CHAPTER 13 MULTIDEGREE OF FREEDOM …

1 CHAPTER 13 MULTIDEGREE OF FREEDOM SYSTEMS A multi degrees of FREEDOM (dof) system is one, which requires two or more coordinates to describe its motion. These coordinates are called generalized coordinates when they are independent of each other and equal in number to the degrees of FREEDOM of the system . The N dof system differs from the single dof system in that it has N natural frequencies, and for each of the natural frequencies there corresponds a natural state of vibration with a displacement configuration known as the normal mode. Mathematical terms associated with these quantities are eigenvalues and eigenvectors. Normal mode vibrations are free vibrations that depend only on the mass and stiffness of the system and how they are distributed. Analytical/closed-form solutions can be established for 2degrees of FREEDOM systems. But for more degree of FREEDOM systems numerical analysis using computer is required to find natural frequencies (eigen-values) and mode shapes (eigen vectors).

2 The equations of motion for these systems in matrix form can be written as (1) 0, MxKx+= where and MK are mass and stiffness matrix respectfully. Premultiplying 1M , the above equation becomes, 0 IxAx+= (2) where . 1 AMK =Assuming harmonic motion , sinxXt =, equation (2) reduces to []AIX0 =, (3) where 2 = For non-trivial solution of X, equation (3) reduces to 0AI =which is known as the characteristic equation. One may find i (i= 1,2,..n), by finding the roots of this characteristic equation or by finding the eigenvalues of A. The eigenvector will correspond to the normal modes of the system .

3 For an n degree of FREEDOM system one will get n natural frequencies (square root of eigen values) and n normal modes (eigen vectors). It is possible to find eigenvectors from adjacent matrix of the system : 232 Let BAI = (4) 11 BadjBB = (5) Premultiplying equation (5) by BB 1 BBBBBadjBB = (6) or, B IBadjB= (7) Hence, ()()AIIAIadjAI = (8) ,0iforAI = =, hence, [][]0 AIadjAI = (9) Comparing equation (3) and (9), one gets []iXadjAI = (10) Hence the normal modes of the system can be obtained by finding the adjoint of[]AI . Example 1 Find the normal modes for torsional vibration of a shaft with two rotors as shown in figure 1.

4 I2 I1 Figure 1: A rotating shaft with two rotors. Solution Let 1 and 2 be the rotation of rotor 1 and 2 respectively. The equation of motion of the system can be given by 2331112220000 IkkIkk += Let us assume 1122sinXtX = So, 11200 IkkAIkk = Hence, 1122 -- kkIIkkII []1122 -- kkIIAIkkII = and solving0AI = one will get 120kkII += Hence 120, or,kkII ==+. So the system is a degenerate system as discussed in the previous CHAPTER . Corresponding to 0, = , for rigid body motion, one may find the normal mode X. []11202221-- kkkk1 IIIIXadjAIkkkkIIII = = == One may note that, the normalized value of each column yield the same value and in this case it is equal to the expected value of 1211XX = , both the rotor will rotate same amount giving rising to rigid-body motion.

5 Now to find the normal mode for, 12kkII =+, one may findAdjAI which is equal to 234 '21121121122 - - - - -kkkkkk2 IIIIIIA djkkkkkkIIIIII == Hence,1122//XkIXkI = To normalize the value, taking 21X= 1212/1 XIIX = Hence both the rotor rotate in opposite directions and their amplitude ratio is inversely proportional to their inertia ratio. Properties of Vibrating Systems: Since the elastic behavior of motion may be expressed in terms of the stiffness or flexibility, the equations of motion may be formulated by either the stiffness matrix []k or the flexibility matrix []a. -In stiffness formulation , the force is expressed in terms of displacement by fx{}[]{}fkx=. (11) Also one may write {}[]{}[]{}1xkfaf == (12) which leads to the flexibility approach.

6 -The choice as to which approach one should adopt depends on the problem. Some problems are more easily pursued as the basis of stiffness, whereas for others the flexibility approach may be desirable. Flexibility matrix: For a three degree-of- FREEDOM system , the displacement as forces are related by flexibility matrix as 11121311221222333313233 aaa2xfxaaafxfaaa = (13) 235 The flexibility influence coefficient is defined as the displacement at i due to unit force applied at j with all other forces equal to zero. Thus the first column represent displacement corresponding to ija1231,0 .fff=== Similarly, second column represents the displacements for 2131 and 0 and so Example 2: Determine the flexibility matrix for the axial displacement of the spring system shown in figure 3.

7 To find , one has to apply unit force at 1, and no force at 2. 1121 and aaa So, and as the second spring will have a rigid body motion. 11a=11/K21a=11/KSimilarly to find , one has to apply only unit force at spring 1 and 2 1222 and a1 1K2K2 Are in series, displacement at 1 , 12a=11/K and 221212/()aKKKK=+ Figure 3 Hence flexibility matrix can be given by 11112121/1/1/() /()KKKKKKK + Example 2: Determine the flexibility influence coefficient for the transverse vibration of a cantilever beam with three equal mass placed at equal interval as shown in figure 4. The flexibility of the rod is EI. l l l 3 2 1 Figure 4 Solution In this the flexibility influence coefficients can be obtained by finding the required displacement, which can easily be obtained by using moment area method. To find the first column of the flexibility matrix, one should apply only unit force at 1. From the bending moment diagram ABC, shown in figure 5(a) 23631132133121/2(3).

8 (3)..393 (=area moment of the ABC/EI about 1)12(2 .. )(. )1423= (= area moment of the trapesum 2 BCE/EI about 2)312(2.)/2(..)4233llllaEIEI lllllllaEIEI lllllllaEI==+=+==+ (= area moment of the trapesum 21 BCD/EI about 3)EI 3l D E C B 31 2 l A l l 1 2 3B C D 2l l l l (c) (b) 3B C l l 2 1 l l (a) Figure 5: Bending moment diagram considering unit load at point (a) 1, (b) 2 , (c) 3 Now to find the second column of the flexibility matrix, one should apply unit force at 2 and draw the corresponding bending moment diagram as shown in Figure 5 (b). From this figure 31232233212(2 .2)(.2)1423 (area moment of the 2about 1)312( ).(2)823(area moment of the 2about 2)312( ).(2)823(area moment of the 3 BCD about 3)3lllllaBEIEI llllaBEIEI llllaEIEI+=========++CC Now to find the third column of the flexibility matrix, one should apply unit force at 3 and draw the corresponding bending moment diagram as shown in Figure 5 (c). From this figure 33112(.)

9 (2)4233lllllaEIEI+== 23733233312(.)()523612(.)233lllllaEIEI llllaEIEI+==== Hence, the flexibility matrix can be written as 327 14 414 8 1laEI = In this example it may be observed that ijjiaa= which is known as reciprocity theorem, which is proved as below. Reciprocity theorem: States that in a linear system ijjiaa= Proof: Consider a linear system and now applying force if, the work done 12force displacement =211()22iiiiiiffaf a= Then applying force jf, the work done =212jjjfa However due to application of force jf, i undergoes further displacement, and the additional work done by ijjafifbecomes . ()ijjiaffSo, total work done 221122iiijjjijjifafaaf=++f (14) Now if one reverses the order application of forces, , first a force jfacts at j followed by a force ifacting at i, the work done will be 221122jjjiiijiijfafaaf=++f (15) Since the work done in the two cases must be equal hence, (16) ijjiaa=Stiffness matrix: For a three dof system , the force and displacements are related by stiffness matrix as 238 11121311221222333313233 kkk2fxfkkkxfxkkk = (17) The stiffnessis defined, as the force required at point i to have unit displacement at point j, displacement at other places being zero.

10 So are the forces required at points 1,2,3 respectively to have unit displacement at 1, , ijk112131, and kkk1231,0xxx===. Example 3: Find stiffness matrix for the spring-mass system shown in Figure 6. 1m 2m 1k 2k 3k 1x4k 3m3x 2x Figure 6 Solution: From the definition of the element of stiffness matrix, the first column can be obtained by finding the forces at mass 1,2, and 3 respectively to have unit displacement for mass 1 and zero displacement for mass 2 and 3, , which is depicted in figure 7. 1(1,2,3iKi=)1221,0xxx=== Figure 7: Freebody diagram considering 1231,0xxx=== From the freebody diagram figure 7(a), to have the above-mentioned displacements, mass 1 will be subjected to a spring force of 1KK2+. Hence to overcome this a force 112fKK=+ is required at 1. Similarly from Figure 7(b) it can be observed that a force 22fK= (negative sign indicate the force to be applied to the left) and from figure 7(c) it may be noted that no force is required at mass 3 ,30f=.