Name: accuracy and approximation in calculations
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 28 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Specialised Topic in Science

112.51.c.1.

112.51.c.2.C.

Accuracy, limits, and approximation

В инженерных расчетах для представления значения используются точность, пределы и аппроксимация.
Рассмотрим два квадратных блока разных размеров. Точные значения их площадей можно получить, округлив их числа до ближайших значений на основе значащих цифр.
Рассмотрим прямоугольный круговой конус, имеющий радиус основания r и высоту h. Если конус имеет круговую нарезку на расстоянии x от его вершины таким образом, что разрезаемый элемент имеет толщину Δx, то его объем может быть определен.
Применяя предел от Δv к dv и Δx к dx и пренебрегая дифференциалами более высокого порядка, получается выражение для объема.
Рассмотрим дугу AB, стягивающую небольшой угол в точке O. Длина дуги может быть аппроксимирована до основания прямоугольного треугольника.
Если длина гипотенузы равна единице, то длина дуги равна синусоидальному тета, примерно равному тета. Кроме того, косинус тета также примерно равен единице.
Для тета одной степени приблизительное значение для тета греха и тан тета почти одинаково.

<ol>
	<li>Мериам, Дж.Л.; Крейдж, Л.Г., и Болтон, Д.Н. (2020). Инженерная механика \u2012 Статика. Хобокен, Нью-Джерси: Джон Уайли. С. – 13-14.</li>
</ol>

Точность, пределы и приближения

Точность, пределы и приближения являются общими во многих областях, особенно в инженерных расчётах. Эти концепции необходимы для того, чтобы обеспечить, чтобы заданное значение было максимально близко к своему истинному значению.

Точность определяется близостью измеренного значения к истинному или фактическому значению. В инженерной механике повторные измерения производятся во время теоретического или экспериментального анализа, чтобы убедиться, что результат точный и достоверный.

Точность любого решения основана на эмпирических данных и результатах, полученных из различных расчётов или измерений. Рассмотрим цилиндрический контейнер, заполненный жидкостью. Объём жидкости можно определить, измерив размеры контейнера и вычислив объём с помощью соответствующей формулы. Однако на практике невозможно измерить размеры контейнера с абсолютной точностью.

Приближения особенно полезны при работе со сложными функциями, которые трудно решить аналитически. Степень приближения относится к количеству значащих цифр, используемых в измерениях. Когда используется больше значащих цифр, результат более точный. Несмотря на полезность приближений, следует быть осторожным при их использовании, поскольку они могут внести ошибки, влияющие на точность расчёта. Это означает, что важно понимать ошибку, допущенную приближением, и как она может повлиять на результат.

Например, для относительно небольшого угла, обычно меньше или равного 15° или 0,26 радиан, можно использовать приближения для упрощения функций синуса, косинуса и тангенса. Особенно точность этих приближений уменьшается, когда угол превышает 15° или 0,26 радиан. Однако для углов меньше или равных 15° (0,26 радиан) эти формулы обеспечивают достаточную точность и дают значения, почти равные точному значению.

Концепция пределов также важна в инженерных расчётах. Пределы используются для определения поведения функции, особенно когда независимая переменная приближается к определённому значению. Рассмотрим функцию, которая стремится к бесконечности при приближении значения x к определённому значению. Предел определяет максимальное значение, которое может быть достигнуто без превышения указанного значения x. Пределы особенно полезны для определения стабильности и производительности систем в критических приложениях.

Точность, пределы и аппроксимации распространены во многих областях, особенно в инженерных расчетах. Эти концепции необходимы для обеспечения того, чтобы ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

An Introduction to Statics

Введение в статику

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

accuracy and approximation in calculations

Accuracy and approximation are important in calculations, especially in engineering. Accuracy means how close a measured value is to the true or actual value. Repeated measurements are often used in engineering mechanics to help make results more precise and accurate.

The accuracy of a solution depends on empirical data and the results from different calculations or measurements. A cylindrical container filled with liquid is a simple example. Its volume can be found from the measured dimensions and a formula, but those dimensions cannot be measured with absolute precision in practice.

Approximations are useful when a function is too complex to solve exactly. The degree of approximation depends on the number of significant figures used in the measurements. More significant figures usually give a result that is closer to the true value.

Approximations can also introduce error, so the effect of the error must be understood. For small angles, usually less than or equal to 15° or 0.26 radians, sine, cosine, and tangent can be approximated with good accuracy. This accuracy becomes worse when the angle is greater than 15° or 0.26 radians.

Limits are another key idea in engineering calculations. A limit describes the behavior of a function as an independent variable approaches a specific value. For a function that grows toward infinity as x approaches a certain value, the limit helps define the maximum value that can be reached without passing that x value. Limits are useful for judging the stability and performance of systems in critical applications.

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accuracy, limits, and approximation

Accuracy and approximation are important in calculations, especially in engineering. Accuracy means how close a measured value is to the true or actual value. Repeated measurements are often used in engineering mechanics to help make results more precise and accurate. The accuracy of a solution depends on empirical data and the results from different calculations or measurements. A cylindrical container filled with liquid is a simple example. Its volume can be found from the measured...

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Learn how accuracy, significant figures, small-angle approximations, and limits help describe real measurements and function behavior in engineering calculations.

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112.51.c.1.E.

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112.51.c.1.F.

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112.51.c.1.G.

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112.51.c.1.H.

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112.51.c.2.A.

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Accuracy and Approximation in Calculations

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