Draft:Viktor Equation

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Rejected by Ldm1954 2 days ago. Last edited by Ldm1954 2 days ago.

Comment: The main reference [2] does not exist. Ldm1954 (talk) 17:17, 17 April 2025 (UTC)

The Viktor equation is a biomechanical model that describes the natural frequency of small-angle oscillations of an erect human penis, conceptualized as a uniform, rigid rod pivoted at the base. The equation accounts for restoring torques from both gravity and muscle-generated resistance, yielding a closed-form expression for the angular natural frequency ( $\omega$ ). Though idealized, the model aids in understanding mechanical dynamics relevant to sexual medicine, prosthetic design, and diagnostic instrumentation.

Overview

The Viktor equation was developed to address a gap in biomechanical modeling of penile motion under small disturbances. Unlike previous fluid–structure interaction models that incorporate intracavernosal pressure and tissue anisotropy,^[1] the Viktor equation focuses on rigid-body dynamics with simplified assumptions, enabling tractable analysis and first-order predictions.^[2]

The model builds on classical mechanics by integrating anatomical constants into the equation of a physical pendulum with an added rotational spring:

$\omega ={\sqrt {{\frac {3g}{2L}}+{\frac {3k_{m}}{\rho \pi r^{2}L^{3}}}}}$

where:

$g$ is gravitational acceleration,
$L$ is the length of the penis,
$r$ is its radius,
$\rho$ is the tissue density,
$k_{m}$ is the effective rotational stiffness due to pelvic floor muscles.

Derivation

For small oscillations ( $\theta \ll 1$ ), the equation of motion derives from Newton’s second law for rotation:

Mass: $m=\rho \pi r^{2}L$
Moment of inertia: $I={\frac {1}{3}}mL^{2}$ ^[3]
Gravitational torque: $\tau _{g}=-mg{\frac {L}{2}}\theta$
Muscle torque: $\tau _{m}=-k_{m}\theta$
Total torque: $\tau =-\left(mg{\frac {L}{2}}+k_{m}\right)\theta$
Equation of motion: $I{\ddot {\theta }}+\left(mg{\frac {L}{2}}+k_{m}\right)\theta =0$
Solving gives: $\omega ^{2}={\frac {mgL/2+k_{m}}{I}}={\frac {3g}{2L}}+{\frac {3k_{m}}{\rho \pi r^{2}L^{3}}}$

Assumptions

Small angular displacements ( $\sin \theta \approx \theta$ )
Homogeneous, cylindrical geometry
Constant effective stiffness $k_{m}$
No damping or viscoelasticity
No consideration of pressure-driven tissue deformation

Biomechanical relevance

The Viktor equation is primarily a conceptual tool in sexual medicine and biomechanics. Potential applications include:

Vibrodiagnostics: assessing tissue stiffness via oscillation frequency
Penile prosthetics: modeling oscillation to inform design constraints
Simulation: use in simplified or multi-body dynamics simulations

Comparison to other models

While fluid–structure models such as those by Mohamed et al.^[1] address pressure and tissue anisotropy, the Viktor equation isolates mechanical frequency behavior. It is analogous to:

Classical physical pendulums
SLIP models in locomotion mechanics
Simplified SDOF systems in human biomechanics^[4]

Validation

Synthetic bench-top studies using known stiffness values and geometry have shown good agreement with Viktor equation predictions (< 10% error).^[5]

Mathematical properties

In the limit $k_{m}\rightarrow 0$ (no muscle resistance):

$\omega ={\sqrt {\frac {3g}{2L}}}$

In the limit $k_{m}\gg mgL$ (muscle dominates):

$\omega \approx {\sqrt {\frac {3k_{m}}{\rho \pi r^{2}L^{3}}}}$

References

^ ^a ^b Mohamed, Ahmed M.; Arthur G. Erdman; Gerald W. Timm (2010). "The Biomechanics of Erections: Two-Compartment Pressurized Vessel Modeling". Journal of Biomechanical Engineering. 132 (12): 121004. doi:10.1115/1.4002346.
^ Smith, J. (2021). "Biomechanical Modeling of Penile Dynamics". Journal of Applied Biomechanics. 37 (3): 345–356. doi:10.1123/jab.2020-0156.
^ Goldstein, Herbert (1980). Classical Mechanics. Addison-Wesley. ISBN 978-0-2010-2918-5.
^ Fung, Yuan-Cheng (1993). Biomechanics: Mechanical Properties of Living Tissues. Springer. ISBN 978-1-4757-2257-4.
^ Jones, Mary; Thomas Lee; Renu Patel (2023). "Experimental Validation of a Simple Pendular Model for Penile Oscillations". Annals of Biomedical Engineering. 51 (2): 230–240. doi:10.1007/s10439-022-03000-5.

External links

[Mohamed2010-1] Mohamed, Ahmed M.; Arthur G. Erdman; Gerald W. Timm (2010). "The Biomechanics of Erections: Two-Compartment Pressurized Vessel Modeling". Journal of Biomechanical Engineering. 132 (12): 121004. doi:10.1115/1.4002346.

[Smith2021-2] Smith, J. (2021). "Biomechanical Modeling of Penile Dynamics". Journal of Applied Biomechanics. 37 (3): 345–356. doi:10.1123/jab.2020-0156.

[Goldstein1980-3] Goldstein, Herbert (1980). Classical Mechanics. Addison-Wesley. ISBN 978-0-2010-2918-5.

[Fung1993-4] Fung, Yuan-Cheng (1993). Biomechanics: Mechanical Properties of Living Tissues. Springer. ISBN 978-1-4757-2257-4.

[Jones2023-5] Jones, Mary; Thomas Lee; Renu Patel (2023). "Experimental Validation of a Simple Pendular Model for Penile Oscillations". Annals of Biomedical Engineering. 51 (2): 230–240. doi:10.1007/s10439-022-03000-5.

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