Name: impact loading on a fixed rod
Uploaded: 2024-05-23T13:00:12.000Z
Duration: 1 min 19 s
Channel: JoVE

K12 Standards

Arizona Science Standards

High School

Physical Science

HS+Phy.P3

HS+Phy.P3U2.5

Impact Loading

Рассмотрим стержень с равномерной площадью поперечного сечения, закрепленный на одном конце. Когда другой движущийся объект ударяется о свободный конец стержня, стержень деформируется, и внутри него возникает напряжение. 
Когда стержень достигает максимального напряжения, он вибрирует примерно в среднем положении. Накопившийся стресс исчезает, как только наступает период отдыха. Эти события известны как ударная нагрузка.
Здесь предполагается, что ударяющее тело передает всю свою энергию стержню, что означает, что не происходит рассеивания тепла, и ударяющее тело не отскакивает от стержня.
Таким образом, энергия деформации, соответствующая максимальному напряжению, равна кинетической энергии ударного тела.
В упругом режиме деформации энергия деформации может быть переписана в терминах максимального напряжения и модуля упругости.
Переставляя члены, получается выражение для максимального напряжения в терминах скорости ударного тела.
Использованное здесь предположение приводит к консервативному проектированию ударной нагрузки, поскольку эти предположения не действительны в реальных системах.

Ударная нагрузка

Ударная нагрузка возникает, когда движущийся объект сталкивается с неподвижной конструкцией, например стержнем с одинаковой площадью поперечного сечения, закрепленным на одном конце. В этих условиях стержень поглощает кинетическую энергию ударяющего объекта, что приводит к деформации и последующему развитию напряжений. Когда стержень возвращается в исходное положение и достигает максимального напряжения, поглощенная энергия, первоначально проявленная как кинетическая энергия, полностью преобразуется в энергию деформации.

В случае упругой деформации, когда материал возвращается в свою первоначальную форму без необратимых повреждений, энергия деформации, накопленная в точке максимальной деформации, эквивалентна кинетической энергии движущегося объекта. Эта эквивалентность предполагает, что энергия не теряется на нагревание или отскок - идеализация, которая обычно не встречается на практике. Из этой зависимости можно определить максимальное напряжение, испытываемое стержнем, исходя из скорости и массы ударяющего объекта, а также модуля упругости стержня.

<img alt="Equation 1" src="/files/ftp_upload/15686/15686_Equation_1v2.svg" style="text-align: center; height: 40px;">

Допущения, сделанные в этом анализе, приводят к консервативному подходу к инженерному проектированию, гарантирующему, что конструкции смогут противостоять неожиданным силам. Этот подход часто приводит к чрезмерному проектированию, включающему коэффициенты безопасности для учета потерь энергии и других динамик, не охваченных теоретической моделью.

Рассмотрим стержень с равномерной площадью поперечного сечения, закрепленный на одном конце. Когда другой движущийся объект ударяется о свободный конец стержня, стержень деформируется, и напряжение ...

Education

JoVE Core

Mechanical Engineering

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Unit 1

Раздел 1

Energy Methods

Энергетические методы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

impact loading on a fixed rod

Impact loading on a fixed rod happens when a moving object strikes a stationary rod that has a uniform cross-sectional area and is fixed at one end. The impact makes the rod deform and begin to develop stress. This setup shows how a sudden hit can store energy in a structure.

As the rod moves back toward its original shape, the stress rises to a maximum. At that point, the energy from the moving object has changed from kinetic energy into strain energy. Kinetic energy is the energy of motion, and strain energy is the energy stored during elastic deformation.

For elastic deformation, the rod returns to its original shape without permanent damage. In that ideal case, the strain energy at maximum deformation is equal to the kinetic energy of the striking object. This relationship assumes that no energy is lost as heat or through rebound, which is rarely true in real situations.

From this energy balance, the maximum stress in the rod can be derived using the mass and velocity of the moving object and the rod’s modulus of elasticity. The result gives engineers a conservative design approach. It can also lead to over-engineering, with safety factors added to account for energy losses and other effects not included in the model.

Impact Loading on a Fixed Rod

strain energy

Load-Deformation and Stored Strain Energy

Strain energy is the energy stored in a material when it deforms under load. It is a key idea in materials science and structural engineering. A loaded rod fixed at one end shows this clearly. When an axial force pulls on the free end, the rod stretches and stress builds inside it. As the axial force increases, the rod elongates more. This creates a direct link between the applied force and the amount of deformation. Engineers can show this relationship with a load-deformation plot. The plot...

strain-energy density

Energy Stored in Loaded Rods

Energy stored in a rod under axial load helps show how a material will respond to stretching and breaking. When the rod elongates, it stores strain energy as potential energy inside the material. Strain energy density is the amount of that energy per unit volume. In the elastic region, stress and strain have a linear relationship that follows Hooke's Law. The strain energy density in this region is found from the area under the stress-strain curve up to the elastic limit. This recoverable...

elastic strain energy for normal stresses

elastic-strain-energy-for-normal-stresses

Strain Energy in Stressed Materials

Strain energy in stressed materials describes the energy stored when a material deforms under load. It is an important idea in materials science and engineering. For axial loading with a uniform stress distribution, the stress is the internal force divided by the cross-sectional area, and the strain follows from the stress through the modulus of elasticity. When the stress is uniform, the strain energy density can be found from the product of stress and strain. That density is then integrated...

elastic strain energy for shearing stresses

elastic-strain-energy-for-shearing-stresses

Shear Energy Stored in Twisted Shafts

Shear energy stored in twisted shafts depends on elastic strain energy in a material. This energy is stored when a solid is deformed within its elastic limit. It comes from internal work done against the cohesive forces inside the material. When a material carries shearing stress, it also develops shearing strain. The strain energy density is the energy stored per unit volume. Within the elastic limit, stress is proportional to strain through the modulus of rigidity, so the strain energy...

impact loading

Impact loading on a fixed rod happens when a moving object strikes a stationary rod that has a uniform cross-sectional area and is fixed at one end. The impact makes the rod deform and begin to develop stress. This setup shows how a sudden hit can store energy in a structure. As the rod moves back toward its original shape, the stress rises to a maximum. At that point, the energy from the moving object has changed from kinetic energy into strain energy. Kinetic energy is the energy of motion,...

impact loading on a cantilever beam

Dropped-Weight Bending of a Cantilever Beam

Dropped-weight bending of a cantilever beam shows how impact loading changes a beam’s shape and stress. The beam has a circular cross-section and is fixed at one end. An object is dropped on the free end from a height h, so its potential energy from gravity becomes kinetic energy at impact. After the object hits the beam, the beam bends and develops a bending moment along its length. The bending is not uniform. The fixed end carries the highest strain energy, which is the energy stored in the...

castigliano&#39;s theorem

Using Strain Energy to Find Deflection

Castigliano's theorem uses strain energy to find deflection and rotation in elastic structures. Strain energy is the work done by external loads during elastic deformation. The theorem links that energy to the movement caused by the load. The key idea is simple. The partial derivative of the total strain energy with respect to a specific force gives the displacement at the point where that force acts. The same method also works for a moment, where the derivative gives the rotation at the point...

castigliano&#39;s theorem: problem solving

Beam Deflection from Central Load

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam. For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support. Castigliano's theorem says the...

key terms and concepts

Learn how a moving object striking a fixed rod creates stress, strain energy, and maximum stress in elastic deformation, with a conservative design model.

Impact Loading on a Fixed Rod | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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Impact Loading on a Fixed Rod

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