Mechanical Vibrations - Chapter 4 - Supplement: Basic Knowledge on Shocks (Course) PDF

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EIGSI Engineering School

Yoann Barranger

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mechanical vibrations shock engineering course material

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This document is a supplement for a course on mechanical vibrations, focusing on the concept of shocks. It covers definitions, examples, representations, and responses of structures to shocks, providing a basic understanding of the topic. It includes mechanical vibrations, shocks, and engineering concepts.

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Mechanical Vibrations – Chapter 4 – Supplement: Basic knowledge on shocks (COURSE) Mechanics & Energy Department Year: 4-FISE Semester: 7 Reference: MEE_TRC_07_074 Lecturer: Yoann Barranger EIGSI – 26 ru...

Mechanical Vibrations – Chapter 4 – Supplement: Basic knowledge on shocks (COURSE) Mechanics & Energy Department Year: 4-FISE Semester: 7 Reference: MEE_TRC_07_074 Lecturer: Yoann Barranger EIGSI – 26 rue Vaux de Foletier – 17041 La Rochelle Cedex 1 https://www.eigsi.fr Published and distributed by EIGSI – All Rights of copying, adaptation and translation reserved Chapter 4 – Supplement: Basic knowledge on shocks (COURSE) Mechanical Vibrations Contents 1. Mechanical shock.......................................................................................................................... 3 1.1. Definitions.......................................................................................................................................... 3 1.2. Examples of mechanisms triggering a shock...................................................................................... 3 1.3. Behaviour in collision shock............................................................................................................... 4 1.3.1. Localised deformation................................................................................................................................................. 4 1.3.2. Propagation of a compression wave............................................................................................................................. 5 1.3.3. Example of the effect of shock wave propagation........................................................................................................ 6 1.3.4. Elastic and inelastic shocks.......................................................................................................................................... 6 2. Representation of a shock............................................................................................................ 7 2.1. Elastic shock using the half-wave sine representation........................................................................ 7 2.2. Inelastic shock using the rectangle representation.............................................................................. 8 2.3. Real shock.......................................................................................................................................... 9 3. Response of structures to shock................................................................................................ 11 3.1. Sensitivity of structures.................................................................................................................... 11 3.2. Damping........................................................................................................................................... 11 Index of illustrations....................................................................................................................... 12 Yoann Barranger – 4-FISE S7 Page 2 / 13 Chapter 4 – Supplement: Basic knowledge on shocks (COURSE) Mechanical Vibrations 1. Mechanical shock 1.1. Definitions What is a shock: A shock is a temporary stimulus (acceleration or force), generally brief and of very high intensity. Shocks are a phenomenon related to vibration, but their transient nature means specific tools are needed to predict the specific forces and deformations they induce. Discontinuity in the speed vector: Mechanical shock occurs when there is an abrupt change in the speed vector. We then talk about a discontinuity in the speed vector, which in theory implies infinite acceleration. Note: A shock corresponds to a Dirac (“infinite”) peak in the force or torque on a body. In reality, the speed change is not instantaneous. The acceleration, and therefore the force or torque, can only exhibit finite values, although these can be very high (the body’s speed and inertia are important factors). In illustration 1.1, we can see the discontinuities caused by a motion involving two shocks: a starting shock and a braking shock. We cannot derivate the position at these points (which are sharp angles), therefore the speed curve exhibits discontinuities and the acceleration curve has Dirac peaks. Illustration 1.1: Discontinuities on a shock The “jolt” notion: A jolt is characterized by a discontinuity in acceleration, distinguishing it from a shock, whose characteristic is a discontinuity in speed. Consequently, with a jolt, the sharp angles are found on the speed graph and not on the position graph (which would then appear differentiable at all points). 1.2. Examples of mechanisms triggering a shock A collision between two objects is an obvious mechanism for triggering a mechanical shock, but it’s not the only phenomenon that can do so. There are quite a few different mechanisms: Yoann Barranger – 4-FISE S7 Page 3 / 13 Chapter 4 – Supplement: Basic knowledge on shocks (COURSE) Mechanical Vibrations An abrupt change in pressure ◦ Example: the “blast” from an explosion is caused by a high pressure spike in the immediate surroundings. As with sound, this localised pressure disturbance propagates in the form of a wave, which in the case of an explosion is more like a shock wave given the brevity and high intensity of the disturbance. ◦ Note: although a sound impulse can exhibit the brevity necessary to a shock, the pressure variations it creates usually lack the intensity required for it to be described as a shock. An abrupt change in fluid flowrate ◦ Example: “water hammer” (when a tap or valve is opened or closed quickly, a high pressure phenomenon can occur, causing an abrupt variation in the speed of a liquid). Sudden braking ◦ Example: the emergency stop buttons on machine tools must stop the rotation almost instantly. Motion of an attached body ◦ Example: a person attached by a rope who falls into a void will be subject to a sudden change of speed as soon as the slack in the rope is taken up. A flow of electrical current ◦ Example: a defibrillator discharges a sudden flow of electrical current through the chest of a person in order to cause a shock, stopping fibrillation of the heart. ◦ Note: although the electrical impulse is obviously abrupt, its “high intensity” nature is probably less obvious. A few numbers: a defibrillator must generate a shock energy of 360 joules through a thorax whose impedance is roughly 50 ohms, corresponding to a discharge of roughly 3000 volts and 60 amps (for “damped sinusoidal” models) or a discharge of 1800 volts and 32 amps (for “truncated exponential decay” models). etc. 1.3. Behaviour in collision shock 1.3.1. Localised deformation A shock is most often the result of a contact action, meaning there will be different speeds inside a given object during a shock. Parts of the object located far from the contact zone will exhibit a different speed from those close to the contact zone. → There is localised deformation of the object. Illustration 1.2: Localised deformation in collision shock Just before impact: the speed is identical at all points in the solid (if we consider it to have only translatory motion). On impact: the speed of the points is identical to that at the previous instant. Yoann Barranger – 4-FISE S7 Page 4 / 13 Chapter 4 – Supplement: Basic knowledge on shocks (COURSE) Mechanical Vibrations Just after impact: the speed of point M’ diminishes sharply and there is local compression of the object in the region of the contact point. Parts of the object far from the contact point maintain their speed and geometry. Note 1: the case represented in illustration 1.2 shows us only what happens up to the instant just after impact (the different behaviours that can occur after this instant will be looked with illustration 1.3). Note 2: the term “impact” is normally reserved for cases where there is dissipation of energy by brittle fracture (which we will not be looking at here). Its meaning here is nevertheless widely accepted. We must understand that the continuous system models in the previous Chapter (including the compression model) can no longer be applied in determining the motion equation, its solution, and its forces. That’s because the constant section assumption is no longer valid (not to mention that the shock may have modified the material’s local properties, potentially invalidating the assumptions of isotropy and linear elastic behaviour). 1.3.2. Propagation of a compression wave Resulting from this phenomenon, a compression wave propagates the disturbance through the matter, rebounding against the object’s edges until it settles down to a new equilibrium (when all the points are back at the same speed). The example of illustration 1.3 represents the propagation of a compression wave through an object coming into collision with an immovable wall. Note 1: To facilitate visualization, the geometry changes are not represented. Note 2: This type of representation can be completed in order to draw a “time-space” graph and a “speed- stress” graph (which will not be covered in this course). Step 1: The object approaches collision. Step 2: Just after impact, a compression wave propagates through the object. Step 3: The wave continues propagating until it rebounds against the opposite free edge. Step 4: The object rebounds and comes back off the wall. Illustration 1.3: Propagation of a wave through an object around a collision Note: this behaviour is not the only one possible; several others also exist: The wave front can propagate in several directions (and therefore rebound against several edges) The wave can travel from end to end several times The object might not rebound (e.g., if the object suffers damage) etc. Yoann Barranger – 4-FISE S7 Page 5 / 13 Chapter 4 – Supplement: Basic knowledge on shocks (COURSE) Mechanical Vibrations 1.3.3. Example of the effect of shock wave propagation The LSP (Laser Shock Processing) method sets out to implant residual stresses in a metal in order to improve its hardness and fatigue strength at certain points by forming a mild plastic region. To do so, a shock is triggered by a laser on the material’s surface. The wave from the laser shock then propagates through the solid, displacing the material within. After several rebounds against the free edges of the solid, the wave is attenuated and finds a new equilibrium that no longer corresponds to the initial state. In the illustration 1.4, we can observe the evolution of stresses in the solid after laser impact and therefore after propagation of the shock wave. Illustration 1.4: Evolution of stresses and propagation of shock wave in a solid after laser impact 1.3.4. Elastic and inelastic shocks If the speed has the same magnitude before and after a rebound, we have a case of elastic shock. If the speed falls after a rebound, there must have been absorption of energy, so we have a case of inelastic shock. Note 1: Elastic shock is unrealistic because it assumes there is no dissipation of energy on contact. It can nonetheless be used in certain cases where energy losses can be considered negligible. Note 2: This has already been partly studied in previous years by considering the conservation of momentum in order to characterise things like The speed of the objects before and after shock The energy losses at the moment of shock The coefficient of restitution (which is 1 for a perfectly elastic shock and 0 for a shock with total dissipation of the impact energy). Several other aspects have not been touched on because of their complexity: The graph of the object’s positions, speeds, and acceleration (especially for a real shock) The sensitivity to shock of materials and structures Potential damage Tools used in engineering and research for studying shocks Retrieval of the forces linked to impact, etc. This part of the course sets out simply to introduce the basic concepts for certain of these issues. Yoann Barranger – 4-FISE S7 Page 6 / 13 Chapter 4 – Supplement: Basic knowledge on shocks (COURSE) Mechanical Vibrations 2. Representation of a shock 2.1. Elastic shock using the half-wave sine representation Consider a non-deformable impactor of mass m colliding with an immovable object. The surfaces in contact undergo perfectly elastic local deformation. The contact is modelled by a fictive spring of stiffness k during contact (x ≤ 0) and null in the absence of contact (x > 0). Illustration 2.1: Representation of an elastic shock Initial conditions (onset of shock): x (0 )=0 ẋ (0 )=V 0

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