Fatigue Damage Spectrum (FDS) is useful for determining the total accumulated potential relative or “pseudo” damage between different vibration signals in the natural frequency domain. It is used in shaker vibration testing to help assess the durability of products and other mechanical objects.
The Fatigue Damage Spectrum (FDS) can be used to:
Compare the damage potential of two or more different testing scenarios.
Accelerate tests that deliver the same amount of fatigue damage in less time.
FDS is calculated from an acceleration time history and involves multiple steps as shown in Figure 1.
This article explains how FDS is calculated conceptually, as well as instructions on calculating FDS with the Simcenter Testlab Mission Synthesis Workbook.
Contents: 1. Terms 2. Fatigue Damage Spectrum Overview 3. Calculation Process 3.1 SDOF Time History Response 3.2 Cycle Counting 3.3 Displacement to Stress 3.4 Damage 3.5 Result 4. Key Calculation Parameters 4.1 Q-Factor 4.2 Woehler Exponent b 4.2 Woehler Constant A 4.3 Woehler Constant A and Stress/Displacement Constant K 5. Calculating Fatigue Damage Spectrum in Testlab 5.1 Getting Started 5.2 Identifying Data 5.3 Performing Calculations 5.4 Sum Versus Envelope 5.5 Display 5.6 Results Storage
DUT – Device Under Test FDS – Fatigue Damage Spectrum SRS – Shock Response Spectrum MRS – Maximum Response Spectrum SDOF – Single Degree of Freedom system Q-factor – Quality Factor
2. Fatigue Damage Spectrum Overview
Fatigue Damage Spectrum (FDS) provides a way of evaluating field data to determine the maximum potential damage versus frequency of any recorded signal. Some examples of input and output plots are shown below in Figure 2:
Figure 2: Three acceleration inputs (top graph: red/green/blue) and their corresponding Fatigue Damage Spectrums (bottom graph: red/green/blue) outputs.
The Fatigue Damage Spectrum clearly indicates that the “accelerate brake” (red) has less damage at all frequencies than the “Belgian block” (green) and “Saw Tooth” (blue) signals. The lower frequencies contribute more damage than the higher frequencies, which is important if trying to recreate the damage on a shaker.
Fatigue Damage Spectrum is usually used as a relative indicator rather than an absolute indicator of the amount of fatigue damage generated from an acceleration signal. The absolute accuracy of an FDS is dependent on the accuracy of the inputs required to calculate it, particularly the material stress-cycle curve (SN-Curve), also known as a Woehler curve.
3. Calculation Process
Fatigue Damage Spectrum (FDS) is calculated by determining the number and amplitude of cycles seen by a virtual series of damped Single Degree of Freedom (SDOF) systems which are tuned to different natural frequencies. The cycles are then converted to a proportional amount of cumulative damage experienced by the Device Under Test (DUT) at each SDOF natural frequency.
Given an acceleration time signal, the process of calculating an FDS follows these steps:
3.1 SDOF Time History Response
The response time history of a set of virtual SDOF systems is calculated relative to a base excitation from the recorded time history (i.e. the acceleration signal from which a FDS is to be calculated) as illustrated in Figure 3.
Figure 3: A recorded acceleration time history (blue, left) is applied to the base of an array of virtual SDOF systems (center) to generate relative displacement responses (z, right side).
Each virtual SDOF system consists of a mass, stiffness, and damping value. There are multiple virtual SDOF systems, each tuned to a specific natural frequency by adjusting the mass versus stiffness. The SDOF natural frequencies are selected to cover the entire frequency range of interest.
The acceleration is converted to displacement by integrating twice. In frequency domain, integration and differentiation simply boils down to respectively division and multiplication by jω, where ω is the frequency in radians per second.
Note: This process is similar to how Shock Response Spectrum (SRS) is calculated, except FDS is based on relative displacement (z) between the base excitation and response. Instead of the relative displacement, SRS is calculated based on the absolute acceleration of the virtual SDOF system response.
More about natural frequencies, resonance, and SRS:
Cycle counting is performed on the calculated displacement time history of each SDOF system to determine the displacement amplitudes and number of cycles at each amplitude. This can be either counted directly (Figure 4) using zero-max or range-pair methods, or via a process called rainflow counting (Figure 5).
Figure 4: Example of direct cycle counting (Range-Pair method) that generates the number of cycles (n) versus cyclic amplitude (z) histogram (bottom graph).
Direct counting methods result in a histogram. The histogram consists of the number of cycles (n) versus the amplitude (z) of the cycles. Direct methods include:
Range-Pair: The amplitude between turning points is used to determine the amplitude.
Zero-Max: Each time the signal crosses zero, the maximum value between crossings is used for the amplitude.
Instead of half cycles as derived from a direct counting method, the rainflow method looks at closed cycles as shown in Figure 5.
Figure 5: Rainflow counting of a time history.
A closed cycle means that two inner points from the time history (in Figure 5, the 2nd and 3rd point in order from left to right) are contained within the amplitude of two outer points (in Figure 5, the 1st and 4th point). The cycle results are stored in a rainflow matrix. A histogram can be derived from a rainflow matrix.
More information on the rainflow counting process can be found in this knowledge article: Rainflow Counting
In the next step, the cycle displacements are converted to stress.
3.3 Displacement to Stress
The displacement amplitude cycles are multiplied by a stress/displacement constant, K, to convert the dataset into stress amplitude cycles. The equation is shown in Figure 6.
Figure 6: The relative displacements (z) are converted to stress (σ) using a stress displacement constant (K).
The K constant has units of stress (example: N/m2) over displacement (example: m). In this example, the K constant units are N/m3.
The resulting stress cycles are used to calculate a damage value for each SDOF frequency using a material property curve (Woehler curve).
The damage total from each SDOF system response is calculated using the equation in Figure 7:
Figure 7: Damage calculation for one SDOF system of a Fatigue Damage Spectrum.
This equation includes using the K constant to convert displacement (z) to stress (σ) which was explained in the previous step. Instead of a separate step, it is included in the damage calculation.
The equation also uses the Woehler material information (exponent b and constant A) to calculate damage. The material information is combined with Miner’s Rule to determine the damage potential of the SDOF signal being analyzed (Figure 8):
Figure 8: Total Accumulated Damage is calculated from the stress cycles and a Woehler curve using Miner’s rule.
Miner’s Rule uses the sum of the ratio of the number of cycles at each amplitude (n) over the number of cycles to failure (N) at that respective amplitude on a Woehler curve. The sum of these ratios yields total damage. Each SDOF system gets its own damage total.
More information on Miner’s Rule for Damage can be found here:
When the damage is plotted for each SDOF frequency, the result is a Fatigue Damage Spectrum (Figure 9):
Figure 9: Fatigue Damage Spectrum shows the total amount of damage (Y-axis) at each frequency (X-axis) for the input acceleration signal.
This “Fatigue Damage Spectrum” plot shows the total damage at each natural frequency of the SDOF systems used in the calculation. A higher amplitude at a given frequency indicates greater damage at that frequency.
The amplitudes in the FDS are not typically scaled as an absolute measure of fatigue damage (i.e., where FDS = 1 means the DUT will fail). The amplitude values are usually proportional to the amount of damage at each frequency, so that relative comparisons between different FDS spectrums can be made. For the FDS to have an accurate lifetime estimation, all parameters (A, b, K, Q-factor, etc.) need to be tuned appropriately.
There are many parameter settings that need to be entered in the software interface of Mission Synthesis to calculate a Fatigue Damage Spectrum (FDS). Some are shown below in Figure 10.
Figure 10: Key calculation parameters include damping, material properties (Woehler curve), and conversion from displacement to stress.
These parameters influence the results of the FDS calculation. This section describes some of these parameters.
In a SDOF system, the damping property controls the peak response as shown in Figure 11 below:
Figure 11: Damping properties of a SDOF system (left) or material controls the amplitude response (right).
In the Simcenter Testlab Mission Synthesis interface, damping for the SDOF systems is set using the Quality Factor, or Q-Factor. For the same input excitation, a high Q-factor results in a higher amplitude Fatigue Damage Spectrum (FDS) whereas a lower Q-factor results in less damage (Figure 12):
Figure 12: For the same excitation signal, a high Q-factor results in a high amplitude FDS (blue). A low Q-factor results in low amplitude FDS (red).
In Mission Synthesis the same Q-factor is used for all SDOF systems. After selecting a Q-factor, it is advisable to use the same factor throughout the subsequent analysis.
A high Q-factor is lightly damped (the response of a steel bell when rung). A low Q-factor is highly damped (the same bell but made of soft rubber). Examples of the frequency response of a SDOF system with two different Q-factors are shown in Figure 13 below.
Figure 13: For a Q-factor of 10, the response of a SDOF is higher than a Q-factor of 2. A Q-factor of 10 has less damping than a Q-factor of 2.
The menu interface of Mission Synthesis allows damping to be entered two different ways: Q-factor or percent damping. These two quantities have a fixed relationship:
Q-factor of 10 corresponds to a damping percentage of 5%.
A Q-factor of 5 equates to 10 percent damping.
Changing one value will change the other in the software interface. Q-factor is inversely proportional (with a factor of 2 thrown in) to damping percentage (%Cr).
The Woehler exponent b is very important in the calculation of damage, even for a relative or “pseudo” damage. The exponent b controls the slope of the material curve as shown in Figure 14:
Figure 14: When the Woehler exponent b is increased, there are more cycles required to reach failure. This results in a lower total amount of damage compared to a smaller exponent b.
A more gradual slope (higher numerical value for exponent b) results in lower damage than a steeper slope (lower numerical value for exponent b). An example of the same acceleration signal processed with only changes in the exponent b are shown in Figure 15 below:
Figure 15: Lower Woehler exponent b values result in higher amount of damage per frequency for the same input signal. The shape of the spectrum versus frequency is also altered.
The selection of the Woehler exponent b has a significant influence on the fatigue damage spectrum. Not only is the amplitude of the resulting spectrum different, but the shape is also changed. The levels of the different frequencies are shifted with respect to each other. Because the shape is changed, and the damage is not simply scaled by a constant, it is very important to have the Woehler exponent b to be correct, even when making relative comparisons.
Using the smallest values of b leads to safest testing. In other words, small b levels generate the highest damage, thus creating the most conservative estimates of damage. The main risk of a small b level is that the Device Under Test (DUT) will be over tested, which requires it to be overdesigned.
Typically, exponent b has a positive value that ranges between 3 and 12. Steel has a value of 5, aluminum has a value of 8. Some sources for determining the value of exponent b:
NATO AECTP-200 Environmental Conditions Guide: Electronics components have an exponent b value of 5.
Mechanical Environment Test Specification by Christian Lalanne: Appendix for “Numerical Values of Parameter b”
When testing a product that has multiple materials, the most conservative approach is to choose the lowest value of b for the materials being tested.
Note: In other material curve definitions, the letter “k” is often used for the slope rather than “b”.
4.3 Woehler Constant A
The Woehler constant A works in conjunction with the Woehler exponent b to define the material properties of the Device Under Test (DUT). The Woehler constant A defines the intercept of the curve while the Woehler exponent b determines the slope of the material curve.
Changing the Woehler constant A shifts the material curve as shown in Figure 16.
Figure 16: When Woehler constant A is increased, more cycles of a given stress level are required for failure.
The Woehler constant A has units of stress.
The amount of damage generated by the same input signal is inversely proportional to the Woehler constant A as shown in Figure 17.
Figure 17: A Woehler constant of 1 (blue curve) generates more damage for a given signal than a Woehler constant of 6 (red curve).
When the Woehler constant is increased, more cycles of a given stress level are required for failure, and thus less damage is generated.
4.4 Woehler Contant A and Stress/Displacement Factor K
In the damage calculation, the Stress/Displacement Factor K and Woehler constant A are in the numerator and denominator respectively.
If the A and K values are the same, the damage is not changed. This is shown in Figure 18 below:
Figure 18: Woehler Constant A and Stress/Displacement Factor K.
If an absolute level for the Fatigue Damage Spectrum is required, then the A and K values must be set appropriately. If only relative comparisons are made, setting both A and K values to one causes the two parameters to cancel each other out. This is often the case when working with the Fatigue Damage Spectrum for comparison purposes only.
If comparisons are being made for the same Device Under Test (perhaps with different acceleration inputs), then relative comparisons are meaningful. For different Devices Under Test (perhaps made with different materials), the FDS will not yield meaningful comparisons unless all parameters (A, H, b, Q-factor, etc) are selected appropriately.
Press the Mission Synthesis workbook at the bottom of the screen. The software has three key sections (Figure 21):
Figure 21: Key areas of the Simcenter Testlab Mission Synthesis software interface.
The key sections include:
Data Selection: Used to durability time histories or power spectral density functions for the fatigue damage calculations.
Calculator: Add repetition factors to the data, calculate the FDS function, and sum or envelope the results.
Display: View and overlay the results of the calculations.
5.2 Identifying Data
Use the data selection navigator to find the data of interest and add it to the calculator as shown in Figure 22.
Figure 22: Four steps for adding data to be analyzed to Mission Synthesis calculator.
Steps illustrated above include:
Finding time history data in Navigator.
Highlighting the channel to be analyzed (in case the recording includes more channels than of interest).
Selecting the key attribute that will identify the data. For example, run name might be “Twist turns” while the Point Id is “Left Front”. Choose which attribute will be prominent.
Click the folder icon to move from data selection to calculator.
This process can be repeated until all environments for the Fatigue Damage Calculation have been selected.
5.3 Performing Calculations
For each event that was selected in the previous step, enter a repetition factor as shown in Figure 23:
Figure 23: Enter a repetition for each event in the Mission Synthesis calculator.
What does the repetition factor mean? For example, if the “Accelerated Brake” recording contained two hours of vibration that the product experiences, a repetition factor of 20 would calculate the equivalent fatigue damage as if the product had experienced 40 hours total of the same vibration.
If no repetition factor is entered, the software automatically assumes a repetition factor of 1.
The Fatigue Damage calculation is initiated by highlighting the events and pressing the “Excitation -> MRS/FDS” button as shown in Figure 24:
Figure 24: Enter a repetition for each event in the Mission Synthesis calculator.
After the calculation is done, a “V” check is added to the event table in the FDS column to indicate the calculation has been performed.
The steps show in Figure 24 above are:
Highlight the recordings/events for which the Fatigue Damage Spectrum calculation is to be done. Each event should have a repetition factor, if required. Not entering a repetition factor will default to a factor of “1” being used by the software.
Press the “Excitation -> MRS/FDS” button. A menu appears for entering several parameters (see below).
A “V” appears in the checkbox when the calculation has been done.
The “Excitation -> MRS/FDS” has several settings as shown in Figure 25.
Figure 25: Settings in the “Excitation -> MRS/FDS” menu.
Some of the key parameters are explained here:
5.3.1 Material parameters:
Q-Factor/Damping: Damping value for each Single Degree of Freedom (SDOF) system. Damping effects the amount of displacement response to the excitation input. Both the Q-factor and Damping percentage are the same quantity and are inter-related – changing one changes the other. The higher the Q-factor, the lower the damage. See previous section for more information.
Woehler constant A and exponent b: Defines material properties, the exponent b effects the damage calculation. The lower the exponent b, the higher the damage. See previous section for more information.
K: Constant to convert displacement to strain.
See previous section for more information on the material properties.
5.3.2 Situation parameters:
Cycle count method: Determines how the damage cycles are determined from time history. Choices are range-pair, zero-max, and rainflow.
See previous section for more information on the counting methods
5.3.3 Frequency axis parameters:
The frequency range and spacing of the calculated results. A virtual SDOF is calculated at each result frequency.
5.4 Sum Versus Envelope
The Fatigue Damage Spectrum of each separate event can be summed or have a peak envelope calculated. This is helpful for determining the total amount of damage from multiple events.
A sum is performed by highlighting the events and pressing the “Sum FDS” button as shown in Figure 26:
Figure 26: FDS sum (blue) of two environments (red/green) performed in Simcenter Testlab Mission Synthesis.
A sum is useful if both environments (for example shipping vibration from a train AND a plane) will be experienced by a product.
On the other hand, if the product will be shipped by either train OR plane, then the “Env FDS” button should be used (Figure 27).
Figure 27: FDS peak envelope (blue) of two environments (red/green) in Simcenter Testlab Mission Synthesis.
The “Env FDS” button calculates a peak envelope of the environments. This way, the Fatigue Damage Sum is the worst case for both environments.
After performing calculations in Simcenter Testlab Mission Synthesis, the results are displayed by highlighting the row in the calculator with the data of interest. Then press the DISPLAY button (with the red data trace icon) to view it (Figure 28).
Figure 28: Highlight the times of interest to display (top middle calculator area) and press the red signal button (just to left of calculator area) to view the excitation, FDS, and MRS data.
There are three types of data that can be displayed (marked with “V” symbol in the calculator if available to display):
Excitation (top): Original input acceleration, in time or Power Spectral Density format
MRS (middle): Maximum Shock Response spectrum, the peak acceleration due to the input to the SDOF systems
FDS (bottom): Fatigue Damage Spectrum
Whenever a calculation button is launched, the display button must be used to see the results. The displays do not automatically refresh without pushing the button.
5.6 Results Storage
The calculated functions (FDS and MRS) are stored in the active project/section in a folder called “Mission Synthesis Data” (Figure 29).
Figure 29: Calculated Fatigue Damage Spectrum is stored in the “Mission Synthesis Data” folder.
There is a subfolder with the Environment Name to distinguish between different datasets.
Hope this gives a good overview of Fatigue Damage Spectrum (FDS) and how to calculate it in Simcenter Testlab. Further information on the FDS can be found in the book “Mechanical Environment Test Specification Development Method” by Christian Lalanne.