Static Analysis: Natural Frequency Analysis

1. Static Analysis: Natural Frequency Analysis

2. Objectives Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 2 The objective of this module is to develop the equations and numerical methods used to compute the natural frequencies and mode shapes of a linear system. • This will be accomplished by specializing the incremental equations developed for general static analysis to linear systems. • The equations for linear static systems will then be extended to include inertia and viscous damping terms. • The free vibration problem will then be used to reveal the eigenvalue problem used to compute the natural frequencies and mode shapes. • Numerical methods for efficiently computing the natural frequencies and mode shapes will then be presented.

3. Incremental Equations for Static Analysis Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 3 • The governing equations used to compute the displacement history of a static system were developed in Modules 1 through 4. These equations are • There is no need to break the solution into small increments if the system is linear. The above equation reduces to the following when there is only one increment

4. Equations of Motion Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 4 • Inertia and viscous damping effects can be added to yield the equation • This equation of equilibrium governs the motion of any linear system undergoing time dependent motion. • Because these equations control the system response they are called the Equations of Motion.

5. Undamped Natural Frequencies Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 5 • The undamped natural frequencies and mode shapes are obtained from the equations of motion by setting the damping and the forcing function to zero. • We seek solutions of the form where describes the deformed shape and defines the magnitude as a function of time.

6. Eigenvalue Problem Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 6 • This is a system of homogeneous equations because the right hand side of the equations are zero. • can only be determined when the determinant is equal to zero. • There are specific values of w, called the natural frequencies, that make the determinant zero. • Once the natural frequencies are determined, the array , called the mode shape, can be found for each natural frequency. • The determinant will yield a polynomial with w as the unknown parameter. • The polynomial will be the same order as the number of equations.

7. Important Parameters Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 7 • The natural frequencies and mode shapes of a system are important parameters. • The natural frequencies describe the frequencies at which the system wants to vibrate. • The mode shape describes the shape the system takes at these preferred frequencies. • The mode shapes form a basis for the system and all solutions to the equations of motion can be written as a linear combination of them. or

8. Constant Stiffness Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 8 • Natural frequencies and mode shapes are computed using the stiffness matrix at a particular instance. • If the stiffness matrix is linear or constant at all instances then the natural frequencies and mode shapes are constants. • In a stressed structure, the stiffness matrix is typically based on the stress at the end of a static solution that is used to compute the stresses. • Natural frequencies and mode shapes are not computed for problems that have intermittent contact, gaps, etc. • When these non-linear effects are present, the natural frequencies and mode shapes are not constants but change as each gap opens or closes.

9. Mass Orthonormal Section II – Static Analysis Module 5 - Natural Frequency Analysis Page9 • The mode shapes can only be determined to within a constant. • They describe the shape that the structure takes while vibrating at a particular natural frequency, but don’t provide the magnitude of the deformation. • It is convenient to scale the mode shapes such that the third equation is satisfied. • This leads to the last equation.

10. Example Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 10 • Simulation was used to compute the first five natural frequencies and corresponding mode shapes of the thin cantilevered beam. 1 inch wide x 12 inch long x 1/8 inch thick. Material - 6061-T6 aluminum. Brick elements with mid-side nodes are used to improve bending accuracy through the thin section. 0.0625 inch element size. Fixed end boundary condition

11. Example – Computed Results Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 11 Mode 1, 28 Hz, 1st bending mode about weak axis Mode 2, 175 Hz, 2nd bending mode about weak axis

12. Example – Computed Results Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 12 Mode 3, 222 Hz, 1st bending mode about the stiff axis Mode 4, 592 Hz, 3rd bending mode about the weak axis

13. Example – Computed Results Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 13 Mode 5, 618 Hz, 1st torsional mode • The shape of each mode is clearly seen in the preceding plots. • Remember that a mode shape represents only the shape that the beam takes as it vibrates. • The mode shape is not the deformation due to an external disturbance.

14. Stressed Systems Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 14 • The natural frequencies and associated mode shapes are sometimes sensitive to the stresses in the system. • The frequency at which a guitar string vibrates can be changed by increasing or decreasing the tension in the string. • In this type of system, the linear stiffness matrix must be augmented with the stress stiffness matrix derived in Module 3. Linear System Stressed System

15. Eigenvalue Solution Methods Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 15 • It is not practical to solve large eigenvalue problems using methods that find the determinant. • A variety of numerical methods have been developed to find the natural frequencies and mode shapes of large systems. • All methods for computing natural frequencies and mode shapes are iterative. Eigenvalue Solution Methods Vector Iteration Methods Transformation Methods Polynomial Iteration Methods Characteristic Polynomial Methods

16. Inverse Iteration Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 16 • Inverse iteration is a vector iteration method. • It is often part of a more advanced method such as Sub-space iteration. • It begins by making an initial estimate for the mode shape and then performing a series of iterations to improve the estimate. • Once the mode shape has converged, the natural frequency is found from Assume Solve for and then make orthogonal to the mass matrix

17. Frequency Shifts Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 17 • Frequency shifts are often used by numerical algorithms to: • extract frequencies in a specific range, • improve poorly conditioned matrices encountered during the solution process, or • improve convergence rate. • When a shift is used, the standard equation becomes • The natural frequencies of the shifted and original systems are related by the equation where s is a shift frequency.

18. Simulation Analysis Parameters Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 18 Number of frequencies and modes to be computed. Used as a search range by the numerical algorithm Check if structure is not constrained. Rigid body modes require special handling in numerical algorithm.

19. Simulation Analysis Parameters Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 19 Lanczos based algorithm that uses multiple processors commonly found on workstations. Percent of RAM used to read the element data and to assemble the matrices. Sparse solver requires less than the Sub-space Iteration solver. Alternate solver Sub-Space Iteration uses inverse iteration and Sturm-sequence properties of characteristic equation. Recommended only for small problems. Use all processors during solution phase  fastest approach. Using a subset of the available processors will let you work on other things while the problem is running.

20. Simulation Analysis Parameters Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 20 If zero, the number of Lanczos vectors used by the Sparse solver is two times the number of frequencies being computed. Used to improve the solution if rigid body modes are included or modes in a particular range are needed. This box needs to be checked if you are not going to use the results in a subsequent analysis (modal superposition). If checked, solver will correct any negative diagonals encountered. Negative diagonals can occur when rigid body modes are included.

21. Module Summary Section II – Static Analysis Module 5 - Natural Frequency Analysis Page 21 • All linear dynamic systems encountered in solid mechanics have natural frequencies at which they prefer to vibrate. • Each natural frequency has a corresponding mode shape that the system takes when vibrating at that natural frequency. • The natural frequencies and corresponding mode shapes are found by solving an eigenvalue problem. • All possible solutions to a forced response problem can be written in terms of the mode shapes. • This property will be used in Module 6: Modal Superposition.