Definitions

# Roughness

[ruhf]

Roughness is a measure of the texture of a surface. It is quantified by the vertical deviations of a real surface from its ideal form. If these deviations are large, the surface is rough; if they are small the surface is smooth. Roughness is typically considered to be the high frequency, short wavelength component of a measured surface (see surface metrology).

Roughness plays an important role in determining how a real object will interact with its environment. Rough surfaces usually wear more quickly and have higher friction coefficients than smooth surfaces (see tribology). Roughness is often a good predictor of the performance of a mechanical component, since irregularities in the surface may form nucleation sites for cracks or corrosion.

Although roughness is usually undesirable, it is difficult and expensive to control in manufacturing. Decreasing the roughness of a surface will usually increase exponentially its manufacturing costs. This often results in a trade-off between the manufacturing cost of a component and its performance in application.

## Measurement

Roughness may be measured using contact or non-contact methods. Contact methods involve dragging a measurement stylus across the surface; these instruments include profilometers. Non-contact methods include interferometry, confocal microscopy, electrical capacitance and electron microscopy.

For 2D measurements, the probe usually traces along a straight line on a flat surface or in a circular arc around a cylindrical surface. The length of the path that it traces is called the measurement length. The wavelength of the lowest frequency filter that will be used to analyze the data is usually defined as the sampling length. Most standards recommend that the measurement length should be at least seven times longer than the sampling length, and according to the Nyquist–Shannon sampling theorem it should be at least ten times longer than the wavelength of interesting features. The assessment length or evaluation length is the length of data that will be used for analysis. Commonly one sampling length is discarded from each end of the measurement length.

For 3D measurements, the probe is commanded to scan over a 2D area on the surface. The spacing between data points may not be the same in both directions.

In some cases, the physics of the measuring instrument may have a large effect on the data. This is especially true when measuring very smooth surfaces. For contact measurements, most obvious problem is that the stylus may scratch the measured surface. Another problem is that the stylus may be too blunt to reach the bottom of deep valleys and it may round the tips of sharp peaks. In this case the probe is a physical filter that limits the accuracy of the instrument.

There are also limitations for non-contact instruments. For example instruments that rely on optical interference cannot resolve features that are less than some fraction of the frequency of their operating wavelength. This limitation can make it difficult to accurately measure roughness even on common objects, since the interesting features may be well below the wavelength of light. The wavelength of red light is about 650 nm, while the Ra of a ground shaft might be 2000 nm.

## Analysis

In the past, surface finish was usually analyzed by hand. The roughness trace would be plotted on graph paper, and an experienced machinist decided what data to ignore and where to place the mean line. Today, the measured data is stored on a computer, and analyzed using methods from signal analysis and statistics.

The first step of roughness analysis is often to filter the raw measurement data to remove very high frequency data since it can often be attributed to vibrations or debris on the part surface. Next, the data is separated into roughness, waviness and form. This can be accomplished using reference lines, envelope methods, digital filters, fractals or other techniques. Finally the data is summarized using one or more of the roughness parameters, or a graph.

## Specification

In the United States, surface finish is usually specified based on the ASME Y14.36M-1996 standard. Other standards also exist, including ISO 1302:2001.

### Lay Patterns

A lay pattern is a repetitive impression created on the surface of a part. It is often representative of a specific manufacturing operation. A product designer may specify a lay pattern on a part because the directionality the lay affects the part's function. Unless otherwise specified, roughness is measured perpendicular to the lay.

## Roughness Parameters

Each of the roughness parameters is calculated using a formula for describing the surface.

There are many different roughness parameters in use, but $R_a$ is by far the most common. Other common parameters include $R_\left\{z\right\}$, $R_q$, and $R_\left\{sk\right\}$. Some parameters are used only in certain industries or within certain countries. For example, the $R_k$ family of parameters is used mainly for cylinder bore linings, and the Motif parameters are used primarily within France.

Since these parameters reduce all of the information in a profile to a single number, great care must be taken in applying and interpreting them. Small changes in how the raw profile data is filtered, how the mean line is calculated, and the physics of the measurement can greatly affect the calculated parameter.

By convention every 2D roughness parameter is a capital R followed by additional characters in the subscript. The subscript identifies the formula that was used, and the R means that the formula was applied to a 2D roughness profile. Different capital letters imply that the formula was applied to a different profile. For example, Ra is the arithmetic average of the roughness profile, Pa is the arithmetic average of the unfiltered raw profile, and Sa is the arithmetic average of the 3D roughness.

Each of the formulas listed in the tables assumes that the roughness profile has been filtered from the raw profile data and the mean line has been calculated. The roughness profile contains $n$ ordered, equally spaced points along the trace, and $y_i$ is the vertical distance from the mean line to the $i^\left\{th\right\}$ data point. Height is assumed to be positive in the up direction, away from the bulk material.

### Amplitude Parameters

Amplitude parameters characterize the surface based on the vertical deviations of the roughness profile from the mean line. Many of them are closely related to the parameters found in statistics for characterizing population samples. For example, Ra is the arithmetic average of the absolute values and Rt is the range of the collected roughness data points.

The amplitude parameters are by far the most common surface roughness parameters found in the United States on mechanical engineering drawings and in technical literature. Part of the reason for their popularity is that they are straightforward to calculate using a digital computer.

Parameter Description Formula
Ra, Raa, Ryni arithmetic average of absolute values R_a = frac{1}{n} sum_{i=1}^{n} left > y_i right |
Rq, RRMS root mean squared $R_q = sqrt\left\{ frac\left\{1\right\}\left\{n\right\} sum_\left\{i=1\right\}^\left\{n\right\} y_i^2 \right\}$
Rv maximum valley depth $R_v = min_\left\{i\right\} y_i$
Rp maximum peak height $R_p = max_\left\{i\right\} y_i$
Rt Maximum Height of the Profile $R_t = R_p - R_v$
Rsk skewness $R_\left\{sk\right\} = frac\left\{1\right\}\left\{n R_q^3\right\} sum_\left\{i=1\right\}^\left\{n\right\} y_i^3$
Rku kurtosis $R_\left\{ku\right\} = frac\left\{1\right\}\left\{n R_q^4\right\} sum_\left\{i=1\right\}^\left\{n\right\} y_i^4$
RzDIN, Rtm average distance between the highest peak and lowest valley in each sampling length, ASME Y14.36M - 1996 Surface Texture Symbols $R_\left\{zDIN\right\} = frac\left\{1\right\}\left\{s\right\} sum_\left\{i=1\right\}^\left\{s\right\} R_\left\{ti\right\}$, where $s$ is the number of sampling lengths, and $R_\left\{ti\right\}$ is $R_\left\{t\right\}$ for the $i^\left\{th\right\}$ sampling length.
RzJIS Japanese Industrial Standard for $R_z$, based on the five highest peaks and lowest valleys over the entire sampling length. $R_\left\{zJIS\right\} = frac\left\{1\right\}\left\{5\right\} sum_\left\{i=1\right\}^\left\{5\right\} R_\left\{pi\right\}-R_\left\{vi\right\}$, where $R_\left\{pi\right\}$ $R_\left\{vi\right\}$ are the $i^\left\{th\right\}$ highest peak, and lowest valley respectively.

### Slope, Spacing, and Counting Parameters

Slope parameters describe characteristics of the slope of the roughness profile. Spacing and counting parameters describe how often the profile crosses certain thresholds. These parameters are often used to describe repetitive roughness profiles, such as those produced by turning on a lathe.
Parameter Description Formula
Rdq, R?q the RMS slope of the profile within the sampling length begin\left\{align\right\} R_\left\{dq\right\} &= sqrt\left\{frac\left\{Delta_i^2\right\}\left\{N\right\}\right\} Delta_i &= frac\left\{1\right\}\left\{60 dx\right\} \left(y_\left\{i+3\right\} - 9y_\left\{i+2\right\} + 45y_\left\{i+1\right\} - 45y_\left\{i-1\right\} + 9y_\left\{i-2\right\} - y_\left\{i-3\right\}\right) end\left\{align\right\}

### Bearing Ratio Curve Parameters

These parameters are based on the bearing ratio curve (also known as the Abbott-Firestone curve.) This includes the Rk family of parameters.

### Fractal theory

The mathematician Benoît Mandelbrot has pointed out the connection between surface roughness and fractal dimension.

## Engineering

In most cases, roughness is considered to be detrimental to part performance. As a consequence, most manufacturing prints establish an upper limit on roughness, but not a lower limit.

It can be difficult to quantify the relationship between roughness and part performance because there are so many different ways to characterize the surface.

### Tribology

Roughness is often closely related to the friction and wear properties of a surface. A surface with a large $R_a$ value, or a positive $R_\left\{sk\right\}$, will usually have high friction and wear quickly.

Deep valleys in the roughness profile are also important to tribology because they may act as lubricant reservoirs.

The peaks in the roughness profile are not always the points of contact. The form and waviness must also be considered.

## Manufacturing

Many factors contribute to the surface roughness in manufacturing. When molding or forming a surface, the impression of the mold or die on the part is usually the principle factor in the surface roughness. In machining, and abrasive processes the interaction of the cutting edges and the microstructure of the material being cut both contribute to the roughness.

Just as different manufacturing processes produce parts at various tolerances, they are also capable of different roughnesses. Generally these two characteristics are linked: manufacturing processes that are dimensionally precise create surfaces with low roughness. In other words, if a process can manufacture parts to a narrow dimensional tolerance, the parts will not be very rough.

### Cost

In general, the cost of manufacturing a surface increases greatly as the roughness tolerance decreases.

## Other Applications

• International Roughness Index (IRI) - a dimensionless quantity used for measuring road roughness and proposed as a world standard by the World Bank. Typically IRI is presented as an average value over 20 m, 100 m, 400 m, 1 mile etc. IRI is not an excellent indicator on ride quality. Consider two 10 cm high and arc-shaped traffic calming speed bumps, one "spinebreaker" being 1 m long and the other being as much as 10 m long and thus too smooth for calming city traffic. Both give an IRI20 of about 8 mm/m. Not being able to distinguish between two bumps that obviously give dramatically different ride quality, one can really question IRI as a pavement performance indicator.
• Manning's n-value - used by geologists to characterise river channels.