Name: differentiating tire stress with logs
Uploaded: 2026-01-12T12:24:15.000Z
Duration: 1 min 28 s
Channel: JoVE

Gdy ciężar i siły napędowe samochodu oddziałują na oponę, wywierają one obciążenie zewnętrzne. Guma opony opiera się temu obciążeniu dzięki siłom wewnętrznym rozłożonym na materiał; Ten opór wewnętrzny nazywany jest naprężeniem.Naprężenie powoduje zmianę kształtu gumy opony, a to odkształcenie jest mierzone jako odkształcenie.Nieliniowa zależność między naprężeniem a odkształceniem w gumie jest dana przez matematyczne przybliżenie opracowane w idealizowanych warunkach, gdzie G pokazuje moduł ścinania, który charakteryzuje odporność materiału na odkształcenia.Naprężenie można wyrazić jako funkcję odkształcenia, zgodnie z następującym równaniem. Aby przeanalizować tę zmieniającą się relację, stosuje się różnicowanie.Standardowe reguły różniczkowania stają się trudne do zastosowania, gdy zarówno baza, jak i wykładnik zawierają zmienne. Różniczkowanie logarytmiczne upraszcza proces poprzez restrukturyzację równania.Wzięcie logarytmu naturalnego dzieli wyrażenie na mniejsze wyrazy, co ułatwia różniczkowanie.Rozróżnienie obu stron za pomocą reguł iloczynu i łańcucha obejmuje zmianę naprężeń w funkcji odkształcenia. Podstawienie pierwotnej funkcji z powrotem do formy różniczkowanej daje szybkość zmiany naprężenia w opony samochodowej.

różniczkowanie logarytmiczne

Gdy ci&#x119;&#x17C;ar samochodu i si&#x142;y nap&#x119;dowe dzia&#x142;aj&#x105; na opon&#x119;, powoduj&#x105; przy&#x142;o&#x17C;enie obci&#x105;&#x17C;enia zewn&#x119;trznego do materia&#x142;u gumowego. Obci&#x105;&#x17C;enie to jest r&#xF3;wnowa&#x17C;one wewn&#x119;trznie przez si&#x142;y roz&#x142;o&#x17C;one w ca&#x142;ej strukturze opony; t&#x119; wewn&#x119;trzn&#x105; odporno&#x15B;&#x107; okre&#x15B;la si&#x119; jako napr&#x119;&#x17C;enie. Powsta&#x142;e odkszta&#x142;cenie gumy spowodowane tym napr&#x119;&#x17C;eniem jest opisywane ilo&#x15B;ciowo jako odkszta&#x142;cenie wzgl&#x119;dne. Zale&#x17C;no&#x15B;&#x107; mi&#x119;dzy napr&#x119;&#x17C;eniem a odkszta&#x142;ceniem wzgl&#x119;dnym okre&#x15B;la spos&#xF3;b odkszta&#x142;cania si&#x119; opony pod obci&#x105;&#x17C;eniem i jest kluczowa dla zrozumienia jej odpowiedzi mechanicznej podczas eksploatacji.Guma wykazuje nieliniowe zachowanie mechaniczne, kt&#xF3;re znacznie r&#xF3;&#x17C;ni si&#x119; od zachowania materia&#x142;&#xF3;w liniowo spr&#x119;&#x17C;ystych. To zachowanie jest powszechnie opisywane za pomoc&#x105; idealizowanych zale&#x17C;no&#x15B;ci konstytutywnych, kt&#xF3;re uwzgl&#x119;dniaj&#x105; parametry materia&#x142;owe odzwierciedlaj&#x105;ce odporno&#x15B;&#x107; na odkszta&#x142;cenie. W obecnej teorii fizycznej zale&#x17C;no&#x15B;&#x107; napr&#x119;&#x17C;enie&#x2013;odkszta&#x142;cenie jest wyra&#x17C;ona jako\begin{equation*} \sigma = G(\varepsilon^2+1)^\varepsilon&#xA0; \end{equation*}Zale&#x17C;no&#x15B;&#x107; napr&#x119;&#x17C;enia od odkszta&#x142;cenia wzgl&#x119;dnego zar&#xF3;wno w podstawie, jak i wyk&#x142;adniku odzwierciedla nieliniowy charakter odpowiedzi materia&#x142;u i pozwala modelowi uwzgl&#x119;dni&#x107; du&#x17C;e odkszta&#x142;cenia typowe dla gumy.Aby oceni&#x107;, jak zmienia si&#x119; napr&#x119;&#x17C;enie wraz ze wzrostem odkszta&#x142;cenia wzgl&#x119;dnego, nale&#x17C;y r&#xF3;&#x17C;niczkowa&#x107; zale&#x17C;no&#x15B;&#x107; napr&#x119;&#x17C;enie&#x2013;odkszta&#x142;cenie wzgl&#x119;dne wzgl&#x119;dem odkszta&#x142;cenia wzgl&#x119;dnego. Bezpo&#x15B;rednie zastosowanie standardowych regu&#x142; r&#xF3;&#x17C;niczkowania jest trudne, poniewa&#x17C; zmienna wyst&#x119;puje jednocze&#x15B;nie w wielu rolach funkcyjnych. R&#xF3;&#x17C;niczkowanie logarytmiczne stanowi systematyczne podej&#x15B;cie poprzez transformacj&#x119; wyra&#x17C;enia za pomoc&#x105; logarytmu naturalnego, co prowadzi do roz&#x142;o&#x17C;enia sk&#x142;adowych funkcji na prostsze wyra&#x17C;enia.R&#xF3;&#x17C;niczkowanie przekszta&#x142;conego r&#xF3;wnania za pomoc&#x105; regu&#x142;y iloczynu i regu&#x142;y &#x142;a&#x144;cuchowej prowadzi do otrzymania wyra&#x17C;enia okre&#x15B;laj&#x105;cego tempo zmiany napr&#x119;&#x17C;enia wzgl&#x119;dem odkszta&#x142;cenia wzgl&#x119;dnego. Podstawienie funkcji wyj&#x15B;ciowej napr&#x119;&#x17C;enia z powrotem do r&#xF3;&#x17C;niczkowanego wyniku zapewnia zwarty opis matematyczny tego, jak op&#xF3;r wewn&#x119;trzny gumy opony zmienia si&#x119; wraz z odkszta&#x142;ceniem, stanowi&#x105;c podstaw&#x119; do analizy zachowania opony pod przy&#x142;o&#x17C;onym obci&#x105;&#x17C;eniem.

Różniczkowanie logarytmiczne

Gdy ci&#x119;&#x17C;ar samochodu i si&#x142;y nap&#x119;dowe dzia&#x142;aj&#x105; na opon&#x119;, powoduj&#x105; przy&#x142;o&#x17C;enie ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Differentiation Rules

Zasady różniczkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

Logarithmic Differentiation

differentiating tire stress with logs

Logarithmic differentiation is used to find how tire stress changes with strain. In this setting, a car’s weight and driving forces act as an external load on the rubber. The rubber resists that load through internal forces called stress. The deformation that results is called strain.

The stress-strain relationship shows how the tire changes shape under load. For rubber, this relationship is nonlinear, so it does not behave like a linear elastic material. The model used here includes material parameters that describe resistance to deformation. It also uses stress and strain in both the base and the exponent, which helps represent the large deformations typical of rubber.

To study how stress evolves as strain increases, the stress-strain equation must be differentiated with respect to strain. Direct differentiation is difficult because the variable appears in more than one role. Logarithmic differentiation solves this by first taking the natural logarithm of the expression. That step breaks the function into simpler parts.

After the expression is transformed, the product rule and chain rule are used to differentiate it. The result gives the rate of change of stress with respect to strain. Substituting the original stress function back into the result gives a compact equation for how the tire rubber’s internal resistance changes as it deforms. This mathematical form helps analyze tire behavior under applied loads.

Differentiating Tire Stress with Logs

derivatives of simple functions

Slope Rules for Simple Functions

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the product rule

Derivative of a Changing Rectangle

The Product Rule helps find the derivative of a product of two changing functions. In calculus, it states that the derivative of one function times another equals the first function times the rate of change of the second, plus the second function times the rate of change of the first. This rule shows how to account for both functions when they vary at the same time. A useful way to see the rule is with a rectangle whose width and height both change over time. If the width and height represent...

the quotient rule

Differentiating Ratios with the Quotient Rule

The quotient rule is a key calculus method for differentiating ratios of two differentiable functions. It is used when a function is written as one function divided by another, and the denominator is not zero. The rule gives a direct way to find the derivative of that ratio. The quotient rule is especially useful for rational functions, trigonometric ratios, and exponential functions. For example, it can be applied to a function in ratio form to find its derivative step by step. This makes it...

derivatives of the trigonometric functions

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limits on trigonometric functions

Trigonometric Limits for Derivatives

Trigonometric limits are key tools in calculus, especially when derivatives are being defined. One of the most important results is the limit of sine over its angle, and this result is used often in math and physics. It helps connect trigonometry with rates of change. A common way to understand this limit uses a sector of a unit circle. In radians, the arc length matches the angle, so the circle gives a clear geometric setting for the proof. The angle also fits inside an inequality that is...

the chain rule

Chain Rule in a Gear System

The Chain Rule appears in a gear system with three connected gears. The first gear is represented by x, the second by z, and the third by y. Each gear’s motion depends on the gear before it, so the system creates a linked change from one stage to the next. This setup shows a composite function, which is a function built from another function. In the first stage, the second gear’s rotation depends on the first gear, so the intermediate variable is written as z = z(x). In the second stage, the...

the chain rule: problem solving

Chain Rule in Thermal Expansion

The Chain Rule appears in thermal expansion when a rod’s length changes because its temperature changes over time. In this example, a metal rod is heated, and its length depends on temperature. The temperature itself changes with time in a quadratic way. For linear thermal expansion, the rod’s length L depends on temperature T with a constant rate of change, using the initial length L0 = 2 m and the coefficient of thermal expansion α. The temperature follows a quadratic relationship in time,...

implicit differentiation

Tangent Lines in Circular Motion

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second derivatives of implicit functions

Slope and Curvature of Elliptical Arches

Elliptical arches connect slope and curvature to the shape of an ellipse. In this geometry, the height y and the horizontal position x are related implicitly, so y cannot be written as an explicit function of x. That is why implicit differentiation is needed to study the arch. The ellipse is centered at the origin and has a semi-major axis a and a semi-minor axis b. Differentiating the ellipse equation with respect to x uses the chain rule. After solving for dy / dx, the first derivative shows...

implicit differentiation: problem solving

Tangent Lines with Implicit Differentiation

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derivatives of logarithmic functions

Natural Log Derivatives and Real-World Uses

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logarithmic differentiation

Logarithmic differentiation is used to find how tire stress changes with strain. In this setting, a car’s weight and driving forces act as an external load on the rubber. The rubber resists that load through internal forces called stress. The deformation that results is called strain. The stress-strain relationship shows how the tire changes shape under load. For rubber, this relationship is nonlinear, so it does not behave like a linear elastic material. The model used here includes material...

the number e as a limit

e from Natural Log Derivatives

The number e is a key constant in calculus, and it helps describe continuous change. It is especially important for exponential growth. Rather than treating e as an arbitrary number, the transcript shows how it can be defined through the natural logarithm and differentiation. The natural logarithm, written as ln x, is the inverse of the exponential function with base e. Its derivative is 1 divided by its input. When that derivative is evaluated at x = 1, the result is exactly 1. This condition...

derivatives of inverse trigonometric functions

derivatives-of-inverse-trigonometric-functions

Inverse Trig Derivatives in Tracking

Inverse trigonometric derivatives can describe how a tracking angle changes as an aircraft moves closer to an observer. In this situation, a ship measures the slant distance to the aircraft and the angle of elevation. These measurements let the observer find the horizontal and vertical parts of the distance using trigonometric relationships. The angle of elevation is treated as the dependent variable, and the ratio of altitude to slant distance is treated as the independent variable. That...

application of rates of change

Velocity and Acceleration in Motion Graphs

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exponential growth

Bacterial Doubling in Exponential Growth

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related rates

Balloon Inflation and Changing Rates

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linearization and approximation

Tangent Line Estimates Near a Point

Linearization uses a tangent line to estimate a nonlinear function near a chosen reference point. It replaces a difficult function with a simpler linear model when the input changes only a little. This makes it useful for small deviations from a known value. A square root example shows how the method works. The value at 4 is known exactly, so 4 is a good reference point. The function value and the rate of change are both easy to find there. Estimating the value at a nearby input, such as 4.1,...

application of linearization and approximation

application-of-linearization-and-approximation

Estimating Altitude Changes with Linearization

Linearization can help a drone estimate small altitude changes from air pressure readings. Atmospheric pressure drops as altitude rises, and this relationship is often modeled with an exponential function. The model is accurate, but the math needed to convert pressure into altitude can be too complex for repeated calculations during flight. To make the calculation easier, the pressure-altitude relationship is approximated locally with a straight line. This line is built near a chosen reference...

hyperbolic functions

Modeling Hanging Cables with Hyperbolic Curves

Hyperbolic functions help describe the shape of a hanging cable. When a flexible cable is suspended between two points at the same height, it naturally forms a catenary. That curve comes from the balance between the cable’s weight and the tension along its length. It shows a state of mechanical equilibrium. The true shape of this hanging cable is not captured well by a simple approximation. Instead, it is described with hyperbolic functions. These functions are related to exponential functions.

inverse hyperbolic functions and their derivatives

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Deriving the Slope of a Catenary Curve

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hyperbolic and inverse hyperbolic functions: problem solving

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Tangent Lines on a Hyperbolic Arch

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key terms and concepts

See how logarithmic differentiation is used to model tire stress and strain in rubber, and how the method simplifies a nonlinear equation.

Differentiating Tire Stress with Logs | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Differentiating Tire Stress with Logs

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