1. Presented by Andrey Kuzmin Mathematical aspects of mechanical systems eigentones Department of Applied Mathematics

2. Joint Advanced Student School St.Petersburg 2006 2 Agenda PART I. Introduction to the theory of mechanical vibrations PART II. Eigentones (free vibrations) of rod systems – Forces Method – Example PART III. Eigentones of plates and shells – Properties of eigentones – The rectangular plate: linear and nonlinear statement – The bicurved shell

3. PART I Introduction to the theory of mechanical vibrations

4. Joint Advanced Student School St.Petersburg 2006 4 1.1 History History of development of the linear vibration theory: –XVIII century “Analytical mechanics” by Lagrange – systems with several degrees of freedom –XIX century Rayleigh and others – systems with the infinite number degrees of freedom –XX century The linear theory has been completed Intro

5. Joint Advanced Student School St.Petersburg 2006 5 Today’s problems of the linear vibration theory: Vibration problems of mechanical systems 1.2 Problems –How correctly to choose degrees of freedom? –How correctly to define external influences? Choice of the calculated scheme Linear statementNonlinear statement Intro

6. Joint Advanced Student School St.Petersburg 2006 6 Role of the nonlinear theory: The phenomena description escaping from a field of vision at any attempt to linearize the considered problem. Approximate solution methods of nonlinear problems: –Poincare and Lyapunov’s Methods –Krylov-Bogolyubov's Method –Bubnov-Galerkin’s Method –and others 1.3 Solution allow making successive approximations allow making any approximations Intro

7. PART II Eigentones (free vibrations) of rod systems

8. Joint Advanced Student School St.Petersburg 2006 8 2.1 Forces Method Consider rod systems in which the distributed mass is concentrated in separate sections (systems with a finite number of degrees of freedom) Define displacements from a unit forces applied in directions of masses vibrations Construct the stiffness matrix of system: the gain matrix depend on the unit forces applied in a direction of masses vibrations in the given system the stiffness matrix of separate elements transposition of the matrix equal to the matrix b, constructed for statically definable system Rod systems

9. Joint Advanced Student School St.Petersburg 2006 9 Construct a diagonal masses matrix M, calculate matrix product D = BM and consider system of homogeneous equations where In the end compute the determinant, eigenvalues and corresponding eigenvectors of matrix D 2.1 Forces Method (1) an amplitudes vector of displacements the unit matrix frequency of free vibrations of the given system Rod systems

10. Joint Advanced Student School St.Petersburg 2006 10 2.2 Example: the problem setup Define frequencies and forms of the free vibrations of a statically indeterminate frame with two concentrated masses т 1 = 2т, т 2 = т and identical stiffnesses of rods at a bending down ( EI = const, where E – Young's modulus; I – Inertia moment of section) Fig. 1, a. Rod system with two degree of freedoms Rod systems

11. Joint Advanced Student School St.Petersburg 2006 11 2.3 Example: the problem solution Fig 1, b. The bending moments stress diagrams depend on the unit forces applied in a direction of masses vibrations Fig 1, c. The stress diagrams depend on the same unit forces in statically determinate system Rod systems

12. Joint Advanced Student School St.Petersburg 2006 12 2.3 Example: the problem solution Calculation of displacements: evaluation of integrals on the Vereschagin's Method Then we construct the stiffness matrix Rod systems

13. Joint Advanced Student School St.Petersburg 2006 13 The masses matrix has the form (at т 1 = 2т, т 2 = т ): To find eigenvalues and eigenvectors of the matrix D = BM we compute the determinant: 2.3 Example: the problem solution Rod systems

14. Joint Advanced Student School St.Petersburg 2006 14 Then we obtain a quadratic equation Thus we can find frequencies of free vibrations of the frame For definition of corresponding forms of vibrations we use (1). Let, for example, X 1 = 1. From the first equation we find Х 2 for each value of λ j : 2.3 Example: the problem solution with roots Rod systems

15. Joint Advanced Student School St.Petersburg 2006 15 1 1 5,569 -0,359 2.3 Example: the problem solution Solving each equations separately, we find eigenvectors ν 1 and ν 2 : Then we obtain forms of the free vibrations Rod systems Fig. 1, d. The main forms of the free vibrations

16. PART III Eigentones of plates and shells

17. Joint Advanced Student School St.Petersburg 2006 17 3.1 Properties of eigentones Properties of linear eigentones (free vibrations): –Plates and shells – systems with infinite number degrees of freedom. That is: number of eigenfrequencies is infinite each frequency corresponds a certain form of vibrations –Amplitudes do not depend on frequency and are determined by initial conditions: deviations of elements of a plate or a shell from equilibrium position velocities of these elements in an initial instant Plates and shells

18. Joint Advanced Student School St.Petersburg 2006 18 3.1 Properties of eigentones Properties of nonlinear eigentones: –Deflections are comparable to thickness of a plate: Rigid plates / shells Flexible plates / shells –Frequency depends on vibration amplitude transform Fig. 2. Possible of dependence between the characteristic deflection and nonlinear eigentones frequency Plates and shells a) Thin systemb) Soft system Skeletal line 1 1 A A

19. Joint Advanced Student School St.Petersburg 2006 19 System with infinite number degrees of freedom System with one degree of freedom 3.2 Solution of nonlinear problems Approximation Plates and shells

20. Joint Advanced Student School St.Petersburg 2006 20 3.3 The rectangular plate, fixed at edges: a linear problem Let a, b – the sides of a plate h – the thickness of a plate Linear equation for a plate: where (2) The rectangular plate w – function of the deflection  – density of the plate material g – the free fall acceleration D – cylindrical stiffness E – Young's modulus  – the Poisson's ratio  4 – the differential functional

21. Joint Advanced Student School St.Petersburg 2006 21 3.4 Solution of the linear problem Integration where Approximation of the deflection on the Kantorovich's Method: Substituting the equation (2) instead of function f(t) : some temporal function The rectangular plate

22. Joint Advanced Student School St.Petersburg 2006 22 3.4 Solution of the linear problem Fig. 3. Character of wave formation of the rectangular plate at vibrations of the different form where The square of eigentones frequency at small deflections has form: The rectangular plate the velocity of spreading of longitudinal elastic waves in a material of the plate m = n = 1 a) the first form m = 2, n = 1 b) the second form m = n = 2 b) the third form

23. Joint Advanced Student School St.Petersburg 2006 23 Examine vibrations of a plate at amplitudes which are comparable with its thickness Assume that the ratio of the plate sides is within the limits of We take advantage of the main equations of the shells theory at k x = k y = 0 : where 3.5 The rectangular plate, fixed at edges: a nonlinear problem (3) (4) The rectangular plate Equilibrium equation differential functional a stress function Deformation equation the main shell curvatures

24. Joint Advanced Student School St.Petersburg 2006 24 Set expression (approximation) of a deflection Substituting (5) in the right member of the equation (4), we shall obtain the equation, which private solution is: Define,, where F x and F y – section areas of ribs in a direction of axes x and y 3.6 Solution of the nonlinear problem where (5) The rectangular plate

25. Joint Advanced Student School St.Petersburg 2006 25 Then the solution of a homogeneous equation will have the form: Finally 3.6 Solution of the nonlinear problem where the stresses applied to the plate through boundary ribs (they are considered as positive at a tensioning) The rectangular plate

26. Joint Advanced Student School St.Petersburg 2006 26 3.7 Solution: the first stage of approximation Apply the Bubnov-Galerkin’s Method to the equation (3) for some fixed instant t Suppose X has the form Generally we approximate functions w(x,y,t) in the form of series some given and independent functions which satisfy to boundary conditions of a problem the parameters depending on t The rectangular plate

27. Joint Advanced Student School St.Petersburg 2006 27 3.7 Solution: the first stage of approximation On the Bubnov-Galerkin’s Method we write out n equations of type In our solution η 1 has the form (6) The rectangular plate

28. Joint Advanced Student School St.Petersburg 2006 28 Hence, integrating (6) and passing to dimensionless parameters, we obtain the equation where the dimensionless parameters K and ζ have the form 3.7 Solution: the first stage of approximation (7) (8) The rectangular plate

29. Joint Advanced Student School St.Petersburg 2006 29 Hence, integrating (6) and passing to dimensionless parameters, we obtain the equation Parameter – the square of the main frequency of the plate eigentones: 3.7 Solution: the first stage of approximation (7) The rectangular plate

30. Joint Advanced Student School St.Petersburg 2006 30 3.7 Solution: the first stage of approximation Bubnov-Galerkin’s Method – the nonlinear differential partial equation of the fourth degree – the nonlinear differential equation in ordinary derivatives of the second degree 2 stage 1 stage  = ? Integration The rectangular plate Thus

31. Joint Advanced Student School St.Petersburg 2006 31 Consider the simply supported plate Let's present temporal function in the form 3.8 Solution: the second stage of approximation from (8) hence The rectangular plate vibration frequency dimensionless amplitude (9) that is ribs are absent

32. Joint Advanced Student School St.Petersburg 2006 32 Let Further integrate Z over period of vibrations We obtain the equation expressing dependence between frequency of nonlinear vibrations ω and amplitude A : 3.8 Solution: the second stage of approximation The rectangular plate

33. Joint Advanced Student School St.Petersburg 2006 33 Define Then 3.8 Solution: the second stage of approximation frequency of nonlinear vibrations frequency of linear vibrations Fig. 4. A skeletal line of the thin type for ideal rectangular plate at nonlinear vibrations of the general form The rectangular plate A

34. Joint Advanced Student School St.Petersburg 2006 34 Now we consider shallow and rectangular in a plane of the shell The main shell curvatures k x, k y are assumed by constants: 3.9 The bicurved shell Fig. 5. The shallow bicurved shell. The bicurved shell where R 1,2 – radiuses of curvature

35. Joint Advanced Student School St.Petersburg 2006 35 The dynamic equations of the nonlinear theory of shallow shells have the form: where the differential functional For full and initial deflections are define by 3.10 The bicurved shell: the problem setup The bicurved shell

36. Joint Advanced Student School St.Petersburg 2006 36 Using the method considered above, we obtain the following ordinary differential equation of shell vibrations Here The square of the main frequency of ideal shell eigentones at small deflections has the form 3.11 The bicurved shell: the problem solution (10) The bicurved shell where

37. Joint Advanced Student School St.Petersburg 2006 37 Here variables , ,  have the form 3.11 The bicurved shell: the problem solution (10) The bicurved shell

38. Joint Advanced Student School St.Petersburg 2006 38 3.11 The bicurved shell: the problem solution Thus we obtain the following equation for definition of an amplitude-frequency characteristic where The bicurved shell Fig. 6. The amplitude-frequency dependences for shallow shells of various curvature A 2 4 6 8 0 1 2 shell at cylindrical shell at plate at

39. Joint Advanced Student School St.Petersburg 2006 39 Applications

40. Joint Advanced Student School St.Petersburg 2006 40 References Ilyin V.P., Karpov V.V., Maslennikov A.M. Numerical methods of a problems solution of building mechanics. – Moscow: ASV; St. Petersburg.: SPSUACE, 2005. Karpov V.V., Ignatyev O.V., Salnikov A.Y. Nonlinear mathematical models of shells deformation of variable thickness and algorithms of their research. – Moscow: ASV; St. Petersburg.: SPSUACE, 2002. Panovko J.G., Gubanova I.I. Stability and vibrations of elastic systems. – Moscow: Nauka. 1987. Volmir A.S. Nonlinear dynamics of plates and shells. – Moscow: Nauka. 1972.