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

Wenn das Gewicht und die Fahrkräfte eines Autos auf einen Reifen wirken, üben sie eine externe Last aus. Der Reifengummi widersteht dieser Last durch innere Kräfte, die über sein Material verteilt sind; Dieser innere Widerstand nennt man Spannung.Spannung führt dazu, dass der Reifengummi die Form ändert, und diese Verformung wird als Dehnung gemessen.Die nichtlineare Beziehung zwischen Spannung und Dehnung im Kautschuk wird durch eine mathematische Näherung unter idealisierten Bedingungen gegeben, wobei G den Schermodul anzeigt, der den Verformungswiderstand des Materials charakterisiert.Spannung kann als Funktion der Dehnung gemäß folgender Gleichung ausgedrückt werden. Um diese sich verändernde Beziehung zu analysieren, wird Differenzierung verwendet.Standard-Differenzierungsregeln werden schwer anzuwenden, wenn sowohl Basis als auch Exponent Variablen enthalten. Die logarithmische Differentiation vereinfacht den Prozess durch Umstrukturierung der Gleichung.Die Verwendung des natürlichen Logarithmus zerlegt den Ausdruck in kleinere Begriffe, was die Differenzierung leichter macht.Die Unterscheidung beider Seiten mittels der Produkt- und Kettenregeln erfasst die Änderung der Spannung als Funktion der Dehnung. Setzt man die ursprüngliche Funktion wieder in die differenzierte Form ein, erhält man die Änderungsrate der Spannung im Autoreifen.

logarithmische differentiation

Wenn das Gewicht eines Autos und die Antriebskr&#xE4;fte auf einen Reifen wirken, &#xFC;ben sie eine &#xE4;u&#xDF;ere Last auf das Gummimaterial aus. Dieser Last wird intern durch in der gesamten Reifenstruktur verteilte Kr&#xE4;fte entgegengewirkt, die als Spannung bezeichnet werden. Die daraus resultierende Verformung des Gummis wird als Dehnung beschrieben. Die Beziehung zwischen Spannung und Dehnung bestimmt, wie sich der Reifen unter Last verformt, und ist zentral f&#xFC;r das Verst&#xE4;ndnis seines mechanischen Verhaltens im Betrieb.Der Gummiwerkstoff zeigt ein nichtlineares mechanisches Verhalten, das sich deutlich von dem linear-elastischer Materialien unterscheidet. Dieses Verhalten wird &#xFC;blicherweise durch idealisierte konstitutive Beziehungen beschrieben, die Materialparameter zur Erfassung des Verformungswiderstands ber&#xFC;cksichtigen. In der vorliegenden physikalischen Theorie wird die Spannungs-Dehnungs-Beziehung wie folgt dargestellt:\begin{equation*} \sigma = G(\varepsilon^2+1)^\varepsilon&#xA0; \end{equation*}Die Abh&#xE4;ngigkeit der Spannung von der Dehnung sowohl in der Basis als auch im Exponenten spiegelt die nichtlineare Natur des Materialverhaltens wider und erm&#xF6;glicht es dem Modell, die f&#xFC;r Gummi typischen gro&#xDF;en Verformungen zu erfassen.Um zu analysieren, wie sich die Spannung mit zunehmender Dehnung ver&#xE4;ndert, muss die Spannungs-Dehnungs-Beziehung nach der Dehnung differenziert werden. Die direkte Anwendung standardm&#xE4;&#xDF;iger Differentiationsregeln ist schwierig, da die Variable gleichzeitig in mehreren funktionalen Abh&#xE4;ngigkeiten auftritt. Die logarithmische Differentiation bietet einen systematischen Ansatz, indem der Ausdruck mithilfe des nat&#xFC;rlichen Logarithmus transformiert wird, sodass die Funktionskomponenten in einfachere Terme aufgespalten werden.Die Differentiation der transformierten Gleichung unter Anwendung der Produkt- und Kettenregel liefert einen Ausdruck f&#xFC;r die &#xC4;nderungsrate der Spannung in Abh&#xE4;ngigkeit von der Dehnung. Durch das R&#xFC;ckeinsetzen der urspr&#xFC;nglichen Spannungsfunktion in das differenzierte Ergebnis entsteht eine kompakte mathematische Beschreibung der Ver&#xE4;nderung des inneren Widerstands des Reifengummis bei Verformung, die als Grundlage f&#xFC;r die Analyse des Reifenverhaltens unter &#xE4;u&#xDF;eren Belastungen dient.

Logarithmische Differentiation

Wenn das Gewicht eines Autos und die Antriebskr&#xE4;fte auf einen Reifen wirken, &#xFC;ben sie eine &#xE4;u&#xDF;ere Last auf das Gummimaterial aus. ...

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Calculus

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Differentiation Rules

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KEY TERMS AND CONCEPTS

SCHLÜSSELBEGRIFFE UND KONZEPTE

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

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the product rule

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the quotient rule

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limits on trigonometric functions

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the chain rule

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implicit differentiation

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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

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derivatives of inverse trigonometric functions

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Inverse Trig Derivatives in Tracking

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application of rates of change

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exponential growth

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related rates

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linearization and approximation

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application of linearization and approximation

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Estimating Altitude Changes with Linearization

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hyperbolic functions

Modeling Hanging Cables with Hyperbolic Curves

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inverse hyperbolic functions and their derivatives

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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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