Chapter 7 Determination of Natural Frequencies and Mode shapes

1. 7 Chapter 7Determination of Natural Frequencies and Mode shapes

2. Chapter Outline 7.1 Introduction 7.2 Dunkerley’s Formula 7.3 Rayleigh’s Method 7.4 Holzer’s Method 7.5 Matrix Iteration Method 7.6 Jacobi’s Method 7.7 Standard Eigenvalue Problem

3. 7.1 7.1Introduction

4. 7.1 Introduction • Computing the natural frequencies and modes by solving a nth degree polynomial equation can be tedious • In this chapter we shall consider several other methods: • Dunkerley’s formula • Rayleigh’s method • Holzer’s method • Matrix iteration method • Jacobi’s method

5. 7.2 7.2Dunkerley’s Formula

6. 7.2 Dunkerley’s Formula • It gives the approx. value of the fundamental frequency of a composite system. • Consider the following general n DOF system: • For a lumped mass system with diagonal mass matrix, the equation becomes:

7. 7.2 Dunkerley’s Formula • i.e. • Expanding:

8. 7.2 Dunkerley’s Formula • Let the roots of this equation be 1/ω12, 1/ω22,…, 1/ωn2. Thus • Equating coefficients of (1/ω2)n-1 in (E.1) and (E.2): • In most cases,

9. 7.2 Dunkerley’s Formula • Thus • Can also be written as • where ωin=(1/aiimi)1/2=(kii/mi)1/2

10. 7.2 Dunkerley’s Formula • Example 7.1 • Fundamental Frequency of a BeamEstimate the fundamental natural frequency of a simply supported beam carrying 3 identical equally spaced masses, as shown below.

11. 7.2 Dunkerley’s Formula • Example 7.1 • Fundamental Frequency of a Beam • SolutionWe have • Since m1=m2=m3=m ,

12. 7.3 7.3Rayleigh’s Method

13. 7.3 Rayleigh’s Method • Based on Rayleigh’s Principle • Kinetic and potential energies of an n-DOF discrete system: • Assume harmonic motion to be where is the mode shape and ω is the natural frequency

14. 7.3 Rayleigh’s Method • Maximum KE: • Maximum PE: • For a conservative system, Tmax=Vmax

15. 7.3 Rayleigh’s Method • Properties of Rayleigh’s Quotient • has a stationary value when is in the vicinity of any eigenvector Proof:

16. 7.3 Rayleigh’s Method • Properties of Rayleigh’s Quotient • If normal modes are normalized,

17. 7.3 Rayleigh’s Method • Properties of Rayleigh’s Quotient where 0(ε) is an expression in ε of the 2nd order or higher. • i.e. differs from by a small quantity of the 2nd order. • i.e. Rayleigh’s quotient has a stationary value in the neighborhood of an eigenvector.

18. 7.3 Rayleigh’s Method • Properties of Rayleigh’s Quotient • The stationary value is a minimum value in the neighborhood of • To see this, let r = 1.

19. 7.3 Rayleigh’s Method • Properties of Rayleigh’s Quotient • In general, Rayleigh’s quotient is never lower than the 1st eigenvalue. • Similarly we can show that Rayleigh’s quotient is never higher than the highest eigenvalue.

20. 7.3 Rayleigh’s Method • Computation of Fundamental Natural Frequency • Rayleigh’s quotient can be used to approximate ω1. • Select a trial vector and substitute into • This will yield a good estimate of • The closer resembles the true mode , the more accurate is the estimated ω1.

21. 7.3 Rayleigh’s Method • Example 7.2Fundamental Frequency of a Three-Degree-of-Freedom SystemEstimate the fundamental frequency of vibration of the system as shown. Assume that m1=m2=m3=m, k1=k2=k3=k, and the mode shape is

22. 7.3 Rayleigh’s Method • Example 7.2Fundamental Frequency of a Three-Degree-of-Freedom System SolutionStiffness matrix • Mass matrix • Substitute the assumed • mode shape into

23. 7.3 Rayleigh’s Method • Fundamental Frequency of Beams and Shafts • Static deflection curve is used to approximate the dynamic deflective curve. • Consider a shaft carrying several masses as shown below.

24. 7.3 Rayleigh’s Method • Fundamental Frequency of Beams and Shafts • Potential energy of the system is strain energy of the deflected shaft, which is the work done by the static loads. • For free vibration, max kinetic energy due to the masses is • Equating Vmax and Tmax,

25. 7.4 7.4Holzer’s Method

26. 7.4 Holzer’s Method • A trial-and-error scheme to find natural frequencies of systems • A trial frequency is first assumed, and a solution is found when the constraints are satisfied. • Requires several trials • The method also gives mode shapes

27. 7.4 Holzer’s Method • Torsional Systems • Consider the undamped torsional semidefinite system shown below. • Equations of motion

28. 7.4 Holzer’s Method • Torsional Systems • Since the motion is harmonic, θi=Θicos(ωt+φ) • Summing these equations gives • This states that the sum of the inertia torques of the system must be zero. • The trial freq must satisfy this requirement.

29. 7.4 Holzer’s Method • Torsional Systems • is arbitrarily chosen as 1. • Substitute these values into to see whether the constraints are satisfied. • If not, repeat the process with a new trial value of ω. • These equations can be generalized for a n-disc system as follows:

30. 7.4 Holzer’s Method • Torsional Systems • The graph below plots the torque Mt applied at the last disc against the chosen ω. The natural frequencies are the ω at which Mt=0. • The amplitudes (i=1,2,…,n) are the mode shapes of the system

31. 7.4 Holzer’s Method • Example 7.4Natural Frequencies of a Torsional SystemSolutionThe arrangement of the compressor, turbine and generator in a thermal power plant is shown below. Find the natural frequencies and mode shapes of the system.

32. 7.4 Holzer’s Method • Example 5.4Natural Frequencies of a Torsional SystemSolutionThis is an unrestrained torsional system. • The table below shows its parameters and the sequence of computations.

33. 7.4 Holzer’s Method • Example 5.4Natural Frequencies of a Torsional SystemSolutionMt3 is the torque to the right of the generator, which must be zero at the natural frequencies. • Closely-spaced trial values of ω are used in the vicinity of Mt3=0 to obtain accurate values of the 1st two flexible mode shapes, as shown.

34. 7.4 Holzer’s Method • Spring-Mass Systems • Holzer’s method is also applicable to vibration analysis of spring-mass systems. • Equations of motion: • For harmonic motion, xi(t)=Xicosωt where Xi is the amplitude of mass mi. Thus

35. 7.4 Holzer’s Method • Spring-Mass Systems • The resultant force applied to the last (nth) mass can be computed as follows: • Repeat for several other trial frequencies ω. • Plot a graph of F vs ω. The natural frequencies are those ω that give F=0.

36. 7.5 7.5Matrix Iteration Method

37. 7.5 Matrix Iteration Method • The method assume that the natural frequencies are distinct and well separated. • Procedure • Select a trial vector • Premultiply it by the dynamical matrix [D]. • Normalize the resultant column vector. • Repeat step 2 and 3 until the successive normalized vectors converge.

38. 7.5 Matrix Iteration Method • Proof: • Expansion theorem • is a known vector selected arbitrarily. • are constant vectors because they depend on the system properties. • ci are unknown numbers to be determined. • Premultiplying by [D]:

39. 7.5 Matrix Iteration Method • Proof: • Recall: • Hence • Repeating the process for r iterations: • Since ω1<ω2<…<ωn, if r is large we have:

40. 7.5 Matrix Iteration Method • Proof: • The only significant we have on the RHS is: • Since ω1 can be found by

41. 7.5 Matrix Iteration Method • Discussion: • A finite number of iterations is sufficient to obtain a good estimate of ω1. • Actual no. of iterations depend on how close resembles • Advantage: Computational errors will not yield incorrect results. • The method fails if is exactly proportional to one of the modes

42. 7.5 Matrix Iteration Method • Convergence to the Highest Natural Frequency • To obtain ωn and the corresponding • Select an arbitrary and premultiply by [D]-1 to obtain an improved trial vector • The sequence of trial vectors will converge to the highest normal mode • Constant of proportionality in this case is ω2 instead of 1/ ω2

43. 7.5 Matrix Iteration Method • Computation of Intermediate Natural Frequencies • Once ω1 and is found, we can find the higher natural frequencies. • Because any premultiplied by [D] would lead to the largest eigenvalue, it is necessary to remove the largest eigenvalue from [D]. • Succeeding λi and can be obtained by eliminating the root λ1 from the characteristic equation |[D] – λ[I]|=0

44. 7.5 Matrix Iteration Method • Computation of Intermediate Natural Frequencies Procedure: • To find normalize wrt mass matrix: • Deflated matrix [Di] is constructed as: • Next the iterative scheme is used, where is an arbitrary trial eigenvector.

45. 7.5 Matrix Iteration Method • Example 7.5 • Natural Frequencies of a Three-Degree-of-Freedom System • Find the natural frequencies and mode shapes of the system as shown for k1=k2=k3=k and m1=m2=m3=m by the matrix iteration method.

46. 7.5 Matrix Iteration Method • Example 7.5 • Natural Frequencies of a Three-Degree-of-Freedom System • Solution • Flexibility matrix [a]=[k]-1= • Dynamical matrix is • Eigenvalue problem:

47. 7.5 Matrix Iteration Method • Example 7.5 • Natural Frequencies of a Three-Degree-of-Freedom System • Solution 1st natural frequency: • Assume , hence • By making the first element equal to unity we obtain

48. 7.5 Matrix Iteration Method • Example 7.5 • Natural Frequencies of a Three-Degree-of-Freedom System • Solution 1st natural frequency: • Subsequent trial eigenvector can be obtained from • Corresponding eigenvalues are given by where is the 1st component of before normalization.

49. 7.5 Matrix Iteration Method • Example 7.5 • Natural Frequencies of a Three-Degree-of-Freedom System • Solution • The various λi and are shown: • The mode shape and natural • frequency converged in • 8 iterations.

50. 7.5 Matrix Iteration Method • Example 7.5 • Natural Frequencies of a Three-Degree-of-Freedom System • Solution 2nd natural frequency: • Deflated matrix • Let the normalized vector • where α must be such that