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

一端に固定された均一な断面積のロッドを考えてみましょう。ロッドの自由端に別の移動物体が当たると、ロッドが変形し、ロッド内で応力が発生します。 
ロッドが最大応力に達すると、平均位置を中心に振動します。溜まったストレスは、いざ休むと消えていきます。これらのイベントは、衝撃荷重と呼ばれます。
ここでは、打撃体が全エネルギーをロッドに伝達するため、熱放散は発生せず、打撃体はロッドに跳ね返らないと仮定しています。
したがって、最大応力に対応するひずみエネルギーは、打撃物体の運動エネルギーと等しくなります。
変形の弾性領域では、ひずみエネルギーは最大応力と弾性率の観点から書き換えることができます。
用語を並べ替えると、打撃物体の速度に関する最大応力の表現が得られます。
ここで使用される仮定は、実際のシステムでは有効ではないため、衝撃荷重の保守的な設計になります。

衝撃荷重

衝撃荷重は、移動する物体が、一端に固定された均一な断面積を持つ棒などの静止構造物に衝突するときに発生します。このような条件下では、棒は打撃物体からの運動エネルギーを吸収し、変形とその後の応力の発生につながります。棒が元の位置に戻り、最大応力に達すると、最初は運動エネルギーとして現れた吸収エネルギーが完全にひずみエネルギーに変わります。

材料が永久的な損傷を受けることなく初期形状に戻る弾性変形の場合、最大変形点に蓄積されるひずみエネルギーは、移動する物体の運動エネルギーに相当します。この等価性は、熱や反発によってエネルギーが失われないことを前提としていますが、これは実際の環境では通常見られない理想化です。この関係から、打撃対象物の速度と質量、および棒の弾性率に基づいて棒が受ける最大応力を導き出すことができます。

<img alt="Equation 1" src="/files/ftp_upload/15686/15686_Equation_1v2.svg" style="height: 40px;">

この分析で行われた仮定は、構造が予期せぬ力に耐えられることを保証する工学設計への保守的なアプローチにつながります。このアプローチでは、多くの場合、理論モデルではカバーされないエネルギー損失やその他の動力学を考慮した安全係数を組み込む過剰設計が発生します。

一端に固定された均一な断面積のロッドを考えてみましょう。ロッドの自由端に別の動く物体が当たるとロッドが変形し、応力が...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Energy Methods

エネルギー法

KEY TERMS AND CONCEPTS

主要用語と概念

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 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

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Impact Loading on a Fixed Rod

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