Material Damping


Internal friction and heat generation cause a deviation from perfect elasticity, resulting in the hysteresis loop in the stress-strain curve.

The damping capacity  ψ is:

ψ =  ΔW/W

where W is the initial strain energy and ΔW is the energy lost during one cycle of oscillation.

Some damping capacities of common materials are:


There is some tendency for damping capacity to increase with at higher stresses. Damping also depends on temperature, strain rate and other parameters.


Maringer, Damping Capacities of Materials

Luxfer, Magnesium Fixtures

See also:    Damping References

–  Tom Irvine

State-Space Method for Systems with Dashpot Damping


Structural dynamics systems can be represented in terms of mass, damping and stiffness matrices.  Each of these matrices may be coupled depending on the model complexity, degrees-of-freedom, etc.   The mass and stiffness matrices in the assembled equation of motion may be uncoupled using the normal modes for the undamped system.  This approach gives real natural frequencies and real mode shapes.

Damping effects can be included in forced response analyses by implicitly assuming that the damping matrix can be diagonalized into modal damping coefficients by the undamped modes.   But systems with dashpots in general have damping matrices which cannot be uncoupled in this manner.

The state-space method is useful for modal and forced response analysis of systems with discrete dashpot damping.  This approach yields complex natural frequencies and mode shapes, with real and imaginary components.

Here is a paper:

Two-Degree-of-Freedom System, State-Space Method:   two_dof_state_space_revC.pdf

More later…

– Tom Irvine

Damping Identification from Shock Data via Wavelet Responses


Structural system & component damping can measured via modal testing with applied force excitation.  One excitation method is an impulse hammer test.  Another method is a  small shaker attached to structure via a stinger rod.

Damping can also be measured by mounting the test unit on a shaker table and applying base excitation.

There is a need to estimate component damping from pyrotechnic or pyrotechnic simulation shock tests where the source energy measurements are incomplete or unavailable.  This need may arise because modal and shaker table test data is unavailable.  Furthermore, damping is nonlinear and may be higher for a pyrotechnic shock event than for a modal or shaker table test

A source shock waveform can be modeled by a series of wavelets per Ferebee R , Irvine T, Clayton J, Alldredge D,  An Alternative Method Of Specifying Shock Test Criteria,  NASA/TM-2008-215253.   This method was original develop to characterize space shuttle solid rocket booster water impact shock.  This method can be extended for natural frequency and damping measurement for shock data.

The Wavelet Response Curve-fit Methodology is appropriate for mid and far field shock measurements where modal responses appear in the accelerometer time history data.

The method is implement via the following steps:

  • Assume a series of wavelet as the base input
  • Calculate the response of one or more SDOF systems to the assumed wavelet series
  • Subtract the resulting response signal from the measured accelerometer data and calculate error
  • The goal is to repeat this process thousands of times where the to minimize the residual error
  • Each of the following parameter are varied randomly with convergence
  • For each base input wavelet: frequency, amplitude, number of half-sines, delay
  • For each SDOF oscillator response: natural frequency, damping

The method yields natural frequency and damping estimates for the response acceleration.  It also gives an estimate of the assumed base input source shock.

The method is demonstrated in the following slides.

Slides:  shock_wavelet_damping_revC.pptx

Matlab scripts & sample shock data: Vibrationdata Signal Analysis Package


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Reference Papers:

An Alternative Method Of Specifying Shock Test Criteria:
NASA/TM-2008-215253 & PowerPoint slide overview


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– Tom Irvine

Damping References


Structural damping depends on materials, joints, boundary conditions, friction, acoustic radiation, etc.  Damping may be nonlinear, increasing as the excitation and response levels rise. The damping for a given structure or component must be measured!  

But here is some empirical background data…


Damping Properties of Materials

Damping Cross-Reference

Damping Values in Aerospace Structures and Components

Free Vibration with Coulomb Damping

Damping Identification Rev A

Related Blog Posts

Honeycomb Sandwich Panels

Vibroacoustics/Statistical Energy Analysis  – empirical formulas for loss factors

Wire Rope Cable Damping

Tall Building Natural Frequencies and Damping

Piezoelectric Shunt Damping

Nonlinear Modeling of Bolted Interfaces & Joints

Convert Modal Damping to a Damping Coefficient Matrix

Vibration Absorbers & Tuned Mass Dampers

Half-Power Bandwidth Method


Isolator Photo Gallery

Avionics Box Isolation

Transmissibility of a Three-Parameter Isolation System

Webinar 43 – Two-degree-of-freedom System, Two-stage Isolation

– Tom Irvine

Wire Rope Cable Damping

Wire rope damping depends on:

1. Material
2. Diameter
3. Length
4. Number of strands
5. Number of turns per length
6. Initial tension or preload
7. Displacement amplitude

The damping can be very nonlinear, particularly depending on displacement amplitude.

The flexing of wire rope involves both coulombic and viscous damping. At very low vibration levels, the wire strands stick together and little sliding occurs. The damping is low and the behavior is viscous.

With higher displacements, coulomb damping predominates as the wires break free and start to slide against each other, absorbing large amounts of energy.

At large displacements the bending and stretching of the wire strands overshadows the sliding friction and viscous behavior again starts to show.

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Here are some viscous damping values for straight cables.

Longitudinal Vibration for 10 mm (0.39 inch) diameter:
Damping = 0.7% to 2.0% (ref 1)

Transverse Vibration for 3/8 inch diameter:
Damping = 0.3% to 0.5% (ref 2)

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For the case of wire rope used in helical isolators, the damping can vary from 5% to 22% (ref 3).

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Wire Rope Isolation of a Camera Video

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1. Feyrer, Wire Ropes: Tension, Endurance, Reliability

2. Vanderveldt, Chung and Reader, “Some Dynamic Properties of Axially Loaded Wire,” Experimental Mechanics, 1973.


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See also:

Initial investigations into the damping characteristics of wire rope vibration isolators

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– Tom Irvine

FRF Substructuring

This post is a work-in-progress…

Frequency Response Function based substructuring (FBS) predicts the dynamic behavior of a coupled system on the basis of free interface FRFs of the uncoupled components.

Here is a tutorial paper:  Frequency Response Function Substructuring 

Modal damping requires special consideration:  Notes on Damping in FRF Substructuring

This script sets up the initial mass and stiffness matrices for the systems in the tutorial paper: spring_mass_series.m

See also:

Transfer Functions from Normal Modes

Joint Receptance for Rigid & Elastically Coupled Subsystems

Matlab script: joint_receptance.m

Reference Papers

K. Cuppens, P. Sas, L. Hermans, Evaluation of the FRF Based Substructuring and Modal Synthesis Technique Applied to Vehicle FE Data:   Download

Dr. Peter Avitabile, Impedance Modeling & Frequency Based Substructuring:  Download

Matthew S. Allen, Randall L. Mayes, Comparison of FRF and Modal Methods for Combining Experimental and Analytical Substructures: Download

Randy L. Mayes, Patrick S. Hunter, Todd W. Simmermacher, Matthew S. Allen , Combining Experimental and Analytical Substructures with Multiple Connections:  Download

Mladen Gibanica, Experimental-Analytical Dynamic Substructuring, A State-Space Approach:  Download

Convert Modal Damping to a Damping Coefficient Matrix

Here is a method for converting modal damping to a damping coefficient matrix for all modes.  It is based on a pseudo inversion of the modal damping matrix.

The resulting coefficient matrix will be fully populated. The damping terms will not necessary corresponding to physical dashpots, but the modal damping will be fully represented. This is useful for the case where a direct solution method will be used, such as the Newmark-beta method.


Here is a Matlab script:

– Tom Irvine

Vibration Absorbers & Tuned Mass Dampers


A vibration absorber is a tuned spring-mass system which reduces the vibration of a harmonically excited system.

Here is a paper for an applied force:  Vibration Absorber

Here is a Matlab script and its supporting function:  vibration_absorber_trans.m


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Here is a paper for base excitation: Vibration Absorber Base Excitation

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See also:

The London Millennium Bridge Vibration

Taipei 101 Building Tuned Mass Damper

The Citicorp Building Tuned Mass Damper

– Tom Irvine

Tall Building Natural Frequencies and Damping


The Transamerica Pyramind has a Fundamental Frequency of 0.3 Hz

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Here are some papers which give natural frequencies and damping values for sample tall buildings:

Damping in Tall Buildings and Towers: tall_building_damping.pdf

The Damping Characteristics of Bolted and Welded Joints: bolted_joint_damping.pdf

Transamerica Pyramid Design: pyramid.pdf

NESC Academy Audio/Visual File: Building Natural Frequencies

– Tom Irvine

Fluid Structure Coupling Damper

Here is a project that my NASA colleagues have been working on:  FSC_damper.pdf

The original goal of the project was to use the Ares I second stage as a LOX damper to mitigate the first stage  thrust oscillation.

See also:

– Tom Irvine