1. ME 440 Intermediate Vibrations Th, March 26, 2009 Chapter 5: Vibration of 2DOF Systems © Dan Negrut, 2009 ME440, UW-Madison

2. Before we get started… Last Time: Response to an arbitrary excitation: The total solution Dynamic Load Factor Response Spectrum Today: HW Assigned (due April 2) 5.1: Assume zero initial velocities 5.4: Assume small oscillations Material Covered: Start Chapter 5: 2DOF systems Next Time: Anonymously, please print out on a sheet of paper Two things that you disliked the most about the class What you would do to improve this class (if you were teaching ME440) 2

3. Two Degree of Freedom Systems Number of Degrees of Freedom The number of generalized coordinates necessary to completely describe the motion of a system So far, we discussed one degree of freedom systems Recall that we had one natural frequency Rule: for a n-degree of freedom system, one has n natural frequencies Associated with each natural frequency, there is a natural mode of vibration The natural vibration modes turn out to be orthogonal (a concept a bit ahead of its time…) 3

4. Multiple DOF Systems Probably the most important thing when trying to derive the equations of motion associated with a mechanical system is this: Make sure you understand how many degrees of freedom you have You will have as many differential equations as many degrees of freedom you have At left, the system has two degrees of freedom Choose  1 and  2 as the 2 DOFs Then x 1, y 1, x 2, y 2 are not independent gen. coordinates, they’re derived based on  1 and  2 Note that you could select y 1 and y 2 to be the independent generalized coordinates Then  1 and  2 become dependent coordinates 4

5. [Text] Example Derive EOMs for system below 5

6. Matrix Notation Serves two purposes Most importantly, it brings sanity to the process of formulating the equations of motion for large systems It clearly shows the parallels that exist between the single and multiple DOF system in relation to their EOMs 6 In compact form, this equation assumes the form Observe the following notation convention: Matrices are in square brackets Vectors are in curly brackets

7. Matrix Notation: Nomenclature Mass Matrix (symmetric!) 7 Damping Matrix (symmetric!) Stiffness Matrix (symmetric!) Displacement Vector Force Vector

8. Final Remarks, Matrix Notation The Mass Matrix Is symmetric Is typically diagonal It is not diagonal if there is dynamic coupling between the generalized coordinates This is the case for instance in Finite Element Analysis The Damping and Stiffness matrices Are symmetric – a consequence of Newton’s Third Law Most of the time, they are not diagonal Note: If [m], [c], and [k] are diagonal, we say that the equations of motion are independent (they are decoupled) 8

9. [Quick Review] Matrix Algebra Definition: A matrix A is singular if its determinant is zero Examples: 9

10. [Quick Review] Inverse of a matrix A Result from Linear Algebra that we rely on heavily: A matrix A has an inverse (denoted by A -1 ) if, and only if, A is not a singular matrix (that is, its determinant is not zero) If A is nonsingular, that is, A -1 exists, then the solution of the linear system A a =b is simply a = A -1 b 10

11. [Quick Review] Dealing with a singular matrix Nomenclature For the linear system Aa=0, the vector a is called a nontrivial solution if a satisfies the equation Aa=0 but a is not zero Example: NOTE: If A is nonsingular, then you cannot find a nontrivial solution a for the problem Aa=0. In other words, to find a nontrivial solution, the matrix A should be singular: det(A)=0 11

12. [Quick Review] On the number of trivial solutions As indicated, you can start looking for a nontrivial solution provided the matrix A is singular Important observation: If a is a trivial solution, then so is 1.23a, 32.908a, -2.128a, etc. For any real number , if a is a nontrivial solution, then so is  a In other words, as soon as you find one trivial solution, you have as many of them you wish So you either don’t have any nontrivial solution at all, or have an infinite number of them Example of trivial solutions for Aa=0 12

13. [Quick Review] Finding a nontrivial solution Example: Find a nontrivial solution for Aa=0, given that 13

14. Free Vibration of Undamped Systems System at left leads to the following EOM 14 Assume a solution of the form: Use the same old trick: substitute back into the EOM and see what conditions A 1, A 2, , and  must satisfy so that x 1 (t) and x 2 (t) verify the EOM

15. [Cntd.] Free Vibration of Undamped Systems Substituting back leads to the following relationship between A 1 and A 2 15 In matrix form: This linear system has a nontrivial solution only if determinant of matrix is zero IMPORTANT: This condition represents the characteristic equation (CE) associated with our 2DOF system Recall from ME340: CE is the equation that provides the natural frequency of system

16. [Cntd.] Free Vibration of Undamped Systems Characteristic Equation: 16 Characteristic Equation, after evaluating the determinant… Two real solutions that lead to two natural frequencies  1 and  2 :

17. [Cntd.] Free Vibration of Undamped Systems A word on notation Why do I have m 1 and  n(1) ? What’s the deal with the parentheses there? I want to emphasize the fact that the “1” in  n(1) doesn’t have anything to do with the “1” in m 1 Rather,  n(1) is a quantity that refers to the *entire* system Specifically, it indicates one of the natural frequencies of the *entire* system It shows how both m 1 *and* m 2 move together in the first vibration mode Note that  n(2) is the other natural frequency at which the bodies move *together* if left alone (free response) 17

18. Recall the meaning of  n(1) and  n(2) : Those values that zero out the determinant of the linear system (1) & (2) Consequently the linear system has an infinite number of solutions A 1 and A 2 What is unique though, it’s the ration between A 2 /A 1 To find the first ratio (associated with  n(1) ), plug back value of  n(1) to obtain 18 [Cntd.] Free Vibration of Undamped Systems To find second ratio (associated with  n(2) ), plug back value of  n(2) to obtain

19. Remember what we assumed: 19 [Cntd.] Free Vibration of Undamped Systems In Matrix/Vector notation: NOTE: At this point,  n(1) and r (1) are known, but not A 1 (1) and  (1) Similarly, for the second natural frequency,

20. 20 [Cntd.] Free Vibration of Undamped Systems Notation – we call modal vectors the following quantities: Solution can be expressed now as Use ICs (two positions and two velocities) to find the following four unknowns:

21. 21 [Short Detour: Notation, FEM related] Free Vibration of Undamped Systems Using again Matrix/Vector notation… Introduce the modal matrix, which is made up of the modal vectors Important: the matrix [u] is constant (doesn’t change, an attribute of the m-k system!!) Then, the solution can be expressed in matrix-vector notation (looks very similar to what you have in FEM) Also, define {f(t)} as

22. Example, 2DOF: m 1 =1kg, m 2 =2kg k 1 =9N/m k 2 =k 3 =18N/m Find modal vectors {u} (1), {u} (2) Find mode ratios Find natural frequencies Find x(t) 22