Longitudinal vibrations of the rod. Longitudinal vibrations of a homogeneous rod

By rod we mean the cylinder П=0х[О, /], when I" diamD. Here D- area on the coordinate plane Ox 2 x 3 (Fig. 62). The material of the rod is homogeneous and isotropic, and the Ox axis passes through the center of gravity of the section D. Field of external mass forces f(r, I)=/(X|, /)e, where e is the unit vector of the Ox axis. Let the external surface forces on the side surface of the cylinder be equal to zero, i.e. Ra= 0 on dD X

Then from (4.8) it follows for 1=0 equality

Own forms X k(j) it is convenient to normalize using the norm of the space /^() to which the function belongs v(s, I), since at each moment of time the kinetic energy functional exists and is limited

Where S- area of ​​the region D. We have

X*(s) = Jj- sin^-l in velocity space I 0 = ji)(s, /): v(s,t)e

As a result, we obtain an orthonormal basis |l r *(^)| ,

Where b to „- Kronecker symbol: Functions X k *(s), k= 1,2 are the normal modes of natural vibrations, and ω*, k= 1, 2, ..., - natural frequencies of oscillations of a system with an infinite number of degrees of freedom.

In conclusion, we note that the function u(s, /) belongs to the configuration space of the system H, = (v(s, t): v(s, t) e e ^(), u(0, 1) = o(1, /) = 0), where U^"OO, / ]) is the Sobolev space of functions summable together with the squares of the first derivatives on the interval. The space I is the domain of definition of the functional of the potential energy of elastic deformations

and contains generalized solutions to the problem under consideration.

ISSN: 2310-7081 (online), 1991-8615 (print) doi: http://dx.doi UDC 517.956.3

PROBLEM ON LONGITUDINAL VIBRATIONS OF AN ELASTICALLY FIXED LOADED ROD

A. B. Beilin

Samara State Technical University, Russia, 443100, Samara, st. Molodogvardeyskaya, 244.

annotation

One-dimensional longitudinal vibrations of a thick short rod fixed at the ends using concentrated masses and springs are considered. An initial boundary value problem with dynamic boundary conditions for a fourth-order hyperbolic equation is used as a mathematical model. The choice of this particular model is due to the need to take into account the effects of deformation of the rod in the transverse direction, neglect of which, as shown by Rayleigh, leads to an error, which is confirmed by the modern nonlocal concept of studying vibrations of solid bodies. The existence of a system of eigenfunctions of the problem under study, orthogonal to the load, is proved and their representation is obtained. The established properties of eigenfunctions made it possible to apply the method of separation of variables and prove the existence of a unique solution to the problem posed.

Key words: dynamic boundary conditions, longitudinal vibrations, orthogonality with load, Rayleigh model.

Introduction. In any working mechanical system, oscillatory processes occur, which can be generated by various reasons. Oscillatory processes can be a consequence of the design features of the system or the redistribution of loads between various elements of a normally operating structure.

The presence of sources of oscillatory processes in the mechanism can make it difficult to diagnose its condition and even lead to disruption of its operating mode, and in some cases to destruction. Various problems associated with disruption of the accuracy and performance of mechanical systems as a result of vibration of some of their elements are often solved experimentally in practice.

At the same time, oscillatory processes can be very useful, for example, for processing materials, assembling and disassembling joints. Ultrasonic vibrations make it possible not only to intensify cutting processes (drilling, milling, grinding, etc.) of materials with high hardness (tungsten-containing steels, titanium carbide steels, etc.),

© 2016 Samara State Technical University. Citation template

Beilin A. B. Problem of longitudinal vibrations of an elastically fixed loaded rod // Vestn. Myself. state tech. un-ta. Ser. Phys.-math. Sciences, 2016. T. 20, No. 2. P. 249258. doi: 10.14498/vsgtu1474. About the author

Alexander Borisovich Beilin (Ph.D., Associate Professor; [email protected]), associate professor, department. automated machine and tool systems.

but in some cases it can become the only possible method for processing brittle materials (germanium, silicon, glass, etc.). The device element (waveguide) that transmits ultrasonic vibrations from the source (vibrator) to the tool is called a concentrator and can have different shapes: cylindrical, conical, stepped, exponential, etc. Its purpose is to convey vibrations of the required amplitude to the instrument.

Thus, the consequences of the occurrence of oscillatory processes can be different, as well as the reasons that cause them, so the need for a theoretical study of oscillation processes naturally arises. The mathematical model of wave propagation in relatively long and thin solid rods, which is based on the second-order wave equation, has been well studied and has long become a classic. However, as shown by Rayleigh, this model does not fully correspond to the study of vibrations of a thick, short rod, whereas many details of real mechanisms can be interpreted as short and thick rods. In this case, the deformation of the rod in the transverse direction should also be taken into account. A mathematical model of longitudinal vibrations of a thick short rod, which takes into account the effects of transverse motion of the rod, is called a Rayleigh rod and is based on a fourth-order hyperbolic equation

^ ^- IX (a(x) e)- dx (b(x))=; (xL (1)

whose coefficients have a physical meaning:

d(x) = p(x)A(x), a(x) = A(x)E(x), b(x) = p(x)u2(x)1p (x),

where A(x) is the cross-sectional area, p(x) is the mass density of the rod, E(x) is Young’s modulus, V(x) is Poisson’s ratio, IP(x) is the polar moment of inertia, u(x, b) - longitudinal displacements to be determined.

Rayleigh's ideas have found their confirmation and development in modern works devoted to oscillation processes, as well as the theory of plasticity. The review article substantiates the shortcomings of classical models describing the state and behavior of solid bodies under load, in which a priori the body is considered an ideal continuum. The current level of development of natural science requires the construction of new models that adequately describe the processes under study, and the mathematical methods developed in the last few decades provide this opportunity. On this path, in the last quarter of the last century, a new approach to the study of many physical processes, including those mentioned above, was proposed, based on the concept of nonlocality (see the article and the list of references in it). One of the classes of nonlocal models identified by the authors is called “weakly nonlocal.” Mathematical models belonging to this class can be implemented by introducing high-order derivatives into the equation describing a certain process, which make it possible to take into account, to some approximation, the interaction of the internal elements of the object of study. Thus, Rayleigh's model is still relevant today.

1. Statement of the problem. Let the ends of the rod x = 0, x = I be attached to a fixed base with the help of concentrated masses L\, M2 and springs, the stiffnesses of which are K\ and K2. We will assume that the rod is a body of rotation about the 0x axis and at the initial moment of time is at rest in an equilibrium position. Then we come to the following initial-boundary value problem.

Task. Find in the area Qt = ((0,1) x (0, T) : 1,T< те} "решение уравнения (1), удовлетворяющее начальным данным

u(x, 0) = (p(x), u(x, 0) = φ(x) and boundary conditions

a(0)ikh(0, r) + b(0)il(0, r) - k^(0, r) - M1ui(0, r) = 0, a(1)ih(1, r) + b(1)uxy(1, r) + K2u(1, r) + M2uy(1, r) = 0. ()

The article examines some special cases of problem (1)-(2) and gives examples in which the coefficients of the equation have an explicit form and M\ = M2 = 0. The article proves the unique weak solvability of the problem posed in the general case.

Conditions (2) are determined by the method of securing the rod: its ends are attached to fixed bases using some devices having masses M\, M2, and springs with stiffnesses K1, K2, respectively. The presence of masses and taking into account transverse displacements leads to conditions of the form (2), containing derivatives with respect to time. Boundary conditions that include time derivatives are called dynamic. They can arise in various situations, the simplest of which are described in the textbook, and much more complex ones in the monograph.

2. Study of the natural vibrations of the rod. Let us consider a homogeneous equation corresponding to equation (1). Since the coefficients depend only on x, we can separate the variables by writing u(x,r) = X(x)T(r). We get two equations:

t""(g) + \2t(g) = 0,

((a(x) - A2b(x))X"(x))" + A2dX(x) = 0. (3)

Equation (3) is accompanied by boundary conditions

(a(0) - \2Ъ(0))Х"(0) - (К1 - \2М1)Х(0) = 0,

(a(1) - \2Ъ(1))Х"(1) + (К2 - \2М2)Х(I) = 0. (4)

Thus, we came to the Sturm-Liouville problem, which differs from the classical one in that the spectral parameter A is included in the coefficient of the highest derivative of the equation, as well as in the boundary conditions. This circumstance does not allow us to refer to results known from the literature, so our immediate goal is to study problem (3), (4). To successfully implement the variable separation method, we need information about the existence and location of eigenvalues, about the qualitative

properties of eigenfunctions: do they have the property of orthogonality?

Let us show that A2 > 0. Let us assume that this is not the case. Let X(x) be the eigenfunction of problem (3), (4), corresponding to the value A = 0. Multiply (3) by X(x) and integrate the resulting equality over the interval (0,1). Integrating by parts and applying boundary conditions (4), after elementary transformations we obtain

1(0) - L2Ъ(0))(a(1) - L2Ъ(1)) I (dX2 + bX"2)yx+

N\X 2(0) + M2X 2(1)

I aX"2<1х + К\Х2(0) + К2Х2(1). Jo

Note that from the physical meaning of the functions a(x), b(x), d(x) are positive, Kr, Mg are non-negative. But then from the resulting equality it follows that X"(x) = 0, X(0) = X(1) = 0, therefore, X(x) = 0, which contradicts the assumption made. Consequently, the assumption that that zero is the eigenvalue of problem (3), (4) is incorrect.

The representation of the solution to equation (3) depends on the sign of the expression a(x) - - A2b(x). Let us show that a(x) - A2b(x) > 0 Vx e (0.1). Let us fix x e (0,1) arbitrarily and find the values ​​of the functions a(x), b(x), d(x) at this point. Let us write equation (3) in the form

X"(x) + VX (x) = 0, (5)

where we designated

at the selected fixed point, and we write conditions (4) in the form

Х"(0) - аХ (0) = 0, Х"(1) + вХ (I) = 0, (6)

where a, b are easy to calculate.

As is known, the classical Sturm-Liouville problem (5), (6) has a countable set of eigenfunctions for V > 0, from which, since x is arbitrary, the required inequality follows.

The eigenfunctions of problem (3), (4) have the property of orthogonality with the load expressed by the relation

I (dХт(х)Хп(х) + БХ"т(х)Х"п(х))<х+ ■)о

M1Xt(0)Xn(0) + M2Xt(1)Xn (I) = 0, (7)

which can be obtained in a standard way (see, for example,), the implementation of which in the case of the problem under consideration is associated with elementary but painstaking calculations. Let us briefly present its derivation, omitting the argument of the functions Xr(x) to avoid cumbersomeness.

Let Am, An be different eigenvalues, Xm, Xn be the corresponding eigenfunctions of problem (3), (4). Then

((a - L2tb)X"t)" + L2tdXt = 0, ((a - L2pb)X"p)" + L2pdXp = 0.

Let us multiply the first of these equations by Xn, and the second by Xm, and subtract the second from the first. After elementary transformations we obtain the equality

(Lt - Lp)YХtХп = (аХтХП)" - ЛП(БХтХ"п)" - (аХ"тХп)" + Lt(БХтХп)",

which we integrate over the interval (0,1). As a result, taking into account (4) and reducing by (Lm - Ln), we obtain relation (7).

The proven statements about the properties of eigenvalues ​​and eigenfunctions of the Sturm-Liouville problem (3), (4) make it possible to apply the method of separation of variables to find a solution to the problem.

3. Solvability of the problem. Let's denote

C(ST) = (u: u e C(St) P C2(St), uikh e C^t)).

Theorem 1. Let a, b e C1, d e C. Then there is at most one solution u e C^t) to problem (1), (2).

Proof. Let us assume that there are two different solutions to problem (1), (2), u1(x,z) and u2(x,z). Then, due to the linearity of the problem, their difference u = u1 - u2 is a solution to the homogeneous problem corresponding to (1), (2). Let us show that its solution is trivial. Let us first note that from the physical meaning of the coefficients of the equation and the boundary conditions, the functions a, b, d are positive everywhere in Qm, and M^, K^ are non-negative.

Multiplying equality (1) by u and integrating over the region Qt, where t e and is arbitrary, after simple transformations we obtain

/ (di2(x,t) + ai2x(x,t) + biHl(x,t))yx+ ./o

K1u2(0, t) + M1u2(0, t) + K2u2(1, t) + M2u2(1, t) = 0,

from which, due to the arbitrariness of m, the validity of the theorem immediately follows. □

We will prove the existence of a solution for the case of constant coefficients.

Theorem 2. Let<р е С2, <р(0) = <р(1) = (0) = ц>"(\) = 0, has a piecewise continuous derivative of the third order in (0.1), φ ε 1, φ(0) = φ(1) = 0 and has a piecewise continuous derivative of the second order in (0.1), f e C(C^m), then a solution to problem (1), (2) exists and can be obtained as a sum of a series of eigenfunctions.

Proof. As usual, we will look for a solution to the problem in the form of a sum

where the first term is the solution to the problem posed for a homogeneous equation corresponding to (1), the second is the solution to equation (1), satisfying the zero initial and boundary conditions. Let us use the results of the research carried out in the previous paragraph and write down the general solution to equation (3):

X(x) = Cr cos A J-+ C2 sin Aw-^rrx.

\¡ a - A2b \¡ a - A2b

Applying boundary conditions (4), we arrive at a system of equations for Cj!

(a - A2b)c2 - (Ki - A2Mi)ci = 0,

(-A(a - A2b) sin Ayja-A¡bl + (K - A2M2) cos A^O-A^l) ci+

Equating its determinant to zero, we obtain the spectral equation

ctg= (a - A4)A2" - (K - A?Mí)(K2 - A"M). (8)

b Va - A2b A^q(a - A2b)(Ki + K2 - A2(Mi + M2))

Let us find out whether this transcendental equation has a solution. To do this, consider the functions on the left and right sides of it and examine their behavior. Without limiting the generality too much, let us put

Mi = M2 = M, Kg = K2 = K,

which will slightly simplify the necessary calculations. Equation (8) takes the form

x I q ​​, Aja - A2b Jq K - A2M ctg A\Z-^l =

a - A2b 2(K - A2M) 2A^^0-A2b" Let us denote

and write the spectral equation in new notation!

aqlß Kql2 + ß2 (Kb - aM)

2Kql2 + 2^2(Kb - aM) 2/j.aql

Analysis of the functions of the left and right sides of the last equation allows us to state that there is a countable set of its roots and, therefore, a countable set of eigenfunctions of the Sturm-Liouville problem (3), (4), which, taking into account the relation obtained from the system with respect to c3, can be written out

v / l l I q K - x2pm. l i q

Xn(x) = COS XnJ-gutx + ----sin XnJ-gutX.

V a - A2b AnVa - ftb^q V a - A2b

Now let's move on to finding a solution that also satisfies the initial conditions. We can now easily find the solution to the problem for a homogeneous equation in the form of a series

u(x,t) = ^ Tn(t)Xn(x),

the coefficients of which can be found from the initial data, using the property of orthogonality of functions Xn(x), the norm of which can be obtained from relation (7):

||X||2 = f (qX2 + bX%)dx + MiX2(0) + M2x2(l). ■Jo

The process of finding the function v(x,t) is also essentially standard, but we still note that, looking for a solution in the traditional form

v(x,t) = ^ Tn(t)Xn(x),

we get two equations. Indeed, taking into account the type of eigenfunctions, let us clarify the structure of the series in the form of which we are looking for a solution:

j(x,t) = ^ (Vn(t)cos Xn^J a b x+

Wn(t) K-XnM~ sin X^HAarx). (9)

v JXnVa - xnb^q V a - xn "

To satisfy the zero initial conditions y(x, 0) = y^x, 0) = 0, we require that Vn(0) = Vn(0) = 0, Wn(0) = W(0) = 0. Expanding f( x,r) into the Fourier series in terms of the eigenfunctions Xn(x), we find the coefficients ¡n(b) and dn(b). Substituting (9) into equation (1), written with respect to y(x, b), after a series of transformations we obtain equations for finding Yn(b) and Wn(b):

yts® + >&pYu =

™ + xn Wn (<) = Xn (-a-iKrW g

Taking into account the initial conditions Vn(0) = Y, (0) = 0, Wn(0) = W, (0) = 0, we arrive at the Cauchy problems for each of the functions Vn(b) and Wn(b), the unique solvability of which guaranteed by the conditions of the theorem. The properties of the initial data formulated in the theorem leave no doubt about the convergence of all series that arose in the course of our research and, therefore, about the existence of a solution to the problem posed. □

Conclusion. The existence of a system of eigenfunctions of the problem under study, orthogonal to the load, is proved and their representation is obtained.

The established properties of the eigenfunctions made it possible to prove the existence of a unique solution to the problem posed. Note that the results obtained in the article can be used both for further theoretical studies of problems with dynamic boundary conditions, and for practical purposes, namely for calculating longitudinal vibrations of a wide range of technical objects.

Alexander Borisovich Beilin: http://orcid.org/0000-0002-4042-2860

BIBLIOGRAPHICAL LIST

1. Nerubay M. S., Shtrikov B. L., Kalashnikov V. V. Ultrasonic machining and assembly. Samara: Samara Book Publishing House, 1995. 191 p.

2. Khmelev V.N., Barsukov R.V., Tsyganok S.N. Ultrasonic dimensional processing of materials. Barnaul: Altai Technical University named after. I.I. Polzunova, 1997. 120 p.

3. Kumabe D. Vibration cutting. M.: Mechanical Engineering, 1985. 424 p.

4. Tikhonov A. N., Samarsky A. A. Equations of mathematical physics. M.: Nauka, 2004. 798 p.

5. Strett J.V. Theory of sound. T. 1. M.: GITTL, 1955. 504 p.

6. Rao J. S. Advanced Theory of Vibration: Nonlinear Vibration and One Dimensional Structures. New York: John Wiley & Sons, Inc., 1992. 431 pp.

7. Fedotov I. A., Polyanin A. D., Shatalov M. Yu. Theory of free and forced vibrations of a solid rod based on the Rayleigh model // DAN, 2007. T. 417, No. 1. pp. 56-61.

8. Bazant Z., Jirasek M. Nonlocal Integral Formulations of Plasticity and Damage: Survey of Progress // J. Eng. Mech., 2002. vol.128, no. 11. pp. 1119-1149. doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119).

9. Beilin A. B., Pulkina L. S. Problem of longitudinal vibrations of a rod with dynamic boundary conditions // Vestn. SamSU. Natural science ser., 2014. No. 3(114). pp. 9-19.

10. Korpusov M. O. Destruction in non-classical wave equations. M.: URSS, 2010. 237 p.

Received by the editor 10/II/2016; in the final version - 18/V/2016; accepted for publication - 27/V/2016.

Vestn. Samar. Gos. Techn. Un-ta. Ser. Fiz.-mat. nauki

2016, vol. 20, no. 2, pp. 249-258 ISSN: 2310-7081 (online), 1991-8615 (print) doi: http://dx.doi.org/10.14498/vsgtu1474

MSC: 35L35, 35Q74

A PROBLEM ON LONGITUDINAL VIBRATION OF A BAR WITH ELASTIC FIXING

Samara State Technical University,

244, Molodogvardeyskaya st., Samara, 443100, Russian Federation.

In this paper, we study longitudinal vibration in a thick short bar fixed by point forces and springs. For mathematical model we consider a boundary value problem with dynamical boundary conditions for a forth order partial differential equation. The choice of this model depends on a necessity to take into account the result of a transverse strain. It was shown by Rayleigh that neglect of a transverse strain leads to an error. This is confirmed by modern nonlocal theory of vibration. We prove existence of orthogonal with load eigenfunctions and derive representation of them. Established properties of eigenfunctions make possible using the separation of variables method and finding a unique solution of the problem.

Keywords: dynamic boundary conditions, longitudinal vibration, loaded orthogonality, Rayleigh's model.

Alexander B. Beylin: http://orcid.org/0000-0002-4042-2860

1. Nerubai M. S., Shtrikov B. L., Kalashnikov V. V. Ul "trazvukovaia mekhanicheskaia obrabotka i sborka. Samara, Samara Book Publ., 1995, 191 pp. (In Russian)

2. Khmelev V. N., Barsukov R. V., Tsyganok S. N. Ul "trazvukovaia razmernaia obrabotka materialov. Barnaul, 1997, 120 pp. (In Russian)

3. Kumabe J. Vibration Cutting. Tokyo, Jikkyou Publishing Co., Ltd., 1979 (In Japanese).

4. Tikhonov A. N., Samarsky A. A. Uravneniia matematicheskoi fiziki. Moscow, Nauka, 2004, 798 pp. (In English)

5. Strutt J. W. The theory of sound, vol. 1. London, Macmillan and Co., 1945, xi+326 pp.

6. Rao J. S. Advanced Theory of Vibration: Nonlinear Vibration and One Dimensional Structures. New York, John Wiley & Sons, Inc., 1992, 431 pp.

Beylin A.B. A problem on longitudinal vibration of a bar with elastic fixing, Vestn. Samar. Gos. Techn. Univ., Ser. Fiz.-Mat. Nauki, 2016, vol. 20, no. 2, pp. 249-258. doi: 10.14498/vsgtu1474. (In Russian) Author Details:

Alexander B. Beylin (Cand. Techn. Sci.; [email protected]), Associate Professor, Dept. of Automation Machine Tools and Tooling Systems.

7. Fedotov I. A., Polyanin A. D., Shatalov M. Yu. Theory of free and forced vibrations of a rigid rod based on the Rayleigh model, Dokl. Phys., 2007, vol.52, no. 11, pp. 607-612. doi: 10.1134/S1028335807110080.

8. Bazant Z., Jirasek M. Nonlocal Integral Formulations of Plasticity and Damage: Survey of Progress, J. Eng. Mech., 2002, vol.128, no. 11, pp. 1119-1149. doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119).

9. Beylin A. B., Pulkina L. S. A promlem on longitudinal vibrations of a rod with dynamic boundary conditions, Vestnik SamGU. Estestvenno-Nauchnaya Ser., 2014, no. 3(114), pp. 919 (In Russian).

10. Korpusov M. O. Razrushenie v neklassicheskikh volnovykh uravneniiakh. Moscow, URSS, 2010, 237 pp. (In English)

Received 10/II/2016;

received in revised form 18/V/2016;

1

A frequency method is proposed for solving the problem of longitudinal vibrations of bars of step-variable cross-section with or without taking into account energy dissipation upon impact with a rigid obstacle. The equation of longitudinal vibrations of the rod is transformed according to Laplace in the presence of non-zero initial conditions. A boundary value problem is solved, which consists in finding Laplace-transformed edge longitudinal forces as functions of edge displacements. Then a system of equilibrium equations for the nodes is compiled, solving which, the amplitude-phase-frequency characteristics (APFC) are constructed for the sections of the rod of interest. By performing the inverse Laplace transform, a transition process is constructed. As a test example, a rod of constant cross-section of finite length is considered. A comparison with the known wave solution is given. The proposed method for the dynamic calculation of a rod in a collision with a rigid obstacle allows generalization to an arbitrary rod system in the presence of an unlimited number of elastically attached masses, with an arbitrary force applied at the ends and along the length of the rod.

Frequency method

longitudinal vibrations of the rod

1. Biderman, V.L. Applied theory of mechanical vibrations / V.L. Biderman. – M.: Higher School, 1972. – 416 p.

2. Lavrentiev, M.A. Methods of the theory of functions of a complex variable / M.A. Lavrentiev, B.V. Shabbat. – M.: Nauka, 1973. – 736 p.

3. Sankin, Yu.N. Dynamic characteristics of viscoelastic systems with distributed parameters / Yu.N. Sankin. – Saratov: Publishing house Sarat. University, 1977. – 312 p.

4. Sankin, Yu.N. Unsteady vibrations of rod systems upon collision with an obstacle / Yu.N. Sankin, N.A. Yuganova; under general ed. Yu.N. Sankina. – Ulyanovsk: Ulyanovsk State Technical University, 2010. – 174 p.

5. Sankin, Y.N. Longitudinal vibrations of elastic rods of step-variable cross-section colliding with rigid obstacle \ Yu. N. Sankin and N.A. Yuganova, J. Appl. Maths Mechs, Vol. 65, no 3, pp. 427–433, 2001.

Let us consider the frequency method for solving the problem of longitudinal vibrations of rods of step-variable cross-section with or without taking into account energy dissipation upon impact with a rigid obstacle, which we will compare with the known wave solution and the solution in the form of a series of vibration modes (14).

The differential equation for the longitudinal vibrations of the rod, taking into account the forces of internal resistance, has the form:

Let us set the following boundary and initial conditions:

. (2)

Let us transform equation (1) and boundary conditions (2) according to Laplace for the given initial conditions (2). Then equation (2) and boundary conditions (2) will be written as follows:

; (3)

,

where are the Laplace-transformed displacements of the points of the rod; p is the Laplace transform parameter.

Equation (3) without taking into account energy dissipation (at = 0) will take the form:

. (4)

For the resulting inhomogeneous differential equation, a boundary value problem is solved, which consists in finding the Laplace-transformed edge longitudinal forces as functions of edge displacements.

To do this, consider the homogeneous equation of longitudinal vibrations of the rod taking into account energy dissipation

(5)

Designating

and passing to a new variable, we get instead of (5)

(6)

If, where is the frequency parameter, then

.

The solution to homogeneous equation (6) has the form:

We find the integration constants c1 and c2 from the initial conditions:

u = u0 ; N = N0,

Those. ;

This solution corresponds to the following transfer matrix:

. (7)

Substituting the resulting expressions for the elements of the transfer matrix into the formulas of the displacement method, we obtain:

; (8)

;

The indices n and k indicate the beginning and end of the rod section, respectively. And geometric and physical constants with indices nk and kn refer to a specific section of the rod.

Dividing the rod into elements, using formulas (8), we will compose equations for the dynamic equilibrium of the nodes. These equations represent a system of equations for unknown nodal displacements. Since the corresponding coefficients are obtained by exact integration, the length of the rod sections is not limited.

By solving the resulting system of equations for , we construct amplitude-phase-frequency characteristics for the sections of the rod that interest us. These AFCs can be considered as a graphical image of a one-way Fourier transform, which coincides with the Laplace transform under pulsed influences. Since all singular points of the corresponding expressions lie to the left of the imaginary axis, the inverse transformation can be carried out by assuming , i.e. using the constructed AFCs. The task of constructing an AFC, where the field of initial velocities multiplied by the density of the rod appears as a force action, is auxiliary. Typically, the AFCs are constructed from the influence of disturbing forces, then the inverse Laplace transform is carried out by numerical integration or some other method.

As a simple example, consider a straight rod of length l, which collides longitudinally with a rigid obstacle at a speed V0 (Fig. 1).

Let us determine the displacement of the points of the rod after the impact. We will assume that after the impact the contact between the obstacle and the rod remains, i.e. there is no rebound of the rod. If the connection is non-containing, then the problem can be considered as piecewise linear. The criterion for moving to another solution option is a change in the sign of the speed at the point of contact.

In the monograph by Lavrentyev M.A., Shabat B.V. the wave solution of equation (4) is given:

and its original was found

, (9)

where is a unit step function.

Another approach to solving this problem can be carried out by the frequency method described in. In relation to this problem we will have:

; ;

; ;

; ;

. (10)

Let's find the original (11)

Let's solve the same problem using the frequency method. From the equilibrium equation of the 1st node:

(12)

we obtain a formula for moving the end of the rod.

Now, if a test rod of constant cross-section is divided into two arbitrary sections of length l1 and l2 (see Fig. 1), then the equilibrium conditions for the nodes will be as follows:

(13)

As a result of solving system (13), we obtain graphs of the phase-frequency response for displacements in the 1st and 2nd sections (U1 and U2, respectively). Thus, the image for the edge displacement in closed form, taking into account energy dissipation, in the case of (12) and (13) coincides and has the form:

. (14)

Let's check the coincidence of the results at the end of the rod. In Fig. Figure 2 shows graphs of solution (10) at x = l0.1 and as a result of solving system (13). They are completely the same.

The discrete Fourier transform can be used to obtain the transient process. The result can be obtained by performing numerical integration at t=0... using the formula

. (15)

In the AFC (see Fig. 2), only one visible turn manifests itself significantly. Therefore, one term of series (15) should be taken. The graphs in Figure 3 show how accurate the solution (9) and the solution for vibration modes (11) coincide with the proposed frequency solution. The error does not exceed 18%. The resulting discrepancy is explained by the fact that solutions (9) and (11) do not take into account energy dissipation in the rod material.

Rice. 3. Transient process for the end of the rod; 1, 2, 3 - graphs constructed according to formulas (9), (11), (15).

As a more complex example, consider the problem of longitudinal vibrations of a stepped rod (Fig. 4) with a load at the end, colliding with a rigid obstacle with a speed V0, and let the mass of the load be equal to the mass of the adjacent section of the rod:.

Rice. 4. Calculation diagram of longitudinal vibrations of a stepped rod with a load at the end

Let us introduce characteristic sections 1,2,3 of the rod in which we will calculate the displacements. Let's create a system of resolving equations:

(16)

As a result of solving system (16), we obtain graphs of the phase-frequency response (Fig. 5) for displacements in the second and third sections (U2() and U3(), respectively). Calculations were carried out with the following constant values: l = 2 m; E = 2.1×1011 Pa; F = 0.06 m2; = 7850 kg/m3; V = 10 m/s. In the obtained AFCs, only two visible turns manifest themselves significantly. Therefore, when constructing the transition process in selected sections, we take two terms of series (16). To do this, you must first determine

Rice. 5. AFC of displacements in the second and third sections of the stepped rod (see Fig. 4)

The transition process is constructed similarly using formula (15).

Conclusion: a method has been developed for calculating longitudinal vibrations of rods upon impact with an obstacle.

Reviewers:

Lebedev A.M., Doctor of Technical Sciences, Associate Professor, Professor of the Ulyanovsk Higher Aviation School (Institute), Ulyanovsk.

Antonets I.V., Doctor of Technical Sciences, Professor of Ulyanovsk State Technical University, Ulyanovsk.

Bibliographic link

Yuganova N.A. LONGITUDINAL VIBRATIONS OF RODS IN COLLISION WITH A HARD OBSTACLE // Modern problems of science and education. – 2014. – No. 2.;
URL: http://science-education.ru/ru/article/view?id=12054 (access date: 01/15/2020). We bring to your attention magazines published by the publishing house "Academy of Natural Sciences"

A rod is a body, one of whose dimensions, called longitudinal, significantly exceeds its dimensions in a plane perpendicular to the longitudinal direction, i.e. transverse dimensions. The main property of the rod is the resistance provided to longitudinal compression (tension) and bending. This property fundamentally distinguishes a rod from a string, which does not stretch and does not resist bending. If the density of the material of the rod is the same at all its points, then the rod is called homogeneous.

Typically, extended bodies bounded by a closed cylindrical surface are considered as rods. In this case, the cross-sectional area remains constant. We will study the behavior of just such a uniform rod of length l, assuming that it is subject only to compression or tension, obeying Hooke's law. When studying small longitudinal deformations of a rod, the so-called hypothesis of plane sections. It lies in the fact that the cross sections, moving under compression or tension along the rod, remain flat and parallel to each other.

Let's direct the axis x along the longitudinal axis of the rod (Fig. 19) and we will assume that at the initial moment of time the ends of the rod are at points x=0 And x=l. Let us take an arbitrary section of the rod with the coordinate x. Let us denote by u(x,t) displacement of this section at the moment of time t, then the displacement of the section with coordinate at the same moment of time it will be equal

Then the relative elongation of the rod in section x will be equal

The resistance force to this elongation according to Hooke's law will be equal to

Where E– elastic modulus of the rod material (Young’s modulus), and S – cross-sectional area. At the boundaries of a section of a rod with a length dx forces act on him Tx And T x + dx, directed along the axis x. The resultant of these forces will be equal to

,

and the acceleration of the section of the rod under consideration is equal to , then the equation of motion of this section of the rod will have the form:

, (67)

Where ρ – density of the rod material. If this density and Young's modulus are constant, then we can enter the quantity through and by dividing both sides of the equation by Sdx, finally get equation of longitudinal vibrations of the rod in the absence of external forces

(68)

This equation has the same form as equation for transverse string vibrations and the solution methods for it are the same, however, the coefficient a These equations represent different quantities. In the string equation, the quantity a 2 represents a fraction, the numerator of which is the constant tension force of the string - T, and in the denominator the linear density ρ , and in the string equation the numerators contain Young’s modulus, and the denominator – volumetric rod material density ρ . Hence the physical meaning of the quantity a in these equations is different. If for a string this coefficient is the speed of propagation of a small transverse displacement, then for a rod it is the speed of propagation of a small longitudinal stretching or compression and is called speed of sound, since it is at this speed that small longitudinal vibrations, representing sound, will propagate along the rod.

For equation (68), initial conditions are set that determine the displacement and displacement speed of any section of the rod at the initial time:

For a limited rod, the conditions for fastening or applying force at its ends are specified in the form of boundary conditions of the 1st, 2nd and 3rd kind.

Boundary conditions of the first kind specify longitudinal displacement at the ends of the rod:

If the ends of the rod are fixed motionless, then under conditions (6) . In this case, as in the problem of oscillation of a clamped string, we apply the method of separation of variables.

In boundary conditions of the second kind, elastic forces are specified at the ends of the rod, resulting from deformation according to Hooke’s law depending on time. According to formula (66), these forces, up to a constant factor, are equal to the derivative u x, therefore, at the ends these derivatives are specified as functions of time:

If one end of the rod is free, then at this end u x = 0.

Boundary conditions of the third kind can be represented as conditions under which a spring is attached to each end of the rod, the other end of which moves along the axis according to a given time law θ (t), as shown in Fig. 20. These conditions can be written as follows

, (72)

Where k 1 and k 2 – spring stiffness.

If an external force also acts on the rod along the axis p(x,t), calculated per unit volume, then instead of equation (50) one should write the inhomogeneous equation

,

Which, after dividing by, takes the form

, (73)

Where . Equation (73) is the equation of forced longitudinal vibrations of the rod, which is solved by analogy with the equation of forced vibrations of the string.

Comment. It should be noted that both the string and the rod are models of real bodies, which in reality can exhibit both the properties of the string and the rod, depending on the conditions in which they are located. In addition, the resulting equations do not take into account environmental resistance forces and internal friction forces, as a result of which these equations describe undamped oscillations. To take into account the damping effect, in the simplest case, a dissipative force is used, proportional to the speed and directed in the direction opposite to the movement, i.e. speed. As a result, equation (73) takes the form

(74)

Let us consider a uniform rod of length, i.e., a body of cylindrical or some other shape, to stretch or bend which a certain force must be applied. The latter circumstance distinguishes even the thinnest rod from a string, which, as we know, bends freely.

In this chapter, we will apply the method of characteristics to the study of longitudinal vibrations of a rod, and we will limit ourselves to studying only such vibrations in which the cross sections, moving along the axis of the rod, remain flat and parallel to each other (Fig. 6). Such an assumption is justified if the transverse dimensions of the rod are small compared to its length.

If the rod is slightly stretched or compressed along the longitudinal axis and then left to itself, then longitudinal vibrations will arise in it. Let us direct the axis along the axis of the rod and assume that in a state of rest the ends of the rod are at the points Let the abscissa of a certain section of the rod when the latter is at rest. Let us denote by the displacement of this section at the moment of time, then the displacement of the section with the abscissa will be equal to

From here it is clear that the relative elongation of the rod in the section with the abscissa x is expressed by the derivative

Now assuming that the rod undergoes small oscillations, we can calculate the tension in this section. Indeed, applying Hooke’s law, we find that

where is the elastic modulus of the rod material, its cross-sectional area. Let us take a rod element enclosed

between two sections, the abscissas of which at rest are respectively equal. This element is acted upon by tension forces applied in these sections and directed along the axis. The resultant of these forces has the magnitude

and is also directed along . On the other hand, the acceleration of the element is equal, as a result of which we can write the equality

where is the volumetric density of the rod. Putting

and reducing by we obtain the differential equation of longitudinal vibrations of a homogeneous rod

The form of this equation shows that the longitudinal vibrations of the rod are of a wave nature, and the speed a of propagation of longitudinal waves is determined by formula (4).

If the rod is also acted upon by an external force calculated per unit of its volume, then instead of (3) we obtain

This is the equation of forced longitudinal vibrations of the rod. As in dynamics in general, the equation of motion (6) alone is not enough to completely determine the motion of the rod. It is necessary to set the initial conditions, i.e. set the displacements of the sections of the rod and their velocities at the initial moment of time

where and are given functions in the interval (

In addition, boundary conditions at the ends of the rod must be specified. For example.