2.1 Harmonic excitation of undamped systems

Transcription

1 2.1 Haronic exciaion of undaped syses Sanaoja 2_ Haronic exciaion of undaped syses (Vaienaaoan syseein haroninen heräe) The following syse is sudied: y x F() Free-body diagra f x g x() N F() In he above syse he applied force is assued o be haronic (sinusoidal) is he inpu frequency, he driving frequency or he forcing frequency. (1) Equaion (1_1.2) 2 reads Based on he free-body diagra and Eq. (2) he following is arrived a: Equaion (3) yields (2) (3) (4) Many exciaions are haronic (roaing achines) If he syse is linear (as above) Principle of superposiion can be used. General periodic (jasollinen) exciaion can be described as a su of haronic exciaions (Fourier series). Non-periodic exciaions are discussed in Chaper 3.

2 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.2 Equaion Of Moion (4) 2 is (liieyhälö) (4) 2 Eq. (2) is divided by he ass and he following is obained: (5) To be haronic he applied force also can have he fors (6) The noaion sands for he iaginary uni (iaginaariluu).

3 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.3 Fourier series If load is periodic, Fourier series give soluion as su of haronic loads. Load on a siply suppored bea OA is given by he Fourier sine series. The load follows he boundary condiions. The load is (7) There is an infinie su. How any ers are enough? The firs four Fourier series approxiaions for a square wave. Eq. (7) is no for a square wave. 1 er 2 ers 3 ers 4 ers

4 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.4 Soluion procedure o obain: General soluion o linear, nonhoogeneous, second-order differenial equaion. The differenial equaion under consideraion is (1) Find he general soluion of he hoogeneous equaion. The hoogeneous differenial equaion is Guess for he general soluion of he hoogeneous equaion. I can be for exaple he following: (8) (9) (10) For derivaives and. Subsiue, and ino Eq. (9). If you ge, Guess (10) is o. Ter general soluion eans ha values of consans and are open. (2) Find he paricular soluion of he nonhoogeneous equaion. There are several sraegies o find a soluion. In his conex here are so few guesses ha hey can be reebered. The guess can be For derivaives and. Subsiue, and ino Eq. (8). Value for is obained. For he forhcoing sudy le. (11) (3) Find general soluion of nonhoogeneous equaion (differenial equaion). The soluions are he su of he above soluions, i.e. (12)

5 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.5 Equaion (12) is (12) (4) Find a soluion of he iniial value proble. Apply iniial values o General Soluion (12). Since here are 2 unnown consans, i.e. and, 2 values are needed. Iniial valuess: and. Rears: The above procedure did no solve any cerain differenial equaion. I was jus a sech for deonsraion of he soluion procedure. Solved exaples follow in his course. Guess for paricular proble can have 2 unnown consans. ( Sec. 2.2) Terinology in Finnish: Guess Yrie The general soluion of he hoogeneous equaion Hoogeeniyhälön yleinen raaisu The paricular soluion of he nonhoogeneous equaion Täydellisen yhälön ysiäisraaisu The general soluion of he nonhoogeneous equaion Täydellisen yhälön yleinen raaisu The soluion of he iniial value proble Aluarvoehävän raaisu

6 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.6 The following syse is sudied: y x F() Free-body diagra f x g x() N F() In he above syse he applied force is assued o be haronic (sinusoidal) (13) Eq. (5) 1 gives he governing differenial equaion for he above syse. I is (14) (1) Find he general soluion of he hoogeneous equaion. The hoogeneous differenial equaion is (15) Guess for he general soluion of he hoogeneous equaion is (16) Guess (16) gives (17) and (18) Subsiuion of Eqs (16) and (18) ino DE (15) yields (19) Equaion (19) yields. Thus, Guess (16) is correc. Noe. In Sec. 1.1 guess o DE (15) was. Boh are correc.

7 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.7 Eq. (14) 1 gives he nonhoogeneous equaion, viz. (20) (2) Find he paricular soluion of he nonhoogeneous equaion. There are several sraegies o find a soluion. The guess is Guess (21) gives Exprs (21) and (22) 2 are subsiued ino Nonhoogeneous Equaion (20). This gives Equaion (23) gives (21) (22) (23) (24) Based on Equaions (21) and (24) he paricular soluion is (25) (3) Find general soluion of he nonhoogeneous equaion (differenial equaion). The soluions are he su of he above soluions, i.e. (26)

8 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.8 The general soluion o nonhoogeneous equaion is given by Eq. (26), viz. (26) (4) Find a soluion of he iniial value proble. Apply iniial values o General Soluion (26). Iniial values are: and. The firs iniial value is subsiued ino Eq. (26). This gives Since and, Equaion (27) reduces o For he second iniial value he derivaive of Eq. (26) is needed. I is (27) (28) (29) The second iniial value is subsiued ino Eq. (29). This gives (30) Since and, Equaion (30) reduces o (31) Based on Eqs (26), (28) and (31) he soluion of he iniial value proble is (32)

9 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.9 Soluion (32) is (32) The figure underneah give he response of he syse, when (33) and furher (34) Response o haronic exciaion

10 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.10 Alernaive soluion 1 In Secion 1.1 he following differenial equaion was sudied: Following Soluion (1_1.6) was given for DE (35): (35) Equaion (35) is he hoogeneous equaion sudied in his secion. Thus, Soluion (36) is he general soluion of he hoogeneous equaion. However, in he presen secion Soluion (16) was used. I is (36) (37) Reason o use Eq. (37) insead of Eq. (36) is ha Eq. (37) leads o sipler algebra. Alernaive soluion 2 The presen secion sudied DE (14) 1, viz. Soluion (32) is of DE (38) reads By seing DE (38) reduces o DE (35). Thus, soluion of DE (35) is obained by seing in Soluion (39). Thus, soluion (38) (39) (40) is he soluion of DE (35).

11 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.11 SUMMARY Free vibraion (vapaa värähely) Equaion Of Moion (1_1.5) is [ Eq. (1_1.11)] (41) The soluion of he Equaion (41) reads (42) where phase and apliude are given by Eqs (1_1.17) and (1_1.19) 2, viz. (43) Soluion (40) is an alernaive soluion of DE (41). I is (44) Vibraion due o haronic exciaion (harooninen paovärähely) Equaion Of Moion (4) 2 is [ (14) 1 ] (45) Soluion (32) is of DE (45) reads (46)

12 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.12 y Beaing (huojuna) x F() f x x () F () The following 1-degree of freedo (equaion of oion) is invesigaed: g N (47) Equaion (47) displays ha he daping er is negleced. In he conex of his secion he syses are conservaive. Thus, no dissipaion. The soluion can be wrien in he for (guess is ) (48) If he iniial values are: and, Soluion (48) reduces o (49) The following expression holds: (50) Equaliy (50) allows Expression (49) o be wrien in he for (51)

13 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.13 Equaion (51) is The following case is sudied: Frequency of he exciaion is close o he naural frequency of srucure. This is. Thus, he following holds: (51) (52) Expression (52) 2 is (53) Since, Expression (53) 2 reduces o (54) Expressions (52) 1, (52) 3 and (54) 2 are subsiued ino Soluion (51). The following is obained: (55)

14 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.14 The following case was sudied: Frequency of he exciaion is close o he naural frequency of srucure. This is. The obained response is given by Equaion 55), viz. (55) The following conclusions can be ade: 1. The period of he coponen is. Apliude 2. The period of he coponen is which is large, since. 3. The response is vibraion which: Period is and Apliude. 4. The ie beween he axiu values of he apliudes in beaing are: Definiion of he period : When. See red spos below. (56) 5. Based on equaion (56) he frequency of he beaing is (57) x() 2 F / 0 2 O 2

15 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.15 Resonance (resonanssi) The following syse is sudied: y x () x F() f x g N F () The above syse leads o he following 1-degree of freedo (equaion of oion) (58) Equaion (58) displays ha he daping er is negleced. The following guess for paricular soluion is used: (59) Guess (2) gives he following soluion: (60) In case of beaing. Soluion (60) o. Soluion (60) is no for he case. Thus, Guess (59) is no correc for. Differen approach has o be aen.

16 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.16 The following syse is sudied: y x () x F() f x g N F () The above syse leads o he following 1-degree of freedo (equaion of oion) (61) The soluion procedure follows. (1) The hoogeneous differenial equaion is (62) The general soluion of Hoogeneous DE (62) is (63) (2) Insead of Guess (59) he following guess o paricular soluion is wrien: (64) Guess (4) gives (65) and (66) Subsiuion of Expressions (65) and (66) ino DE (61) yields (67)

17 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.17 Expression (67) is (67) Since he special case is sudied, he following holds: (68) Expression (68) is (69) Guess (64) is (70) (3) Soluion o he iniial value proble, DE (61), is he su of and. Based on Expressions (63) and (70) he following is arrived a: (4) Soluion o ini. val. probl., DE (1), is obained by applying iniial values. Iniial values and. The ie derivaive of Soluion (71) reads Subsiuion of iniial values ino Expressions (71) and (72) yields Subsiuion of Resuls (73) ino Soluion (71) gives (71) (72) (73) (74)

18 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.18 Soluion (74) is (74) A special case is evaluaed. This is and. Soluion (74) reduces o (75) The period and he apliude of response (75) are (76) According o Equaion (76) he apliude grows in ie. Dangerous!! Unbounded apliude. Displaceen () , f 0 2 n f 0 2 n Tie ( s)

19 2.1 Haronic exciaion of undaped syses Sanaoja 2_1.19 In real srucures differen coponens have differen naural angular freqs. In a oorcycle he handlebar (ohjausano) and fooress (jalaapi) ay vibrae. A he sae ie he frae (runo) of he oorbie can be cal. Soe anufacures use bar end weighs (angonpainoja) o avoid resonance. Naural angular frequency is changed by bar end weighs. I is also possible o buy bar end weighs.