Longitudinal vibrations. Methods for solving longitudinal vibrations of a rod
Consider a rod with a length l, which in the equilibrium position is along the x-axis. Its longitudinal vibrations are described by the function Q(x, t), which at each moment of time t is the longitudinal displacement of the rod point, the coordinate of which in the equilibrium position was equal to x. It is assumed that the tension in the rod obeys Hooke's law. Then the equation describing the longitudinal vibration of the rod has the form:
where a is the wave speed, m/s;
f (x, t) - specific force, m / s 2.
The wave velocity of the rod is determined according to the expression:
,
(2.16)
where k is the coefficient of elasticity, N;
ρ – linear density (mass per unit length of the rod), kg/m.
The elasticity coefficient k can be found as follows:
,
(2.17)
E - Young's modulus (stress arising in the sample when its length is doubled (reduced) under other unchanged conditions), N / m 2.
For a homogeneous rod k=const, ρ=const. Otherwise k(х), ρ(х).
The specific force, in turn, can be represented as:
,
(2.18)
where g(x,t) is the linear density of the longitudinal external force (force acting per unit length), N/m.
The initial conditions are given in the form:
– profile of initial displacements:
– initial speed profile:
.
(2.20)
Boundary conditions can be specified for the following cases:
1) The first boundary value problem (boundary conditions of the 1st kind):
where μ 1 (t), μ 2 (t) are given time functions describing the law
movement of the end of the rod.
For a rigidly fixed end μ(t)=0.
2) The second boundary value problem (boundary conditions of the 2nd kind):
;
(2.23)
,
(2.24)
where T 1, T 2 - tension force applied to the end of the rod, N.
In the case of a free end, there is no tension of the rod near it (g(t)=0).
3) The third boundary value problem (boundary conditions of the 3rd kind):
.
(2.25)
These conditions are formulated in the case of elastic fastening of the rod, in which the end of the rod can move, but an elastic force arises that tends to return the displaced end to its previous position.
Formulate a boundary value problem on the longitudinal vibrations of a homogeneous cylindrical rod, one end of which is fixed, and a force F(t)=A·sin(ωt) is applied to the other end, the direction of which coincides with the axis of the rod.
The function Q(x,t), describing the longitudinal oscillations of the rod, is determined by the equation:
.
Initial conditions are zero:
;
.
Boundary conditions are given as:
;
,
where S is the cross-sectional area of the rod, m 2;
E is Young's modulus of the rod material, Pa (see Appendix).
General remarks.
1) If we consider the oscillatory process of a string (rod), whose ends are far enough away and the influence of the ends does not yet have time to manifest itself for a short time interval, then we can consider the string to be infinite. In this case, a problem is considered in which -∞ 2) If the section of the string (rod) under consideration is close to one of its ends and far from the other, then the problem of a semi-infinite string is considered when 0≤x<+∞
и граничные условия формулируются
только на одном ее конце. ISSN: 2310-7081 (online), 1991-8615 (print) doi: http://dx.doi UDC 517.956.3 PROBLEM OF LONGITUDINAL VIBRATIONS OF 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 with the help of concentrated masses and springs are considered. As a mathematical model, an initial-boundary value problem with dynamic boundary conditions for a fourth-order hyperbolic equation is used. 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, the neglect of which, as shown by Rayleigh, leads to an error, which is confirmed by the modern non-local concept of studying the vibrations of solids. The existence of a system of eigenfunctions of the problem under study orthogonal with the load is proved and their representation is obtained. The established properties of the eigenfunctions made it possible to apply the method of separation of variables and prove the existence of a unique solution to the problem. Key words: dynamic boundary conditions, longitudinal vibrations, load orthogonality, 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 a violation of its operation mode, and in some cases to destruction. Various problems associated with a violation of the accuracy and performance of mechanical systems as a result of the 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 allow not only to intensify the cutting processes (drilling, milling, grinding, etc.) of materials with high hardness (tungsten-containing, titanium-carbide steels, etc.), © 2016 Samara State Technical University. Citation Sample Beilin, A.B., The problem of longitudinal vibrations of an elastically fixed loaded rod, Vestn. Myself. state tech. university Ser. Phys.-Math. Nauki, 2016. V. 20, No. 2. P. 249258. doi: 10.14498/vsgtu1474. About the author Alexander Borisovich Beilin (Ph.D., Assoc.; [email protected]), Associate Professor, Dept. automated machine and tool systems. but in some cases become the only possible method for processing brittle materials (germanium, silicon, glass, etc.). The element of the device (waveguide), which transmits ultrasonic vibrations from the source (vibrator) to the tool, is called a concentrator and can have a different shape: cylindrical, conical, stepped, exponential, etc. . Its purpose is to convey fluctuations of the required amplitude to the instrument. Thus, the consequences of the occurrence of oscillatory processes can be different, as well as the causes that cause them, therefore, the need naturally arises for a theoretical study of the processes of oscillation. The mathematical model of wave propagation in relatively long and thin solid rods, which is based on a second-order wave equation, has been well studied and has long become a classic. However, as shown by Rayleigh, this model is not quite consistent with 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 deformations of the rod in the transverse direction should also be taken into account. The mathematical model of longitudinal oscillations of a thick short rod, which takes into account the effects of the transverse motion of the rod, is called the 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: g(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, 1P(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 the processes of vibrations, as well as the theory of plasticity. The review article substantiates the shortcomings of classical models describing the state and behavior of solids under load, in which a priori the body is considered an ideal continuum. The modern 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 was proposed to the study of many physical processes, including those mentioned above, based on the concept of nonlocality (see the article and the list of references in it). One of the classes of non-local models identified by the authors is called “weakly non-local”. 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, in some approximation, the interaction of the internal elements of the object of study. Thus, the Rayleigh model is relevant in our time. 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 N1, M2 and springs, the stiffnesses of which are K1 and K2. We will assume that the rod is a body of revolution about the 0x axis and the initial moment of time is at rest in the equilibrium position. Then we come to the following initial-boundary value problem. Task. Find in the area Qt \u003d ((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)ux(0, r) + b(0)uu(0, r) - k^(0, r) - M1u(0, r) = 0, a(1)ux(1, r) + b(1)uu(1, r) + K2u(1, r) + M2uu(1, r) = 0. () The article considers 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 unambiguous weak solvability of the problem in the general case. Conditions (2) are determined by the method of fixing the rod: its ends are attached to fixed bases with the help of some devices having masses M1, M2, and springs with stiffnesses K1, K2, respectively. The presence of masses and allowance for transverse displacements leads to conditions of the form (2) containing time derivatives. Boundary conditions that include time derivatives are called dynamic. They can arise in various situations, the simplest of which are described in a textbook, and much more complex ones in a monograph. 2. Study of natural oscillations of the rod. Consider a homogeneous equation corresponding to equation (1). Since the coefficients depend only on x, we can separate the variables by representing u(x, z) = X(x)T(z). We get two equations: m""(r) + \2m(r) = 0, ((a(x) - A2b(x))X"(x))" + A2dX(x) = 0. (3) Equation (3) is accompanied by boundary conditions (a(0) - \2b(0))X"(0) - (K1 - \2M1)X(0) = 0, (a(1) - \2b(1))X"(1) + (K2 - \2M2)X(I) = 0. (4) Thus, we have arrived at the Sturm-Liouville problem, which differs from the classical one in that the spectral parameter Λ 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). For the successful implementation of the method of separation of variables, we need information about the existence and location of eigenvalues, about 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 an eigenfunction of problem (3), (4) corresponding to the value A = 0. We 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) - A2b(0))(a(1) - A2b(1)) I (dX2 + bX"2)dx+ N\X 2(0) + M2X 2(1) I aX "2<1х + К\Х2(0) + К2Х2(1). Jo We note that from the physical meaning of the functions a(x), b(x), g(x) are positive, Kr, Mr are non-negative. But then it follows from the resulting equality that X "(x) \u003d 0, X (0) \u003d X (1) \u003d 0, therefore, X (x) \u003d 0, which contradicts the assumption made. Therefore, the assumption that that zero is an eigenvalue of problem (3), (4) is false. The representation of the solution of 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). We fix arbitrarily x e (0, 1) and find the values at this point of the functions a(x), b(x), g(x). We write equation (3) in the form X "(x) + VX (x) \u003d 0, (5) where we marked at the chosen fixed point, and conditions (4) can be written in the form X "(0) - aX (0) \u003d 0, X" (1) + bX (I) \u003d 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, whence, due to the arbitrariness of x, the desired inequality follows. The eigenfunctions of problem (3), (4) have the property of orthogonality with the load , expressed by the relation I (dXm (x) Xn (x) + bX "m (x) X" p (x))<х+ ■)о M1Xm(0)Xn(0) + M2Xm(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) in order to avoid cumbersomeness. Let λm, λn be different eigenvalues, λm, λn be the eigenfunctions of problem (3), (4) corresponding to them. Then ((a - L2mb)X"t)" + L2tdXm = 0, ((a - L2nb)X"n)" + L2pdXp = 0. We 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) YHtXp \u003d (aXtXP) "- LP (bXtX" p) "- (aX "tXp)" + Rt (bXtXp)", which we integrate over the interval (0,1). As a result, taking into account (4) and reducing by (Лт - Лп), we obtain relation (7). Proved statements about the properties of eigenvalues and eigenfunctions of the Sturm-Liouville problem (3), (4) allow us to apply the method of separation of variables to find a solution to the problem. 3. Solvability of the problem. Denote C(CT) = (u: u e C(St) P C2(St), uixx e C^m)). Theorem 1. Let a, b e C1 , e C. Then there exists at most one solution u e C(m) of 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. We note in advance that, from the physical meaning of the coefficients of the equation and the boundary conditions, the functions a, b, q are positive everywhere in Qm, while M^, K^ are non-negative. Multiplying equality (1) by u and integrating over the domain Qt, where t e and arbitrarily, after simple transformations, we obtain / (di2(x, m) + au2x(x, m) + buXl(x, m)) ux + ./o K1u2(0, m) + M1u2(0, m) + K2u2(1, m) + M2u2(1, m) = 0, whence, by virtue of the arbitrariness of m, the assertion of the theorem immediately follows. □ Let us 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 third-order piecewise continuous derivative in (0,1), φ e C 1, φ(0) = φ(1) = 0, and has a second-order piecewise continuous derivative in (0,1), f e C(C^m), then the solution to problem (1), (2) exists and can be obtained as the sum of a series of eigenfunctions. Proof. We will, as usual, look for a solution to the problem in the form of a sum where the first term is the solution of the formulated problem for the homogeneous equation corresponding to (1), the second is the solution of equation (1) that satisfies zero initial and boundary conditions. Let us use the results of the studies carried out in the previous paragraph and write down the general solution of 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 \u003d (a - A4) A2 "- (K - A? Mí) (K2 - A "M). (eight) 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 that are in its left and right parts, and examine their behavior. Without limiting the generality too much, we set 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" and write the spectral equation in new notation! aqlß Kql2 + ß2 (Kb - aM) 2Kql2 + 2^2(Kb - aM) 2/j.aql An analysis of the functions of the left and right parts of the last equation allows us to assert 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 c¿, can be written v / l l I q K - x2pm. l i q Xn(x) = COS XnJ-myx + ----sin XnJ-myx. V a - A2b AnVa - ftb^q V a - A2b Now we turn to finding a solution that also satisfies the initial conditions. We can now easily find the solution of the problem for the homogeneous equation in the form of a series u(x,t) = ^Tn(t)Xn(x), whose coefficients can be found from the initial data using the orthogonality property of the functions Xn(x), whose norm 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 notice 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 form of the eigenfunctions, let us specify the structure of the series in which we are looking for a solution: j(x,t) = ^ (Vn(t)cos Xn^J a b x+ Wn(t) K-XnM~sin X^ GAirx). (nine) v JXnVa - xnb^q V a - xn" To satisfy the zero initial conditions y(x, 0) = y^x, 0) = 0, we require that Yn(0) = Yn(0) = 0, Wn(0) = W(0) = 0. Expanding f( x, d) into a Fourier series with respect to 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 Shn(b): uc® + >&pYu = ™ + xn Wn (<) = Xn (-a-iKrW g Taking into account the initial conditions Yn(0) = Y,(0) = 0, Shn(0) = W,(0) = 0, we arrive at the Cauchy problems for each of the functions Yn(b) and Shn(b), whose unique solvability 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 the series that have arisen in the course of our research and, therefore, about the existence of a solution to the problem. □ Conclusion. The existence of a system of eigenfunctions of the problem under study orthogonal with 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. 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 the longitudinal vibrations of a wide range of technical objects. Alexander Borisovich Beilin: http://orcid.org/0000-0002-4042-2860 REFERENCES 1. Nerubay M. S., Shtrikov B. L., Kalashnikov V. V. Ultrasonic mechanical processing 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 im. I.I. Polzunova, 1997. 120 p. 3. Kumabe D. Vibration cutting. M.: Mashinostroenie, 1985. 424 p. 4. A. N. Tikhonov and A. A. Samarskii, 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. The theory of free and forced vibrations of a solid rod based on the Rayleigh model// DAN, 2007. V. 417, no. 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. A. B. Beilin and L. S. Pulkina, “Problem of Longitudinal Vibrations of a Rod with Dynamic Boundary Conditions,” Vestn. SamGU. Natural Science Ser., 2014. No. 3 (114). pp. 9-19. 10. M. O. Korpusov, Fracture in nonclassical wave equations. M.: URSS, 2010. 237 p. Received 10/II/2016; in the final version - 18/V/2016; accepted for publication - 27/V/2016. Vestn. Samar. gos. Techn. Unta. Ser. Phys.-mat. science 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 the 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 Russian) 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. Technology. Univ., Ser. Phys.-Mat. Science, 2016, vol. 20, no. 2, pp. 249-258. doi: 10.14498/vsgtu1474. (In English) 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 uravneniakh. Moscow, URSS, 2010, 237 pp. (In Russian) Received 10/II/2016; received in revised form 18/V/2016; DEFINITION Longitudinal wave- this is a wave, during the propagation of which the displacement of the particles of the medium occurs in the direction of the wave propagation (Fig. 1, a). The cause of the occurrence of a longitudinal wave is compression / extension, i.e. the resistance of a medium to a change in its volume. In liquids or gases, such deformation is accompanied by rarefaction or compaction of the particles of the medium. Longitudinal waves can propagate in any media - solid, liquid and gaseous. Examples of longitudinal waves are waves in an elastic rod or sound waves in gases. DEFINITION transverse wave- this is a wave, during the propagation of which the displacement of the particles of the medium occurs in the direction perpendicular to the propagation of the wave (Fig. 1b). The cause of a transverse wave is the shear deformation of one layer of the medium relative to another. When a transverse wave propagates in a medium, ridges and troughs are formed. Liquids and gases, unlike solids, do not have elasticity with respect to layer shear, i.e. do not resist shape change. Therefore, transverse waves can propagate only in solids. Examples of transverse waves are waves traveling along a stretched rope or along a string. Waves on the surface of a liquid are neither longitudinal nor transverse. If you throw a float on the surface of the water, you can see that it moves, swaying on the waves, in a circular fashion. Thus, a wave on a liquid surface has both transverse and longitudinal components. On the surface of a liquid, waves of a special type can also occur - the so-called surface waves. They arise as a result of the action and force of surface tension. EXAMPLE 1 Let's draw the surface of the wave near the float after a certain time interval , considering that during this time the float went down, since it was directed down at the moment of time. Continuing the line to the right and to the left, we show the position of the wave at time . Comparing the position of the wave at the initial moment of time (solid line) and at the moment of time (dashed line), we conclude that the wave propagates to the left. MECHANICS UDC 531.01/534.112 LONGITUDINAL VIBRATIONS OF A PACKAGE OF RODS A.M. Pavlov, A.N. Temnov MSTU im. N.E. Bauman, Moscow, Russian Federation e-mail: [email protected]; [email protected] In questions of the dynamics of liquid-propellant rockets, an important role is played by the problem of the stability of the rocket's motion in the event of the occurrence of longitudinal elastic oscillations. The appearance of such oscillations can lead to the establishment of self-oscillations, which, if the rocket is unstable in the longitudinal direction, can lead to its rapid destruction. The problem of longitudinal oscillations of a package rocket is formulated; a package of rods is used as a calculation model. It is assumed that the liquid in the rocket tanks is "frozen", i.e. proper fluid motions are not taken into account. The total energy balance law for the problem under consideration is formulated and its operator statement is given. A numerical example is given, for which the frequencies are determined, and the eigenmodes are constructed and analyzed. Keywords: longitudinal vibrations, frequency and shape of vibrations, rod package, total energy balance law, self-adjoint operator, vibration spectrum, POGO. SYSTEM OF RODS LONGITUDINAL VIBRATIONS A.M. Pavlov, Al. Temnov Bauman Moscow State Technical University, Moscow, Russian Federation e-mail: [email protected]; [email protected] In questions of dynamics of liquid fuel rockets the problem of motion stability for this rocket has an important role with the appearance of longitudinal elastic vibrations. An occurrence of such kind of vibrations can evoke self-vibrations which may cause rapid destruction of the rocket in case of rocket instability within longitudinal direction. The problem on longitudinal vibrations of the liquid fuel rocket based on the packet scheme has been formulated using package rods as a computational model. It is assumed that the liquid in the rocket tanks is "frozen", i.e. proper motions of the liquid are not included. For this problem energy conservation principle was formulated and its operator staging is given. There is a numerical example, for which the frequencies have been determined, forms of Eigen vibration were built and analyzed. Keywords: longitudinal modes vibrations, eigen modes and frequencies, rods model, energy conservation principle, selfadjoint operator, vibration spectrum, POGO. Introduction. At present, in Russia and abroad, in order to launch a payload into the required orbit, launch vehicles (LV) of a package layout with identical side blocks evenly distributed around the central block are often used. Studies of oscillations of package structures encounter certain difficulties associated with the dynamic action of the side and central blocks. In the case of symmetry of the layout of the launch vehicle, the complex, spatial interaction of the blocks of a package design can be divided into a finite number of vibration types, one of which is the longitudinal vibrations of the central and side blocks. The mathematical model of longitudinal vibrations of a similar design in the form of a package of thin-walled rods is considered in detail in the work. Rice. 1. Scheme of the central significant vibrations of a package of rods, supplementing the study carried out by A.A. Pitiful. Formulation of the problem. Consider other longitudinal vibrations of a package of rods, consisting of a central rod of length l0 and N side rods of the same length j = l, (l0 > lj), j = 1, 2,..., N, fastened at point A (xA = l) (Fig. 1) with central spring elements of stiffness k. Let us introduce a fixed frame of reference ОХ and assume that the rigidity of the rods EFj (x), the distributed mass mj (x) and the perturbation q (x,t) are bounded functions of the coordinate x: 0 0 < mj < mj (x) < Mj; (1) 0 Let displacements Uj (x, t) appear in the cross sections of the rods with coordinate x, which are determined by the equations mj (x) ^ - ¿(eFj (x) ^ = qj (x,t), j = 0,1, 2,..., N, (2) boundary conditions for the absence of normal forces at the ends of the rods 3 \u003d 0, x \u003d 0, ^ \u003d 1, 2, 0, x = 0, x = l0; conditions of equality of normal forces arising in the rods, EF-3 = F x = l elastic forces of spring elements FпPJ = k (u (ha) - u (¡,)); (4) EUodX (xa - 0) - EFodX (xa + 0) = , x = xa; the condition of equality of displacements at the point xa of the central rod W (ha-o) \u003d W (ha + o) and initial conditions W y (x, 0) - W (x); , _ u(x, 0) = u(x), where u(x, 0) = "q^1(x, 0). The law of total energy balance. We multiply equation (2) by u(x, t), integrate over the length of each rod, and add the results using boundary conditions (3) and matching condition (4). As a result, we get (( 1 ^ [ (diL 2 tz (x) "BT" (x + dt | 2 ^ J 3 w V dt N x „ h 2 .. N „ i. 1 ⩽ Г „„ , f dn3\ , 1 ⩽ Гj 1 N /* i dpl 2 1 N fl j EF3 dx +2^Yo N (x - -)(no - Uj)2 dx = / ^ (x, t) ux y (x, t) (x, (6) where 8(x - y) is the Dirac delta function. In equation (6), the first term in curly brackets is the kinetic energy T (¿) of the system, the second is the potential energy Pr (£) due to the deformation of the rods, and the third is the potential energy Pk (£) of the spring elements, which in the presence of elastic deformations rods can be written as Pk (*) = 2 £ / Cy (¡y) 8 (x - ¡1) E^ (¡y) (ddit (¡1)) 2 (x, Cy = Ey. Equation (6) shows that the change in the total energy per unit time of the considered mechanical system is equal to the power external influence. In the absence of an external perturbation q (x,t), we obtain the law of conservation of the total energy: T (t) + Pr (t) + Pk (t) = T (0) + Pr (0) + Pk (0). Operator setting. The energy balance law shows that for any time t the functions Uj (x, t) can be considered as elements of the Hilbert space L2j(; m3 (x)), defined on length ¡i by the scalar product (us,Vk)j = J mj (x) usVkdx 0 and the relevant regulation. Let us introduce the Hilbert space H, which is equal to the orthogonal sum L2j, H = L20 Φ L21 Φ... Φ L2N, the vector function U = (uo, Ui,..., uN)m, and the operator A acting in the space H according to the relation AU = diag(A00U0, A11U1, ..., Annun). mj(x)dx\jdx" operators defined on set B (A33) C H of functions satisfying conditions (3) and (4). The original problem (1)-(5) together with the initial conditions can be written as Au = f(*), u(0) = u0, 17(0) = u1, (7) where f (*) = (to (*) ,51 (*),..., Yam (¿)) i.e. Lemma. 1. If the first two conditions (1) are satisfied, then the operator A in the evolution problem (7) is an unbounded, self-adjoint, positive-definite operator in the space H (Au, K)n = (u, AK)n, (Au, u)n > c2 (u, u)n. 2. The operator A generates an energy space HA with a norm equal to twice the value of the potential energy of oscillations of the package of rods 3 \ ^ I h)2 = 2n > 0. (8) IIUIIA = £/ EF^^J dx + k £ (uo - U)2 = 2П > 0. < Оператор А неограничен в пространстве Н, поскольку неограничен каждый диагональный элемент А33. Самосопряженность и положительная определенность оператора А проверяются непосредственно: (AU, v)h =/m (x) (-^| (EFo (x) ^j) Vo (x) dx+ +£ jm(x) (- jx) | (ef-(x) dndxa))v-(x) dx=... = EFo (x) uo (x) vo (x) dx - EFo (x) U) (x) vo (x) J E Fo (x) uo (x) vo (x) dx - E Fo (x) uo (x) ?o (x) + ^^ / EF- (x) u- (x) vo (x) dx - ^^ EF- (x) u- (x) v- (x) J E Fo (x) uo (x) v" (x) dx - E Fo (xa - 0) uo (xa - 0) vo (xa) + 0 EFo (xa + 0) uo (xa + 0) vo (xa) - £ EF- (/-) u- (/-) v- (/-) + J EF- (x) u- (x) v- (x) dx = J EFo (x) uo (x) vo (x) dx+ -=100 + £ / EF.,- (x) u- (x) r?- (x) dx+ o O(xa)- £ EF- (/-) u- (/-) v?"- (/-) = EFo (x) uo (x) v?"o (x) dx+ -=10 + £ / EF- (x) u- (x) v- (x) dx+ -=1 0 - + £ k (uo (xa) - u- (/-)) (vo (xa) - v- (/-)) = (U, A?)H (AU, U)H \u003d ... \u003d I EF0 (x) u "2 (x) dx - EF0 (x) u0 (x) u0 (x) J EF0 (x) u "0 (x) dx - EF0 (x) u0 (x) u0 (x) + ^^ / EFj (x) u"2 (x) dx - ^^ EFj (x) uj (x) u3 (x) "J EF°(x) u"2 (x) dx 4EF0 (x) u"2 (x) dx+£ JEFj (x) u"2 (x) dx Y^k (u0 (l) uj (l) - u2 (/)) + u0 (l) ^ k (u0 (l) - uj (l)) = EF0 (x) u "2 (x) dx + / EF0 (x) u" 0 (x) dx + S / EFj (x) u"2 (x) dx + k ^ (u0 (l) - uj (l))2 > c2 (U, U)H It follows from the above results that the energy norm of the operator A is expressed by formula (8). Solvability of the evolutionary problem. We formulate the following theorem. Theorem 1. Let the conditions U0 £ D (A1/2) , U0 £ H, f (t) £ C (; H), then problem (7) has a unique weak solution U (t) on the segment defined by the formula U (t) = U0 cos (tA1/2) + U1 sin (tA1/2) +/sin ((t - s) A1/2) A-1/2f (s) ds. 5 in the absence of an external perturbation f (£), the law of conservation of energy is satisfied 1 II A 1/2UИ2 = 1 1 II A1/2U 0|H. < Эволюционная задача (7) - это стандартная задача Коши для дифференциального операторного уравнения гиперболического типа, для которого выполнены все условия теоремы о разрешимости . Natural vibrations of a package of rods. Let us assume that the field of external forces does not act on the rod system: f (t) = 0. In this case, the motion of the rods will be called free. The free motions of the rods, which depend on the time t according to the law exp (iwt), will be called eigenoscillations. Taking in equation (7) U (x, t) = U (x) eiWU, we obtain the spectral problem for the operator A: AU - AEU \u003d 0, L \u003d w2. (nine) The properties of the operator A allow us to formulate a theorem on the spectrum and properties of eigenfunctions. Theorem 2. The spectral problem (9) on natural oscillations of a package of rods has a discrete positive spectrum 0 < Ai < Л2 < ... < Ak < ..., Ak ^ то and a system of eigenfunctions (Uk (x))^=0, complete and orthogonal in the spaces H and HA, and the following orthogonality formulas hold: (Ufe, Us)H = £ m (xj UfejMSjdx = j=0 0 (Uk= £/U^) d*+ K ("feo - Mfej) (uso -) = Afeífes. j=i Investigation of the spectral problem in the case of a homogeneous package of rods. Representing the displacement function m-(x, t) in the form m-(x, t) = m-(x), after separating the variables, we obtain spectral problems for each rod: ^0u + LM = 0, ^ = 0,1,2,..., N (10) which we write in matrix form 4 £ + Li = 0, A = -,-,-,...,- \ t0 t1 t2 t« u = (u0, u1, u2,..., u') i.e. Solution and analysis of the obtained results. Let us designate the displacement functions for the central rod in the section as u01 and in the section as u02 (g). In this case, for the function u02, we move the origin of coordinates to the point with coordinate /. For each rod, we represent the solution of equation (10) in the form To find the unknown constants in (11), we use the boundary conditions formulated above. From homogeneous boundary conditions, some constants can be determined, namely: C02 = C12 = C22 = C32 = C42 = ... = CN 2 = 0. As a result, it remains to find N + 3 constants: C01, C03, C04, C11, C21, C31, C41,..., CN1. To do this, we solve N + 3 equations for N + 3 unknowns. We write the resulting system in matrix form: (A) (C) = (0) . Here (C) = (C01, C03, C04, C11, C21, C31, C41,..., Cn 1)m is the vector of unknowns; (A) - characteristic matrix, cos (L1) EF0 L sin (L1) + L sin (L (Zo - 1)) L cos (L (Zo - 1)) 0 00 0 \ -1 0 0000 0 y 00 00 0 000Y a \u003d k coe ^ ^A-L^; c \u003d -k co8 ((.40-01L) 1 / 2 ^; 7 \u003d (A4 "-1 l) 1/2 ap ((A" 1l) 1/2 + to owls ((A "1l) 1/2; (~ \ 1/2 ~ A = ^ A] ; A-- : 3 = 0. To find a non-trivial solution, we take the constant C01 € M as a variable. We have two options: C01 = 0; C01 = 0. Let С01 = 0, then С03 = С04 = 0. In this case, a nontrivial solution can be obtained if 7 = 0 from (12) under the additional condition £ c-1 = 0, (13) which can be obtained from the third equation of system (12). As a result, we obtain a simple frequency equation EP (A "1 L) 1/2 w ((A" 1 ^ 1/2 P + zz y \ V zz K cos ^ (A-/a) 1/2 ^ = 0, j G , coinciding with the frequency equation for a rod elastically fixed at one end, which can be considered as the first partial system. In this case, all possible combinations of movements of the side rods that satisfy condition (13) can be conditionally divided into groups corresponding to different combinations of phases (in the case under consideration, the phase is determined by the sign of S.d). If we take the side rods identical, then we have two options: 1) Cd \u003d 0, then the number of such combinations n for different N can be calculated by the formula n \u003d N 2, where is the division function without a remainder; 2) any (or any) of the C- constants are equal to 0, then the number of possible combinations increases and can be determined by the formula £ [(N - m) div 2]. Let Coi = 0, then Cn = C21 = C31 = C41 = ... = CN1 = C01 (-v/t), where c and y are complexes in (12). From system (12) we also have: C03 = C01 cos (L/); C04=C03 tg (L (/0 - /)) = C01 cos (A/) x x tg (L (/0 - /)), i.e. all constants are expressed through C01. The frequency equation takes the form EFo U-o1 L tg A-1 L) "(lo - l)) - K2 cos | ía!-,1 L As an example, consider a system with four side rods. In addition to the method described above, for this example, you can write the frequency equation for the entire system by calculating the determinant of the matrix A and equating it to zero. We present its form Y4 (L sin (L (/o - /)) cos (L/) EFoL+ L cos (L (/ o - /)) (EFoL sin (L /) + 4v)) - 4avt3L cos (L(/0 - /)) = 0. Graphs of transcendental frequency equations for the cases considered above are shown in fig. 2. The following data were taken as initial data: EF = 2109 N; EF0 = 2.2 109 N; k = 7 107 N/m; m = 5900 kg/m; mo = 6000 kg/m; /=23; /o = 33 m. The values of the first three oscillation frequencies of the scheme under consideration are given below: n..................................... and, rad/s...................... 1 2 3 20,08 31,53 63,50 Rice. 2. Plots of transcendental frequency equations for Coi = 0 (i) and Coi = 0 (2) Let us present the vibration modes corresponding to the solutions obtained (in the general case, the vibration modes are not normalized). The waveforms corresponding to the first, second, third, fourth, 13th and 14th frequencies are shown in fig. 3. At the first oscillation frequency, the side rods oscillate with the same shape, but in pairs in antiphase Fig.3. Vibration modes of the side (1) and central (2) rods corresponding to the first V = 3.20 Hz (a), the second V = 5.02 Hz (b), the third V = 10.11 Hz (c), the fourth V = 13.60 Hz (d), 13th V = 45.90 Hz (d) and 14th V = 50.88 Hz (e) frequencies (Fig. 3, a), at the second - the central rod oscillates, and the lateral ones oscillate in the same form in phase (Fig. 3, b). It should be noted that the first and second oscillation frequencies of the considered rod system correspond to the oscillations of a system consisting of solid bodies. When the system oscillates with the third natural frequency, nodes appear for the first time (Fig. 3c). The third and subsequent frequencies (Fig. 3d) correspond to already elastic oscillations of the system. With an increase in the frequency of oscillations associated with a decrease in the influence of elastic elements, the frequencies and forms of oscillations tend to be partial (Fig. 3, e, f). Curves of functions, the points of intersection of which with the abscissa axis are solutions of transcendental equations, are shown in fig. 4. According to the figure, the natural oscillation frequencies of the system are located near the partial frequencies. As noted above, as the frequency increases, the convergence of natural frequencies with partial ones increases. As a result, the frequencies at which the entire system oscillates are conditionally divided into two groups: those close to the partial frequencies of the side rod and frequencies close to the partial frequencies of the central rod. Findings. The problem of longitudinal vibrations of a package of rods is considered. The properties of the formulated boundary value problem and the spectrum of its eigenvalues are described. A solution of the spectral problem for an arbitrary number of homogeneous side rods is proposed. For a numerical example, the values of the first oscillation frequencies are found and the corresponding forms are constructed. Some characteristic properties of the constructed modes of vibrations were also revealed. Rice. 4. Curves of functions, the points of intersection of which with the abscissa axis are solutions of transcendental equations, for Cox = 0 (1), Cox = 0 (2) coincide with the first partial system (side rod fixed on the elastic element at the point x = I) and of the second partial system (5) (central rod fixed on four elastic elements at point A) LITERATURE 1. Kolesnikov K.S. Rocket dynamics. M.: Mashinostroenie, 2003. 520 p. 2. Ballistic missiles and launch vehicles / O.M. Alifanov, A.N. Andreev, V.N. Gushchin et al. M.: Drofa, 2004. 511 p. 3. Rabinovich B.I. Introduction to the dynamics of spacecraft carrier rockets. M.: Mashinostroenie, 1974. 396 p. 4. Parameter study on POGO stability of liquid rockets / Z. Zhao, G. Ren, Z. Yu, B. Tang, Q. Zhang // J. of Spacecraft and Rockets. 2011 Vol. 48. Is. 3. P. 537-541. 5. Balakirev Yu.G. Methods for analysis of longitudinal oscillations of carrier rockets with a liquid engine // Cosmonautics and Rocket Engineering. 1995. No. 5. S. 50-58. 6. Balakirev Yu.G. Peculiarities of the mathematical model of a packaged liquid-propellant rocket as a control object // Selected problems of the strength of modern mechanical engineering. 2008. S. 43-55. 7. Dokuchaev L.V. Improving methods for studying the dynamics of a package design launch vehicle, taking into account their symmetry // Cosmonautics and Rocket Engineering. 2005. No. 2. S. 112-121. 8. Pozhalostin A.A. Development of Approximate Analytical Methods for Calculating Natural and Forced Vibrations of Elastic Shells with Fluid: Cand. ... Dr. tech. Sciences. M., 2005. 220 p. 9. Krein S.G. Linear differential equations in Banach spaces. M.: Nauka, 1967. 464 p. 10. Kopachevsky I.D. Operator methods of mathematical physics. Simferopol: OOO "Forma", 2008. 140 p. Kolesnikov K.S. Dinamika missile. Moscow, Mashinostroenie Publ., 2003. 520 p. Alifanov O.N., Andreev A.N., Gushchin V.N., eds. Ballisticheskie rakety i rakety-nositeli. Moscow, Drofa Publ., 2003. 511 p. Rabinovich B.I. Vvedenie v dinamiku raket-nositeley kosmicheskikh apparatov. Moscow, Mashinostroenie Publ., 1974. 396 p. Zhao Z., Ren G., Yu Z., Tang B., Zhang Q. Parameter study on POGO stability of liquid fuel rocket. J. Spacecraft and Rockets, 2011, vol. 48, iss. 3, pp. 537-541. Balakirev Yu.G. Methods of analysis of longitudinal vibrations of launch vehicles with liquid propellant engine. Kosm. i rockettostr. , 1995, no. 5, pp. 50-58 (in Russ.). Balakirev Yu.G. Osobennosti matematicheskoy modeli zhidkostnoy rakety paketnoy komponovki kak ob "ekta upravlenii. Sb. "Izbrannye problemy prochnosti sovremennogo mashinostroeniya" . Moscow, Fizmatlit Publ., 2008. 204 p. (cited pp. 4355). Dokuchaev L.V. Improvement of methods for studying the dynamics of clustered launch vehicle considering their symmetry. Kosm. i rockettostr. , 2005, no. 2, pp. 112-121 (in Russ.). Pozhalostin A.A. Razrabotka priblizhennykh analiticheskikh metodov rascheta sobstvennykh i vynuzhdennykh kolebaniy uprugikh obolochek s zhidkost "yu. Diss. doct. tekhn. nauk . Kreyn S.G. Lineynye differentsial "nye uravneniya v Banakhovykh prostranstvakh. Moscow, Nauka Publ., 1967. 464 p. Kopachevskiy I.D. Operatornye metody matematicheskoy fiziki. Simferopol", Forma Publ., 2008. 140 p. The article was received by the editors on April 28, 2014 Pavlov Arseniy Mikhailovich - student of the department "Spacecraft and launch vehicles" of Moscow State Technical University. N.E. Bauman. Specializes in the field of rocket and space technology. MSTU im. N.E. Baumash, Russian Federation, 105005, Moscow, 2nd Baumanskaya st., 5. Pavlov A.M. - student of "Spacecrafts and Launch Vehicles" department of the Bauman Moscow State Technical University. Specialist in the field of rocket-and-space technology. Bauman Moscow State Technical University, 2-ya Baumanskaya ul. 5, Moscow, 105005 Russian Federation. Temnov Alexander Nikolaevich - Ph.D. Phys.-Math. Sci., Associate Professor, Department of Spacecraft and Launch Vehicles, Moscow State Technical University. N.E. Bauman. Author of more than 20 scientific papers in the field of fluid and gas mechanics and rocket and space technology. MSTU im. N.E. Baumash, Russian Federation, 105005, Moscow, 2nd Baumanskaya st., 5. Temnov A.N. - Cand. sci. (Phys.-Math.), assoc. professor of "Spacecrafts and Launch Vehicles" department of the Bauman Moscow State Technical University. Author of more than 20 publications in the field of fluid and gas mechanics and rocket-and-space technology. Bauman Moscow State Technical University, 2-ya Baumanskaya ul. 5, Moscow, 105005 Russian Federation. Consider a homogeneous rod of length l, i.e. a body of cylindrical or other shape, for stretching or bending of which it is necessary to apply a known force. The latter circumstance distinguishes even the thinnest rod from the string, which, as everyone knows, bends freely. In the work I have presented, I will show the application of the method of characteristics to the study of longitudinal vibrations of a rod, and I will limit myself to studying only such vibrations in which the cross section pq, moving along the axis of the rod, remains flat and parallel to each other. Such an assumption is justified if the transverse dimensions of the rod are small compared to its length. If the rod is somewhat stretched or compressed along the longitudinal axis, and then left to itself, then longitudinal vibrations will occur in it. Having directed the x-axis along the axis of the rod, I will assume that at rest the ends of the rod section are at the points x=0 and x=l. Let x be the abscissa of some section of the rod when the latter is at rest. Let me introduce the notation, through u(x,t) the displacement of this section at time t; then the offset of the section with abscissa x+dx 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 makes small vibrations, we can calculate the tension T. Applying Hooke's law, we get: Where E is the modulus of elasticity of the rod material, and S is its cross-sectional area. Let me take an element of a rod enclosed between two sections, the abscissas of which at rest are respectively equal to x and x + dx. Tension forces applied in these sections and directed along the Ox axis act on this element. The resultant of these forces has the value ES - ES?ES (2) (Lagrange's theorem) And 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 bulk 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 oscillations of the rod are of a wave nature, and the propagation velocity of longitudinal waves is determined by formula (4). If the rod is also subject to an external force calculated per unit of its volume, then instead of (3) we get This is the equation of forced longitudinal vibrations of the rod. As in dynamics in general, one equation of motion (6) is not enough to completely determine the motion of the rod. It is necessary to set the initial conditions, i.e. set the displacement of the rod sections and their speed at the initial moment of time where and F(x) are given functions in the interval (0,l). In addition, the boundary conditions at the ends of the bar must be specified. For example: 1) The rod is fixed at both ends. In this case u(0,t)=0, u(l,t)=0 (8) at any time t. 2) One end of the rod is fixed, the other is free, i.e. u(0,t)=0,=0 (9) at any time t. At the free end x=l, the tension T=ES is equal to zero (no external forces) and, therefore, =0 3) Both ends of the rod are free. At any moment in time Thus, the problem of longitudinal vibrations of a homogeneous bounded rod is reduced to solving equation (6) that satisfies the initial condition (7) and one of the boundary conditions (8), (9), (10), etc. longitudinal oscillation differential wave Consider the problem of longitudinal vibrations of a homogeneous elastic rod of length l, when its end x=0 is fixed, and the other x=l is free. This problem is reduced to solving the wave equation. Under boundary conditions and initial conditions F(x) (0?x?l) (3) According to the Fourier method, we are looking for particular solutions to equation (1) in the form u(x,t)=X(x) T(x) (4) I substitute equation (4) into (1) and get whence we get two equations In order for function (4), which is different from the identical zero, to satisfy the boundary conditions (2), it is obviously necessary to require the fulfillment of the conditions X(x)=0, X(l)=0 (6) Thus, I came to the eigenvalue problem for equation (5) under boundary conditions (7). Integrating equations (5) we obtain From the boundary conditions (6) we have Counting I find =0, whence where k is an integer Thus, nontrivial solutions to problem (4), (5) are possible only for the values ??: Eigenvalues correspond to eigenfunctions (x)= (k=1,2,…..) Defined up to a constant factor, which we set equal to one (k- will not be negative) For ??= the general solution to equation (5) has the form Where are arbitrary constants. Due to (3) we get Satisfy (1) and boundary conditions (2) for any. I make a row. to fulfill the initial conditions (2) it is necessary that Assuming that the series (8), (9) converge uniformly, we can determine the coefficients by multiplying by both parts of the equalities by and integrating over x in the range from x=0 to x=l. Considering, Substituting the found values of the coefficients in the series (7), I, perhaps, will get a solution to the problem obtained from it by a two-fold term-by-term differentiation with respect to x and t, uniformly converge. Having considered the solution (7), it can be seen that the oscillatory motion of the rod is the result of the addition of simple harmonic vibrations Committed with amplitude and with frequencies The fundamental tone obtained at k=0 has an oscillation period Since the amplitude of the fundamental tone is It is obvious that a node is formed at the fixed end of the rod x=0, and at the free end x=l-antinode.transverse waves
Examples of problem solving
Exercise
Determine the direction of propagation of the transverse wave if the float at some point in time has the direction of velocity indicated in the figure.
Decision
Let's make a drawing. 
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