing link

Additional reference papers are posted at:

Response of an SDOF System to an Applied Force PSD or Acoustic SPL

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Matlab script: Vibrationdata Signal Analysis Package

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Python version:

webinar_17_SDOF_force_python.pptx

Python script: Python Signal Analysis Package GUI

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See also: Vibrationdata Webinars

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– Tom Irvine

Semidefinite Two-degree-of-freedom System Subjected to a Sinusoidal Force

two_dof_semidefinite

Here is a paper which gives a derivation of the equations of motion:  semidefinite_force.pdf

It also covers the transfer functions which can be calculated if the rigid-body mode is omitted.

Here is a Matlab script:  semidefinite_force.zip

The main script is:  semidefinite_force.m

The remaining scripts are supporting functions.

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See also:  Two-degree-of-freedom Systems

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– Tom Irvine