Load Spectra in the FVA-Workbench
Image: © FVA GmbH
Realistic Load Modeling: The Role of Load Spectra
In the design and validation of mechanical systems, especially in complex drivetrain applications, engineers are often faced with the challenge of accurately representing operational loads. Conventional design methods typically rely on nominal, static, or extreme load cases to ensure component safety. While these approaches provide a baseline level of robustness, they often lead to overly conservative designs, increased material use, and limited insight into actual component behavior under realistic conditions.
Load spectra offer a more sophisticated alternative by statistically describing the frequency and magnitude of varying loads that occur during operation. This enables fatigue-based design, precise lifetime predictions, the possibility to tailor components to multiple load cases, and even the ability to calculate the reliability of the entire drivetrain system. In many modern applications, such as electric drives with highly dynamic load profiles, this level of detail is not only beneficial but essential.
The relevance of load spectra is reflected in a number of engineering standards and guidelines, including ISO 281 [1] and ISO 16281 [2] for rolling bearings, ISO 6336 [3] for cylindrical gears, ISO 10300 [4] for bevel gears, and the FKM Guideline [5] for a universal approach. All of these are based on methods such as the rainflow counting algorithm [6] and Palmgren-Miner rule. These methods form the backbone of reliability-based design, which is increasingly expected in regulated or safety-critical industries.
Despite these advantages, load spectra are still underutilized in many industrial settings. Reasons for this include the perceived complexity of integrating them into development processes, uncertainties regarding the operational loads, and limited tool support. The FVA-Workbench addresses this gap by offering a simulation platform specifically designed for mechanical power transmission analysis, with full support for load spectrum-based evaluation.
This white paper introduces the concept of load spectra, outlines their practical relevance, and demonstrates how they are implemented and utilized within the FVA-Workbench. It is intended for engineers seeking a deeper understanding of accurate load capacity calculations as well as decision-makers interested in the strategic value of realistic load evaluations.
Engineering Advantages of Load Spectra Over Nominal Loads
Real operating conditions rarely involve steady, uniform loads. Instead, they fluctuate due to external factors like environment, usage behavior, and system dynamics. Examples:
Electric vehicles: Rapid torque changes from acceleration, braking, and regeneration
Industrial gearboxes: Load variations across duty cycles or production shifts
Wind turbines: Continuous fluctuations due to changing wind speeds
Designs based on nominal loads often lead to oversized components to cover infrequent extremes. While safe, this leads to inefficiencies in material use, weight, and cost, and provides no insight into the actual lifespan or failure risk.
More importantly, nominal design approaches do not allow for a fatigue-based lifetime prediction. They only answer the question: “Will this part survive the worst-case scenario?” - but not “How long will it last under real usage conditions?” or “What’s the probability of failure within the warranty period?” This is where the concept of load spectra becomes essential.
A load spectrum (sometimes also referred to as a load collective) is a condensed statistical representation of varying mechanical loads occurring over time. Rather than describing a single nominal load value, a load spectrum reflects the frequency and intensity of multiple load cases, typically using discrete load levels with associated cycle counts or durations. This is particularly relevant for the analysis of load capacity in components and systems subjected to dynamic, non-stationary, or varying stationary operating conditions.
At its most basic, a load spectrum answers two questions:
How strong was the load? (e.g., torque level, force amplitude)
How often did it occur? (e.g., number of cycles or operating time within a given load range)
Load spectra can take different forms, depending on the application and data source:
Time series: Load data over time, converted into spectra using cycle counting methods
Histogram-based spectra: Discrete cases representing load level and frequency
Rainflow matrices: Two-dimensional representations commonly used in fatigue analysis, derived from the rainflow counting algorithm
Each format offers advantages in terms of precision, computational efficiency, or compatibility with specific analysis methods.
In drivetrain engineering, load spectra are critical for:
Predicting the lifetime of gears, shafts, bearings, and other rotating (and even non-rotating) parts
Optimizing geometry and flank modifications under multiple load scenarios
Assessing system reliability, failure probability, and expected service life
Avoiding overdesign by replacing worst-case assumptions with data-driven insight
Transforming Time Signals into Load Spectra
In many engineering workflows, the starting point for generating a load spectrum is a time signal: a series of force, torque, or acceleration values measured or simulated over time. These time series reflect the dynamic behavior of the system under specific operating conditions, such as a drive cycle, test scenario, or a specific real-world usage pattern.
Transforming this raw time data into a usable load spectrum involves several steps:
Step 1: Preprocessing the Time Signal
The time signal must be prepared before any cycle counting can occur:
Filtering: Remove irrelevant high-frequency noise or non-relevant vibrations
Normalization: Adjust values to consistent units or reference levels
Segmentation: Divide signals into distinct operating modes, if needed
Truncation: Exclude startup/shutdown or idle periods that distort statistics
Step 2: Cycle Counting
Once the time signal has been processed, the next step is to identify load cycles - that is, the repeated stress or strain reversals that contribute to material fatigue. This is the core step in building a load spectrum.
The most widely used method here is rainflow counting. The core idea is to break down complex, irregular load sequences into individual, closed cycles. Rainflow counting is based on a material mechanical foundation and can be interpreted as an algorithm which searches for closed stress-strain hysteresis (see Figure 1).
The area inside these hystereses can be interpreted as damage-relevant energy equivalents. The result of the rainflow counting method is the so-called rainflow matrix, which contains either the upper and lower reversal points or the mean and amplitude values of all occurring hystereses.
Figure 1: The rainflow counting algorithm is based on a hysteresis simulation of the elastic-plastic material behavior. The (extended) rainflow algorithm transforms the applied load into a stress-strain series and counts closed hystereses. Most of the rainflow algorithms used resemble this algorithm, but with improved performance
Step 3: Spectrum Construction
After counting, the cycles are either used in the form of the rainflow matrix or transformed into a histogram representation. The resulting spectrum can then be scaled to a total number of operating hours or a specific lifetime target (e.g., 100,000 km, 10,000 h, 1 million load cycles).
The transformation from time signal to load spectra does not just enable the evaluation of a single specific design variant; well-processed spectra can be also applied to multiple variants if the signal is divided into distinct operating modes and well normalized.
Applications in Gearbox and Drivetrain Engineering
The drivetrain comprises a chain of interacting components such as gears, shafts, bearings, and couplings, all of which are exposed to complex, time-varying loads throughout their service life. Therefore, one can apply the load spectra at the component level to calculate the load capacity of those individual components without accounting for interactions. However, applying load spectra at the system level allows optimization of the entire gearbox; e.g., NVH optimization of multiple gears.
Component-Level Applications
Gears
Fatigue assessment using S-N curves and safety factors (e.g., via ISO 6336)
Supports micro-geometry optimization (e.g., tip relief, crowning) across varying loads
Shafts
Calculation of damage-equivalent nominal stresses (e.g., for component tests)
Fatigue assessment using S-N curves and safety factors (e.g., per FKM Guideline)
Identification of critical load ranges which influence fatigue life
Bearings
Bearing life calculations under variable loads (e.g., ISO 281)
Prevent under- or overestimation of bearing life by reflecting real load variations
System-Level Applications
Multiple-load-case design (e.g., optimizing flank modifications with consideration of all occurring load levels instead of the nominal design level)
Fatigue life calculation of the whole system, taking component interactions into account
This is essential for reliability planning, warranty strategies, and predictive maintenance.
Creating and Managing Load Spectra in the FVA-Workbench
The FVA-Workbench is a simulation and calculation platform specifically developed for the analysis, design, and optimization of mechanical power transmission systems.
What makes the FVA-Workbench unique is that these research-based methods are not hidden in expert modes or academic add-ons. They are accessible as standard calculations: clearly documented, workflow-integrated, and supported by intuitive user guidance.
This means engineers benefit from the most advanced methodologies without being research specialists. The complexity stays under the hood; the application remains clear, structured, and usable for everyday engineering tasks.
Working With Load Spectra in the FVA-Workbench
The starting point for any load spectrum-based evaluation in the FVA-Workbench is the availability of representative load data. The FVA-Workbench includes a broad range of methods to transform a load signal into a load spectrum, such as:
Multi-parameter time-at-level-counting
Single-parameter time-at-level-counting
Synthetic spectrum calculation
Standard spectrum
Rainflow counting
The software manages these steps with a guided interface, including suitable defaults based on the application context (see Figure 2).
The resulting spectra can then be imported into the local database for the current project.
Figure 2: The user is guided through the process of transforming load data (time-series, spectrum, etc.) via a wizard – this figure shows the process for rainflow counting.
Applying Load Spectra in System Calculations
Once a load spectrum has been generated in the FVA-Workbench, it can immediately be used in system calculations. The FVA-Workbench offers two distinct modes for analyzing load spectra: scaled and flexible. The scaled load spectrum resembles the classical histogram-based load spectra, meaning that each load case is based on the nominal load case and scaling factors (e.g., for speed and applied torque). On the other hand, the flexible load spectrum allows any combination of loads, speeds, and selected gears (meaning the active gears change in different load cases). It offers the unique ability to define fully independent load cases and to perform a separate system calculation for each of them. This means that the user can subsequently evaluate the effects of the various operating states separately, as if all load cases were independent nominal load cases. This enables the possibility of extended optimizations and postprocessing using the FVA-Workbench’s scripting environment.
Calculation Methods and Supported Components
The following calculations support spectrum-based evaluation in the FVA-Workbench:
ISO 6336 for cylindrical gears
Gear excitation according to FVA 338 I for cylindrical gears
ISO 10300 for bevel gears
Local damage accumulation for bevel gears according to FVA 586 I
Fatigue calculation of shafts and shaft notches using the FKM Guideline
ISO 281 and 16281 for rolling bearings
For each calculation, the load cycles and their damage contribution are considered cumulatively (independent of whether the scaled or flexible load spectrum mode is used). This enables component-level lifetime prediction under realistic operational conditions.
Visualization and Interpretation of Results
Once the load spectrum calculation has been performed, the FVA-Workbench provides a wide range of visualization and interpretation tools to help engineers understand the impact of varying loads on component durability.
One of the most fundamental and widely used visualizations in load spectrum analysis is the S-N curve, along with the corresponding stress spectrum (see Figure 3). In the FVA-Workbench, two safety factors are always calculated: the static safety factor, which represents the ratio between the maximum load in the load spectrum and the static limit of the S-N curve, and the spectrum safety factor, which indicates how much the entire spectrum can be scaled while keeping the cumulative damage below 1.
Figure 3: Load spectrum and SN curves for tooth flank and root according to ISO 6336
Figure 4: Partial damage of load cases in the load spectrum to identify the most damaging load case
The partial damage of all load cases can also be represented (see Figure 4). This helps to optimize the gearing for the load cases where the most damage occurs.
The software calculates and displays lifetime metrics based on the accumulated damage across all load cycles. Depending on the calculation method and settings, results may include:
Estimated service life (e.g., in hours, km, or load cycles)
Damage sum (e.g., Miner’s D value)
Utilization/safety factor (e.g., how close the component is operating to its fatigue limit)
Since the FVA-Workbench supports parametric studies and optimization, users can compare the results of different design variants or load scenarios. Figure 5 shows an example where two flank modifications are applied using the same load spectrum. The flank modification on the left is optimized for the nominal load case, whereas the flank modification on the right is optimized for the entire spectrum. In this case, the resulting lifetime differs by up to approx. 10% with regard to the tooth flank load capacity. This means that neglecting the influence of the load spectrum in the optimization of flank modifications can (in a worst-case scenario) lead to significant reduction of the resulting load capacity of the gear.
For detailed analysis purposes, the FVA-Workbench provides comparative tables as well as extended analysis capabilities using the integrated scripting environment to support systematic engineering decisions.
All results can be:
Exported as tables, images, or raw data
Integrated into the automated reporting system (HTML and PDF)
Annotated with project metadata and calculation settings
Versioned and archived for traceability and certification
This ensures that spectrum-based evaluations can be seamlessly integrated into documentation and review processes, both internally and for customers or regulatory submissions.
Figure 5: Resulting damage sums of a cylindrical gear stage when optimizing for nominal load (left) and when optimizing for the load spectrum (right) – the damage sum can be significantly lowered by considering the load spectrum instead of just the nominal load.
Integrating Load Spectra into Automated Workflows
In engineering practice, repeatability and efficiency are just as important as technical accuracy. Especially in organizations that develop multiple product variants or work across large teams, the handling of load spectra must be reproducible, transparent, and capable of being automated.
The FVA-Workbench is designed with these needs in mind. Its approach to load spectrum integration goes far beyond “single use” functionality—it provides a framework for systematic, scalable engineering.
Engineers can define spectrum templates based on typical operating profiles, such as:
Standardized duty cycles
Customer-specific usage data
Internal load definitions from previous projects
These templates can be saved, customized, and reused across systems and projects. This standardization ensures consistency in evaluation and reduces setup time for recurring workflows.
FVA-Workbench models are fully parametric, enabling automated generation and calculation of design variants with different geometries, materials, or load conditions. Combined with load spectrum evaluation, this allows for:
Batch simulations, where multiple design candidates are evaluated under identical load spectra
Automated comparisons of lifetime, damage, and safety factors
Optimization studies, targeting improved durability or reduced overdesign
Application Example
The use case shown in Figure 6 illustrates how load spectra can be used in the FVA-Workbench as part of a real-world gearbox development scenario.
This example deals with a ball mill from a lime plant. The principal data originates from FVA project 48 I [7]. The ball mill is used to finely grind burnt lime. For this example, we are focusing on the 1-stage reduction gearbox, the varying torques resulting from the grinding process, and the different loads. Key technical details of the gearbox and drivetrain are summarized in Table 1.
Figure 6: Application example - Mill (left) and applied torque spectrum at the input of the gearbox (right)
| Input speed | 985 min-1 |
| Transmission ratio of the gearbox | 6,54 |
| Number of teeth of the pinion | 26 |
| Number of teeth of the wheel | 170 |
| Gear ratio of the sprocket gear | 6,96 |
| Number of teeth of the sprocket pinion | 23 |
| Number of teeth of the sprocket wheel | 160 |
| Speed of the grinding drum | 21.6 min-1 |
The mill gearbox is to be designed for an operating life of 100,000 hours. For this example, we will consider the resulting service life of the cylindrical gear teeth in accordance with ISO 6336, the rolling bearings in accordance with ISO 281 and ISO 16281, and the degree of utilization of the shaft notches in accordance with the FKM Guideline. This shows concrete savings potential for the individual components as well as the overall system, all made possible by using load spectra in the design instead of the nominal load.
The FVA-Workbench's pre-dimensioning module was used for the initial design of the mill gearbox. This makes it possible to create a gearbox design that ensures a sufficiently high load-bearing capacity for the gearing without further adjustments for the nominal load case based solely on the information in Table 1.
The initial gearing, which has a safety factor >=1, is very bulky due to the design being based on the nominal value. The gears account for approx. 1,100 kg. Provided that the number of teeth remains constant, the reduction of the normal module offers great potential for weight savings. In this case, the normal module can be reduced from the original 6 to 3.5 by using the load spectrum calculation, while maintaining the same resulting safety. This reduces the mass of the spur gear stage by a factor of 5.
Once the gearing has been reduced in size, the shafts and rolling bearings must be adapted to the new gearbox size. The FVA-Workbench provides very powerful functionality for this purpose with the 2D modeler. Shaft edges and components can be adapted and moved quickly and precisely by ”snapping.” Intuitive shaft modeling using a polygon path also allows the shaft contour to be adapted to the new gearbox size within seconds. The optimized design is shown in Figure 7.
The rolling bearings once again demonstrate how much savings potential lies in the design with a load spectrum. Table 2 shows the results of all bearings used in this example when calculated with nominal load and with the load spectrum. It can be seen that the required service life of 100,000 hours is significantly exceeded by taking the load spectrum into account, while only a fraction of the required service life can be achieved with the nominal load.
Figure 7: Optimized mill gearbox
Table 2: Resulting calculated lifetimes in operating hours for the mill application when calculating with just the nominal case and when calculating with the load spectrum
| Modified rating life in h using nominal design | Modified rating life in h using load spectrum | |||
| ISO 281 | ISO 16281 | ISO 281 | ISO 16281 | |
| Rolling bearing 1 (SKF 30216) |
1.095 | 3.435 | 1.481.170 | 2.453.468 |
| Rolling bearing 2 (SKF 30216) |
706 | 1.746 | 917.652 | 1.315.214 |
| Rolling bearing 3 (SKF 30216) |
2.002 | 5.108 | 1.473.792 | 2.709.666 |
| Rolling bearing 4 (SKF 30216) |
2.992 | 9.691 | 2.364.789 | 5.152.745 |
To achieve the service life of 100,000 hours under nominal load, a size of 32316 is required instead of 30216. This alone increases the mass of the installed bearings by a factor of ≈ 4.
Finally, Figure 8 shows the safety factors according to the FKM Guideline for the critical shaft notches. The figure on the left shows the calculation using only the nominal load, and the figure on the right shows the calculation with load spectrum using the Miner modification “Miner consistent.”
Figure 8: Resulting notch safeties of the optimized gearbox design
Figure 9: Gearbox mass before and after optimization with consideration of load spectra
This optimized design would not have been possible without the consideration of load spectra. Figure 9 compares the composition of the gearbox masses for the design under nominal load and for the design with the load spectrum. It can be seen that the consideration of load spectra offers considerable potential for material savings, thus producing more resource-efficient gearboxes.
Conclusion and Outlook
The consideration of load spectra has become essential in gearbox design in order to exploit open optimization potential and significantly increase resource efficiency. The FVA-Workbench offers a comprehensive range of features for evaluating measurement signals, generating load spectra, and calculating load capacities using these load spectra, as well as functions for load spectrum-based optimization.
Therefore, the consideration of load spectra in gearbox design with the FVA-Workbench is also suitable for inexperienced users and enables non-experts to exploit the shown optimization potential.
References
[1] ISO 281: Rolling bearings - Dynamic load ratings and rating life, 2007.
[2] ISO 16281: Rolling bearings - Methods for calculating the modified reference rating life for universally loaded rolling bearings, 2025.
[3] ISO 6336: Calculation of load capacity of spur and helical gears, 2019.
[4] ISO 10300: Calculation of load capacity of bevel gears, 2023.
[5] R. Rennert, E. Kullig, M. Vormwald, A. Esderts and M. Luke, Rechnerischer Festigkeitsnachweis für Maschinenbauteile aus Stahl, Eisenguss- und Aluminiumwerkstoffen: FKM Guideline, Frankfurt am Main: VDMA Verlag, 2020.
[6] U. Clormann and T. Seeger, "Rainflow-HCM. Ein Zählverfahren für Betriebsfestigkeitsnachweise auf werkstoffmechanischer Grundlage," Stahlbau, Vol. 55, No. 3, 1986.
[7] H. D. Eisbrecher, F. W. Griese, Laumann, D. Wünsch, Kuitzsch and W. Stühler, FVA 48 I: Ermittlung und Auswertung von Lastkollektiven - Systematische, detaillierte Auswertung der Literatur zu beanspruchungsorientierten Untersuchungen an Antriebssystemen, Frankfurt am Main: Drive Technology Research Association e.V., 1976.
Author
Dr.-Ing. Ralf Wuthenow
Head of Modeling and Simulation
FVA GmbH