Calculation of Conveyor-Belt Sliding Distance^[4]

Calculating the sliding distance on a conveyor belt over a pulley starts from the idea that sliding occurs whenever the belt speed differs from the pulley surface speed. As a belt element wraps around the pulley through a contact angle 𝛼 (in radians), it travels along an arc of length

${\displaystyle s=\alpha R}$,

where ${\displaystyle R}$ is the pulley radius.

Let

${\displaystyle v_{b}}$ be the belt velocity (along the belt),

${\displaystyle v_{p}}$ be the tangential velocity of the pulley surface (for example, ${\displaystyle v_{p}=R\omega }$).

The relative sliding speed is then

${\displaystyle \left|v_{b}-v_{p}\right|.}$

The time a belt element spends in contact with the pulley is

${\displaystyle t_{c}={\frac {\alpha R}{v_{b}}}.}$

The sliding distance experienced by that belt element over the contact zone is therefore^[4]

${\displaystyle L=\left|v_{b}-v_{p}\right|t_{c}=\left|v_{b}-v_{p}\right|{\frac {\alpha R}{v_{b}}}=\alpha R\left|1-{\frac {v_{p}}{v_{b}}}\right|.}$

Here, ${\displaystyle L}$ is the total distance that the belt surface slides relative to the pulley surface while passing once through the contact angle ${\displaystyle \alpha }$.

Contact zone complication case

The so-called twin-disc setup is widely used to investigate the wear resistance of materials across a broad range of engineering systems^[6](Fig. 1).

In this configuration, the effect of the kinematic variables s, u, and v on L can be defined as follows:

${\displaystyle dL=N\cdot \operatorname {d} \!s\cdot {\operatorname {d} \!u \over \operatorname {d} \!v}}$

Let L denote the sliding distance of disc A accumulated over N revolutions while in contact with rotating disc B. Provided that the angular velocities of both discs are constant and unequal, integrating the differential equation above yields^[7]

${\displaystyle L=N\,s\,{\frac {u}{v}}}$

When the measuring system for finding the contact length (s) is unavailable, the Hertzian elastic contact model serves as a suitable substitute.

${\displaystyle s={\sqrt {\dfrac {8FR_{\mathrm {eff} }}{\pi LE_{\mathrm {eff} }}}}}$

Where:

F = Applied normal force (in N)

L = Length (thickness in axial direction) of disc (in m)

R_eff = effective radius (in m)

E_eff = effective elastic modulus (in Pa)

For two discs with radii R_A and R_B :

${\displaystyle R_{\mathrm {eff} }={\dfrac {R_{A}R_{B}}{R_{A}+R_{B}}}}$

${\displaystyle {\frac {1}{E_{\mathrm {eff} }}}={\frac {1-\nu _{A}^{2}}{E_{A}}}+{\frac {1-\nu _{B}^{2}}{E_{B}}}}$

${\displaystyle E_{A}}$ and ${\displaystyle E_{B}}$ are Young’s moduli, and ${\displaystyle \nu _{A}}$ and ${\displaystyle \nu _{B}}$ are Poisson’s ratios, for discs A and B, respectively.

The above Hertzian approximation for s can be used in a laboratory setting where "twin-disc" tribometers are often employed for the evaluation of engineering materials.

Instead of using the above approximation, in scenarios such as cylindrical grinding, the estimation of the variable s is more accurate because of the geometric relationship between the depth of the cut and the contact length (i.e., the arc s). It is significant to note that in conventional cylindrical grinding, the workpiece tangential velocity v (sliding distance L) is normally kept much lower (higher) than that of the grinding wheel. This ensures optimal grinding efficiency.^[8]

A galactic-scale case

Let examine how the equation

${\displaystyle L=N\,s\,{\frac {u}{v}}}$

can be applied at a larger scale. Assuming ideally equal other properties (e.g., mass, size, and composition), for two galaxies on a collision course, the galaxy with lower angular velocity—indicating longer sliding distance, i.e. exposure—would generally experience more pronounced effects during the merger.

Simulation models^[9][10] support this conclusion. The slower-rotating systems undergo more severe orbital decay and tidal disruption due to stronger inelastic effects in close encounters. The slower of two colliding galaxies experiences greater violent relaxation, angular momentum redistribution, and structural disruption, often leading to fuller disassembly of one component compared to the ‘faster’ counter-galaxy where rotation is preserved with minimal central change.^[11][12][13]

A more complex case^[7][14]

A more down-to-Earth example is related to configurations such as met in the rolling deformation zone, where the variable u is a function of s (u = f(s)), and thus the equation for sliding distance assumes the form: ${\displaystyle L(s_{1})=\int _{0}^{s_{1}}{\frac {u(s)}{v}}\,ds}$

The relation u(s) remains a subject of advanced research in the rolling deformation zone^[15][16]. Several models offer acceptable approximations, provided that their selection is based on understanding the differences between “hot”, “warm”, and “cold” rolling; “breaking”, “roughing”, and “finishing” passes; as well as “continuous”, “semi-continuous”, and “line” rolling mills. In conclusion, sliding distance models allow for more realistic and flexible comparison of the rolling process, individual passes, and especially the reliability of the tools, i.e. the rolls themselves. The criteria based on rolled tonnes of product, or the number of roll revolutions, blur or even yield misleading comparisons in the case of intermediate and finishing rolling sectors. Even in the case of roughing stands, the sliding distance must at least be included in analyses along with the other criteria.^[5][7]

In the case of rolling deformation zone, ${\displaystyle u(s)=V_{r}-v_{\text{bar}}}$.
Where (assuming that the density of the rolled solid is constant, and the lateral spread is zero):
${\displaystyle V_{r}}$ = peripheral velocity of the roll; in most cases, a constant value
${\displaystyle V_{r}=V_{\mathrm {roll} }\cdot R}$
${\displaystyle V_{\mathrm {roll} }}$ = roll's constant angular velocity (in rad/s)
${\displaystyle v_{\text{bar}}}$ = theoretical velocity of a point on the surface of the rolled solid within the deformation zone
${\displaystyle s=R\phi }$, ${\displaystyle R}$ = roll radius, ${\displaystyle \phi }$ = angle from the exit point along the arc of the contact.
${\displaystyle \phi =0}$ at exit, ${\displaystyle \phi =\alpha }$ at entry,
${\displaystyle \alpha =1-0.5\,(h_{0}-h_{f})/R}$

The horizontal component of the roll's surface velocity at angle ${\displaystyle \phi }$ is ${\displaystyle V_{r}\cos \phi }$.

Friction (with coefficient ${\displaystyle \mu =0.3-0.6}$)^[17][7][18] enables deformation by determining the neutral point (angle ${\displaystyle \phi _{n}}$ from the exit), where the strip velocity equals the roll's horizontal component: ${\displaystyle v(\phi _{n})=V_{r}\cos \phi _{n}}$. Before ${\displaystyle \phi _{n}}$ (entry side), the strip is slower (${\displaystyle v<V_{r}\cos \phi }$), and friction pulls it forward. After ${\displaystyle \phi _{n}}$ (exit side), the strip is faster, and friction opposes motion.

Volume constancy still holds: ${\displaystyle v(\phi )\,h(\phi )=v_{\text{entry}}h_{0}=v_{\text{exit}}h_{f}}$.
Entry thickness ${\displaystyle h_{0}}$, exit thickness ${\displaystyle h_{f}}$

Step 1: Calculate the bite angle ${\displaystyle \alpha }$

${\displaystyle \alpha =\arccos \left(1-{\frac {h_{0}-h_{f}}{2R}}\right)}$.

Step 2: Calculate the Neutral Angle φ_n Using Ekelund's Formula^[19]

The exact form (valid for general α):

${\displaystyle \sin \phi _{n}={\frac {1}{2}}\sin \alpha -{\frac {1-\cos \alpha }{2\mu }}}$

Solve for ${\displaystyle \phi _{n}}$ (in radians, ${\displaystyle 0<\phi _{n}<\alpha }$). For viability, ${\displaystyle \phi _{n}>0}$; if negative, rolling may not be possible without slipping (requires higher ${\displaystyle \mu }$ or smaller reduction).

For small ${\displaystyle \alpha }$ (common in practice, ${\displaystyle \alpha \ll 1}$):

${\displaystyle \alpha \approx {\sqrt {\frac {h_{0}-h_{f}}{R}}},\quad \phi _{n}\approx {\frac {\alpha }{2}}-{\frac {\alpha ^{2}}{4\mu }}}$

The point along the contact arc is at arc length ${\displaystyle s_{n}=R\phi _{n}}$ from the exit (or projected horizontal distance ${\displaystyle \approx R\sin \phi _{n}}$).

Step 3: Thickness at Neutral Point^[20]

${\displaystyle h(\phi _{n})=h_{f}+2R(1-\cos \phi _{n})}$

Step 4: Entry Velocity ${\displaystyle v_{\text{entry}}}$ (Calculated Backwards)

At the neutral point: ${\displaystyle v(\phi _{n})=V_{r}\cos \phi _{n}}$

From volume constancy: ${\displaystyle v_{\text{entry}}=v(\phi _{n})\cdot {\frac {h(\phi _{n})}{h_{0}}}=V_{r}\cos \phi _{n}\cdot {\frac {h(\phi _{n})}{h_{0}}}}$

Step 5: Velocity Along the Contact Arc

${\displaystyle v(\phi )=v_{\text{entry}}\cdot {\frac {h_{0}}{h(\phi )}}=V_{r}\cos \phi _{n}\cdot {\frac {h(\phi _{n})}{h(\phi )}}}$

• At entry (${\displaystyle \phi =\alpha }$): ${\displaystyle v=v_{\text{entry}}}$ (minimum).
• At neutral (${\displaystyle \phi =\phi _{n}}$): ${\displaystyle v=V_{r}\cos \phi _{n}}$.
• At exit (${\displaystyle \phi =0}$): ${\displaystyle v=v_{\text{exit}}=V_{r}\cos \phi _{n}\,{\frac {h(\phi _{n})}{h_{f}}}}$ (maximum, with forward slip ${\displaystyle s_{f}={\frac {v_{\text{exit}}-V_{r}}{V_{r}}}}$).

This model requires specific values of ${\displaystyle R}$, ${\displaystyle h_{0}}$, ${\displaystyle h_{f}}$, and ${\displaystyle V_{\text{roll}}}$ for numerical computation. If provided, ${\displaystyle \phi _{n}}$ can be solved numerically (e.g., via iteration on the sine equation), and velocities evaluated.

Calculation Procedure for Roll Sliding Distance During One Revolution

In the deformation zone of flat rolling, assuming constant density of the strip and no lateral spread, the local sliding velocity between the roll surface and the strip surface is ${\displaystyle u(\phi )=V_{r}-v_{\text{bar}}(\phi ),}$ where

• ${\displaystyle V_{r}}$ is the (constant) peripheral velocity of the roll, and
• ${\displaystyle v_{\text{bar}}(\phi )}$ is the tangential (peripheral) surface velocity of the strip at angle ${\displaystyle \phi }$, measured from the exit plane along the arc of contact.

The dimensionless sliding distance across the contact arc is then defined by ${\displaystyle L=\int _{0}^{\alpha }{\frac {u(\phi )}{V_{r}}}\,\mathrm {d} \phi =\int _{0}^{\alpha }\left[\,1-{\frac {v_{\text{bar}}(\phi )}{V_{r}}}\right]\mathrm {d} \phi ,}$ so that the physical sliding distance along the roll surface is ${\displaystyle s=R\,L,}$ with ${\displaystyle R}$ the roll radius.

Conclusions

In summary, the kinematics of sliding defines the relationships between relative velocity, sliding distance, and other factors governing the interactions between components exposed to abrasion, adhesion, erosion, and other interfacial processes, such as convection and induction. Practical applications of the discussed models range from conveyor transport and rolling technology to heat exchangers and cutting operations—or, more generally, chemo-physical systems involving contact between entities moving at different velocities.^{[23][24][25][26]}