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

Quando o peso e as forças motrizes de um carro atuam sobre um pneu, eles aplicam uma carga externa. A borracha do pneu resiste a essa carga por meio de forças internas distribuídas pelo seu material; Essa resistência interna é chamada de tensão.O estresse faz com que a borracha do pneu mude de forma, e essa deformação é medida como deformação.A relação não linear entre tensão e deformação na borracha é dada por uma aproximação matemática desenvolvida sob condições idealizadas, onde G mostra o módulo de cisalhamento, que caracteriza a resistência do material à deformação.A tensão pode ser expressa como uma função da deformação, de acordo com a seguinte equação. Para analisar essa relação em mudança, é usada a diferenciação.Regras padrão de diferenciação tornam-se difíceis de aplicar quando tanto a base quanto o expoente contêm variáveis. A diferenciação logarítmica simplifica o processo ao reestruturar a equação.Tomar o logaritmo natural divide a expressão em termos menores, tornando a diferenciação mais gerenciável.Diferenciar ambos os lados usando as regras do produto e da cadeia captura a variação de tensão em função da deformação. Substituir a função original de volta na forma diferenciada fornece a taxa de variação do estresse no pneu do carro.

diferenciação logarítmica

Quando o peso e as for&#xE7;as de tra&#xE7;&#xE3;o de um carro atuam sobre um pneu, imp&#xF5;em uma carga externa ao material de borracha. Essa carga &#xE9; resistida internamente por for&#xE7;as distribu&#xED;das por toda a estrutura do pneu, definidas como tens&#xE3;o. A deforma&#xE7;&#xE3;o resultante da borracha devido a essa tens&#xE3;o &#xE9; quantificada como deforma&#xE7;&#xE3;o. A rela&#xE7;&#xE3;o entre tens&#xE3;o e deforma&#xE7;&#xE3;o governa como o pneu se deforma sob carga e &#xE9; fundamental para a compreens&#xE3;o de sua resposta mec&#xE2;nica durante a opera&#xE7;&#xE3;o.A borracha apresenta um comportamento mec&#xE2;nico n&#xE3;o linear que difere significativamente do de materiais el&#xE1;sticos lineares. Esse comportamento &#xE9; comumente descrito por meio de rela&#xE7;&#xF5;es constitutivas idealizadas que incorporam par&#xE2;metros do material que representam a resist&#xEA;ncia &#xE0; deforma&#xE7;&#xE3;o. Na presente teoria f&#xED;sica, a rela&#xE7;&#xE3;o tens&#xE3;o&#x2013;deforma&#xE7;&#xE3;o &#xE9; expressa como\begin{equation*} \sigma = G(\varepsilon^2+1)^\varepsilon&#xA0; \end{equation*}A depend&#xEA;ncia da tens&#xE3;o em rela&#xE7;&#xE3;o &#xE0; deforma&#xE7;&#xE3;o, tanto na base quanto no expoente, reflete a natureza n&#xE3;o linear da resposta do material e permite que o modelo leve em conta grandes deforma&#xE7;&#xF5;es t&#xED;picas da borracha.Para avaliar como a tens&#xE3;o evolui &#xE0; medida que a deforma&#xE7;&#xE3;o aumenta, a rela&#xE7;&#xE3;o tens&#xE3;o&#x2013;deforma&#xE7;&#xE3;o deve ser diferenciada com rela&#xE7;&#xE3;o &#xE0; deforma&#xE7;&#xE3;o. A aplica&#xE7;&#xE3;o direta das regras de diferencia&#xE7;&#xE3;o padr&#xE3;o &#xE9; desafiadora porque a vari&#xE1;vel aparece simultaneamente em m&#xFA;ltiplos pap&#xE9;is funcionais. A diferencia&#xE7;&#xE3;o logar&#xED;tmica fornece uma abordagem sistem&#xE1;tica, transformando a express&#xE3;o por meio do logaritmo natural, o que resulta na separa&#xE7;&#xE3;o dos componentes da fun&#xE7;&#xE3;o em termos mais simples.A diferencia&#xE7;&#xE3;o da equa&#xE7;&#xE3;o transformada, usando as regras do produto e da cadeia, fornece uma express&#xE3;o para a taxa de varia&#xE7;&#xE3;o da tens&#xE3;o em rela&#xE7;&#xE3;o &#xE0; deforma&#xE7;&#xE3;o. A substitui&#xE7;&#xE3;o da fun&#xE7;&#xE3;o de tens&#xE3;o original de volta no resultado diferenciado fornece uma descri&#xE7;&#xE3;o matem&#xE1;tica concisa de como a resist&#xEA;ncia interna da borracha do pneu varia com a deforma&#xE7;&#xE3;o, constituindo a base para a an&#xE1;lise do comportamento do pneu sob cargas aplicadas.

Diferenciação Logarítmica

Quando o peso e as for&#xE7;as de tra&#xE7;&#xE3;o de um carro atuam sobre um pneu, imp&#xF5;em uma carga externa ao material de borracha. Essa carga ...

Education

JoVE Core

Calculus

Cálculo

Unit

Unidade

Differentiation Rules

Regras de Derivação

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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

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the quotient rule

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derivatives of the trigonometric functions

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limits on trigonometric functions

Trigonometric Limits for Derivatives

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the chain rule

Chain Rule in a Gear System

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the chain rule: problem solving

Chain Rule in Thermal Expansion

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implicit differentiation

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implicit differentiation: problem solving

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derivatives of logarithmic functions

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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

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

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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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Deriving the Slope of a Catenary Curve

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hyperbolic and inverse hyperbolic functions: problem solving

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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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