# Surface roughness

Surface roughness, often shortened to 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. On the other hand, roughness may promote adhesion.

Although roughness is often 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.

Roughness is typically measured in "RMS" microinches and is often only measured by manual comparison against a "surface roughness comparator", a sample of known surface roughnesses.

## Parameters

A roughness value can either be calculated on a profile (line) or on a surface (area). The profile roughness parameter (Ra, Rq,...) are more common. The area roughness parameters (Sa, Sq,...) give more significant values.

### Profile 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 $R_\text{a}$ is by far the most common. Other common parameters include $R_\text{z}$, $R_\text{q}$, and $R_\text{sk}$. Some parameters are used only in certain industries or within certain countries. For example, the $R_\text{k}$ 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 $i^\text{th}$ 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, $R_\text{a}$ is the arithmetic average of the absolute values and Rt is the range of the collected roughness data points.

The roughness average, $R_\text{a}$, is the most widely used one dimensional roughness parameter.

Parameter Description Formula
Ra,[1] Raa, Ryni arithmetic average of absolute values $R_\text{a} = \frac{1}{n} \sum_{i=1}^{n} \left | y_i \right |$[1]
Rq, RRMS[1] root mean squared $R_\text{q} = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} y_i^2 }$[1]
Rv maximum valley depth $R_\text{v} = \min_{i} y_i$
Rp maximum peak height $R_\text{p} = \max_{i} y_i$
Rt Maximum Height of the Profile $R_\text{t} = R_\text{p} - R_\text{v}$
Rsk skewness $R_\text{sk} = \frac{1}{n R_\text{q}^3} \sum_{i=1}^{n} y_i^3$
Rku kurtosis $R_\text{ku} = \frac{1}{n R_\text{q}^4} \sum_{i=1}^{n} y_i^4$
RzDIN, Rtm average distance between the highest peak and lowest valley in each sampling length, ASME Y14.36M - 1996 Surface Texture Symbols $R_\text{zDIN} = \frac{1}{s} \sum_{i=1}^{s} R_{\text{t}i}$, where $s$ is the number of sampling lengths, and $R_{\text{t}i}$ is $R_\text{t}$ for the $i^\text{th}$ sampling length.
RzJIS Japanese Industrial Standard for $R_\text{z}$, based on the five highest peaks and lowest valleys over the entire sampling length. $R_\text{zJIS} = \frac{1}{5} \sum_{i=1}^{5} R_{\text{p}i}-R_{\text{v}i}$, where $R_{\text{p}i}$ and $R_{\text{v}i}$ are the $i^\text{th}$ 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$\Delta$q the RMS slope of the profile within the sampling length \begin{align} R_{dq} &= \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \Delta_i^2} \end{align}
Rda, R$\Delta$a the average absolute slope of the profile within the sampling length \begin{align} R_{da} &= \frac{1}{N} \sum_{i=1}^{N} |\Delta_i| \end{align}
$\Delta$i where delta i is calculated according to ASME B46.1 \begin{align} \Delta_i &= \frac{1}{60 dx} (y_{i+3} - 9y_{i+2} + 45y_{i+1} - 45y_{i-1} + 9y_{i-2} - y_{i-3}) \end{align}

#### 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.

Sketches depicting surfaces with negative and positive skew. The roughness trace is on the left, the amplitude distribution curve is in the middle, and the bearing area curve (Abbott-Firestone curve) is on the right.

#### Fractal theory

The mathematician Benoît Mandelbrot has pointed out the connection between surface roughness and fractal dimension.[2]

### Areal roughness parameters

Areal roughness parameters are defined in the ISO 25178 series. The resulting values are Sa, Sq, Sz,... . At the moment many optical measurement instruments are able to measure the surface roughness over an area.

## Practical effects

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. An exception is in cylinder bores where oil is retained in the surface profile and a minimum roughness is required.[citation needed]

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 $R_{sk}$, will usually have high friction and wear quickly. The peaks in the roughness profile are not always the points of contact. The form and waviness must also be considered.