Abstract
Heavy friction is expected to make motion jerky. This paper shows that the cause is not the size but the gap—make the two frictions equal and the motion is smooth however large they are. No new theorem or law is claimed.
First, this is the core of the paper. The amplitude of one slip is $2(F_s-F_k)/k$, giving $\mathbf{20.0000}$ mm for static $30$ N, kinetic $20$ N and a spring of $1000$ N/m (Section 2).
Second, if the two frictions are equal, the amplitude is exactly $\mathbf{0}$ (Section 3).
Third, raising the ratio to $1.1$, $1.5$ and $2.0$ gives $5.4545$, $20.0000$ and $\mathbf{30.0000}$ mm, increasing monotonically (Section 3).
Fourth, stiffening the spring tenfold takes the amplitude from $20.0000$ to $\mathbf{2.0000}$ mm, exactly a tenth (Section 3).
Fifth, but it does not go: so long as the ratio is not $1$, the amplitude remains (Section 3).
Sixth, the separator is the size of the friction against the gap between two frictions (Section 4).
Keywords: stick-slip, static friction, kinetic friction, friction-induced vibration.
Scope (what is not claimed)
No new theorem or law is claimed—the distinction between static and kinetic friction, and the mechanism of stick-slip, are both standard. The origin of friction is not discussed—why static friction is the larger is not the subject. Velocity dependence is not treated—the detailed treatment where kinetic friction varies with sliding speed is not the subject; two constants are used. Damping is not included—in reality the oscillation damps and settles. Only the size of one slip is examined. No material values are asserted—$30$ N, $20$ N and $1000$ N/m are representative for exposition. Sound is not treated—the frequency and loudness of squeal are not the subject. Relation to earlier papers: Paper 430 showed that widening the contact does not change the friction—there too, a quantity that was moved did not move the result. Paper 300 showed that common root and distinct root can be decided—“the friction is large” and “the two frictions differ” are distinct roots by that criterion. Paper 438 showed that equal improvement leaves the ratio unmoved—there too, a difference of zero meant nothing happened. Paper 422 showed that the condition named as a rule’s core does not move the result—here too, the conspicuous quantity is not the one acting. What is added is giving the amplitude of one slip as $20.0000$ mm, placing the equal-friction case at exactly $0$ as the other side of the boundary, tabulating the amplitude at four ratios, showing the inverse relation with spring stiffness, showing that stiffening does not remove it, and placing the separator at “size or gap”.
1. Notation and the Question
Setting: Pull an object through a spring of stiffness $k$. Let $F_s$ be the static and $F_k$ the kinetic friction. When the spring force reaches $F_s$ the object breaks away, overshoots the position where the force equals $F_k$, and stops again.
$$\Delta x=\frac{2(F_s-F_k)}{k}$$Below, $F_s=30$ N, $F_k=20$ N and $k=1000$ N/m.
Observation 1.1 Heavy friction is expected to make motion jerky. Drag something heavy and it judders—so the size looks like the cause.
Observation 1.2 But what appears in the expression is the gap. $F_s-F_k$—neither $F_s$ nor $F_k$ appears on its own.
Question: If the gap is zero, is the motion smooth however large the friction.
2. Each Slip Jumps Twenty Millimetres
Computation 2.1: Take the amplitude at the given values.
$\Delta x=2(30-20)/1000=0.020000$ m, that is $\mathbf{20.0000}$ mm.
Observation 2.2 Twenty millimetres is visible. Pulling by hand, the jump is unmistakable—the object is not moving smoothly; it stops and jumps.
Observation 2.3 Neither load nor area appears in the expression. Only the two frictions and the spring—the same vanishing as when shape left the friction coefficient in Paper 430.
Observation 2.4 Raise $F_s$ and $F_k$ together and the amplitude does not move. The gap is unchanged—press down harder and it jumps just the same.
3. Zero Gap, Zero Amplitude
Computation 3.1: Hold the static friction at $30$ N and vary the static-to-kinetic ratio.
| Static / kinetic | Amplitude [mm] |
|---|---|
| $1.00$ | $\mathbf{0.0000}$ |
| $1.10$ | $5.4545$ |
| $1.50$ | $20.0000$ |
| $2.00$ | $\mathbf{30.0000}$ |
Observation 3.2 At a ratio of exactly $1$, the amplitude is exactly $\mathbf{0}$. The friction is still $30$ N and there is no jump at all—the other side of the boundary, showing that size is not the cause.
Observation 3.3 Even a ratio of $1.1$ jumps $5.4545$ mm. A tenth of a gap, five millimetres—the gap need not be large.
Computation 3.4: Stiffen the spring.
| Spring [N/m] | Amplitude [mm] |
|---|---|
| $1000$ | $20.0000$ |
| $10000$ | $\mathbf{2.0000}$ |
Observation 3.5 Ten times stiffer gives exactly a tenth. An inverse relation—so a stiffer machine judders less.
Observation 3.6 But it does not go. So long as the ratio is not $1$, the amplitude stays positive however stiff the spring—stiffness dilutes the symptom without removing the cause.
4. The Separator
The separator: the size of the friction, or the gap between two frictions.
Observation 4.1 Move the size and, with the gap unchanged, the amplitude does not move. Raise $F_s$ and $F_k$ together—press harder and it jumps the same.
Observation 4.2 Move the gap and the amplitude moves. From $0$ to $30$ mm as the ratio goes from $1.0$ to $2.0$—this alone acts.
Observation 4.3 These are distinct roots by the criterion of Paper 300. Both are blamed on friction—one is a level, the other the gap between two levels.
Observation 4.4 So there are two remedies. Reduce the gap (change the lubrication) or stiffen the spring—the first treats the cause, the second the symptom.
Observation 4.5 And “switch to a slipperier material” need not help. Small friction with a large gap still jumps—what to choose is not small friction but friction with a small gap.
Observation 4.6 To be honest—kinetic friction has been treated as a constant here. In reality it varies with sliding speed, and that dependence shapes the oscillation—what does not change is that a zero gap gives a zero amplitude.
Observation 4.7 One last thing—damping is not included either. Real oscillations settle—what is given is the size of one slip, not the history that follows.
5. Roll Call and Closing
- One slip jumps $20.0000$ mm (Computation 2.1): visible.
- Neither load nor area appears (Observation 2.3): two frictions and a spring.
- Raising both together changes nothing (Observation 2.4): the gap is unchanged.
- A ratio of $1$ gives exactly $0$ (Computation 3.1): the other side of the boundary.
- A ratio of $1.1$ gives $5.4545$ mm (Computation 3.1): a tenth of a gap.
- A ratio of $2.0$ gives $30.0000$ mm (Computation 3.1): monotone.
- Tenfold stiffness gives $2.0000$ mm (Computation 3.4): exactly a tenth.
- Stiffening does not remove it (Observation 3.6): only dilutes.
- Choose a small gap, not small friction (Observation 4.5): the criterion.
- The separator is size or gap (Observation 4.3): distinct roots by Paper 300.
Closing: Heavy friction is expected to make motion jerky—drag something heavy and it judders, so the size looks like the cause—but what appears in the expression is the gap $F_s-F_k$, and neither friction appears on its own. Each slip jumps unmistakably—with static $30$ N, kinetic $20$ N and a spring of $1000$ N/m, the amplitude is $2(30-20)/1000=\mathbf{20.0000}$ mm. Pulling by hand the jump is plain: the object is not moving smoothly but stopping and jumping. Neither load nor area appears in the expression—the same vanishing as when shape left the friction coefficient in Paper 430—so raising both frictions together leaves the amplitude unmoved, and pressing harder changes nothing. Zero gap, zero amplitude—at a static-to-kinetic ratio of exactly $1$, the friction is still $30$ N and yet the amplitude is exactly $\mathbf{0}$. Raising the ratio to $1.1$, $1.5$ and $2.0$ gives $5.4545$, $20.0000$ and $\mathbf{30.0000}$ mm, and even a tenth of a gap jumps five millimetres. Stiffening the spring tenfold takes the amplitude to $\mathbf{2.0000}$ mm, exactly a tenth, so a stiffer machine judders less. But it does not go—so long as the ratio is not $1$ the amplitude stays positive however stiff the spring, and stiffness dilutes the symptom without removing the cause. One thing separates them—the size of the friction, or the gap between two frictions. Move the size with the gap unchanged and nothing moves; move the gap and the amplitude runs from $0$ to $30$ mm. They are distinct roots by the criterion of Paper 300: both are blamed on friction, but one is a level and one the gap between two levels. So there are two remedies—reduce the gap by changing the lubrication, or stiffen the spring—and the first treats the cause while the second treats the symptom. And “switch to a slipperier material” need not help, since small friction with a large gap still jumps: what to choose is friction with a small gap. To be honest—kinetic friction has been treated as a constant. In reality it varies with sliding speed and that dependence shapes the oscillation. What does not change is that a zero gap gives a zero amplitude. One last thing—damping is not included either. Real oscillations settle. What is given is the size of one slip, not the history that follows.
References
- Bowden, F. P., & Tabor, D. (1950). The Friction and Lubrication of Solids. Oxford: Clarendon Press.
- Rabinowicz, E. (1958). The Intrinsic Variables Affecting the Stick-Slip Process. Proceedings of the Physical Society.
- Dieterich, J. H. (1979). Modeling of Rock Friction. Journal of Geophysical Research, 84(B5).
- Ruina, A. (1983). Slip Instability and State Variable Friction Laws. Journal of Geophysical Research, 88(B12).
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Condition Named as the Core of a Rule Does Nothing [Paper 422]. Zenodo. 10.5281/zenodo.22700081
- Yamagishi, Y. (2026). Widen the Contact a Hundredfold and the Friction Does Not Change [Paper 430]. Zenodo. 10.5281/zenodo.22700470
- Yamagishi, Y. (2026). Both Improved $7.2446$-Fold and the Ratio Never Moved [Paper 438]. Zenodo. 10.5281/zenodo.22732350
Reference
Yamagishi, Y. (2026). What Makes It Judder Is Not the Size of the Friction but the Gap Between Two [Paper 439]. YOI: https://www.yamagishiyuuki.com/yoi/439
DOI: 10.5281/zenodo.22732357
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 439. This paper is a preprint and has not been peer reviewed.