Pulse - health, life expectancy, aging and immortality. Impact force - impulse, speed, equipment and explosive force exercises for fighters from Pavel Badyrov An impact can be considered instantaneous if its duration

MECHANICAL SHOCK

Nizhny Novgorod
  year 2013

Laboratory work No. 1-21

Mechanical shock

purpose of work: Get familiar with the elements of the theory of mechanical shock and experimentally determine the impact time, the average impact force Frecovery factor E, as well as study the basic characteristics of the impact and familiarize yourself with digital instruments for measuring time intervals.

Theoretical part

  Impact is called a change in the state of movement of the body, due to its short-term interaction with another body. During impact, both bodies undergo shape changes (deformation). The essence of elastic shock lies in the fact that the kinetic energy of the relative motion of colliding bodies, in a short time, is converted into the energy of elastic deformation or, to one degree or another, into the energy of molecular motion. In the process of impact, energy is redistributed between colliding bodies.

Let a ball with a certain speed V 1 fall on a flat surface of a massive plate and bounce off it with a speed V 2.

We denote   Are the normal and tangential components of the velocities and, and, and are the angles of incidence and reflection, respectively. In the ideal case, with absolutely elastic impact, the normal components of the rates of incidence and reflection and their tangential components would be equal; . Upon impact, a partial loss of mechanical energy always occurs. The ratio of both normal and tangential velocity components after impact to velocity components before impact is a physical characteristic that depends on the nature of the colliding bodies.

This characteristic Ecalled the recovery coefficient. Its numerical value lies between 0 and 1.

Determination of average impact force,

Initial and final ball velocities upon impact

  The experimental setup consists of a steel ball A suspended on conductive threads and a motionless body B of a larger mass with which the ball collides. Suspension angle α is measured on a scale. At the moment of impact, a ball of mass m is affected by gravity from the side of the Earth, the reaction force from the side of the thread, and the average force of impact from the side of body B (see Fig. 2).

Based on the theorem on the change in momentum of a material point:

where and are the velocity vectors of the ball before and after the impact; τ is the duration of the impact.

After designing equation (2) on the horizontal axis, we determine the average impact force:

(3)

Ball speeds V 1 and V 2 are determined on the basis of the law of conservation and conversion of energy. The change in the mechanical energy of the system formed by the ball and the stationary body B in the Earth's gravitational field is determined by the total work of all external and internal non-potential forces. Since the external force is perpendicular to the movement and the thread is inextensible, this force does not work, the external force and the internal force of the elastic interaction are potential. If these forces are much larger than other non-potential forces, then the total mechanical energy of the selected system does not change. Therefore, the energy balance equation can be written as:

(4)

From the drawing (Fig. 2) it follows that   , then from equation (4) we obtain the values \u200b\u200bof the initial V 1 and final V 2 ball speeds:

(5)

where and are the angles of deflection of the ball before and after the collision.

Method for determining the duration of an impact

In this paper, the duration of a ball hitting a plate is determined by the frequency meter Ch3-54, the functional diagram of which is shown in Fig. 3. From the generator, pulses with a period T are fed to the input of the control system of the control system. When during the collision of the metal plate B, the electric circuit formed by the control system, the conductive ball suspension threads, the ball, the plate B and the pulse counter C h, turns out to be closed, and the control system passes at the counter input C h, electric current pulses only in the time interval equal to the duration of the shock. The number of pulses recorded during the time is equal to where.

To determine the duration of the impact, it is necessary to multiply the number of pulses recorded by the counter by the period of pulses taken from generator G.

experimental part

Initial data:

1. The mass of the ball m \u003d (16.7 ± 0.1) * 10 -3 kg.

2. Thread length l \u003d 0.31 ± 0.01 m.

3. Acceleration of gravity g \u003d (9.81 ± 0.005) m / s 2.

4. Experience for each corner is performed 5 times.

The results of the experiment are listed in the table:

Calculations

=20 0   μs

=30 0   μs

=40 0   μs

If the products have shock absorbers, then when choosing the duration of the impact acceleration, the lower resonant frequencies of the products themselves, and not the protective elements, are taken into account.

The parameters to be checked are the parameters by the change of which one can judge the impact stability of the CEA as a whole (distortion of the output signal, stability of the functioning characteristics, etc.).

When developing a test program, the directions of impacts are set depending on the specific properties of the tested CEA. If the properties of CEA are unknown, then the test should be carried out in three mutually perpendicular directions. It is recommended to choose (from the range specified in the TU) the duration of the shocks that cause the resonance excitation of the CEA under test.

Impact strength is evaluated by the integrity of the structure (for example, the absence of cracks, the presence of contact). Products are considered to have passed the impact test if, after testing, they meet the requirements of standards and PI for this type of test.

An impact test is recommended after an impact test. Often they are combined. In contrast to the impact test, the impact test is carried out under electric load, the nature and parameters of which are set in TU and PI. At the same time, control of the REA parameters is performed during the impact to check the operability of the products and identify false positives. Products are considered to have passed the test if, in the process and after it, they meet the requirements established in the standards and PI for this type of test.

2.3. The third task.

Explore devices for testing CEA for impact / 1. p. 263-268. 2.p.171-178. 3.p.138-143 /

Devices for testing.Impact stands are classified according to the following criteria:

By the nature of reproduced strokes - stands of single and multiple strokes;

By the method of obtaining shock overloads - stands for free fall and forced acceleration of the platform with the tested product;

According to the design of braking devices - with a hard anvil, with a springy anvil, with shock-absorbing rubber and felt pads, with crumpled deformable braking devices, with hydraulic braking devices, etc.

Depending on the design of the shock stand and, in particular, the brake device used in it, shock pulses of a semi-sinusoidal, triangular and trapezoidal shape are obtained.

To test REA for single impacts, shock-type shock stands are used, and for multiple ones, cam-type stands that reproduce half-sine wave strokes are used. These stands use the principle of free fall of the platform with the tested product on shock absorbing pads.

The main elements of the impact stand of the copra type (Fig. 2.3.1.) Are: table 3; base 7, used to damp the speed of the table at the time of impact; guide 4, providing a horizontal position of the table at the time of impact; pads 5 forming a shock pulse.

The energy necessary to create a shock is accumulated as a result of lifting the table with the tested product fixed on it to a predetermined height. To lift the table and then drop it, the stand is equipped with a drive and a reset mechanism. Kinetic energy acquired by the body in the process

Soundproofing, reducing the sound pressure level to established standards;

Earthing circuit, resistance not 40 m;

Concrete foundation.

4. During operation, the shock stand must be

installed on the foundation.

5. Power supply of the unit from AC

voltage 220 ± V, frequency 50 Hz.

6. Power consumption (maximum) not

more than 1kW.

7. Installation provides combinations of accelerations and

In mechanics, shock refers to the mechanical effect of material bodies, leading to a finite change in the velocities of their points over an infinitely small period of time. Impact motion is movement resulting from a single interaction of a body (medium) with the system under consideration, provided that the smallest period of natural vibrations of the system or its time constant is comparable or longer than the interaction time.

During impact interaction, shock accelerations, velocity or displacement are determined at the points under consideration. Together, these effects and reactions are called shock processes. Mechanical shocks can be single, multiple and complex. Single and multiple impact processes can affect the apparatus in the longitudinal, transverse and any intermediate directions. Complex shock loads affect an object in two or three mutually perpendicular planes simultaneously. Shock loads on aircraft can be both non-periodic and periodic. The occurrence of shock loads is associated with a sharp change in the acceleration, speed or direction of movement of the aircraft. Most often, in real conditions, a complex single shock process occurs, which is a combination of a simple shock pulse with superimposed oscillations.

The main characteristics of the shock process:

• the laws of change in time of shock acceleration a (t), velocity V (t) and displacement X (t) \\ duration of shock acceleration m is the time interval from the moment of appearance to the moment of disappearance of shock acceleration, satisfying the condition a\u003e an, where an peak shock acceleration;
• the duration of the shock acceleration front Tf is the time interval from the moment the shock acceleration appears to the moment corresponding to its peak value;
• the coefficient of superimposed oscillations of shock acceleration is the ratio of the total sum of the absolute values \u200b\u200bof the increments between adjacent and extreme values \u200b\u200bof shock acceleration to its doubled peak value;
• impact acceleration momentum is the integral of impact acceleration over a time equal to the duration of its action.

According to the shape of the curve of the functional dependence of motion parameters, shock processes are divided into simple and complex. Simple processes do not contain high-frequency components, and their characteristics are approximated by simple analytical functions. The name of the function is determined by the shape of the curve approximating the dependence of acceleration on time (semi-sinusoidal, cosanusoidal, rectangular, triangular, sawtooth, trapezoidal, etc.).

A mechanical shock is characterized by a rapid release of energy, resulting in local elastic or plastic deformations, excitation of stress waves and other effects, sometimes leading to disruption of the functioning and destruction of the aircraft structure. The shock load applied to the aircraft excites rapidly damped natural oscillations in it. The value of the overload upon impact, the nature and speed of the stress distribution along the aircraft structure are determined by the strength and duration of the impact, and the nature of the change in acceleration. Impact, acting on the aircraft, can cause its mechanical destruction. Depending on the duration, complexity of the shock process and its maximum acceleration during testing, the degree of rigidity of the structural elements of the aircraft is determined. A simple blow can cause destruction due to the occurrence of strong, albeit short-term overvoltages in the material. A complex blow can lead to the accumulation of microstrains of a fatigue nature. Since the design of the aircraft has resonance properties, even a simple shock can cause an oscillatory reaction in its elements, which is also accompanied by fatigue phenomena.

Mechanical overloads cause deformation and breakdown of parts, loosening of joints (welded, threaded and riveted), loosening of screws and nuts, movement of mechanisms and controls, as a result of which the adjustment and adjustment of devices change, and other malfunctions appear.

The harmful effects of mechanical overloads are controlled in various ways: by increasing the strength of the structure, using parts and elements with increased mechanical strength, using shock absorbers and special packaging, and rational placement of devices. Protection measures from the harmful effects of mechanical overload are divided into two groups:

1. measures aimed at ensuring the required mechanical strength and rigidity of the structure;
2. measures aimed at isolating structural elements from mechanical stress.

In the latter case, various shock absorbing means, insulating gaskets, compensators and dampers are used.

The general task of testing aircraft for impact loads is to test the ability of the aircraft and all its elements to perform their functions during and after impact, i.e. maintain their technical parameters during impact and after impact within the limits specified in the regulatory and technical documents.

The main requirements for impact tests in laboratory conditions are the maximum proximity of the result of a test impact on an object to the effect of a real impact in full-scale operating conditions and reproducibility of impact.

When reproducing shock loading modes under laboratory conditions, restrictions are imposed on the pulse shape of the instantaneous acceleration as a function of time (Fig. 2.50), as well as on the permissible limits of the pulse shape deviations. Almost every shock pulse at the laboratory bench is accompanied by a pulsation, which is a consequence of resonance phenomena in drum sets and auxiliary equipment. Since the spectrum of the shock pulse is mainly a characteristic of the destructive effect of the shock, even a small pulsation superimposed can make the measurement results unreliable.

Testing installations simulating individual impacts with subsequent vibrations make up a special class of equipment for mechanical testing. Impact stands can be classified according to various criteria (Fig. 2.5!):

I - on the basis of the formation of a shock pulse;

II - by the nature of the tests;

III - by the type of reproducible shock loading;

IV - according to the principle of action;

V - by energy source.

In general terms, the shock stand scheme consists of the following elements (Fig. 2.52): a test object mounted on a platform or container together with a shock overload sensor; acceleration means to inform the object of the required speed; brake device; control systems; recording equipment for recording the studied parameters of the object and the law of change in shock overload; primary converters; auxiliary devices for adjusting the functioning modes of the test object; power sources necessary for the operation of the test object and recording equipment.

The simplest bench for shock tests in laboratory conditions is a bench working on the principle of dropping a test object fixed to the carriage from a certain height, i.e. using to disperse the force of Earth's gravity. The shape of the shock pulse is determined by the material and the shape of the colliding surfaces. Acceleration up to 80,000 m / s2 can be provided at such stands. In fig. 2.53, a and b show fundamentally possible schemes of such stands.

In the first version (Fig. 2.53, a), a special cam 3 with a ratchet tooth is driven into rotation by a motor. When the cam reaches the maximum height H, the table 1 with the test object 2 falls on the brake devices 4, which give him a blow. Shock overload depends on the height of the fall H, the stiffness of the braking elements k, the total mass of the table and the test object M and is determined by the following dependence:

By varying this value, various overloads can be obtained. In the second version (Fig. 2.53, b), the stand works according to the method of dropping.

Test benches using a hydraulic or pneumatic drive to disperse the carriage are practically independent of gravity. In fig. 2.54 shows two options for pneumatic impact stands.

The principle of operation of the stand with air gun (Fig. 2.54, a) is as follows. Compressed gas is supplied to the working chamber /. When the specified pressure is reached, which is controlled by a pressure gauge, the automaton 2 for releasing the container 3 is activated, where the test object is located. Upon exiting the barrel 4 of the air gun, the container contacts the device 5, which allows you to measure the speed of movement of the container. The air gun through the shock absorbers is attached to the supporting posts b. The predetermined braking law on the shock absorber 7 is implemented by changing the hydraulic resistance of the flowing fluid 9 in the gap between the specially profiled needle 8 and the hole in the shock absorber 7.

The structural diagram of another pneumatic impact stand, (Fig. 2.54, b) consists of test object 1, carriage 2, on which the test object is installed, gaskets 3 and brake device 4, valves 5, allowing to create specified gas pressure drops on piston b, and gas supply systems 7. The brake device is activated immediately after the collision of the carriage and the gasket to prevent the carriage from returning and distorting the shapes of the shock pulse. Management of such stands can be automated. They can reproduce a wide range of shock loads.

As an acceleration device, rubber shock absorbers, springs, and, in some cases, linear induction motors can be used.

The capabilities of almost all shock stands are determined by the design of the brake devices:

1. The impact of the test object with a rigid plate is characterized by inhibition due to the emergence of elastic forces in the contact zone. This method of inhibition of the test object allows to obtain large values \u200b\u200bof overloads with a small front of their growth (Fig. 2.55, a).

2. To obtain overloads in a wide range, from tens to tens of thousands of units, with their rise time from tens of microseconds to several milliseconds, deformable elements are used in the form of a plate or gasket lying on a rigid base. The materials for these gaskets can be steel, brass, copper, lead, rubber, etc. (Fig. 2.55, b).

3. To ensure any specific (predetermined) law of variation of n and t in a small range, deformable elements are used in the form of a tip (crash), which is installed between the plate of the shock stand and the test object (Fig. 2.55, c).

4. To reproduce an impact with a relatively large path of braking, a braking device is used, consisting of a lead, plastically deformable plate located on the rigid base of the stand and a hard tip embedded in it of the corresponding profile (Fig. 2.55, d), mounted on the object or platform of the stand . Such braking devices make it possible to obtain overloads in a wide range of n (t) with a short rise time, reaching tens of milliseconds.

5. As a braking device, an elastic element in the form of a spring (Fig. 2.55, e) mounted on the moving part of the shock stand can be used. This type of braking provides relatively small overloads of a semi-sinusoidal shape with a duration measured in milliseconds.

6. A punched metal plate fixed along the contour at the base of the installation, in combination with a rigid tip of the platform or container, provides relatively small overloads (Fig. 2.55, f).

7. Deformable elements mounted on a movable platform of the stand (Fig. 2.55, g), in combination with a rigid conical trap, provide long-term overloads with a rise time of up to tens of milliseconds.

8. The brake device with a deformable washer (Fig. 2.55, h) allows to obtain large braking paths of an object (up to 200 - 300 mm) with small washer deformations.

9. The creation in laboratory of intense shock pulses with large fronts is possible using a pneumatic brake device (Fig. 2.55, s). Among the advantages of a pneumatic damper are its reusable effects, as well as the ability to reproduce shock pulses of various shapes, including those with a significant predetermined front.

10. In the practice of conducting shock tests, the brake device in the form of a hydraulic shock absorber has been widely used (see Fig. 2.54, a). When the test object hits the shock absorber, its rod is immersed in liquid. The fluid is pushed out through the stem point according to the law determined by the profile of the control needle. By changing the profile of the needle, it is possible to implement a different kind of law of inhibition. The needle profile can be obtained by calculation, but it is too difficult to take into account, for example, the presence of air in the piston cavity, the friction forces in the sealing devices, etc. Therefore, the calculated profile must be experimentally adjusted. Thus, by the calculation-experimental method, it is possible to obtain the profile necessary for the implementation of any law of inhibition.

Conducting shock tests in laboratory conditions also puts forward a number of special requirements for the installation of the facility. So, for example, the maximum permissible movement in the transverse direction should not exceed 30% of the nominal value; both when testing for impact resistance, and when testing for impact strength, the product should be able to be installed in three mutually perpendicular positions with the reproduction of the required number of shock pulses. The single characteristics of the measuring and recording equipment should be identical in a wide frequency range, which ensures the correct registration of the ratios of the various frequency components of the measured pulse.

Due to the variety of transfer functions of various mechanical systems, the same shock spectrum can be caused by a shock pulse of various shapes. This means that there is no one-to-one correspondence between some temporal acceleration function and the shock spectrum. Therefore, from a technical point of view, it is more correct to set the technical conditions for impact tests containing requirements for the impact spectrum, and not for the temporal characteristic of acceleration. This primarily relates to the mechanism of fatigue failure of materials due to the accumulation of loading cycles, which may vary from test to test, although the peak values \u200b\u200bof acceleration and stress will remain constant.

When modeling the shock processes of the system of determining parameters, it is advisable to compile according to the identified factors necessary for a sufficiently complete determination of the desired value, which can sometimes be found only experimentally.

Considering the impact of a massive, freely moving rigid body on a deformable element of relatively small size (for example, on the brake device of the stand), mounted on a rigid base, it is necessary to determine the parameters of the shock process and establish the conditions under which such processes will be similar to each other. In the general case of the spatial motion of a body, six equations can be composed, three of which give the law of conservation of momentum, two the laws of conservation of mass and energy, and sixth is the equation of state. These equations include the following quantities: three velocity components Vx Vy \\ Vz\u003e density p, Pressure p and entropy. Neglecting dissipative forces and considering the state of the deformable volume to be isentropic, one can exclude entropy from the number of determining parameters. Since only the motion of the center of mass of the body is considered, it is possible not to include velocity components Vx, Vy among the determining parameters; Vz and the coordinates of the points A, Y, Z inside the deformable object. The state of the deformable volume will be characterized by the following determining parameters:

• the density of the material p;
• pressure p, which is more appropriate to take into account through the maximum local strain and Otmax, considering it as a generalized parameter of the force characteristic in the contact zone;
• the initial impact velocity V0, which is directed normal to the surface on which the deformable element is mounted;
• current time t;
• body weight t;
• acceleration of gravity g;
• the elastic modulus of materials E, since the stress state of the body upon impact (with the exception of the contact zone) is considered elastic;
• characteristic geometric parameter of the body (or deformable element) D.

In accordance with the mc-theorem, of eight parameters, among which three have independent dimensions, five independent dimensionless complexes can be composed:

The dimensionless complexes composed of the determined parameters of the impact process will be independent of some functions] dimensionless complexes P1 - P5.

The parameters to be determined include:

• current local deformation a;
• body speed V;
• contact force P;
• tension inside the body a.

Therefore, we can write the functional relationships:

The form of the functions / 1, / 2, / e, / 4 can be established experimentally, taking into account a large number of determining parameters.

If during impact in the sections of the body outside the contact zone no residual deformations appear, then the deformation will be local in nature, and, therefore, the complex H5 \u003d pU ^ / E can be excluded.

The complex Jl2 \u003d Pttjjjax) ~ Cm is called the coefficient of relative body mass.

The coefficient of resistance to plastic deformation Cp is directly related to the force characteristic N (the material compliance coefficient, which depends on the shape of the colliding bodies) by the following dependence:

where p is the reduced density of materials in the contact zone; Cm \u003d m / (pa?) Is the reduced relative mass of the colliding bodies, which characterizes the ratio of their reduced mass M to the reduced mass of the deformable volume in the contact zone; xV is a dimensionless parameter characterizing the relative work of deformation.

The function Cp - / s (R1 (R1, R3, R4) can be used to determine the overloads:

If we ensure the equality of the numerical values \u200b\u200bof the dimensionless complexes IJlt Я2, Я3, Я4 for two shock processes, then these conditions, i.e.

will be similarity criteria for these processes.

When these conditions are fulfilled, the numerical values \u200b\u200bof the functions f / g./z »» »te will be the same at similar instants of time -V CtZoimax- const; ^ r \u003d const; Cp \u003d const, which allows us to determine the parameters of one shock process by simply recalculating the parameters of another process. The necessary and sufficient requirements for physical modeling of shock processes can be formulated as follows:

1. The working parts of the model and the full-scale object should be geometrically similar.
2. Dimensionless complexes composed of defining steam, meters, must satisfy the condition (2.68). Introducing scale factors.

It must be borne in mind that when modeling only the parameters of the shock process, the stressed states of bodies (natures and models) will be necessarily different.

Estimate the time of elastic impact of solids by considering the collision of a rod, which faces endlessly on a fixed, non-deformable wall (Fig.).

Most often in tasks it is believed that the elastic impact of solids occurs instantly, but it is clear that this assumption is idealization.
  Collision of real bodies always takes a finite period of time τ . In fact, if the change in momentum of the body in a collision occurred instantly,
F \u003d mΔv / t → 0 → ∞
  then the force of interaction of bodies upon impact would be infinitely large, which, of course, does not happen.
  What can the duration of a collision depend on? Suppose that we are considering the reflection of an elastic body from an undeformable wall. In a collision, the kinetic energy of the body during the first half of the collision turns into the potential energy of the elastic deformation of the body. During the second half, the deformation energy is converted back into the kinetic energy of the bouncing body.

This idea was laid in the testing task. 2005 year. Solve this problem to comprehend this moment.
 Task. Two completely elastic washers in mass m 1 \u003d m 2 \u003d 240 g   each slide progressively on a smooth horizontal surface towards each other with speeds whose modules v 1 \u003d 21 m / s   and v 2 \u003d 9.0 m / s. Maximum value of potential energy E   the elastic deformation of the washers in their central collision is ... j.

Therefore, it is obvious that the elastic properties of the body play a role in a collision. So, we can expect that the duration of the impact depends on the Young's modulus of the body material Eits density ρ   and its geometric dimensions. It is possible that the duration of the stroke τ   depends on speed v, with which the body hits an obstacle.
  It is easy to verify that it is not possible to estimate the collision time using dimensional considerations alone. Indeed, even if we take a ball as an incident body, the dimensions of which are characterized by only one parameter - the radius R, then from the quantities E, ρ , R   and v   You can compose countless expressions having a dimension of time:
τ \u003d √ (ρ / E) × f (ρv 2 / E), (1)
  Where f   - arbitrary function of dimensionless quantity ρv 2 / E. Therefore to find τ dynamic consideration is needed.
  The easiest way to do this is for a body that has the shape of a long rod.
  Let the rod moving with speed v, flies end on a motionless wall. When the end section of the rod touches the wall, the velocities of the rod particles lying in this section instantly vanish. At the next moment of time, particles located in an adjacent section stop, etc. The portion of the rod whose particles have already stopped at this moment is in a deformed state. In other words, at that moment in time, the part of the rod that is reached by the wave of elastic deformation propagating along the rod from the point of contact with the barrier is deformed. This deformation wave propagates along the rod at the speed of sound. u. If we assume that the rod came into contact with the wall at a time t \u003d 0then at time t   the length of the compressed part of the rod is ut. This part of the rod in fig. a   hatched.

  In the unshaded part of the rod, the velocities of all its particles are still equal v, and in the compressed (shaded) part of the rod, all particles are at rest.
  The first stage of the collision process between the rod and the wall will end at the moment when the entire rod is deformed and the velocities of all its particles become zero (Fig. b).

  At this moment, the kinetic energy of the incident rod is completely transformed into the potential energy of elastic deformation. Immediately after this, the second stage of the collision begins, in which the rod returns to the undeformed state. This process begins at the free end of the rod and, propagating along the rod with the speed of sound, gradually approaches the barrier. In fig. in

  the rod is shown at the moment when the unshaded part is no longer deformed and all its particles have a velocity vdirected to the left. The hatched area is still deformed, and the velocities of all its particles are zero.
  The end of the second stage of the collision will come at a time when the entire rod is undeformed, and all the particles of the rod gain speed vdirected opposite to the speed of the rod before impact. At this moment, the right end of the rod is separated from the obstacle: the undeformed rod bounces off the wall and moves in the opposite direction with the same modulus of speed (Fig. g).

  The energy of the elastic deformation of the rod in this case is completely transferred back to kinetic energy.
  From the foregoing it is clear that the duration of the collision τ equal to the travel time of the front of the elastic deformation wave along the rod there and back:
τ \u003d 2l / u, (2)
  Where l   - the length of the rod.
  The speed of sound in the rod u can be determined as follows. Consider the rod at time t   (fig. a) when the deformation wave propagates to the left. The length of the deformed part of the rod at this moment is equal to ut. In relation to the undeformed state, this part shortened by vtequal to the distance traveled by this moment still undeformed part of the rod. Therefore, the relative deformation of this part of the rod is v / u. Based on Hooke's Law
v / u \u003d (1 / E) × F / S, (3)
  Where S   - the cross-sectional area of \u200b\u200bthe rod, F   - the force acting on the rod from the side of the wall, E   - Young's modulus.
  Since the relative deformation v / u   the same at all times, while the rod is in contact with the barrier, then, as can be seen from formula (3), the force F   constant. To find this force, we apply the law of conservation of momentum to the stopped part of the rod. Before contact with the obstacle, the considered part of the rod had an impulse ρSut.v, and at time t   its momentum is zero.
  therefore
ρSut.v \u003d Ft. (4)
  Substituting Strength From Here F   into formula (3), we obtain
u \u003d √ (E / ρ). (5)
  Now expression for time τ . The collision deformation of the rod with the wall (2) takes the form
τ \u003d 2l√ (ρ / E). (6)
  Collision time τ   can be found differently, using the law of conservation of energy for this. Before the collision, the rod is undeformed and all its energy is the kinetic energy of the translational motion mv 2/2. After some time τ / 2   from the beginning of the collision, the velocities of all its particles, as we have seen, vanish, and the entire rod is deformed (Fig. b) Rod length decreased by Δl   in comparison with its undeformed state (Fig. d).

  At this moment, all the energy of the rod is the energy of its elastic deformation. This energy can be written as
W \u003d k (Δl) 2/2,
  Where k   - coefficient of proportionality between force and deformation:
F \u003d kΔl.
  Using Hooke's law, this coefficient is expressed in terms of Young's modulus E   and rod sizes:
σ \u003d F / S \u003d (Δl / l) E,
F \u003d SEΔl / l and F \u003d kΔl,
  from here
k \u003d ES / l. (7)
  Maximum deformation Δl   equal to the distance that the particles of the left end of the rod move over time τ / 2   (fig. d) Since these particles moved at a speed vthen
Δl \u003d vτ / 2. (8)
  Equate the kinetic energy of the rod to impact and the potential energy of deformation. Given that the mass of the rod
m \u003d ρSl,
  and using relations (7) and (8), we obtain
ρSlv 2/2 \u003d ES / (2l) × (vτ / 2) 2,
  where for τ   we again obtain formula (6).
  This collision time is usually very short. For example, for a steel bar ( E \u003d 2 × 10 11 Pa, ρ \u003d 7.8 × 10 3 kg / m 3) length 28 cm   calculation by formula (6) gives τ \u003d 10 −4 s.
  Force Facting on the wall during the impact can be found by substituting the speed of sound in the rod (5) in the formula (4):
F \u003d Sv√ (ρE). (9)
  It can be seen that the force acting on the wall is proportional to the speed of the rod before impact. But for the applicability of the above solution, it is necessary that the mechanical stress of the rod F / sdid not exceed the elastic limit of the material of which the rod is made. For example, for steel, the elastic limit
(F / S) max \u003d 4 × 10 8 Pa.
  Therefore the maximum speed v   steel rod, in which its impact with the barrier can still be considered elastic, according to formula (9) is equal 10 m / s. This corresponds to the rate of free fall of the body from a height of only 5 m.
  We indicate for comparison that the speed of sound in steel u \u003d 5000 m / si.e. v<< u .
  The time of the collision of the rod with a fixed barrier (in contrast to the force) turned out to be independent of the speed of the rod. This result, however, is not universal, but is associated with the specific form of the body under consideration. For example, for an elastic ball, the collision time with the wall depends on its speed. Dynamic consideration of this case is more complicated. This is due to the fact that both the area of \u200b\u200bcontact of the deformed ball with the wall and the force acting on the ball during the collision do not remain constant.

A shock is a mechanical phenomenon in which a short-term interaction of bodies causes a finite change in the velocity vector of all or some points of the material system with a negligible change in the position of the points of the system. The time interval during which a strike occurs is indicated by a letter and is called the stroke time.

Impact is a common phenomenon when considering the motion of both macroscopic bodies and microscopic particles, such as gas molecules. Thus, the impact phenomenon plays an essential role in a number of technical and physical problems. The nature of the impact substantially depends on the physical structure of the colliding bodies.

Instant power

Since the time during which the impact occurs is small, the final change in velocity during the impact corresponds to very large accelerations of the points of the system. Therefore, the forces acting in the process of impact are many times higher than conventional forces.

These forces are called instantaneous forces. The direct measurement of instantaneous forces is very difficult, since the impact time is usually expressed in thousandths or ten thousandths of a second. In addition, during this extremely short period of time, the instantaneous forces do not remain constant: they increase from zero to a certain maximum, and then decrease again to zero. Thanks to this, the forces causing the blow have to be characterized using some special concepts.

Shock impulse

Consider a point of mass moving under the action of some finite force. Then, at the moment, then apply an instantaneous force P to it, the action of which ceases at the moment. We denote the speeds of the point at the moments and, accordingly, applying the momentum theorem to these moments, we obtain:

The first of these integrals represents the momentum of a finite force in time and therefore is a small quantity of the same order as. Therefore, the speed of the point in question can receive a finite change only if the momentum of the instantaneous force P is finite, denoting by which we have:

where it is called shock, or instantaneous impulse, it characterizes the action of instantaneous force upon impact.

The basic equation of shock theory

Since the impulse of the final force is of the order of small magnitude, it can be neglected in comparison with the final impulse. Therefore, when studying the action of instantaneous forces during an impact, one can ignore the effects of finite forces, and the impulse theorem for a point during an impact has the form:

The velocities of the point corresponding to the beginning and end of the impact are called before the shock and after the shock speed. The resulting equality connecting the speed of the point before and after the impact with an instantaneous momentum is called the basic equation of the theory of impact. It in this theory plays the role of the basic law of dynamics.

Point offset on impact

The speed of the point during the impact remains finite, varying from to From here the movement of the point will be or it will be a small value of the order of m. Thus, during the impact the point does not have time to shift in any noticeable way. Neglecting this negligible movement, we can say that the only consequence of the action of instantaneous force is a sudden change in the speed of the point. Since the velocity vector can change in this case not only in magnitude, but also in direction, the trajectory of a point at the moment of impact can get a kink (an angular point forms on the trajectory) (Fig. 131).

Material system impact equations

Consider a mechanical system consisting of material points. Suppose that among the external and internal forces acting on the points of the system there are instantaneous forces, which we denote accordingly. For each point of the system, we can write the basic equation of impact:

We multiply each of these equalities by r, vector, where is the radius vector of the point corresponding to the moment of impact (or an infinitely small interval of time of impact). Then we get the equality:

To exclude internal instantaneous forces acting on the system, we add each group of these equalities term by term. As a result, we get:

as previously proved that for internal forces

where P is the momentum of the system.

Moreover,

where is the shock impulse of an external force acting on a point in the system. Therefore, the first of the obtained equalities can be written in the form:

Since they will be the amount of movement of the system before and after the impact, we have: the change in the amount of movement of the system during the time is equal to the sum of the instantaneous pulses of all external forces acting on the system.