1 / 14

Vibration

Vibration. Eigen Value and Eigen Vector Problem in free vibration 1D-Bar Element. Eigen Value Analysis or Modal Analysis. Prof.R.N.Dehankar Anjuman CET. CO802.3.

arthurreyes
Download Presentation

Vibration

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Vibration Eigen Value and Eigen Vector Problem in free vibration 1D-Bar Element Eigen Value Analysis or Modal Analysis Prof.R.N.Dehankar Anjuman CET

  2. CO802.3 • Solve the basic finite elements formulation for static and dynamic conditions for evaluation of frequency response problems.(L1,2,3)

  3. Eigen Value Analysis or Modal Analysis (Free Vibration) In many engineering applications, the natural frequencies of vibration are of interest. This is probably the most common type of dynamic analysis and is referred to as an ‘eigenvalue analysis’. In addition to the frequencies, the mode shapes of vibration which arise at the natural frequencies are also of interest. These are the undamped free vibration responseof the structure caused by an initial disturbance from the static equilibrium position. This solution derives from the general equation of motion by zeroing the damping and applied force terms. Thereafter, it is assumed that each node is subjected to a sinusoidal functions of the peak amplitude for that node. *Displacement Function (vector) *The general equation of motion for multiple degrees of freedom is; Mass Matrix Damping Matrix Stiffness Matrix

  4. For a 1D bar member in state of free vibration the damping and applied force is considered to be zero, which would yield the “Eigent Value Equation” Where; Φ: Eigen Vector ω: Circular Natural Frequency λ: Eigen Value Where, i=1,2,3…. (mode shapes)

  5. The total number of eigenvalues (λ) or natural frequencies is equal to the total number of degrees of freedom in the model. Each eigenvalue (λ) or frequency has a corresponding eigenvector (ω) or mode shape. Since each of the eigenvectors (ω) cannot be null vectors, the equation which must be solved to get the eigenvalues (λ) is; * Natural Frequency of bar member * Siffness Matrix * Mass Matrix

  6. Determine the Eigen Value and Eigen Vector for the step bar as shown in the fig. A1=500mm2, A2=250mm2, E=200GPa,ρ=7.8x10-6kg/mm3 A1 A2 1000mm 600mm K1 K2 2 1 3 U3 U2 U1

  7. *Step2: Element Stiffness Matrix *Element 2: *Element 1: 1 2 2 3 *Step3: Global Stiffness Matrix 1 2 3

  8. *Step4: Element Mass Matrix *Element 2: *Element 1: 1 2 2 3 *Step5: Global Mass Matrix 1 2 3

  9. *Step6: Using the Eigen Value Equation A Applying the B.C; U1=0 B To evaluate the Eigen Values (λ) consider;

  10. Eigen Values *Step7: Calculating the Eigen Vectors CASE I: When λ=1.83x105 (rad/s)2

  11. i.e Therefore; Normalizing the problem to get a definite solution (eigen vector); When λ=1.83x105 (rad/s)2

  12. CASE II: When λ=0.307x105 (rad/s)2 i.e Normalizing the problem to get a definite solution (eigen vector);

  13. When λ=0.307x105 (rad/s)2 Natural Frequency of bar member

  14. Mode Shape U At f1=68.08Hz 0.325 3 2 1 -1.123 At f2=27.88Hz U 0.678 0.538 3 2 1

More Related