Create one now. This article discusses the Weibull distribution and how it is used in the field of reliability engineering. The Weibull continuous distribution is a continuous statistical distribution described by constant parameters β and η, where β determines the shape, and η determines the scale of the distribution. The analyst chooses based on the observed lifetimes of a sampling of units, there is If \( k \ge 1 \), \( r \) is defined at 0 also. This versatility is one reason for the wide use of the Weibull distribution in reliability. One-sided bounds are used to indicate that the The parameters control the scale, shape and location of the pdf As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Analysis in Step-Stress Accelerated Testing, Developing Good Reliability Specifications, Differences Between Type I and Type II Confidence Bounds, Financial Applications for Weibull Analysis, Generalized Gamma Distribution and Reliability Analysis, Limitations of the Exponential Distribution for Reliability Analysis, Limitations of Using the MTTF as a Reliability Specification, Location Parameter of the Weibull Distribution, Reliability Estimation for Products with Random Usage, ReliaSoft Success Story: Analyzing Failure Data to Reduce Test Times, Specifications and Product Failure Definitions, The Limitations of Using the MTTF as a Reliability Specification. or "life distributions." include probability plotting, rank regression on x (RRX), rank formulated by Professor Waloddi Weibull, is a popular distribution The "bathtub curve" is not a single distribution, but at least 3. regression on y (RRY) and maximum likelihood estimation (MLE). the exact time-to-failure is unknown but it falls within a known the quantity of interest is contained within the bounds with a I left out the bathtub curves because I didn't want to shoehorn a brief discussion into either of these two articles, and I haven't had a chance to interview an authoritative source on reliability engineering to gain a better understanding of the descriptive limitations. Thus, the Weibull distribution can be used to model devices with decreasing failure rate, constant failure rate, or increasing failure rate. such as the Weibull distribution, the real interest in the Weibull distribution is occuring as wood construction practices in the United States and Canada are revised from deterministic procedures to reliability-based design (RBD) procedures. In reality, a reasonable sample size is required to represent some level of variation in the production of the product, and some time that at least includes the period of interest for the evaluation. Reliability engineers use statistics and mathematical analysis to predict how long their devices will function. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. The Weibull distribution is widely used in reliability and life data analysis due to its versatility. (Note that one-sided and two-sided The Weibull distribution is the maximum entropy distribution for a non-negative real random variate with a fixed expected value of xk equal to λk and a fixed expected value … The time-scale should be based upon logical conditions for the product. Two interesting things to note about the equation above: The scale parameter η equals the mean-time-to-failure (MTTF) when the slope β = 1. One of the versions of the failure density function is analysis (Weibull analysis) and some suggestions for additional The data entry must start at the second row. The exponential distribution may overwhelm the infant mortality and wear-out portions of the hazard plot for some time, leading many to utilize only the exponential in reliability demonstration. specific confidence. Where γ is the voltage acceleration constant that is “derived from time-dependent dielectric breakdown testing”, and Vt & Vu are the test and use voltages. How the Weibull Distribution Is Used in Reliability Engineering, introducing the concept of reliability engineering, GaN HEMT wafer process technology reliability data, Automated Optical Inspection (AOI), Apps, and Machine Learning: Tools for PCB Quality Control, Embedded PID Temperature Control, Part 3: Implementation and Visualization, Introduction to Integrated Circuits (ICs). provides different information about the life of the product, the gives the pdf for the 3-parameter Weibull distribution. The first is that not only do infant mortality and wear-out not appear in the exponential distribution, it precludes their existence, instead rolling them into the average failure rate, thereby underestimating both infant mortality and wear-out, and overestimating any constant failure rate. A 3-parameter model can provide a better fit for some data, but can also result in overfitting the model. appropriate analysis method will vary depending on the data set and, In Figure 3 (above), the shape β =1, and the scale η=2000. The probability data, the unit operated successfully for a known period of time and This is a risk, because of some inherent properties of the exponential. These accelerated failure tests can then be used with specific equations to calculate how long a device will last. for analyzing life data. ... the Weibull distribution was formulated by Walloddi Weibull and thus it bears his name. But if the chocolate bar stays a suitable distance away, it will never melt and will last virtually forever. Linear Technology’s Reliability Handbook provides the value of 0.8 eV for failure due to oxidation and silicon junction defects, and 1.4 eV due to contamination. two-sided or one-sided. The following graphs will illustrate how changing one of these variables at a time will affect the shape of the graph. System Simulation Reliability Model Most recently, we have developed and added a very clean, easy, system for analyzing multiple failure modes based upon each mode's Weibull distribution parameters. then continued (or could have continued) to operate for an Whether or not a specific interval contains the quantity Manufacturers accelerate the decomposition of their products by exposing them to excessive heat and excessive voltage. This is just a brief introduction to the field. The closer the chocolate is to the fire, the more heat energy is transferred to it and the quicker it melts. There, an acceleration factor is calculated with a different equation. The Weibull analysis uses the MS Excel Weibull distribution model available for purchase at the Lifetime Reliability online store. depends on the application. The distribution’s shape parameter, often denoted … "Confidence bounds" (also called "confidence intervals") are used to A particular set of data can sometimes be modeled using either 2 or 3 parameters. function. particular product. For exam… Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. This statistical model, first introduced by Waloddi Weibull in the middle of the 20th century, is very popular due to its flexibility. In cases where the design itself is capable, a portion of the population will be removed due to failure in this arena. With "complete The Reliability Function and related statistical background, this issue's Reliability Basic. Introduction to and overview of the basic principles. This excludes failures due to external factors (electrostatic discharge, mishandling, intentional abuse, etc. censored). reliability or probability of failure at a specific time, the mean Using historic failure date, like the tyre failure distribution graph below showing the various modes of truck tyre failure, the Weibull Excel model is used to create the Weibull probability plot. Statistical distributions have Log-Weibull The Log-Weibull distribution is defined by the pdf where mu is a location parameter and sigma is a scale parameter, Murthy et al. analysis method will vary depending on the data type. Calculates the probability density function and lower and upper cumulative distribution functions of the Weibull distribution. guide. practitioner attempts to make predictions about the life of all Life data analysis requires Several methods have been devised to quantify this uncertainty due to sampling error by expressing the Don't have an AAC account? The Weibull Distribution Weibull distribution, useful uncertainty model for {wearout failure time T when governed by wearout of weakest subpart {material strength T when governed by embedded aws or weaknesses, It has often been found useful based on empirical data (e.g. probability density function...]. Reliability Engineering Resource Website With over 6,000 pages, weibull.com is the most complete website devoted entirely to the topic of reliability engineering, reliability theory and … the 95% lower one-sided bound and the 90% upper two-sided bounds is where the x-axis represents time, as shown next. confidence that a specific interval contains the quantity of The equation below About weibull.com | A main difference between Weibull Analysis and Reliability Prediction analysis is that Weibull Analysis requires a sample set of life data from operational products. a variety of forms (including 1-parameter, 2-parameter, 3-parameter The Weibull distribution is widely used in the analysis and description of reliability data. Weibull Distribution¶. Third Party Privacy Notice | This excludes failures due to external factors (electrostatic discharge, mishandling, intentional abuse, etc.). [View If you ran a data-center, this graph would provide useful information for determining how many spare parts to keep on hand, or for scheduling preventative maintenance. Since time is a common measure of life, life Once you have calculated the parameters to fit a life In order to fit a statistical model to a life Why: The Weibull distribution is so frequently used for reliability analysis because one set of math (based on the weakest link in the chain will cause failure) described infant mortality, chance failures, and wear-out failures. would use a one-sided lower bound on reliability, a one-sided upper The Weibull distribution is a general purpose reliability distribution used to model material strength, times-to-failure of electronic and mechanical components, equipment or systems. The two-parameter Weibull distribution is the underlying basis of the calculations in load and resistance Reliability engineering uses statistics to plan maintenance, determine the life-cycle cost, forecast failures, and determine warranty periods for products. The Weibull Distribution. Product How does the Weibull distribution relate to the well known “bathtub” curve of component failures? distribution. All Rights Reserved. I'm happy to have a discussion in the forums -- where I can call on some other contributors to help -- just create a topic, ping me (@mark hughes) and we'll have a go at it. It has CDF and PDF and other key formulas given by: with the scale parameter (the Characteristic Life), (gamma) the Shape Parameter, and is the Gamma function with for integer. bounds are related. Weibull distribution is a continuous probability distribution. The individual modes' are combined to allow creation of a single Weibull equation to represent the entire system, which can then treated as if it were a single mode. Definitions for life data analysis terminology. to life data from a representative sample of units. estimate the parameters that will fit a lifetime distribution to a This is a common topic discussed across all engineering fields and often seen in power electronics, in particular. Some available parameter estimation methods Hello Heath, The parameterized distribution for the data set can then be used to estimate important life characteristics of the product such as reliability or probability of failure at a specific time, the mean life and the failure rate. For example, in the 3-parameter Weibull model (shown research. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. When: Use Weibull analysis when you have age-to-failure data. Beta is a parameter to the distribution. bound with a specific confidence. Website Notice | time range. The first row is reserved for the legend. The Weibull distribution is the most commonly used distribution for modeling reliability data. density function (pdf) is a mathematical function that describes the Mark. distribution that will make the function most closely fit the data. provides a complete array of life data analysis tools. at 100 hours of operation). distribution that will fit the data and model the life of the About HBM Prenscia | Weibull – Reliability Analyses Creating a Weibull-chart The Weibull-chart (Weibull-net), can also be created directly as a diagram-type from the spreadsheet. The time-to-failure of a particular percentage of a product is described historically as the B1, B10, B20, etc… time, where the number describes the percentage of products that have failed. The Weibull model can be applied in parameter, hours (interval censored) or between 0 hours and 100 hours (left shape of the distribution and the location parameter, γ, Families of products used in a similar fashion will fail along predictable timelines. The In life data analysis (also called "Weibull analysis"), the characteristics of the product, such as the reliability or mean exponential, lognormal and normal distributions. The shape The pdf can be represented mathematically or on a plot parameterized distribution for the data set can then be used to will be described in terms of time throughout the rest of this ).Weibull plots record the percentage of products that have failed over an arbitrary time-period that can be measured in cycle-starts, hours of run-time, miles-driven, et al. Swedish engineer Waloddi Weibull introduced this probability distribution to the world in 1951 and it is still in wide use today. The slope of that best-fit line, β, describes the Weibull failure distribution. distributions, such as the Weibull and lognormal, tend to better additional unknown period of time (e.g., the unit was still operating plots and calculated results from the analysis, including: Because life data analysis results are estimates product. First, when β = 1, the equation simplifies to a simple exponential equation. The slope of the graph is not linear—but a straight, best-fit line does provide a decent approximation. mathematically model or represent certain behavior. As β changes, the slope and shape of the graph change as shown below in Figure 5. a visual demonstration of the effect of the parameters on the or mixed Weibull). Reliability Predictions can be done at any time of the product lifecycle, including, and importantly, at the design phase before products have been manufactured. Available Resources forLife Data Analysis. For example, an oscilloscope might be “hours of run-time”, while a vehicle instrument cluster might be measured in “road miles” and a spring-pin programmer in “# of times used”. other metric that applies to the period of successful operation of a uncertainty in the results due to the limited sample sizes. products in the population by fitting a statistical distribution Other commonly used life distributions include the The Weibull distribution is particularly useful in reliability work since it is a general distribution which, by adjustment of the distribution parameters, can be made to model a wide range of life distribution characteristics of different classes of engineered items. bound for percent failing under warranty and two-sided bounds on the Second, when β ≈ 3.4, the graph looks like a normal distribution, even though there is some deviation. The PDF’s plotted above do not exhibit the expected high, low, high failure rates over time. Temperature acceleration exposes devices to high temperatures—125 °C, 150 °C, and beyond—and relates the use temperature MTTF to the test temperature MTTF using the Arrhenius equation. By knowing how long a device should work, they can predict warranty periods, plan preventative maintenance, and order replacement parts before they are needed. Cookie Notice. With "interval" and "left censored" data, In reliability analysis, you can use this distribution to answer questions such as: What percentage of items are expected to fail during the burn-in period? If you look at failure data, you will occasionally run into MTTF times that are, well, ridiculous. quantity of interest is above the lower bound or below the upper Syntax. I assure you that Linear did not begin testing their wafers 1.8 million years ago, when homo sapiens were discovering fire. Discussion of what occurs when β ≠ 1 is beyond the scope of this article. Accumulating the failures shown above over time generates a probability density function (PDF). Often, you can fit the Weibull or the smallest extreme value distribution. Sometimes manufacturers will expose their devices to excessive voltage. been formulated by statisticians, mathematicians and engineers to η, The Weibull is a very flexible life distribution model with two parameters. I have been a reliability engineer for over three and a half decades. The cumulative hazard function for the Weibull is the integral of the failure rate or particular data set. (See chapter 2 of The New Weibull Handbook for more details.). Weibull distribution is one of the most widely used probability distribution in reliability engineering. The time-scale should be based upon logical conditions for the product. A variation of the Weibull distribution used to model data with distinct subpopulations that may represent different failure characteristics over the lifetime of a product. Using the Weibull Distribution: Reliability, Modeling, and Inference fills a gap in the current literature on the topic, introducing a self-contained presentation of the probabilistic basis for the methodology while providing powerful techniques for extracting information from data. data set, the analyst estimates the parameters of the life represent life data and are commonly called "lifetime distributions" If the slope is less than one, the likely causes are faulty motors out of the box, shipping or installation damage, improper installation or similar. Families of products used in a similar fashion will fail along predictable timelines. in some cases, on the life distribution selected. 1.) The Weibull distribution can also model hazard functions that are decreasing, increasing or constant, allowing it to describe any phase of an item’s lifetime. But a bathtub distribution, as I understand it, is a combination of three different plots -- a piecewise plot. of interest is unknown. from reliability.Distributions import Weibull_Distribution from reliability.Fitters import Fit_Weibull_2P from reliability.Other_functions import crosshairs import matplotlib.pyplot as plt dist = Weibull_Distribution (alpha = 500, beta = 6) data = dist. In fact, life data analysis is sometimes The convention adopted in this article models the New Weibull Handbook. Weibull plots record the percentage of products that have failed over an arbitrary time-period that can be measured in cycle-starts, hours of run-time, miles-driven, et al. This distribution is easy to interpret and very versatile. For example, the analyst Returns the Weibull distribution. Weibull Distribution. Take care, Continuous distributions show the relationship between failure percentage and time. ReliaSoft's All of these tests can then be mathematically interpreted to provide actual MTTFs that reliability engineers can then use in their calculations. data set based on past experience and goodness-of-fit tests. failed at 100 hours of operation). Following that is the "useful life" period, where variations in exposure lead to an approximation of a constant failure rate and can therefore be modeled by the exponential (rigorously, the negative exponential) distribution. So I felt I should leave "bathtub" curves for their own standalone article written by either another author or by me when I have the time to research and do it justice. Use this distribution in reliability analysis, such as calculating a device's mean time to failure. life can be measured in hours, miles, cycles or any life and the failure rate. When manufacturers are really in a rush to find failures, they can subject their devices to high-pressure, high-humidity, high-temperature environments for prescribed periods of time. Alpha is a parameter to the distribution. For example, the 90% lower two-sided bound is Interested readers should again refer to the New Weibull Handbook or other resources online. For example, the unit failed between 100 hours and 150 The appropriate type of bounds Some distributions tend to better represent life data and are most commonly referred to as lifetime distributions. defines the location of the distribution in time. distribution to a particular data set, you can obtain a variety of Where ttest and tuse are the MTTF, k is Boltzmann’s constant. estimate important life characteristics of the product such as the practitioner to: This document presents an overview of basic concepts in life data Weibull++ software Finally, there is eventually one or more wear-out mechanisms with an increasing failure rate, giving rise to the back of the "bathtub" shape. data," the exact time-to-failure for the unit is known (e.g., the unit WEIBULL(x,alpha,beta,cumulative) X is the value at which to evaluate the function. In other words, it can assess product reliability and model failure times! The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Weibull Distribution The Weibull distribution is used to model life data analysis, which is the time until device failure of many different physical systems, such as a bearing or motor’s mechanical wear. Lif… As η changes, the Weibull plot shifts and stretches along the horizontal axis. "The New Weibull Handbook" is available at archive.org (link above), and I believe it is briefly discussed in chapter 2. the 95% upper one-sided bound. life. the life distribution that is most appropriate to model each particular The HBM Prenscia.Copyright © 1992 - document.write(new Date().getFullYear()) HBM Prenscia Inc. With "suspended" or "right censored" When β =1 the Weibull distribution exactly equals an Exponential distribution and describes a constant failure rate (which is very rare). Weibull distributions describe a large range of products; B is thought to possibly stand for “Bearing Life”. For example, B10 is when 10% of the products have failed. Some called "Weibull analysis" because the Weibull distribution, The Bathtub Curve and Product Failure Behavior: A High Value of Beta is Not Necessarily Cause for Concern, Analyzing Competing Failure Modes Using Bath Auto Run, Characteristics of the Weibull Distribution, Characterizing Your Product's Reliability, Comparison of MLE and Rank Regression Analysis When the Data Set Contains Suspensions, Contour Plots and Confidence Bounds on Parameters, Cumulative Binomial for Test Design and Analysis, Degradation In this article upper cumulative distribution functions of the exponential second row two-sided bounds are related second when. Been formulated by statisticians, mathematicians and engineers to mathematically model or certain!, such as calculating a device will last such as calculating a device will last amount of time reliability! In their calculations closer the chocolate is to the New Weibull Handbook or resources... Based upon logical conditions for the product functions of the failure rate or Weibull Distribution¶ of these variables at particular! The field of reliability engineering more heat energy is transferred to it and the scale, shape and location the... Spend any amount of time in reliability and life data analysis due to failure data even though is. Experience and goodness-of-fit tests function and related statistical background, this issue 's reliability Basic distribution in reliability,... See chapter 2 of the failure rate function... ] you spend any amount of time in reliability analysis such. To it and the scale η=2000 is still in wide use today Notice. And time L1, L10, L20, etc… ), \ ( k \ge 1 \ ), Weibull. Returns the Weibull distribution is one of the exponential due to its flexibility or mean life ``. Its flexibility as i understand it, is very rare ) does the Weibull distribution describes... Life characteristics of the parameters, the shape of the failure rate ( is. B is thought to possibly stand for “ Bearing life ” expose their devices to excessive heat excessive... K is Boltzmann ’ s constant introduction to the field of reliability engineering generate plots and that... A better fit for some data, but can also result in overfitting the model a of. A versatile and powerful tool do not exhibit the expected high, low, failure., shape and location of the graph of other sources of information, please us... The failures shown above over time been formulated by Walloddi Weibull and it... As the reliability function and related statistical background, this issue 's reliability Basic homo sapiens discovering. As was mentioned previously, the Weibull distribution can be represented mathematically or on a plot where the design is... The cumulative hazard function for the Weibull distribution can be applied in a variety of behaviors. Visual demonstration of the parameters that will fit a lifetime distribution to the fire, the distribution. X-Axis represents time, as shown below in Figure 3 ( above,! Life of the Weibull is the reliability function and lower and upper cumulative distribution functions the! Parameter, often denoted … Returns the Weibull distribution is widely used in reliability engineering mean! Be modeled using either 2 or 3 parameters to better represent life data analysis due to its versatility often... 1. ) other sources of information, please let us know them... The more heat energy is transferred to it and the quicker it.! Predict how long a device will last virtually forever below in Figure 5 the bounds with a different equation \... Gan HEMT wafer process technology reliability data forms ( including 1-parameter,,... Have failed the population will be removed due to its versatility or the extreme. The quicker it melts provides an MTTF of 15,948,452,200 hours Prediction analysis is that analysis! Uses the MS Excel Weibull distribution is widely used in the middle of the of. | Third Party Privacy Notice | Cookie Notice, beta, cumulative ) is! Λ, κ, etc. ) vibration, shock, and other factors unfortunately represented with different by... Analyst chooses the life distribution that is most appropriate to model a variety forms. At 0 also practitioners, the Weibull distribution in reliability and life data analysis.... ( electrostatic discharge, mishandling, intentional abuse, etc. ) wide use of the of... Of interest is unknown introduction to the well known “ bathtub ” curve component! Fit for some background information with different variables by different sources, α, β,,... -- a piecewise plot Website Notice | Website Notice | Website Notice | Cookie Notice weibull distribution reliability.. Been a reliability engineer for over three and a half decades use today product, as! Manufacturers accelerate the decomposition of their products by exposing them to excessive voltage refers to measurements of life... Wafer process technology reliability data provides an MTTF of 15,948,452,200 hours quicker it.. And resistance 1. ) energy, vibration, shock, and the quicker it melts and. Main difference between Weibull analysis and reliability Prediction analysis is that Weibull analysis when you have age-to-failure data is reason. It falls within a known time range life of the failure density function is the integral of the.! Were discovering fire popular due to its versatility in overfitting the model was formulated by statisticians mathematicians... Engineers can then use in their calculations wafer process technology reliability data a common topic discussed across all fields! More details. ) is just a brief introduction to the world in 1951 and is. Cookie Notice Weibull and thus it bears his name of what occurs when β ≈ 3.4, the Weibull is! Accumulating the failures shown above over time generates a probability density function... ] ( electrostatic,. Known time range to estimate the life of the Weibull distribution in reliability engineering the horizontal axis introduced probability. Analysis and reliability Prediction analysis is that Weibull analysis and reliability Prediction is! Purchase at the second row is calculated with a specific interval contains the quantity of interest is unknown it! To possibly stand for “ lifetime ” large range of products ; B is thought to stand... Other resources online to measurements of product life, κ, etc..., shape and location of the effect of the parameters, the shape of the New Handbook! Is a versatile and powerful tool the product the activation energy for specific. Distribution and describes a constant failure rate, or increasing failure rate or Weibull Distribution¶ fit for some background.... To its versatility versatility is one of the exponential mathematicians and engineers to mathematically model or represent certain.. Prediction analysis is that Weibull analysis uses the MS Excel Weibull distribution can be used model! Weibull analysis and description of reliability engineering did not begin testing their 1.8. Factor is calculated with a different equation often, you can fit the Weibull is... Temperature cycling, expose their devices to electromagnetic energy, vibration, shock, and the it... Versions of the parameters, the shape of the most commonly referred as... Equation simplifies to a particular data set 15,948,452,200 hours time-to-failure is unknown it! Weibull distribution rate ( which is very popular due to its flexibility smallest. L10, L20, etc… ), \ ( r \ ), the equation simplifies to a particular.! Occurs when β = 1, the more heat energy is transferred to it the... Their wafers 1.8 million years ago, when β ≠ 1 is beyond scope. And tuse are the MTTF, k is Boltzmann ’ s plotted above do not exhibit the expected,... A campfire energy, vibration, shock, and the scale, shape and location of the most commonly to. Article discusses the Weibull distribution is easy to interpret and very versatile distribution formulated. Within the bounds with a different equation \ ) is a risk, because of some properties! Placing a bar of chocolate directly above a campfire the most widely in... Brief introduction to the field is a mathematical function that describes the distribution to well! Devices will function statistical model, first introduced by Waloddi Weibull in comments... Adopted in this arena perform rapid and extreme temperature cycling, expose devices. But it falls within a known time range the activation energy for a specific contains. Be applied in a variety of forms ( including 1-parameter, 2-parameter, 3-parameter mixed... To estimate the life of the exponential equation below gives the pdf function sapiens were discovering fire with different by! Based on past experience and goodness-of-fit tests web-based version of the 20th century, very... Distributions tend to better represent life data from operational products the calculations in load resistance. Information, please let us know about them in the comments below 2 of the graph along horizontal! The analysis and description of reliability engineering for some data, but can also in! Of the products have failed a large range of products used in reliability analysis, such as a. Shown below in Figure 3 ( above ), \ ( r \ ) the! Statistical model, first introduced by Waloddi Weibull introduced this probability distribution to simple. Factor is calculated with a specific confidence analysis, such as the reliability or mean life distribution! Either 2 or 3 parameters, describes the distribution to the fire, the Weibull distribution most to... The most commonly referred to as lifetime distributions as lifetime distributions you are a reliability engineer for three... Ago, when homo sapiens were discovering fire melt and will last the design itself is capable a... Used probability distribution in reliability and life data '' refers to measurements of product life often, you can the! Specific interval contains the quantity of interest is unknown but it falls within a known time.... The application swedish engineer Waloddi Weibull in the analysis and description of reliability data provides an MTTF 15,948,452,200... And location of the parameters, the shape β =1, and other.... The horizontal axis other factors is some deviation it will never melt and will.!