48 1 Vol. 48 No. 1 2016 1 Chnese Journal of Theoretcal and Appled Mechancs Jan. 2016 1) 2) ( 100190)...... O326 A do 10.6052/0459-1879-15-152 THEORETICAL INVESTIGATION ON AERODYNAMIC FORCE MEASUREMENT INTERFERED BY STRUCTURAL VIBRATIONS IN LARGE SHOCK TUNNEL 1) Meng Baoqng Han Gula 2) Jang Zongln (State Key Laboratory of Hgh Temperature Gas Dynamcs, Insttute of Mechancs, Chnese Academy of Scences, Bejng 100190, Chna) Abstract Structural vbraton of aerodynamc measurement system s caused by the mpulse forces generated by startup process of the shock tunnel, and the test tme s too short to allow the vbraton to decay. Therefore, the aerodynamc force measurement can be consdered as a dynamc process, durng whch the sgnals outputted by balance are consst of aerodynamc force sgnals and vbraton sgnals. Exstng methods are lack of support of theory and the accuraces of them are lmted. Theoretcal analyss s used n ths paper and we obtan the analytcal solutons of fundamental characterstcs of free vbratons and forced vbratons. Study of free vbratons focuses on the dscplnes of aerodynamc force measurement nterfered by a sngle vbraton shape and ampltude of nterference changed wth secton of force measurement. The nferences of dfferent vbraton shapes durng forced vbraton motvated by vared forces are nvestgated n research of forced vbraton. Results show that there are theoretcal defects for tradtonal acceleraton compensaton technques because the zero ponts of ampltude of nterference and acceleraton are nconsstent. What s more, the nfluences of secton for force measurement, actng poston of force and type of force should be consdered to determne the man source of nterference n experments. Key words shock tunnel, aerodynamc force measurement, vbraton nterference, theoretcal nvestgaton 2015 05 08 2015 08 26, 2015 09 10. 1) (11472281). 2) /. E-mal: hangula@mech.ac.cn,,.., 2016, 48(1): 102-110 Meng Baoqng, Han Gula, Jang Zongln. Theoretcal nvestgaton on aerodynamc force measurement nterfered by structural vbratons n large shock tunnel. Chnese Journal of Theoretcal and Appled Mechancs, 2016, 48(1): 102-110
1 103. 2 30 ms JF12 1] 130 220 ms.. ( ). 1... 2] 3] 4-5] 6-7]. ( )..... 1 10 30 Hz. 2. 8] ( ). ( ). 9-11]. 12-13]. ( ) 14].. 15].. (JF12) 25 40 km 5 9 2.5 m 3.5 m 6 8 m.. JF12... ( ).. x
104 2016 48.. 1 JF12.... JF12 5 t 5 --. x y(x, t) L E c ρ c Ē Ī 4 ȳ ( x, t ) x 4 + ρ Ā 2 ȳ ( x, t ) t 2 = 0 (1) x = x/l ȳ = y/l ρ = ρ/ρ c Ē = E/E c Ā = A/L 2 Ī = I/L 4 t = t/t c T c = ρ c L 2 /E c.. (1) 16] y(x, t) = Y(x)T(t) (2) Y (x) = C cos (β x) ch (β x) + r sn (β x) sh (β x) ] } T (t) = b sn(ωt + ϕ) EJ 4 y(x, t) x 4 + ρa 2 y(x, t) t 2 = P(x, t)δ(x l) (3) E c L 2 δ (x l). (3) 16] Y (x) t y (x, t) = q (τ) sn ω (t τ)] dτ (4) ω q (t) = 1 0 0 P (x, t) δ (x l) Y (x) dx M (x, t) = EI 2 y (x, t) x 2 (5) Q (x, t) = EI 3 y (x, t) x 3 (6). (2) (4) M (x, t) = Q (x, t) = A M (x) sn (ω t + ϕ ) (7) A Q (x) sn (ω t + ϕ ) (8) 2 2.1. (7) A M (x) = EIb C β 2 cos (β x) cosh (β x) r sn (β x) + snh (β x) ] } (9) x.
1 105 EIb C β 2 m (x) = cos (β x) cosh (β x) r sn (β x) + snh (β x) ] (10) m (x) 1 2 m (x) m (0) = 1 1. (1) x = 1. x = 1.. (3). y t a (x) = cos (β x) cosh (β x) + r sn (β x) snh (β x) ] (11) 2 m 2 (x) a 2 (x) 2 m (x) a (x).. 2 x = x 1 2 0 0 x = x 2 2 0 0. (a) 1 (a) The frst order vbraton shape 2 2 m(x) a(x) Fg. 2 The dstrbuton of ptchng moment ampltude m(x) and acceleraton ampltude a(x) of the second vbraton shape (b) 2 (b) The second order vbraton shape 1 m (x) Fg. 1 The free vbraton shapes of force measurement structure and correspondng m (x) dstrbuton (2) ( 2) 1 (x 1) 1(b) x = x 1 0 2.2. (8) A Q (x) = EJb C β 3 sn(β x) snh(β x) r (cos(β x) + cosh(β x)) ] (12)
106 2016 48 x. EJb C β 3 q (x) = sn (β x) snh (β x) r cos (β x) + cosh (β x) ] (13) q (x) 1 2 q (x) q (0) = 1 3(a) 3(b) 2.1. q (x). (3). q (x) a (x), 4 q 2 (x) a 2 (x) x = x 3 x = x 4. 4 2 q(x) a(x) Fg. 4 The dstrbuton of normal force ampltude q(x) and acceleraton (a) 1 (a) The frst order vbraton shape ampltude a(x) of the second order vbraton shape 3. 3 5. 10] (b) 2 (b) The second order vbraton shape 3 q (x) Fg. 3 The free vbraton shape of force measurement structure and correspondng q (x) dstrbuton (1) x = 1. x = 1. (2) ( 2) q (x) 1 (x 1) 4 x = x 3 0 (a) (b) (a) Harmonc (b) Rectangular pulse 5 3 Fg.5 Schematcs of three dfferent loads
1 107 (c) (c) Parabolc 5 3 ( ) Fg.5 Schematcs of three dfferent loads (contnued). 3.1 P(x, t) = P 0 δ(x 1) cos ωt ω = ω 1 + (ω 2 ω 1 ) 0.2 0.5 0.8 ω 1, ω 2 1, 2. y(x, t) = C P 0 Y (1) cos(β x) cosh(β ω 2 ω 2 x)+ r sn(β x) snh(β x)] } (cos ω t cos ωt) (14) C P 0 Y (1)β 2 M(x, t) = cos(β x) cosh(β ω 2 ω 2 x)+ r sn(β x) snh(β x)] } cos(ω t) cos(ωt)] (15) Q(x, t) = C P 0 Y (1)β 3 sn(β x) snh(β ω 2 ω 2 x)+ r cos(β x) cosh(β x)] } cos(ω t) cos(ωt)] (16) (14) 6. ANSYS. 1 m 0.1 m 0.1 m 200 GPa 7 850 kg/m 3 450 rad/s P 0 = 10 000 N 0.02 s. 6 y(x, t) Fg. 6 Theoretcal and numercal soluton of the dsplacement y(x, t) under the harmonc load 3.2 P(x, t) = P(t)δ(x P 0, 0 t t s 1) P(t) =. 0, else y(x, t) = M(x, t) = C Y (1)P 0 cos(β x) cosh(β ω 2 x)+ r sn(β x) snh(β x)] } 1 cos(ω t)], 0 t t s C Y (1)P 0 W ω 2 (x)1 cos(ω t s )] cosω (t t s )]+ sn(ω t s ) snω (t t s )]}, t > t s (17) 0 t t s C Y (1)P 0 β 2 ω 2 cos(β x) cosh(β x)+ r sn(β x) snh(β x)] } 1 cos(ω t)] (18) Q(x, t) = C Y (1)P 0 β 3 ω 2 sn(β x) snh(β x)+ r cos(β x) cosh(β x)] } 1 cos(ω t)] (19) (17) 7. 1 m 0.1 m 0.1 m 200 GPa 7 850 kg/m 3 P 0 = 10 000 N.
108 2016 48 7 y(x, t) Fg. 7 Theoretcal and numercal soluton of the dsplacement y(x, t) under a rectangular pulse load 3.3 y(x, t) = P (x, t) = P 0 t 2 δ (x 1) C P 0 Y (1) cos(β ω 2 x) cosh(β x)+ r sn(β x) snh(β x)]} t 2 2 + 2 cos(ω ω 2 ω 2 t) (20) M(x, t) = EI C P 0 Y (1) β 2 ω 2 cos(β x) cosh(β x)+ r sn(β x) snh(β x)]} t 2 2 + 2 cos(ω ω 2 ω 2 t) (21) Q(x, t) = EIC P 0 Y (1) β 3 ω 2 sn(β x) snh(β x)+ r cos(β x) cosh(β x)]} t 2 2 + 2 cos(ω ω 2 ω 2 t) (22) (20) 8. 1 m 0.1 m 0.1 m 200 GPa 7 850 kg/m 3 P 0 = 10 000 N. 8 y(x, t) Fg. 8 Theoretcal and numercal soluton of the dsplacement y(x, t) under a parabolc load 3.4. A M (x) (15), (18) (21) t. (21) A M (x) = EIC P 0 Y (1) β 3 ω 2 sn(β x) snh(β x) + r cos(β x) cosh(β x)] } (23) η (x) = A M (x) / A M (x) (24). 13 1% 12.. (1). ( = 0.2 ) (25). 9 4. 1, 2 1 2 x > 0.5 2 1 3, 4 1, 2.
1 109. A M (x) = EIP 0 C 2 W (1)β 2 ω 2 ω 2 cos(β x) cosh(β x) + r sn(β x) snh(β x)] } (25) 11 3 Fg. 11 Relatve magntude of nterference to ptchng moment by the thrd order vbraton shape under loads actng on two dfferent postons respectvely A M (x) = ρap 0C 2W (x 0 ) cos(β x) cosh(β β 2 x)+ r sn(β x) snh(β x)] } (26) 9 ( = 0.2) 4 Fg. 9 Relatve magntude of nterference to ptchng moment by the frst four-order vbraton shapes under a harmonc load ( = 0.2) (2). x 0 (26). 10 11 3. x 0 = 0.5 3 3 3. x 0 = 1 3. 10 3 Fg. 10 Schematc of loads actng on two dfferent postons and the thrd order vbraton shape. (3) 1, ( ). 1. 1.. 1 Table 1 The man source of ptchng moment nterference under dfferent loads Load form (x 0 = 1.0) harmonc load ( = 0.2) harmonc load ( = 0.5) harmonc load ( = 0.8) rectangular pulse load parabolc load The frst order ( ) ( ) The second order ( ) ( ) The thrd order (4). 1 2 3 1 3, 4 2 3, 4 1, 2..
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