Army Research Laboratory multiaxial vibration experiment

Vibration fatigue is a

mechanical engineering Mechanical engineering is the study of physical machines and mechanism (engineering), mechanisms that may involve force and movement. It is an engineering branch that combines engineering physics and engineering mathematics, mathematics principl ...

term describing

material fatigue In materials science, fatigue is the initiation and propagation of cracks in a material due to cyclic loading. Once a fatigue crack has initiated, it grows a small amount with each loading cycle, typically producing striations on some parts of ...

, caused by forced vibration of

random In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. ...

nature. An excited structure responds according to its natural-dynamics modes, which results in a dynamic stress load in the material points. The process of

is thus governed largely by the shape of the excitation profile and the response it produces. As the profiles of excitation and response are preferably analyzed in the

frequency domain In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...

it is practical to use

fatigue life In materials science, fatigue is the initiation and propagation of cracks in a material due to cyclic loading. Once a fatigue crack has initiated, it grows a small amount with each loading cycle, typically producing striations on some parts of ...

evaluation methods, that can operate on the data in

frequency-domain In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...

, s power spectral density (PSD). A crucial part of a vibration fatigue analysis is the

modal analysis Modal analysis is the study of the dynamic properties of systems in the frequency domain. It consists of mechanically exciting a studied component in such a way to target the Normal mode, modeshapes of the structure, and recording the vibration ...

, that exposes the natural modes and frequencies of the vibrating structure and enables accurate prediction of the local stress responses for the given excitation. Only then, when the stress responses are known, can vibration fatigue be successfully characterized. The more classical approach of fatigue evaluation consists of cycle counting, using the rainflow algorithm and summation by means of the Palmgren-Miner linear damage hypothesis, that appropriately sums the damages of respective cycles. When the time history is not known, because the load is

(''e.g.'' a

car A car, or an automobile, is a motor vehicle with wheels. Most definitions of cars state that they run primarily on roads, seat one to eight people, have four wheels, and mainly transport people rather than cargo. There are around one billio ...

on a rough

road A road is a thoroughfare used primarily for movement of traffic. Roads differ from streets, whose primary use is local access. They also differ from stroads, which combine the features of streets and roads. Most modern roads are paved. Th ...

or a wind driven turbine), those cycles can not be counted. Multiple time histories can be simulated for a given

random process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...

, but such procedure is cumbersome and

computationally expensive In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that r ...

. Vibration-fatigue methods offer a more effective approach, which estimates

based on moments of the PSD. This way, a value is estimated, that would otherwise be calculated with the

time-domain In mathematics and signal processing, the time domain is a representation of how a signal, function, or data set varies with time. It is used for the analysis of function (mathematics), mathematical functions, physical signal (information theory), ...

approach. When dealing with many material nodes, experiencing different responses (''e.g.'' a model in a FEM package), time-histories need not be simulated. It then becomes viable, with the use of vibration-fatigue methods, to calculate

in many points on the structure and successfully predict where the failure will most probably occur.

Vibration-fatigue-life estimation

Random load description

In a random process, the amplitude can not be described as a function of time, because of its

probabilistic Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

nature. However, certain statistical properties can be extracted from a signal sample, representing a realization of a random process, provided the latter is

ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...

. An important characteristics for the field of vibration fatigue is the amplitude

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

, that describes the statistical distribution of peak amplitudes. Ideally, the probability of cycle amplitudes, describing the load severity, could then be deduced directly. However, as this is not always possible, the sought-after probability is often estimated empirically.

Effects of structural dynamics

Random excitation of the structure produces different responses, depending on the natural dynamics of the structure in question. Different natural modes get excited and each greatly affects the stress distribution in material. The standard procedure is to calculate

frequency response function In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...

s for the analyzed structure and then obtain the stress responses, based on given loading or excitation. By exciting different modes, the spread of

vibration Vibration () is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the os ...

energy over a frequency range directly affects the durability of the structure. Thus the structural dynamics analysis is a key part of vibration-fatigue evaluation.

Vibration-fatigue methods

Calculation of damage intensity is straightforward once the cycle amplitude distribution is known. This distribution can be obtained from a time-history simply by counting cycles. To obtain it from the PSD another approach must be taken. Various vibration-fatigue methods estimate damage intensity based on moments of the PSD, which characterize the statistical properties of the random process. The formulas for calculating such estimate are empirical (with very few exceptions) and are based on numerous simulations of random processes with known PSD. As a consequence, the accuracy of those methods varies, depending on analyzed response spectra, material parameters and the method itself - some are more accurate than others. The most commonly used method is the one developed by T. Dirlik in 1985. Recent research on frequency-domain methods of fatigue-life estimation compared well established methods and also recent ones; conclusion showed that the methods by Zhao and Baker, developed in 1992 and by Benasciutti and Tovo, developed in 2004 are also very suitable for vibration-fatigue analysis. For narrow-band approximation of random process analytical expression for damage intensity is given by Miles. There are some approaches with adaptation of narrow-band approximation; Wirsching and Light proposed the empirical correction factor in 1980 and Benasciutti presented _0.75 in 2004. In 2008, Gao and Moan published a spectral method that combines three narrow-band processes. Implementation of those method is given in the

Python Python may refer to: Snakes * Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia ** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia * Python (mythology), a mythical serpent Computing * Python (prog ...

open-source FLife package.

Applications

Vibration fatigue methods find use wherever the structure experiences loading, that is caused by a

. These can be the forces that bumps on the road extort on the

chassis A chassis (, ; plural ''chassis'' from French châssis ) is the load-bearing framework of a manufactured object, which structurally supports the object in its construction and function. An example of a chassis is a vehicle frame, the underpart ...

, the wind blowing on the

wind turbine A wind turbine is a device that wind power, converts the kinetic energy of wind into electrical energy. , hundreds of thousands of list of most powerful wind turbines, large turbines, in installations known as wind farms, were generating over ...

, waves hitting an

offshore construction Offshore construction is the installation of structures and facilities in a marine environment, usually for the production and transmission of electricity, oil, gas and other resources. It is also called maritime engineering. Construction a ...

or a

marine vessel A watercraft or waterborne vessel is any vehicle designed for travel across or through water bodies, such as a boat, ship, hovercraft, submersible or submarine. Types Historically, watercraft have been divided into two main categories. *Raf ...

. Such loads are first characterized statistically, by measurement and analysis. The data is then used in the

product design Product design is the process of creating new Product (business), products for businesses to sell to their customers. It involves the generation and development of ideas through a systematic process that leads to the creation of innovative products ...

process. The computational effectiveness of vibration-fatigue methods in contrast to the classical approach, enables their use in combination with FEM software packages, to evaluate fatigue after the loading is known and the dynamic analysis has been performed. Use of the vibration-fatigue methods is well-suited, as structural analysis is studied in the

. Common practice in the

automotive industry The automotive industry comprises a wide range of company, companies and organizations involved in the design, Business development, development, manufacturing, marketing, selling, Maintenance, repairing, and Custom car, modification of motor ve ...

is the use of accelerated vibration tests. During the test, a part or a product is exposed to

, that are in correlation with those expected during the service-life of the product. To shorten the testing time, the amplitudes are amplified. The excitation spectra used are broad-band and can be evaluated most effectively using vibration-fatigue methods.

References

{{Reflist, refs= {{cite journal, last=Mršnik, first=Matjaž, author2=Slavič, Janko , author3=Boltežar, Miha , title=Frequency-domain methods for a vibration-fatigue-life estimation - application to real data, journal=International Journal of Fatigue, date=31 July 2012, doi=10.1016/j.ijfatigue.2012.07.005, url=http://lab.fs.uni-lj.si/ladisk/?what=abstract&ID=75, volume=47, pages=8–17, url-access=subscription {{cite book, last=Nuno Manuel Mendes, first=Maia, title=Theoretical and experimental modal analysis, year=1998, publisher=Research Studies Press, location=Baldock, isbn=0863802087, edition=Reprinted. {{cite book, last=Varoto, first=Kenneth G. McConnell, Paulo S., title=Vibration testing : theory and practice, year=2008, publisher=John Wiley & Sons, location=Hoboken, N.J., isbn=978-0-471-66651-6, edition=2nd {{cite book, last=Sarkani, first=Loren D. Lutes, Shahram, title=Random vibrations analysis of structural and mechanical systems, year=2004, publisher=Elsevier, location=Amsterdam, isbn=9780750677653, edition= nline-Ausg.} {{cite journal, last=Benasciutti, first=D, author2=Tovo, R , title=Spectral methods for lifetime prediction under wide-band stationary random processes, journal=International Journal of Fatigue, date=1 August 2005, volume=27, issue=8, pages=867–877, doi=10.1016/j.ijfatigue.2004.10.007 {{cite journal, last=Zhao, first=W, author2=Baker, M , title=On the probability density function of rainflow stress range for stationary Gaussian processes, journal=International Journal of Fatigue, date=1 March 1992, volume=14, issue=2, pages=121–135, doi=10.1016/0142-1123(92)90088-T {{cite thesis , type=Ph.D. , first=Turan , last=Dirlik , title=Application of computers in fatigue analysis , publisher=University of Warwick , year=1985 {{cite book, last=Slavič, first=Janko , author2=Boltežar, Miha, author3=Mršnik, Matjaž , author4=Česnik, Martin , author5=Javh, Jaka , title=Vibration Fatigue by Spectral Methods: From Structural Dynamics to Fatigue Damage – Theory and Experiments , year=2020, publisher=Elsevier, location=Amsterdam, Netherlands, isbn=9780128221907, doi=10.1016/C2019-0-04580-3, s2cid=243156155 , url=https://repozitorij.uni-lj.si/Dokument.php?id=166493&dn= , edition=1st {{cite journal, last=Miles, first=John W. , title=On structural fatigue under random loading, journal=Journal of the Aeronautical Sciences, date=1954, volume=21, issue=11, pages=753–762, doi=10.2514/8.3199 {{cite journal, last=Wirsching, first=Paul H., author2=Light, Mark C., title=Fatigue under wide band random stresses, journal=Journal of the Structural Division, date=1980, volume=106, issue=7, pages=1593–1607, doi=10.1061/JSDEAG.0005477 {{cite report , last=Benasciutti, first=Denis, author2=Tovo, Roberto , title=Rainflow cycle distribution and fatigue damage in Gaussian random loadings, publisher=Department of Engineering, University of Ferrara, date=2004 {{cite journal, last=Gao, first=Zhen, author2=Moan, Torgeir, title=Frequency-domain fatigue analysis of wide-band stationary Gaussian processes using a trimodal spectral formulation, journal=International Journal of Fatigue, date=2008, volume=30, issue=10–11, pages=1944–1955, doi=10.1016/j.ijfatigue.2008.01.008 {{cite web , url=https://github.com/ladisk/FLife , title=FLife, website=

GitHub GitHub () is a Proprietary software, proprietary developer platform that allows developers to create, store, manage, and share their code. It uses Git to provide distributed version control and GitHub itself provides access control, bug trackin ...