Lecture 14_Multi-Degree of Freedom System(2)-Mode Shape_180503.pdf - SE382 Engineering Vibrations 2018 Spring Lecture 14 Multi-Degree of Freedom

Lecture 14_Multi-Degree of Freedom System(2)-Mode Shape_180503.pdf

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1 SE382: Engineering Vibrations Lecture 14: Multi-Degree of Freedom System(2): Mode Shape Dr. YongSeob Lim School of Undergraduate Studies | School of Transdisciplinary Studies Daegu Gyeongbuk Institute of Science and Technology (DGIST) 2018, Spring
2 Last Lecture… Multiple Degree of Freedom Systems (1) 2-DOF system First!! How to obtain the solution to the 2-DOF.
3 Today’s Lecture: Multiple Degree of Freedom Systems (2) Mode Shape Derivation of Modal Matrix. For What?
4 The Solution of 2-DoF System ( Review ) Thus, the solution to the algebraic matrix eq. is: 2 - M K u 0 1 2 , = , and ?? u where u u 1 3 1,2 1 1 3 3,4 2 2, has mode shape 1 2, has mode shape 1     u u ( ) j t t e x u Questions: What is the Mode Shape !!
5 Mode Shape of 2-DOF System Questions: Then, how to calculate amplitude and phase of vibration ?
6 Mode Shape of 2-DOF System (cont’d) Return now to the time response: We have computed four solutions: 1 1 2 2 1 1 2 2 ( ) , , , j t j t j t j t t e e e e x u u u u ( ) j t t e x u Since linear, we can combine as: 1 2 1 2 where, , , , and are constants determined by initial c o nd itions. A A 1 1 2 2 1 1 2 2 1 1 2 2 1 2 1 1 1 1 2 2 2 2 ( ) ( ) sin( ) sin( ) j t j t j t j t j t j t j t j t t a e b e c e d e t ae be ce de A t A t x u u u u x u u u u
7 Mode Shape of 2-DOF System (cont’d) What Physical Interpretation of all that Math ? Each of the TWO masses is oscillating at TWO natural frequencies ω 1 and ω 2 . The relative magnitude of each sine term, and hence of the magnitude of oscillation of m 1 and m 2 is determined by the value of A 1 u 1 and A 2 u 2 . The vectors u 1 and u 2 are called mode shapes because the describe the relative magnitude of oscillation between the two masses. 1 1 1 1 2 2 2 2 ( ) sin( ) sin( ) t A t A t x u u
8 Mode Shape of 2-DOF System (cont’d) Then, How to obtain the Specific Mode Shapes ? First note that A 1 , A 2 , Φ 1 and Φ 2 are determined by the initial conditions . Choose them so that A 2 = Φ 1 = Φ 2 = 0 . Then, 1 1 1 1 2 2 2 2 ( ) sin( ) sin( ) t A t A t x u u x ( t ) x 1 ( t ) x 2 ( t ) A 1 u 11 u 12 sin 1 t A 1 u 1 sin 1 t Thus, each mass oscillates at (one) frequency ω 1 with magnitudes proportional to u 1 the 1 st Mode Shape .
9 Mode Shape of 2-DOF System (cont’d)