Mathematical aspects of mechanical systems eigentones

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

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. Intro 1.1 History • History of development ofthe 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

5. Intro 1.2 Problems • Today’s problems of the linear vibration theory: • Vibration problems of mechanical systems • How correctly to choose degrees of freedom? • How correctly to define external influences? Choice of thecalculated scheme Linear statement Nonlinear statement

6. Intro 1.3 Solution • 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 allow making successive approximations allow making any approximations

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

8. Rod systems 2.1Forces 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

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

10. Rod systems 2.2 Example: the problem setup • Define frequencies and forms of the free vibrations of a statically indeterminate framewith two concentrated masses т1 = 2т, т2 = тand 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

11. Rod systems 2.3Example: 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

12. Rod systems 2.3Example: the problem solution • Calculation of displacements: evaluation of integrals on the Vereschagin's Method • Then we construct the stiffness matrix

13. Rod systems 2.3Example: the problem solution • The masses matrix has the form (at т1 = 2т, т2 = т): • To find eigenvalues and eigenvectors of the matrix D = BM we compute the determinant:

14. Rod systems 2.3Example: the problem solution • 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, X1 = 1. From the first equation we find Х2for each value ofλj: with roots

15. Rod systems 1 -0,359 1 5,569 2.3Example: the problem solution • Solving each equations separately, we find eigenvectors ν1 and ν2: • Then we obtain forms of the free vibrations Fig. 1, d. The main forms of the free vibrations

16. PART III Eigentones of plates and shells

17. Plates and shells 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 fromequilibrium position • velocities of these elements in an initial instant

18. Plates and shells 3.1 Properties of eigentones • Properties of nonlinear eigentones: • Deflections are comparable to thickness of a plate: Rigid plates / shellsFlexibleplates / shells • Frequency depends on vibration amplitude transform A A Fig. 2. Possible of dependence between the characteristic deflection and nonlinear eigentones frequency Skeletal line 1 1   a) Thin system b) Soft system

19. Plates and shells 3.2Solution of nonlinear problems System with infinite number degrees of freedom System with one degree of freedom Approximation

20. The rectangular plate 3.3The 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) 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. The rectangular plate 3.4Solution of the linear problem • Approximation of the deflection on the Kantorovich's Method: • Substituting the equation (2) instead of function f(t): Integration some temporal function where

22. The rectangular plate 3.4Solution of the linear problem • The square of eigentones frequency at small deflections has form: Fig. 3. Character of wave formation of the rectangular plate at vibrations of the different form where 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. The rectangular plate 3.5The rectangular plate, fixed at edges: a nonlinear problem • 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 kx= ky=0: where a stress function Equilibrium equation (3) the main shell curvatures Deformation equation (4) differential functional

24. The rectangular plate 3.6Solution of the nonlinear problem • 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 Fx and Fy – section areas of ribs in a direction of axes x and y (5) where

25. The rectangular plate 3.6Solution of the nonlinear problem • Then the solution of a homogeneous equation will have the form: • Finally where the stresses applied to the plate through boundary ribs (they are consideredas positive at a tensioning)

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

27. The rectangular plate 3.7Solution: the first stage of approximation • On the Bubnov-Galerkin’s Method we write outn equations of type • In our solution η1 has the form (6)

28. The rectangular plate 3.7Solution: the first stage of approximation • Hence, integrating (6) and passing to dimensionless parameters, we obtain the equation where the dimensionless parametersK and ζ have the form (7) (8)

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

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

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

32. The rectangular plate 3.8Solution: the second stage of approximation • Let • Further integrateZover period of vibrations • We obtain the equation expressing dependence between frequency of nonlinear vibrations ω and amplitudeA:

33. The rectangular plate 3.8Solution: the second stage of approximation frequency of nonlinear vibrations frequency of linear vibrations • Define • Then A Fig. 4. A skeletal line of the thin type for ideal rectangular plate at nonlinear vibrations of the general form 

34. The bicurved shell 3.9The bicurved shell • Now we consider shallow and rectangular in a plane of the shell • The main shell curvatureskx, ky are assumed by constants: Fig. 5. The shallow bicurved shell. where R1,2 – radiuses of curvature

35. The bicurved shell 3.10The bicurved shell: the problem setup • 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

36. The bicurved shell 3.11The bicurved shell: the problem solution • 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 (10) where

37. The bicurved shell 3.11The bicurved shell: the problem solution Here variables , ,  have the form (10)

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

39. Applications

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.