GENERAL THEORY 17 Eqs 2.34a and 2.34b may be written in the form of expressions for dynamic factors #, and p, as follows with appropriate substitutions Me and He where — lal a nỷ (nƒ—n‡ nỆ) Gat ad) (nỉ ni =ninЗ vi — tal _ ning cau 3á (2.34¢) : Gad a) Gt af eae ni) nêm đạt = , 9.34d 2.34) P Ee _—0m _ m and V Km tìm t For the particular case when V K,/m, K, _ X, z= : : (considered in Section 1m 2.3a), : : that is, n,=7, (= say) the Eqs 2.34c and 2.34d may be further simplified as by = and ee (2.34e) (a?) (1 aa?) Be = (1-7?) (1+ a7?) —« (2.34f) Fig 2.5 shows the variation of and gy (given by Eqs 2.34e and 2.34f) with for the case when a=0.2 Two points worth noting from this figure are: 1, There are two values of y at which p, or py is oo The values of wm corresponding to these infinite ordinates are the natural frequencies wp, and ong When y=1, i.e @m=Gpp, 4,=0 In other words, when the values my and k, are such that lo is equal to the fre- Tạ quency (m) of exciting force acting on mass mm, then the amplitude of mass m, will be zero When Gn2=©m, while a4,=0, the amplitude of mass m, may be obtained (from Eq 2.34b) as (2.352) The amplitude of mass m, is thus equal to its static displacement under the static influence of P,) Application: The (displacement of m, above theoretical treatment will be useful in the application of an undamped vibration neutralizer for a rigid block foundation as explained in Section 7.3c 18 HANDBOOK OF MACHINE FOUNDATIONS - 6.0 T 6.0 A ] ‡ 5.0 4.0 4.0 1_| 3.0 a3.0 20 1.0 JT ` : | i 04 iN 081012 —~ ? / +oL—— " 1.6.2.0 I | | iy | 04 | \ | | 081.012 — ? hay 16 KY 20 (b) Response Curves for an Undamped Two-Degree Freedom Fig 2.5: At it TN (a) for the Case when «=0.2 and J a 5.0 x 2.3.2 = & m System ` Damped Case Frec Vibrations Consider the system shown in Fig 2.3b Viscous dampers with damping coefficients C, and C, are additionally introduced here It is difficult to precisely assess the values of C, and C, in practice and consequently they are not generally considered in practical designs based on multiple degree freedom systems However, the following theoretical treatment will be helpful in cases where the influence of damping cannot be neglected, and this data can be obtained from field measurements or otherwise The equations of motion for the system shown in Fig 2.3b may be written as under: m &+C,%4+ Z¡ + Ky (21-22) + Cy (41%) = (2.36a) mạ 3; -E Œy (—a) + Ấs (za—) =0 (2.36b) Both z, and z, are harmonic functions and can be represented by vectors Writing the vec- tors as complex numbers and substituting Z¡ = £tant (2.37a) Zp — da etant (2.37b) in Eqs 2.37a and 2.37b, and solving, the following governing equation is obtained for the natural frequencies of the system Om) (Ita) }?=0 ( F(o2,) }?4-402,(G Gar(@2,—08)o/ TF at Gatna(@e— (2.38) GENERAL THEORY l9” where, F(a, )=of, a m(1-†-ø) (9 +o? „+# & Cefn ®4T+ &ato, a? x(1-E#) (2.39) where ta, @ ®,, and @ are already defined in Eqs, 2.23a, 2.23b and 2.24 respectively; %, and ¢, are damping ratios defined by GQ _ ® 21 RI T C, TH =2 (2.40a) Ñ Ẩm (2.40b) Corollary: When (=0, and €,=0, Eq 2.39 reduces to the form given for the undamped case b Forced Case 1: by Eq 2.22 Vibrations When the harmonic force P, sin wmf acts on mass mj The equations of motion for the system’ may be written as my 2+ Cy& + Kyat Ky (21-22) + G (4—-%&) = P, sin amt mẹ -Ƒ Ôy (Êy— Š1) + Ấy (z—z¡) =0 (2.41a) (2.41b) Since the system moves at the frequency of the exciting force under steady-state conditions, the solution may be assumed in the form: 2, = a, e'omt and (2.42a) Zp = ay elon (2.42b) Substituting these relations in Eqs 2.41a and 2.41b and solving, the following relations are obtained for a and a, mi “ d my (G2a— em) + 2/Va tổn; tm (e2) +2i@n[ (2.43a) em (6ã—0m) 4/T-+a+be One (2,—a2) (1a) fy” a= - a (fk +& Cate) _ K—m,a? + Cia (2.43b) where F(w2.) is given by Eq 2.39 Using the principles of complex algebra, the modulus of a, and a, may be written as (ope Om)? +403 Why aN (oa) Paar ea, (02,— vet eat “ND {F(w2) 4-402, Sale eatsath Ges ee 2g)0+aj t5 oF) (1+0) }? (2.44b) ‘ 20 HANDBOOK OF MACHINE FOUNDATIONS Particular Case: When %=0 (i.e., the damping in the lower system is neglected) ¢,=6, the amplitude of mass m, subjected to a harmonic force P, sin Ont is given by nà +40? 2, «2, =_ PomL TOF 1-4(at o2— 02)? Coson (Fe) (a R0]vl and Po đạ== mỊ mã tân, Còng ni +4 (Ff (ot)p+ _ 402 Pat, (+0)? oe ni : and (2.45a) (2.45b) al (2.46a) where f (w2)is given by Eq 2.32 or in terms of basic parameters, substituting Cj=0 and C2=C a=— Pal Mg @e,) Ca {(K,—-m,of) (Ky—m, a2)(K2— —Kym,o2}? Th dã [mot and “ =P KR4C | a? {(Ky—m wf) (Kg—m, @2)— K,m,02,}? ‘ Oe al 46Gb (2.460) Expressed in non-dimensional form, Eqs 2.46a and 2.46b may be further written as By = lal ~Ƒ [ [xn? nf (1-98)? +40 + 2E nề (䧗1)]? —(nï—] and tạ =1] đạt _— =[ [xn? — (n?—] (ni—1+ 1442293 (nˆ—Ð]? + 4Œn (n†—1 sua y 7a) ba(2.4 | } + an?)? | (2.47b) where (2.48a) Gat = =e ` m= Jun Tịa = Vi a = m/my §=(ŒIŒ For the case = (2-48b) (2.48c) (2.484) (2.48e) = AL (considered in preceding case), y,=7, (=, say), Fig 2.6 H shows the variation of u, with q for various damping values (€¢) It is interesting to note from Fig 2:6 that irrespective of the degree of damping, all the response curves pass through two fixed points S, and S,, the abscissa of which may be obtained as roots of the following equation yẻ—9 ”Í b ae 202 =) + sa + =0 (2.49) GENERAL ale8 SJ | 5.0 :[ = eB ate 3i: tị | s I 6.0° li t 4.0 TH Ị lộ I7 iN 3.9 i 2.0 \ ] PeReke \ \ 00 02 04 Fig 2.6: 06 \ Nị rl 1.0 i= 0-2 Ko Ks mim || 21 M2 q| a THEORY NỈ; : 08 10 / HS SN = / ——p.=m Qn =- man 14 12 om ong 14 16 SS | 18 30 Response of Mass m, for Various Damping Ratios (€) where B = min; For the particular case considered above since 4, = y2, = 1.0 ; Substituting, the abscissae of the fixed points S, and S, in Fig 2.6 are given by Tịi!—2 4_ Tị 9x8 +( “ = +a ) ( 2.50 ) with z = 0.2 in this case Tables 2.2 and 2.3 give the values of ụạ and y, for various mass x, — Ms ratio « and damping x, =—+ my values of frequency ratio a ratio { for the particular case when y,=7, or the relation is satisfied Application: The theory explained in the above particular case is used in the design of auxiliary mass-vibration dampers, which will be explained in Section 7.3c The data con- tained in Tables 2.2 and 2.3 will be useful in the choice of appropriate parameters for the design of auxiliary mass-vibration dampers for a rigid block foundation 2.4 Multiple-Degree Freedom System Although the vibration analysis of a multiple-degree freedom system is relatively more complicated and often necessitates the use of a digital computer, the theoretical approach for the analysis of such a system for the undamped case is given in this section for the benefit of interested readers Matrix notation* is used here for a concise presentation *Readers not familiar with this notation may refer to standard Vol of Ref G 1.6 books on matrix algebra or Section 28, tố IEBOS 08/0 PIL 6PETI Z8 6901 SIO /0/0% Z1/21 GET Z9UI: G001 SIOT LBL% | BESS - Z0 FILL | SIOT €EEI | 80EI |6EII Z9 S901 |0901 GI0I |EIOI —- - COWS OP'S | 2BO°S S0E0 |9960 9870 |9720 Z090 |G020 Z8/0 |0190 /0L11|6€/0 /BB1 |Z660 //69 |/6Ú1 |98E1 Z6 [SII |Z6ET 6/#0 |6611 0000 |8EZØ1 0990 |6561 885 |Đy8G 9S6/ |9ZBE 1291 ZZ91 |#ESI 6061 60ÿ1.|//7Ø1 6PUL 6EUI |961I 0901 0901 |SG01 FIOT #101 |EIOI 7667 0291 GOST 8ÿII 0901 ÿI0I zo : SIG IGS PEBl S860 |EGE0.69E0 E900 9060 |ZIFO0 #Z#0 /Zÿ0 #ÿ6G0 |98E0 S060 IIG0 6690 |SBG0 8190 8780 /E60 |ZZ/0 E820 FORO 80WI|6I60 0801 E011 BLAS | GOVT YEST 65/1 EEL | COST IGEð% PSOE 9ÿBE1 |66/I #Z65 PSG 6590 |BE/1 £/EI Z911 0000 |/6BI IBBED 860 8160 |E55E //0Z 0/91 GOL'S |9B8%3 £090196/61 605% |#WZB6ð #/0E 9ZIE |//BI EOF LeU EðII ISO ZIOT ð616 EBEO EGEO0 GEC'O 1690 6160 6ÿE1 6EÿZ 97%ÿ Z€/1 6ZL°0 ZEE0 9101 8//G IEEY E9#I Leo EðII IEUI ZIOT 6117 8661 |I9ÿ[ LEON | Lee '96EI |EšIT GG0I |0601 SION | ZIOL 6/E0 /20 GES'O 1/90 6/U0 #IZ1 BZB1 86E7 0861 L880 ÿS90 E971 B6/G SOY BEST //61 96ET E01 EIOI LEST //ZI SET] SSO" EIOT = S00 JO NOLLVIUVA z0 I0 £00 Su, Ry Z0 L0 E0 Su ty #z 2IqEL "¬ Hứ SSVW MOA (Tr) WOLOVA OINVNAG S00 I0 +0 Gš?0 -6z#0 |9/E0 /6E0 EOF 88E'0 EIG0 1720 |6EEO0 Z/E0 E80 5IEP'0: 8ÿ0°'0 669'0 |ZZ6'0 //20 66G0 :96E'0 ÿ980 88810 |IE90 IE/0 89/0 98g80 E971 761 |9//0 F160 890°T “ $8L'0 IffØZ /GĐZ |9€60 06E1 60/1 ;0660 Z6P'G 9E#91|/ZIH1 I/61 Z9 zz01 [PLOT OLB'T | TAIT 6081 929% EPG0 BEB'O 96E'0 |IS0I S0EI 9611 0600 LI¥'0 #/E0 |0I60 E690 IE60 ZEL0 "6610 0000 |8Z60 060 6ÿZ0 ÿE£0 69G0 GIG'0 |†6E'1 9060 06/20 „0801 Leet 0601 |GBR9G O6O'E I16Z OKA POOL GOSEL VSS SLLegl LVE'S #09 /66'/ 90/1 ZS E01 6901 SIOT - S00 sơ I0 KT th 90 0% 61 81 z1 ST #1 E1 al Vl 01 60 g0 £0 SILT 99E0|#f60 8PEO 6E0 6EEO 6220 |/690 ÿ0P0 Z0P0 S0YO0 #2160 |89ÿ0 ÿ/ÿ0 9/0 L/E0 1E90|8600 1/0 ÿ/60 9/00 Z180 |S890 LOL'O ÿ120 910 11 | S980 Z160 BZ60 SE60 €IB1|E#II 6GØ1 E0E1 EZET 966ÿ |ÿ6G] 7#61 616 (E5 199% | LAUD LEE 5IEG l/BB 1901 | 96'S ZLBe 98/š IE/5 00010 |#I⁄E 1961 S660 0000 0191 |95809 FES PLS EZ © | ORI) 688% 686G Bers 9LE |ISEZ 96% IIW6 IFS g0 y0 E0 z0 vo ZZ S6ET BIẾT tml 9ÿ01 1101 TILT G6EI 8121 TTT 901 TIƠT |90/T G6E'I BỊZI LIVE 9ÿ01 TIÔT S00 9681 EOP | 466° Leet | BIZI eZee | UL 1601 |9P0T 5101 |[TI0T E0 T0 |Z0 , 6SE0 8070 T8Z'O 68E0 #660 #810 É60I -ð8BZT Đ#ELI VOT 000'2 05012 OSS #1/'6I LEO’) I6 60861 SLOT GOUT 9201 s0 L0 ĐBPLO £0Z0 0670 BEEO 61/0 6EZI I06I I06I /Z/1 GEL 000% 6E/7 9IESG 606'6Z GOGH PLES 96G LEO GOVT 9201 S00 #P?P!Ú0 00Z0 E670 T9F'O /IB0 76/11 SENS #0Wð 0°BT TLel 0002 L5 697G 016'# ZEEV E8SZ BEST /F/2XI GOTT 9Z0T "0 eo SSTO SPIO] |£E0Z0 0020 |9/70 #620 |08E0 0/70 |9ÿG0 /980 98722 |ÿ0E0 ESTA ZOLT I1L'S | Let |06/1 0061 EB/1|0P026 |0067 0002 |If9E (1/2 |6E0B ø626 62299] HPSS GCE} | SESE EBS OBST] 86G HI |OGG'I |19ZL //Z1 GOTT | FOIE |#Z01 9201 /EL0 SEI'O | ISTO |/B10 8B8I0 9810 |ÿ920 6970-6820 IIPO IIEO0 |#I?0 11E60 6890 60/0 |0080 Z/ÿL GOST CONE LOVL | GEST 1//E E0O'S | LOBl |00E2 6067 S192 |S896 E666 6/20 |EEEE 0062 OOS |OIIG YBEE E/b€ II6)/ | SE/'51 OFGL E6/ ĐI 00961589 SOLE | LELE BLE PINS ZVS! 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It is also called the “real eigen value problem” to distinguish it from the complex eigen value problem obtained when the damping matrix is also considered in the equations of motion (Eq 2.51) Eq 2.56 represents a set of homogeneous equations (right-hand side equal to zero), the condition for obtaining a non-trivial solution being that the determinant formed by the coefficients of the left-hand side of the equation system should vanish gives the relation in its general form as Kay —Cmó3 Ấqy, Tân Ky sa —m Kun MAAMI Ky This =0 (2.58) na - Xnn —fn wo? Eq 2.58 on expansion gives n roots for w%, say œñŸ, ö$ cŸ such that öŸ