# Inverse power law relationship advice

### Accelerated Testing with the Inverse Power Law - ReliaSoft

Does the inverse square law proposed by Coulomb seem to hold so that the relationship Fcoul = A/r2 holds within the limits of experimental uncertainty? What is. A power law is any polynomial relationship that exhibits the property of scale scaling by a constant c simply multiplies the original power-law relation by the. The Inverse Square Law relates the intensity of a light source to the illumination it but to the relationship between the light source and the background as well. The best advice I can give about the Inverse Square Law is to simply be aware.

It can be deflected. This means that over time and distance its intensity can and will diminish. What does that mean for your photography? It means that doubling the flash-to-subject distance reduces the light falling on the subject to one-quarter.

Logically, we might assume that doubling the distance would reduce the power by half. We are looking at a blank wall approximately ten feet long, illuminated with a single light source.

Meter readings along the wall show the progression of one-stop increments. The Inverse Square Law relates the intensity of a light source to the illumination it produces at any given distance. One-stop increments are spread over a wider area the farther the light travels. Now that we understand what the Inverse Square Law is and how it affects the intensity of light, how do we apply it to our photography?

Those closer to the light source catch the brunt of the light and may be overexposed, while those further from it could be underexposed. The variance in the light over such a short distance means the light falling on our subjects will be very uneven. If, on the other hand, we move our family down the wall to the 7- or 8-feet mark, we have a wider area in which to achieve a more even exposure across the group.

Remember, though, that the same principles apply not only to our subjects, but to the relationship between the light source and the background as well.

If we are photographing our imaginary family with a plain white wall for a background, simply moving them closer to or farther away from the wall will affect whether the wall appears white, gray, or even black.

But what about natural light? The same concept applies, whether you are using window light, a reflector, a sunset, or any other non-electrical light source. The principles of how light travels do not change just because the light in question has no batteries.

The best advice I can give about the Inverse Square Law is to simply be aware of it and understand its potential impact on your photos and lighting setups.

These failure modes may interact mask or enhance each otherand are generally thought of as "competing" failure modes. In these cases, it is best to focus on the end user's experience. A very effective system model can be derived if all failure modes that lead to the same outcome are lumped together for analysis purposes. The motorist may not care about the details as to why the car will not start but will certainly notice when it does not. Life-Stress Relationships Generally speaking, the "life" of a component will go down as stress goes up.

While this is not a hard and fast rule, very few systems do not behave in this intuitive fashion. This allows for shorter test times at higher levels of stress. With solid knowledge of the life-stressor relationship, effective predictions of life at normal or usage conditions can be made. Probably the most important and widely used model for mechanical systems is the inverse power law IPL. The most critical factor is n, the life-stressor slope with s being stress applied to the system.

A is a convenient mathematical constant; in reality it relates the basic mechanical strength of the design to resist the stress applied to it. More generally, if multiple stressors need to be considered, the model is given by: Since most fatigue testing generally fits a Weibull distribution, it may be best to consider the life being modeled as the mean or B10 life of the component. An important measure often used in life testing is the acceleration factor AFwhich relates how one cycle or hour at an elevated stress level S relates to the usage, or base, level SBase.

There are many other life-stress relationships that are commonly employed based on the underlying physics of failure. See references [ 1 ] and [ 3 ] for more information. Inverse Power Law in Action The practical application of the IPL allows "scaling," through the acceleration factor, of data and predictions from one stress level to another. In general, this is used to predict the life of a component at a stress level lower than was tested.

The following plot depicts a typical IPL relationship with a life — stress slope of 4 that could represent the life of a component or a full system. In general, the data are plotted using a log-log representation, with the stressor as the abscissa horizontal axis and the life as the ordinate vertical axis. With this example, if testing were conducted at 4 g of input acceleration, the expected life would be about 35 hours.

This can be related back to a prediction at the operating conditions 1 g to a life of about 10, hours. The acceleration factor isor 1 hour at 4 g is equivalent to hours at 1 g.

In practice though, acceleration factors much above can prove problematic by producing false failures that would not be seen in the field. In most cases, there are low — high — low periods of loading. Likewise in testing, it may not be practical to always run at fixed levels of input. At lower levels, closer to field conditions, the time required to run until failure may be prohibitive.

In these cases, it may be advantageous to "step" or increase the stress over time.

## An Introduction to the Inverse Square Law

The problem comes down to analyzing the resulting data and relating back to field or base conditions. One way of dealing with these sorts of situations is to use a damage model. The foundation of most damage models is that each cycle or hour of stress exposure accumulates damage leading to eventual failure.

The failure point is predicted to occur when the total damage DTotal — summed from each step reaches unity.

Using the slope and data model from the previous plot as a simple three-step example: The total damage sums to 0. Another way of looking at the same data is that 30 hours of testing time 3 x 10 hour steps would accumulate a similar amount of damage as if the unit had run for hours at the lowest level of 2 g. This large acceleration factor comes from the sensitivity of the component to stress being quite significant — n is 4 in this example.

Obviously, this simple tool has limitations but it can be used surprisingly effectively to aid in developing test plans and in relating results of similar tests run at different levels. In general, not all samples in any given fatigue-based test will fail at the same time.

Determining the IPL Slope The three methods outlined next can be used to determine with reasonable accuracy the slope of the life-stressor line for a particular design or material.

### Arrhenius, Cumulative Damage, Eyring, Log-Linear, HALT, HASSAccelerated Life Testing Terms

IPL Slope from Datasheets The easiest means of determining the life-stressor slope is to simply "look it up. In general practice, S-N curves can be quite limited to generic materials and may not adequately represent the stress state of a component in use. While a significant data set is available for most metals, often little data will exist for other materials.

In cases where data can be found, it most likely will not incorporate effects of mean stress and probably was determined through the use of bending or uniaxial tension on standard coupons. The following plot is representative of what is typically available from open literature. While we are interested in life- stress relationships, most of the data to be found is plotted in the "opposite" direction, as stress-life or S-N data.

Depending on the application of the system under investigation, choosing which portion of the data to use for your estimate can be critical. Because the life-stress slope is usually quite different in low cycle fatigue regions than in high, it is important to have an idea of the stress state high vs.

Consider the plot shown next. There are many possible ways to fit a straight slope to the data for this plot; each will result in significant differences. For example, if the whole range of the presented data is used red line that results in a S-N slope of about 0. On the other hand if we know that the operational and test regime will be in the high cycle fatigue range then the blue line is a better choice from this data.

With that assumption, a life-stress slope of about This is quite a large difference, especially when you recall that this is the exponent in the equation. Since very few systems are designed to survive only a few cycles, when in doubt bias the fit towards the lower stress, high cycle data. It is best to use textbook data only as a starting point or for rough estimates.

There are just too many factors at work for accurate predictions, even in non safety-critical systems. IPL Slope from Material Testing The second method for determining the life-stress slope is to perform simple experiments with samples of the actual material in use. This has the advantage of being able to closely resemble the stress state in the field and may be the only choice if "textbook" data is not available for the material of concern. One other advantage is that it presents the opportunity to gather data using field equivalent types of loads, such as GRMS values for vibration over a more generic mechanical stress.

This may also be the only choice if the life-limiting stress is unconventional, such as the exposure of aluminum to mild acid solutions previous discussed. The following plot is based on data taken from an experiment with an aluminum brazing sheet used at Dana Corporation to make oil cooler brackets — little fatigue data existed for this material in post-brazed conditions.

### Difference between power law distribution and exponential decay - Mathematics Stack Exchange

It was most expedient to construct a small series of tests with weighted strips of material attached to the shaker table — hence they were called the "flapping strip" experiments. The experiment was quite simple. Four levels of input were used from 2. These levels and the frequencies involved do not differ much from some of the inputs seen from actual road load testing of full cooler systems. Due to the abstract nature of the experiment, little more than the slope of the life-stress line 3.

One interesting observation is that if textbook data had been used to estimate the slope for this material, a slope of closer to 10 would have been found — a great deal different than what the experiments showed.

Had the higher estimate been used, gross errors in predictions would have been made that could have led to debate on the validity of the test, or worse, the suitability of the component. The investment of a day or so in effort to organize and a couple days of shaker time was well worth the results. IPL Slope from System Testing The final method to determine the life-stressor relationship is to perform system or sub-system level testing.

- Inverse Power Law Relationship
- Inverse Square Law, General
- Reliability Glossary

This tends to be the most costly and time-consuming method but also produces data that most closely represents field usage. If the components have a significant design margin especially a safety-critical item then running at operation conditions may be impractically long but unfortunately provides the most useful data point.

Working with components as close as possible to production design generally produces results and uses inputs that most closely resemble field conditions. The inputs and outputs do not need to be abstract concepts such as stress and strain, but rather engineering and user inputs such as temperature, button pushes, speeds and on-off-on cycles.

Having easy-to-interpret results, such as hours or days to failure, also helps in relating testing and predictions to full system analysis and field conditions. Some tips to consider for system testing: You can never have too much data.