1SE382: Engineering Vibrations Lecture 14: Multi-Degree of Freedom System(2):Mode ShapeDr. YongSeob LimSchool of Undergraduate Studies | School of Transdisciplinary StudiesDaegu Gyeongbuk Institute of Science and Technology (DGIST)2018, Spring
2Last Lecture…Multiple Degree of Freedom Systems (1)2-DOF system First!!How to obtain the solution to the 2-DOF.
3Today’s Lecture:Multiple Degree of Freedom Systems (2)Mode ShapeDerivation of Modal Matrix. For What?
4The Solution of 2-DoF System (Review)Thus, the solution to the algebraic matrix eq. is:2-MKu012,=, and ??uwhereuu131,21133,422, has mode shape 12, has mode shape 1 uu( )jttexuQuestions:What is the Mode Shape!!
5Mode Shape of 2-DOF SystemQuestions:Then, how to calculate amplitude and phase of vibration?
6Mode Shape of 2-DOF System (cont’d) Return now to the time response:We have computed four solutions:11221122( ), , ,jtjtjtjtteeeexuuuu( )jttexuSince linear, we can combine as:1212where, ,,, and are constants determined by initial conditions.AA1122112211221211112222( )( )sin()sin()jtjtjtjtjtjtjtjttaebecedetaebecedeAtAtxuuuuxuuuu
7Mode 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 ω1and ω2.The relative magnitude of each sine term, and hence of the magnitude of oscillation of m1and m2is determined by the value of A1u1and A2u2.The vectors u1and u2are called mode shapesbecause the describe the relative magnitude of oscillation between the two masses.11112222( )sin()sin()tAtAtxuu
8Mode Shape of 2-DOF System (cont’d) Then, How to obtain the Specific Mode Shapes? First note that A1, A2, Φ1and Φ2are determined by the initial conditions.Choose them so that A2= Φ1= Φ2= 0. Then,11112222( )sin()sin()tAtAtxuux(t)x1(t)x2(t)A1u11u12sin1tA1u1sin1tThus, each mass oscillates at (one) frequency ω1with magnitudes proportional to u1the 1stMode Shape.