This time, I'm going to replace the battery in my digital caliper (a cheap Chinese-made one) because it ran out of power and became unusable.
In addition, looking at the calipers reminded me of Abbe's principle, one of the fundamental principles of measurement engineering and the basis of understanding measurement errors in measuring instruments.
Since we're at it, I'd like to take this opportunity to explain the measurement errors inherent in calipers (which is actually the main point) while we're changing the batteries in the calipers.
The main content of this lesson might be particularly relevant for mechanical engineering students and engineers (it's guaranteed to appear on the test for obtaining credits in measurement engineering).
First, I'll replace the batteries in the calipers.
Replacing the battery in a digital caliper
The calipers I'm using are shown in the picture below.
It's quite useful because it's reasonably accurate despite being inexpensive (accuracy verification will be discussed later).
Digital calipers
It's reasonably accurate for the price. However, it's on the larger side.
The battery is dead and it's unusable, so I'll replace it.
Incidentally, the device comes with a separate battery in addition to the built-in test battery. Even the initial test battery lasted quite a long time.
Open the battery compartment.
The cover on the picture is quite stiff, but you have to work hard to open it and replace the batteries with the ones included.
Now that it's working again, I tried measuring a 6mm BB pellet.
I'm not sure if this is accurate, so I'll measure the same bullet with my own micrometer.
The micrometer reading was 5.931mm, which was surprisingly accurate for a caliper (with a difference of approximately 0.01mm).
Structurally, a micrometer is more accurate, so this digital caliper is off by 0.01 mm.
It's not that the calipers are bad, but regardless of whether they're digital or analog, calipers inherently (logically) have a large margin of error. In my experience, the limit of reliable accuracy is 0.05mm (micrometers are 0.005mm).
One of the advantages and disadvantages of digital calipers is that they display measurements in 0.01mm increments, but structurally, the reliable accuracy is only 0.05mm (when using them accurately, you must always mentally convert to 0.00mm or 0.05mm increments).
Furthermore, the difference in error between micrometers and calipers isn't due to physical factors like the use of a precise lead screw, but rather to differences in logical measurement methods.
Many people probably intuitively know that a micrometer is more accurate than a caliper, but in this article, I will explain the theory behind this difference in accuracy.
Now let's consider "why calipers have a large logical margin of error."
Calculate the measurement error of the caliper.
Since we're on the subject, let's learn the English terms for calipers: caliper, or vernier (the vernier scale). (These are seriously likely to appear on tests.)
This is a photo of a typical analog caliper.
I will omit the names of the parts and how to use an analog caliper here (I will write about it if requested; it is a question that is always included in introductory lab tests for engineering students).
Next, let's draw a diagram showing how to measure an object using calipers.
Here, we define the main scale (the main marking) as the scale axis, and define the distance from the axis to the center of the object being measured as 'a'.
An ideal (perfect) caliper should be able to measure the length A of the object being measured, as shown in the picture.
However, in reality, some play occurs, so measurements cannot be taken exactly as ideal.
Next, we consider the type of play or misalignment that results in the largest measurement error among the multiple types of play and misalignment.
The most influential factor in the measurement is the tilt of the measuring unit (defining the tilt angle as θ).
Using the angle of inclination θ and the measured distance a, the measurement result can be expressed as follows (simple trigonometric functions, geometry problem).
$$ Tilting measurement result = A + a \times tanθ $$
Next, the error that occurred can be calculated simply by subtracting the incorrect measurement result from the ideal measurement result (let's call the measurement error ε: epsilon).
$$ Measurement error ε = Ideal measurement result - Deflection measurement result = a × tanθ $$
will be important.
Since the formula alone may not give you a clear idea of the error, let's create a graph with θ on the horizontal axis and the measurement error ε on the vertical axis.
As the graph shows, the error is the tangent, so as the tilt angle θ increases, the error also increases rapidly. In extreme cases, the error for a tilt angle of 90° (which doesn't exist in reality or logically) is infinite.It will be ∞.
Up to this point, many people would probably think, "The actual tilt angle θ is quite small, so the error probably isn't that significant."
Therefore, next we will consider in detail the magnitude of the error when θ is very small (minimal) (θ is around 0°).
Since the change when the angle θ is small is difficult to understand if the formula is left as tan, we will perform a Maclaurin expansion of tan (the outline of the Maclaurin expansion will be explained later, so for now just think of it as "that's how it is").
Using all the terms expanded here would make it complicated, so we will only use up to the third-order component $(θ^3)$ (in engineering, the second to third order is generally sufficient).
Furthermore, in the expanded equation, θ is not an angle (°) but a radian (radian measure, angle 180° =We'll calculate it using π.
We find the error ε by swapping the first three terms of the tangent expansion.
In other words, the measurement error ε in a caliper increases in proportion to the first and third orders of the tilt angle θ, so you can imagine that it is quite large.
This represents the magnitude of the logical error inherent in the caliper measurement method.
When choosing calipers, paying attention to their tilt angle will help you get accurate ones. So, when buying calipers at a home improvement store, it's a good idea to check the tilt angle of several of the same type before buying (don't underestimate the sharpness of human senses).
Next, I'd like to move on to micrometers, but first I'll give a very brief overview of Maclaurin series (something that might get me in trouble with my math teacher).
Image and overview of the Maclaurin series.
First, let's review functions, which you learned in middle and high school. Simply put, a function represents the relationship between variables (like x and y).
I'd like you to recall this, but in the graph above...
$$ y = f(x) = Ax, meaning that when x is 1, y is A, and when x is 2, y is 2A $$
$$ y = f(x) = Ax^2, meaning that if x is 1, y is A; if x is 2, y is 4A $$
.
For such a simple function, both the formula and the value are relatively easy to find.
However, extremely complex functions (most of nature is complex) are not so easily understood.
It would be a shame to give up on deciphering the function just because it's too complex and difficult, so we'll use one of humanity's greatest strengths here.
It's understandable if the entire function is too difficult to understand, so let's focus on the range we're interested in (a specific small range).
Even if the overall function is complex, extracting only a specific, extremely small range can reveal a surprisingly simple function that becomes much easier to understand.
This process of transforming a complex function into a simpler function by focusing on a specific, extremely small range is called Taylor expansion.
Naturally, since it focuses on a very small range, it cannot be applied to the entirety of a complex function, but it is sufficient for local use.
Among the various Taylor expansions, the one that expands to a simple function around the origin (x,y)=(0,0) is called the Maclaurin expansion.
I'll include the definition of the Maclaurin series (which uses differentiation) just in case (sorry if it's confusing).
In this way, humanity has evolved by breaking down complex things into localized, part-by-part components and simplifying them. One such method is the Maclaurin series (there are also other methods such as the Fourier transform).
As you can see, mathematics is extremely important in engineering, so if there are any students or young engineers reading this, please do your best. Also, mathematics plays a very big role in everyday tools like calipers (all the tools around us), so even if you were bad at math in school, I think it will be beneficial in life if you take an interest in it (I'll try to do a math course or something similar to relearn it sometime).
Now, let's return to measurement errors.
Micrometer measurement error (Abbe's principle)
This is what a micrometer looks like. Incidentally, the English word is also "micrometer."
Similar to calipers, I will omit the names of the parts and how to use a micrometer here (I will write about it if requested; it is a question that engineering students are guaranteed to encounter on their introductory lab tests).
Next, let's illustrate the state of measuring an object with a micrometer.
An ideal (perfect) micrometer will produce the ideal measurement result A (the same as a caliper).
What's important here is that in a micrometer, the measuring axis and the scale axis are coaxial (satisfying Abbe's principle). This is the crucial difference in the logical structure of measurement between a micrometer and a caliper.
However, in reality, some play occurs, so measurements cannot be taken as ideally as with calipers.
Next, we consider the type of play or misalignment that results in the largest measurement error among the multiple types of play and misalignment.
The most influential factor in the measurement is the tilt of the scale axis (defining the tilt angle as θ) (this is an important difference from calipers).
Using the tilt angle θ, the measurement result can be expressed as follows (simple trigonometric functions, geometry problem).
$$ Tilting measurement result = A \times cosθ $$
Next, the error that occurred can be calculated simply by subtracting the incorrect measurement result from the ideal measurement result (let's call the measurement error ε: epsilon).
$$ Measurement error ε = Ideal measurement result - Fall measurement result = AA\times cos θ=A(1-cos θ) $$
It will be the same as a caliper.
Since the formula alone may not give you a clear idea of the error, let's create a graph with θ on the horizontal axis and the measurement error ε on the vertical axis.
As the graph shows, the error is A(1-cosθ), so it increases gradually as the tilt angle θ increases. To take an extreme example, the error for a tilt angle of 90° (which doesn't exist in reality or logical structure) is A.(Unlike calipers, it doesn't become infinite; it converges.)
Just like with calipers, many people probably think, "The actual tilt angle θ is quite small, so the error won't be that significant."
Next, let's consider in detail the magnitude of the error when θ is very small (minimal) (θ is around 0°).
Since the change when the angle θ is small is difficult to understand if the formula is left as cosine, we will perform a Maclaurin expansion of cosine.
Using all the terms expanded here would make it complicated, so we will only use up to the third-order component $(θ^3)$ (in engineering, the second to third order is generally sufficient).
Furthermore, in the expanded equation, θ is not an angle (°) but a radian (radian measure, angle 180° =We'll calculate it using π.
We find the error ε by swapping the cosine term with the third-order terms in the expansion.
In other words, the measurement error ε in a micrometer increases proportionally to the quadratic angle of tilt θ, so you can imagine that it is a fairly large error.
Therefore, compared to the third order of a caliper, the error of a micrometer is proportional to the second order, resulting in a significantly smaller error.
The principle that errors are small when the scale (measurement axis) is on the same axis as the measurement axis is called Abbe's principle.
This represents the magnitude of the logical error inherent in the micrometer's measurement method (this is also something that actually appears on exams).
Comparison of measurement errors between calipers and micrometers
Now that we've calculated the errors between the caliper and the micrometer, let's compare them.
As you can see from the diagram, even with the same tilt angle θ, the error with a micrometer is overwhelmingly smaller.
Although unlikely, if we calculate assuming the tilt angle of the caliper and micrometer is 1° (approximately 0.017 [rad]) then...
The caliper's error is approximately 0.017
The error of the micrometer is approximately 0.000145
This means that for approximately the exact same tilt angle of 1°, the accuracy changes by about 100 times (this depends entirely on the method and has nothing to do with the manufacturing precision of the equipment).
This logical characteristic makes it clear that no matter how precisely a caliper is manufactured, it can never match a micrometer (the difference between a micrometer's accuracy of 0.005mm and a caliper's accuracy of 0.05mm is easily 10 times greater).
Furthermore, the accuracy of all measuring instruments, not just calipers and micrometers, is critically determined by whether the scale is coaxial or not, that is, whether it follows Abbe's principle (even with the exact same instrument, the accuracy can easily change by about 100 times depending on the method used).
Therefore, when looking at measuring instruments (all types, including physical and electronic measuring instruments), it's interesting to pay attention to where the scale is located (where the scale is taken!).
To broaden the discussion, measurement engineering, including this theory, is roughly equivalent to sensor engineering, so it's not limited to measurement but is a crucial foundation for the resolution and accuracy of optical sensors (FCU sensors in electric guns), magnetic sensors or Hall element sensors (magnetic sensors in electric gun FCUs), and potential sensors or potentiometers necessary for controlling all kinds of products (I'll explain sensors in more detail if I feel like it).
Therefore, I think it's important for engineering students and engineers (including mechanical and electrical engineers, as well as SEs working on trendy embedded software).
Summary
The conversation started with changing the batteries in my own digital calipers and then jumped to explaining the basics of measurement engineering.
Based on Abbe's principle, it's safe to imagine that the limiting accuracy of a caliper is 0.05 mm, and that of a micrometer is 0.005 mm (a 10-fold difference due to the difference in principles).
In other words, the common diameter of airsoft bullets is 5.95 ± 0.01 mm. The nominal 5.95 can be measured with calipers, but the tolerance of 0.01 cannot be measured without using a micrometer (it's meaningless without a micrometer).
Also, not just with bullets, but the 0.01mm dimension on digital calipers is just a rough indication (to put it extremely, it's almost completely inaccurate), and you need to think of it as 0.05mm to get accurate measurements.
However, just because calipers are not very accurate doesn't mean they're completely useless. They have good characteristics such as being easy to use, being able to measure relatively large objects, and being inexpensive, so they are suitable for certain situations.
Thank you for bearing with me through this long and somewhat technical explanation (this content is guaranteed to appear on exams for engineering students, and I think it's essential for engineers).
• High-precision digital caliper (Mitutoyo)
Unlike the caliper introduced at the beginning of the article, this is a high-precision caliper. It's a professional-grade Mitutoyo. For normal use, this level of precision is unnecessary; the one at the beginning of the article is sufficient.
• Analog caliper (pocket caliper)
Its reliability and compactness (digital cameras are bulky) make it incredibly convenient to have one. I always kept one in my pocket while working.
Micrometer
While you may not use it often in everyday life, it becomes necessary when measuring bullets or when entering the world of microns (0.005mm). The one introduced here is analog, but digital versions are also available.
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