Transfer power flow calculation Generally, the vibration isolation system has at least one geometric and stiffness symmetry plane. The air spring vibration isolation system analyzed in this paper is a plane vibration isolation system model with a symmetry plane as shown in Fig. 1, and the XY plane is a symmetry plane. The air springs kii=1, 2, 23 are regarded as vibration isolator with lateral stiffness and vertical stiffness; the foundation kf is regarded as a spring with two degrees of freedom fixed at one end connected to the isolator; foundation and isolator The vibration isolation device Ki is formed in series. The vibration isolation device is regarded as a rigid body whose center of gravity deviates from the geometric center, and the center of gravity is G. Under the condition that the initial shape variables of the vibration isolation device are the same, the static equilibrium equation is: K10a1+K20a2+K30a3=0(1)K10+K20+K30=4π2f2m (2) where Ki0 is the vertical static stiffness of the vibration isolation device; Aii = 1, 2, 23 are the lateral distances of the vibration isolating device from the center of gravity of the machine; f is the natural frequency of the system, and m is the mass of the machine. For small displacement linear plane vibration, there is the following dynamic equation: mx bite G+3i=1ΣKixxG+3i=1ΣKixdγ=Fx(3)my bite G+3i=1ΣKiyyG-3i=1ΣKiyaiγ=Fy(4)Izγ bite+3i=1ΣKixdxG -3i=1ΣKiyaiyG+3i=1ΣKixd2+Kiyai2Σγ=Mz(5) where Kix and Kiy are the lateral and vertical dynamic stiffness of the vibration isolation device; Iz is the equipment area moment; d is the vertical distance of the vibration isolator from the machine center of gravity ;xG, yG, γ is the displacement at the center of gravity of the machine; Fx, Fy, Mz are the forces (moments) at the center of gravity of the machine. For the force Fx=0, Fy=Fejωt, Mz=20 in 1 , solve the above dynamic equation: YG=mω2d2AA-mω2-D-ω2Iz2ΣFω2mD+Iz2B-mIzω4-E+mω2d2AA-mω2B-mω2(6)Γ=- CYGmω2d2AA-mω2-D+ω2Iz(7)XG=-AdΓA-mω2(8) where XG, YG, Γ are the complex amplitudes of the machine's center of gravity, and ω is the vibration circle frequency. The coefficients only related to position and stiffness are as follows: A=3i=1ΣKix, B=3i=1ΣKiy, C=3i=1ΣKiyai, D=3i=1ΣKiyai21E=K1yK2ya1-a2+K2yK3ya2-a32+K1yK3ya1-a32 can derive the basic displacement Substituting the power flow formula to obtain a single frequency transfer power flow is: P=12ω2Rejωkfx3i=1ΣkixXG+dΓkix+kfx2+12ω2Rejωkfy3i=1ΣkiyYG-aiΓkiy+kfy2(9) where kfx,kfy is the fundamental transverse and vertical dynamic stiffness, j is Imaginary unit.
Calculation Analysis of Optimal Installation Position and Load Distribution From the above power flow derivation, it can be seen that under the condition of certain machine and basic parameters, the lateral stiffness, longitudinal stiffness, structural damping and installation position of the isolator affect the transmission of power flow. If the foundation and the isolator structure have the same damping factor, and (1), C is zero; at this time, the system vibration is decoupled.
(1) System vibration independence Because of the relationship between Ki0 and Kiy: Ki01+jg=Kiy, g is the structural damping factor when the foundation is the same as the vibration isolator. The single frequency transfer power flow can be reduced to the following formula: P=12ω2F2RejωkfyΣΣ1+g2kfy24π2f2m1+gi-mω2Q where kfy is the fundamental vertical dynamic stiffness, and the coefficient Q is only related to the vibration of the isolator, as follows: Q=K102+ K202+K302 adjusts the air spring stiffness to minimize the transmitted power flow P, that is, the value of ki0 that satisfies (1), (2), and minimizes Q. 2 is the Qmin surface of different installation positions, from which the optimal installation position and the corresponding minimum transfer power flow can be determined, and it is not difficult to find the Qmin and the corresponding ki0 when the installation position is different (every Qmin surface The coordinates of the point -a1/a3 and -a2/a3 are substituted into the formula (1) and the formula (2) is combined to find ki0).
(2) The vibration coupling of the system vibration coupling system is mainly due to the difference between the vibration of the vibration isolator and the foundation; in order to study the influence of the vibration of the vibration isolator on the transmitted power flow, the transmission power flow with the same vibration isolator and the base damping is used as the reference. The study was conducted by comparing the transmitted power flow with the vibration of the isolator. A large number of comparisons are clear, the damping only affects the transmitted power flow in the 1 Hz band near the formant, and the power flow transmission in other frequency regions is basically unchanged. The vibration isolation design generally avoids the frequency resonance peak. Therefore, the optimal installation position and load distribution selected in (1) are also applicable to the system vibration coupling. 3 is the power flow contrast when changing the vibration of the isolator.
Conclusion In this paper, the optimal selection of the installation position and load distribution of the planar air spring vibration isolation system is theoretically derived; the specific solution steps are given; it is proved that the static balance posture of the vibration isolation equipment is kept constant, and the air spring load can be adjusted. Affects the transmission of total power flow; there is an optimal air spring load distribution scheme and corresponding installation position that minimizes the total transmitted power flow; and indicates that the vibration coupling effect due to the vibration of the isolator does not affect the total power flow. Large, can be calculated according to the independent vibration of the system.
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