EFFECTIVE MODA MASS & MODA PARTICIPATION FACTORS Revision I B To Irvine Eail: to@vibrationdata.co Deceber, 5 Introduction The effective odal ass provides a ethod for judging the significance of a vibration ode. Modes with relativel high effective asses can be readil ecited b base ecitation. On the other hand, odes with low effective asses cannot be readil ecited in this anner. Consider a odal transient or frequenc response function analsis via the finite eleent ethod. Also consider that the sste is a ulti-degree-of-freedo sste. For brevit, onl a liited nuber of odes should be included in the analsis. How an odes should be included in the analsis? Perhaps the nuber should be enough so that the total effective odal ass of the odel is at least 9% of the actual ass. Definitions The equation definitions in this section are taen fro Reference. Consider a discrete dnaic sste governed b the following equation where M K M K F () is the ass atri is the stiffness atri is the acceleration vector is the displaceent vector F is the forcing function or base ecitation function
A solution to the hoogeneous for of equation () can be found in ters of eigenvalues and eigenvectors. The eigenvectors represent vibration odes. et be the eigenvector atri. The sste s generalized ass atri ˆ is given b ˆ T M () et r be the influence vector which represents the displaceents of the asses resulting fro static application of a unit ground displaceent. The influence vector induces a rigid bod otion in all odes. Define a coefficient vector as T M r (3) The odal participation factor atri i for ode i is The effective odal ass i i (4) ˆ ii eff, i for ode i is i eff, i (5) ˆ i i Note that ˆ ii = for each inde if the eigenvectors have been noralized with respect to the ass atri. Furtherore, the off-diagonal odal ass ( ˆ, i j ) ters are zero regardless of the i j noralization and even if the phsical ass atri M has distributed ass. This is due to the orthogonalit of the eigenvectors. The off-diagonal odal ass ters do not appear in equation (5), however. An eaple for a sste with distributed ass is shown in Appendi F.
Eaple Consider the two-degree-of-freedo sste shown in Figure, with the paraeters shown in Table. 3 Figure. Table. Paraeters Variable 3 Value. g. g N/ N/ 3 N/ The hoogeneous equation of otion is 3 3 3 3 (6) The ass atri is M g (7) 3
The stiffness atri is 4 3 K N / 3 5 (8) The eigenvalues and eigenvectors can be found using the ethod in Reference. The eigenvalues are the roots of the following equation. det K M (9) The eigenvalues are 9.9 rad / sec () 3.3 rad /sec () f 4.78 Hz () 698 rad /sec (3) (4) 78.9 rad /sec f.4 Hz (5) The eigenvector atri is.68.4597.35.888 (6) 4
The eigenvectors were previousl noralized so that the generalized ass is the identit atri. ˆ T M (7).68.4597.68.35 ˆ (8).35.888.4597.888.68.4597.56.65 ˆ (9).35.888.4597.888 ˆ () Again, r is the influence vector which represents the displaceents of the asses resulting fro static application of a unit ground displaceent. For this eaple, each ass sipl has the sae static displaceent as the ground displaceent. r () The coefficient vector is T M r ().68.4597 (3).35.888 5
.68.4597 (4).35.888.757 g.379 (5) The odal participation factor i for ode i is i i (6) ˆ ii The odal participation vector is thus.757.379 (7) The coefficient vector and the odal participation vector are identical in this eaple because the generalized ass atri is the identit atri. The effective odal ass For ode, eff, i for ode i is i eff, i (8) ˆ ii.757 g eff, (9) g eff,.944 g (3) 6
For ode, eff,.379 g (3) g eff,.56 g (3) Note that eff, eff,.944 g.56 g (33) eff, eff, 3 g (34) Thus, the su of the effective asses equals the total sste ass. Also, note that the first ode has a uch higher effective ass than the second ode. Thus, the first ode can be readil ecited b base ecitation. On the other hand, the second ode is negligible in this sense. Fro another viewpoint, the center of gravit of the first ode eperiences a significant translation when the first ode is ecited. On the other hand, the center of gravit of the second ode reains nearl stationar when the second ode is ecited. Each degree-of-freedo in the previous eaple was a translation in the X-ais. This characteristic siplified the effective odal ass calculation. In general, a sste will have at least one translation degree-of-freedo in each of three orthogonal aes. iewise, it will have at least one rotational degree-of-freedo about each of three orthogonal aes. The effective odal ass calculation for a general sste is shown b the eaple in Appendi A. The eaple is fro a real-world proble. 7
Aside An alternate definition of the participation factor is given in Appendi B. References. M. Papadraais, N. agaros, V. Plevris; Optiu Design of Structures under Seisic oading, European Congress on Coputational Methods in Applied Sciences and Engineering, Barcelona,.. T. Irvine, The Generalized Coordinate Method For Discrete Sstes, Vibrationdata,. 3. W. Thoson, Theor of Vibration with Applications nd Edition, Prentice Hall, New erse, 98. 4. T. Irvine, Bending Frequencies of Beas, Rods, and Pipes, Rev M, Vibrationdata,. 5. T. Irvine, Rod Response to ongitudinal Base Ecitation, Stead-State and Transient, Rev B, Vibrationdata, 9. 6. T. Irvine, ongitudinal Vibration of a Rod via the Finite Eleent Method, Revision B, Vibrationdata, 8. 8
APPENDIX A Equation of Motion, Isolated Avionics Coponent z z, z z3 z4 3 3 4 4 Figure A-. Isolated Avionics Coponent Model The ass and inertia are represented at a point with the circle sbol. Each isolator is odeled b three orthogonal DOF springs. The springs are ounted at each corner. The springs are shown with an offset fro the corners for clarit. The triangles indicate fied constraints. indicates the origin. 9
z a a C. G. b c c Figure A-. Isolated Avionics Coponent Model with Diensions All diensions are positive as long as the C.G. is inside the bo. diension will be negative otherwise. At least one
The ass and stiffness atrices are shown in upper triangular for due to setr. (A-) K = z z z z z z a a b 4 b c c a a c c c c a a b a a c c b 4 a a b 4 4 a a c c 4 b 4 c c 4 (A-) z M
The equation of otion is M z K z (A-3) The variables, β and represent rotations about the X, Y, and Z aes, respectivel. Eaple A ass is ounted to a surface with four isolators. The sste has the following properties. M z z a a b c c = 4.8 lb = 44.9 lb in^ = 39.9 lb in^ = 8.8 lb in^ = 8 lbf/in = 8 lbf/in = 8 lbf/in = 6.8 in = -.68 in = 3.85 in = 3. in = 3. in
et r be the influence atri which represents the displaceents of the asses resulting fro static application of unit ground displaceents and rotations. The influence atri for this eaple is the identit atri provided that the C.G is the reference point. r (A-4) The coefficient atri is T M r (A-5) The odal participation factor atri i for ode i at dof j is i j i j (A-6) ˆ i i Each ˆ ii coefficient is if the eigenvectors have been noralized with respect to the ass atri. The effective odal ass eff, i vector for ode i and dof j is i j eff, i j (A-7) ˆ ii The natural frequenc results for the saple proble are calculated using the progra: si_dof_iso.. The results are given in the net pages. 3
si_dof_iso. ver. March 3, 5 b To Irvine Eail: toirvine@aol.co This progra finds the eigenvalues and eigenvectors for a si-degree-of-freedo sste. Refer to si_dof_isolated.pdf for a diagra. The equation of otion is: M (d^/dt^) + K = Enter (lb) 4.8 Enter (lb in^) 44.9 Enter (lb in^) 39.9 Enter z (lb in^) 8.8 Note that the stiffness values are for individual springs Enter (lbf/in) 8 Enter (lbf/in) 8 Enter z (lbf/in) 8 Enter a (in) 6.8 Enter a (in) -.68 Enter b (in) 3.85 Enter c (in) 3 4
Enter c (in) 3 The ass atri is =....63.34.487 The stiffness atri is =.e+4 *.3.3.3 -.48.3 -.3.48 -.3.763 -.5458.48 -.5458.4.3 -.48.3 Eigenvalues labda =.e+5 *.3.57.886.98.5699.738 5
Natural Frequencies =. 7.338 Hz.. Hz 3. 7.4 Hz 4. 7.47 Hz 5. 63.6 Hz 6. 83.9 Hz Modes Shapes (rows represent odes) z alpha beta theta. 5.9-6.8 -.4. 8.69.954 -.744 3. 7.7 6.3 4..4 -.6 -.95 5. -3.69.6 -.3 6..96 -.5 4.3 Participation Factors (rows represent odes) z alpha beta theta..656 -.755 -.693..963. -.769 3..795.69 4..5 -.63 -. 5. -.49.87 -.38 6..7 -.5. Effective Modal Mass (rows represent odes) z alpha beta theta..43.569.48..98.3.59 3..63.477 4..33.69.48 5..68.35.566 6..47.63.439 Total Modal Mass....6.3.487 6
APPENDIX B Modal Participation Factor for Applied Force The following definition is taen fro Reference 3. Note that the ode shape functions are unscaled. Hence, the participation factor is unscaled. Consider a bea of length loaded b a distributed force p(,t). Consider that the loading per unit length is separable in the for The odal participation factor where p(, t) Po p()f (t) (B-) i for ode i is defined as i p() i ()d (B-) i () is the noral ode shape for ode i 7
APPENDIX C Modal Participation Factor for a Bea et Y n () = ass-noralized eigenvectors () = ass per length The participation factor is n () Y n ()d (C-) The effective odal ass is () Y n ()d eff, n (C-) () Y d n () The eigenvectors should be noralized such that Y d () n () (C-3) Thus, eff, n n () Yn()d (C-4) 8
APPENDIX D Effective Modal Mass Values for Bernoulli-Euler Beas The results are calculated using forulas fro Reference 4. The variables are E I = is the odulus of elasticit = is the area oent of inertia = is the length = is (ass/length) Table D-. Bending Vibration, Bea Sipl-Supported at Both Ends Mode Natural Frequenc EI n Participation Factor Effective Modal Mass 8 4 EI 3 9 EI 3 8 9 4 6 EI 5 5 EI 5 8 5 6 36 EI 7 49 EI 7 8 49 95% of the total ass is accounted for using the first seven odes. 9
Table D-. Bending Vibration, Fied-Free Bea Mode Natural Frequenc n Participation Factor Effective Modal Mass.875 EI.783. 63 4.6949 EI.4339.883 3 5 EI.544.6474 4 7 EI.88.336 9% of the total ass is accounted for using the first four odes.
APPENDIX E Rod, ongitudinal Vibration, Classical Solution The results are taen fro Reference 5. Table E-. ongitudinal Vibration of a Rod, Fied-Free Mode Natural Frequenc n Participation Factor.5 c /.5 c / 3 3.5 c / 5 Effective Modal Mass 8 8 9 8 5 The longitudinal wave speed c is c E (E-) 93% of the total ass is accounted for b using the first three odes.
APPENDIX F This eaple shows a sste with distributed or consistent ass atri. Rod, ongitudinal Vibration, Finite Eleent Method Consider an aluinu rod with inch diaeter and 48 inch length. The rod has fied-free boundar conditions. A finite eleent odel of the rod is shown in Figure F-. It consists of four eleents and five nodes. Each eleent has an equal length. E E E3 E4 N N N3 N4 N5 Figure F-. The boundar conditions are U() = (Fied end) (F-) du d (Free end) (F-) The natural frequencies and odes are deterined using the finite eleent ethod in Reference 6.
The resulting eigenvalue proble for the constrained sste has the following ass and stiffness atrices as calculated via Matlab script: rod_fea.. Mass =.6.4.4.6.4.4.6.4.4.8 Stiffness =.e+6 *.39 -.6545 -.6545.39 -.6545 -.6545.39 -.6545 -.6545.6545 The natural frequencies are n fn(hz) 9.9 348.8 3 59.6 4 8534.3 The ass-noralized eigenvectors in colun forat are 5.547 4.8349 8.6-9.435.496.354-3.783 6.895 3.398-6.448-7.4584 -.744 4.495-6.57 9.4897 3.893 3
et r be the influence vector which represents the displaceents of the asses resulting fro static application of a unit ground displaceent. The influence vector for the saple proble is r = The coefficient vector is T M r (F-3) where T = transposed eigenvector atri M = ass atri The coefficient vector for the saple proble is =.867.33.86 -. The odal participation factor atri i for ode i is i i (F-4) ˆ i i Note that ˆ ii = for each inde since the eigenvectors have been previousl noralized with respect to the ass atri. 4
Thus, for the saple proble, i i (F-5) The effective odal ass eff, i for ode i is i eff, i (F-6) ˆ i i Again, the eigenvectors are ass noralized. Thus eff, i i (F-7) The effective odal ass for the saple proble is eff =.75.5.. The odel s total odal ass is.8 lbf sec^/in. This is equivalent to 3.4 lb. The true ass or the rod is 3.77 lb. Thus, the four-eleent odel accounts for 83% of the true ass. This percentage can be increased b using a larger nuber of eleents with corresponding shorter lengths. 5
APPENDIX G Two-degree-of-freedo Sste, Static Coupling Figure G-. ( - - ) ) ( - + ) ) Figure G-. The free-bod diagra is given in Figure G-. 6
The sste has a CG offset if. The sste is staticall coupled if. The rotation is positive in the clocwise direction. The variables are i z i is the base displaceent is the translation of the CG is the rotation about the CG is the ass is the polar ass oent of inertia is the stiffness for spring i is the relative displaceent for spring i 7
8 Sign Convention: Translation: upward in vertical ais is positive. Rotation: clocwise is positive. Su the forces in the vertical direction F (G-) ) ( ) ( (G-) ) ( ) ( (G-3) (G-4) ) ( ) ( (G-5) Su the oents about the center of ass. M (G-6) ) ( ) ( (G-7) ) ( ) ( (G-8) (G-9) (G-) The equations of otion are (G-)
9 The pseudo-static proble is (G-) Solve for the influence vector r b appling a unit displaceent. r r (G-3) r r (G-4) Define a relative displaceent z. z = (G-5) = z + (G-6) z z (G-7) z z
3 (G-8) The equation is ore forall r r z z (G-9) Solve for the eigenvalues and ass-noralized eigenvectors atri using the hoogeneous proble for of equation (G-9). Define odal coordinates z (G-) r r (G-) Then preultipl b the transpose of the eigenvector atri T. r r T T T (G-)
3 r r T (G-3) The participation factor vector is T r r (G-4) T T (G-5) Eaple Consider the sste in Figure G-. Assign the following values. The values are based on a slender rod, aluinu, diaeter = inch, total length=4 inch. Table G-. Paraeters Variable Value 8.9 lb 97 lb in^, lbf/in, lbf/in 8 in 6 in The following paraeters were calculated for the saple sste via a Matlab script. The ass atri is
=.49.3497 The stiffness atri is = 4 6 6 64 Natural Frequencies = 33.8 Hz 67.9 Hz Modes Shapes (colun forat) = -4.4.9.486.635 Participation Factors =.56.54 Effective Modal Mass.465.5 The total odal ass is.49 lbf sec^/in, equivalent to8.9 lb. 3
APPENDIX H Two-degree-of-freedo Sste, Static & Dnaic Coupling Repeat the eaple in Appendi G, but use the left end as the coordinate reference point. Figure H-. ( - ) ( + ) Figure H-. The free-bod diagra is given in Figure H-. Again, the displaceent and rotation are referenced to the left end. 33
Sign Convention: Translation: upward in vertical ais is positive. Rotation: clocwise is positive. Su the forces in the vertical direction F (H-) ( ) ( ) (H-) ( ) ( ) (H-3) (H-4) ( ) (H- 5) (H-6) ( ) (H-7) ( ) (H-8) Su the oents about the left end. M (H-9) ( - + ) - (H-) - ( - + ) (H-) - - (H-) - - (H-3) 34
35 (H-4) - - (H-5) - (H-6) The equations of otion are (H-7) Note that (H-8) (H-9) The pseudo-static proble is (H-) Solve for the influence vector r b appling a unit displaceent. r r (H-)
r r (H-) The influence coefficient vector is the sae as that in Appendi G. The natural frequencies are obtained via a Matlab script. The results are: Natural Frequencies No. f(hz). 33.79. 67.93 Modes Shapes (colun forat) ModeShapes = 5.5889 4.57.486.635 Participation Factors =.55 -.539 Effective Modal Mass =.464.539 The total odal ass is.49 lbf sec^/in, equivalent to8.9 lb. 36