CN107451355B - Shock absorber design method - Google Patents

Shock absorber design method Download PDF

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CN107451355B
CN107451355B CN201710628133.7A CN201710628133A CN107451355B CN 107451355 B CN107451355 B CN 107451355B CN 201710628133 A CN201710628133 A CN 201710628133A CN 107451355 B CN107451355 B CN 107451355B
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王继强
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a method for designing a shock absorber, which comprises the following steps: step 1, establishing a shock absorber model: writing out a shock absorber motion equation and redefining parameters to obtain a standard form model; step 2, evaluating the feasibility of simultaneous vibration reduction of the vibration reduction system and the isolation system: the method can obtain the characteristics of a series of vibration reduction systems, and the feasibility, the vibration reduction amount and other information of simultaneous vibration reduction of the vibration reduction systems and the isolation systems; step 3, isolation system parameter design: according to the proposed-field method, the transmission of vibration to the vibration reduction system and the isolation system is ensured to be simultaneously attenuated, and the attenuation performance is kept consistent; and 4, evaluating the damping performance of the damper: and (3) selecting a corresponding value according to the method in the step (3), then carrying out feasibility evaluation according to the method in the step (2), and finally determining the final parameters of the isolation system or giving different alternative schemes according to the evaluation result. The invention can obviously improve the vibration damping performance of the existing vibration damper.

Description

Shock absorber design method
Technical Field
The invention belongs to the field of design of vibration dampers, and particularly relates to a design method capable of improving the performance of a vibration damper through global optimization.
Background
Since the invention of the classical undamped vibration absorber, the vibration absorber has been widely used, for example, for vibration absorption of high-rise and tower-type buildings, vibration absorption of train and airplane systems, and the like. This type of damping to absorb the conduction of vibrations is commonly referred to as passive damping. While passive dampers are primarily intended to reduce the amount of steady state vibration, they may be undamped when used at a single frequency, but they only function at this single fixed frequency and have the disadvantage of an infinitely long transient response time. To overcome this drawback, it is often necessary to add damping, although this will result in less than complete absorption of the amount of vibration, but the transient response can be greatly improved and performance gains can be achieved over a larger frequency band. However, if the frequency of the vibration source exceeds the effective frequency band of the vibration absorber, the performance of the passive vibration absorber with damping will be reduced dramatically. Meanwhile, the passive damper cannot be used in the case of system characteristic variation and low frequency band. In this case, an active or semi-active (e.g., active-passive, adaptive-passive, etc.) method is usually adopted to deal with the corresponding vibration problem.
Aiming at an active or semi-active vibration control technology, Williams, K.A. et al introduced a design method for achieving adaptive vibration isolation by using shape memory alloy in "dynamic modeling of shape memory adaptive tuned vibration absorber" in 2005. In fact, the adaptive adjustment technique has been implemented by Kim, J.et al in "New shock parameter tuning method for piezoelectric estimated based on measured electrical impedance" in 2000. However, from design to designFrom the perspective, whether active, passive, or adaptive dampers, the design goal is to find a set of damper parameters such that the vibration transfer is isolated or damped. Specifically, the model of any shock absorber can be simplified into a two-degree-of-freedom mass-spring-damper system, see fig. 1, where d denotes external vibration, u denotes design input, and x denotes1And x2Displacement of two masses (relative to equilibrium position), m1,k1And c1Parameter representing the damping system, m1Representing the mass of the damping system, k1Spring rate of the damping system, c1Representing a damper coefficient of the damping system; m is2,k2And c2Representing an isolated system parameter to be designed, m2Denotes the quality of the insulation system, k2Showing the spring rate of the isolation system, c2Representing a damper coefficient of the isolation system; it is the appropriate m that the damper is designed to pass2,k2and c2The design of the (so-called isolation system) is such that the transmission of the vibrations d to the base (shaded part, so-called damping system) is minimal or even isolated.
To achieve this, there are generally two types of methods, one being the iso-highly optimized method described by Den Hartog in its classic work, Mechanical publications (4 th edition) 1956; another type is a method that is widely adopted at present to find the optimal parameters by forming a Nonlinear constrained optimization problem (Nonlinear constrained optimization problem) and then using a numerical solution. Passive damper designs are described in the review article "Optimal vibration reduction over a frequency range" by Pilkey, Kitis and Wang; active dampers can be found in The atto and furuisi "Active on reactive dampers", Shaw "Active visual adaptive control", Herzog "Active visual passive dampers", and Zuo and Nayfeh "The two-dimensional-coarse-spatial-mass for rendering The two-dimensional-modal visual adaptive excitation".
However, as a result of the above review and numerous other documents and reports, the vibration damper design methods involved at present only consider the transmission of vibration from external disturbances to the base damping system, and do not simultaneously consider the vibration damping problem of vibration to the isolation system formed by the mass. That is, the existing design method does not consider the problem of damping or isolating the vibrations (damping system and isolation system) at both ends of the damper at the same time. In practice, it often happens that although the vibration conduction to the base is suppressed, the vibration at the other end (mass end) is increased, which in turn leads to premature fatigue and other undesirable consequences. Therefore, the design of the damper requires consideration of both ends of the vibration damping. The invention aims to provide a design method for designing a shock absorber, so that the shock absorber can simultaneously attenuate the vibration at two ends, and the problem of vibration enhancement and conduction which may be caused is effectively solved.
Disclosure of Invention
The purpose of the invention is as follows: the invention provides and verifies a novel design method aiming at the problems of performance attenuation, structural fatigue and the like caused by the fact that the existing shock absorber only considers the vibration transmission of a shock absorption system but not simultaneously considers the vibration attenuation of an isolation system, so that the designed shock absorber can simultaneously inhibit the vibration conduction of the shock absorption system and the isolation system.
The technical scheme is as follows: in order to achieve the purpose, the invention adopts the technical scheme that: a method of designing a shock absorber comprising the steps of:
step 1: establishing a shock absorber model;
step 2: evaluating the feasibility of damping the vibration of the vibration damping system and the isolation system simultaneously;
and step 3: designing isolation system parameters;
and 4, step 4: and (5) evaluating the damping performance of the damper.
Further, the specific method of step 1 is as follows: the model of any vibration damper can be simplified into a two-degree-of-freedom mass block-spring-damper system, wherein d represents external vibration, u represents design input, and x represents1And x2Displacement of the two masses relative to the equilibrium position, m1,k1And c1Parameter representing the damping system, m1Representing the mass of the damping system, k1Spring rate of the damping system, c1Representing a damper coefficient of the damping system; m is2,k2And c2Representing an isolated system parameter to be designed, m2Denotes the quality of the insulation system, k2Showing the spring rate of the isolation system, c2Representing a damper coefficient of the isolation system; it is through m that the shock absorber passes2,k2And c2The design of (3) makes the vibration transmission of the external vibration d to the base reach the minimum even isolation;
the system equation of motion is:
Figure BDA0001363268230000031
wherein M represents a mass matrix, C is a damping coefficient matrix, K is a spring stiffness wash matrix, F is an external force vector, and x represents a state variable and is defined as
Figure BDA0001363268230000032
Accordingly, the number of the first and second electrodes,
Figure BDA0001363268230000033
and
Figure BDA0001363268230000034
the derivative and the second derivative of the state variable are respectively;
at any frequency ω, equation of motion (1) is written as an expression of frequency:
x(jω)＝(K-Mω2+jCω)-1F(jω) (2)
and define
Figure BDA0001363268230000035
Is composed of
Figure BDA0001363268230000036
Writing equation (2) in the form of:
Figure BDA0001363268230000037
wherein:
Figure BDA0001363268230000038
Figure BDA0001363268230000039
Figure BDA00013632682300000310
further, the step 2 defines:
Figure BDA00013632682300000311
g is expressed by the right formula, and the feasibility of damping the vibration of the vibration damping system and the isolation system simultaneously is evaluated according to the formula (4):
a. at frequency ω ═ ω0，ω0Is a constant, physical parameter m1,m2,k1,k2,c1And c2Satisfy when Im (g (j omega)0) Re (g (j ω)) when equal to 00) 0, i.e. when g (j ω)0) When the imaginary part is zero and the real part is greater than zero, a compensator k (j omega) exists0) So that u (j ω)0)＝k(jω0)y(jω0) After compensation, the vibration amount y (j ω0) And z (j ω)0) Simultaneously attenuating;
b. at frequency ω ═ ω0，ω0Is constant, let z (j ω)0) Decays to zero while ensuring y (j ω)0) Without increasing, then the physical parameter m1,m2,k1,k2,c1And c2Must satisfy g (j ω0) Is true and (-g (j omega)0) Is located within a unit circle of (-1, 0);
c. at frequency ω ═ ω0，ω0Is constant, let y (j ω0) Decays to zero while ensuring z (j ω)0) Without increasing, then the physical parameter m1,m2,k1,k2,c1And c2Re (g (j ω) must be satisfied0) 0.5, i.e., g (j ω)0) The real part of (a) is greater than 0.5.
Further, the step 3 adopts a domain method, and the minimum positive real number is adopted to ensure that:
|g-1|≤ (5)
converting the design problem of the isolation system into a search parameter m2，c2And k2Satisfies the formula (5).
Further, the specific method of step 4 is as follows: redesigning the Mass m of the isolation System according to equation (5)2Then, there are:
Figure BDA0001363268230000041
will be the parameter m1＝1,k1＝4,k2＝2,c 11 and c2Substituting equation (6) for 0.5 can calculate m2The value range of (A):
Figure BDA0001363268230000042
due to the existence of square roots, it is necessary to satisfy: > 0.78.
Compared with the prior art, the invention has the following beneficial effects:
the invention not only considers the performance of the vibration damping system, but also considers the damping performance of the isolation system to the vibration. Therefore, the performance of the shock absorber is considered, the shock absorption characteristic of the isolation system is improved, and a series of adverse consequences such as structural fatigue and even damage caused by overlarge vibration amplitude of the end of the isolation system are effectively avoided.
Drawings
FIG. 1 is a simplified model of a shock absorber;
FIG. 2 is a schematic diagram of evaluating the simultaneous damping feasibility of a damping system and an isolation system;
FIG. 3 is a schematic view of a damping system and an isolation system that are not simultaneously damping;
FIG. 4 is a schematic illustration of simultaneous damping of the damping system and the isolation system.
Detailed Description
The present invention is further illustrated by the following description in conjunction with the accompanying drawings and the specific embodiments, it is to be understood that these examples are given solely for the purpose of illustration and are not intended as a definition of the limits of the invention, since various equivalent modifications will occur to those skilled in the art upon reading the present invention and fall within the limits of the appended claims.
Fig. 1 shows a method for designing a shock absorber, comprising the steps of:
step 1: establishing a shock absorber model; the model of any vibration damper can be simplified into a two-degree-of-freedom mass block-spring-damper system, wherein d represents external vibration, u represents design input, and x represents1And x2Displacement of the two masses relative to the equilibrium position, m1,k1And c1Parameter representing the damping system, m1Representing the mass of the damping system, k1Spring rate of the damping system, c1Representing a damper coefficient of the damping system; m is2,k2And c2Representing an isolated system parameter to be designed, m2Denotes the quality of the insulation system, k2Showing the spring rate of the isolation system, c2Representing a damper coefficient of the isolation system; it is through m that the shock absorber passes2,k2And c2The design of (3) makes the vibration transmission of the external vibration d to the base reach the minimum even isolation;
as shown in fig. 1, the system equation of motion is first written:
Figure BDA0001363268230000051
wherein M represents a mass matrix, C is a damping coefficient matrix, K is a spring stiffness wash matrix, F is an external force vector, and x represents a state variable and is defined as
Figure BDA0001363268230000052
Accordingly, the number of the first and second electrodes,
Figure BDA0001363268230000053
and
Figure BDA0001363268230000054
the derivative and the second derivative of the state variable are respectively;
Figure BDA0001363268230000055
and
Figure BDA0001363268230000056
then, at any frequency ω, the equation of motion of the time domain representation described above can be written as a frequency representation:
x(jω)＝(K-Mω2+jCω)-1F(jω), (2)
it is assumed that the inversion operation is established.
Is defined as follows
Figure BDA0001363268230000057
Is composed of
Figure BDA0001363268230000058
Equation (2) can be written as a "standard" form as follows for the damper model:
Figure BDA0001363268230000059
wherein:
Figure BDA00013632682300000510
Figure BDA00013632682300000511
Figure BDA00013632682300000512
step 2: evaluating the feasibility of damping the vibration of the vibration damping system and the isolation system simultaneously;
first, the following definitions are made:
Figure BDA00013632682300000513
substitution into g11(jω)、g22(jω)、g12(j ω) and g21(j ω) gives:
Figure BDA00013632682300000514
g is defined as the expression of the right formula.
It can be demonstrated that:
conclusion 1: at frequency ω ═ ω0，ω0Is constant if the physical parameter m1,m2,k1,k2,c1And c2Satisfy when Im (g (j omega)0) Re (g (j ω)) when equal to 00) 0 (i.e., when g (j ω)0) When the imaginary part is zero, the real part is greater than zero), then a compensator k (j ω) exists0) So that u (j ω)0)＝k(jω0)y(jω0) After compensation, the vibration amount y (j ω0) And z (j ω)0) While attenuating.
Conclusion 2: at frequency ω ═ ω0，ω0Is constant if z (j ω) is made0) Decays to zero while ensuring y (j ω)0) Without increasing, then the physical parameter m1,m2,k1,k2,c1And c2Must satisfy g (j ω0) Is positive real (positive real) and (-g (j omega)0) Is located within the unit circle of (-1, 0).
Conclusion 3: at frequency ω ═ ω0，ω0Is constant if y (j ω0) Decays to zero while ensuring z (j ω)0) Without increasing, then the physical parameter m1,m2,k1,k2,c1And c2Re (g (j ω) must be satisfied0) 0.5 (i.e., g (j ω)0) The real part of (a) is greater than 0.5).
The above conclusions can be collectively expressed by a schematic diagram as shown in fig. 2. In the figure, α -plan represents a complex plane with the abscissa representing the real part (denoted Re) and the ordinate representing the imaginary part (denoted ImShown in (a); unit alpha-circle is a Unit circle; the center of the beta-circle is (-g (j omega)0) With a radius of | g (j ω)0) A circle of |. Alpha optIs a connecting line connecting the centers of the two circles, and the shaded area represents the compensator k (j ω)0) Such that u (j ω) is within a range of values0)＝k(jω0)y(jω0) After compensation, the vibration amount of the vibration damping system and the vibration isolating system are simultaneously attenuated.
And step 3: designing isolation system parameters;
as can be seen from fig. 2, if the transmission of vibrations to the damping system and the isolation system is simultaneously damped and the damping performance is completely consistent, the conditions must be satisfied:
-g＝-1 (6)
namely:
Figure BDA0001363268230000061
the above formula can be written as:
m1m2ω4-(m1k2+m2k2+m2k1+c1c2)ω2+k1k2＝0 (8-1)
and
(m2c1+m1c2+m2c2)ω2-c1k2-c2k1＝0 (8-2)
in a real system, a damping system parameter m is given1,k1And c1And isolation system parameter m2,c2Or k2One, due to m2,c2And k2Must be positive, so it is not always possible to find a set of two other parameters such that equations (8-1) and (8-2) hold. Therefore, a domain method is usually adopted in the design, that is, the minimum positive real number is taken so that:
|g-1|≤ (9)
that is, the design problem of the isolation system translates into finding the parameter m2,c2And k2To be full ofThe foot inequality (9). The vibration damper designed in this way can ensure that the transmission of vibration to the vibration damping system and the isolation system is simultaneously damped and the damping performance is kept consistent.
And 4, step 4: evaluating the vibration reduction performance of the vibration absorber;
taking a certain set of parameters as an example for explanation, let m1＝1,m2＝5,k1＝4,k2＝2,c11 and c20.5. At natural frequency
Figure BDA0001363268230000071
Here, it can be calculated that:
Figure BDA0001363268230000072
the corresponding simultaneous damping feasibility diagram is shown in fig. 3. Fig. 3 clearly shows that the two circles have no intersection area except the origin. It can therefore be seen that there is no compensator, so that y (jw)0) And z (j ω)0) At the same time, vibration damping is obtained, i.e. the transmission of vibrations to the damping system and the isolation system cannot be damped at the same time.
The proposed method is now used for design, i.e. using condition (9), assuming that the quality of the isolation system needs to be redesigned m2Then, there are:
Figure BDA0001363268230000073
carry over the value to get m2The value range of (A):
Figure BDA0001363268230000074
since square roots exist, it is satisfied that:
＞0.78. (13)
although this indicates that it is impossible to design the centers of the two circles to be completely coincident (the minimum distance is 0.78), since 0.78 is located within the unit circle, z (j ω) can be made as seen from conclusion 20) Attenuation ofTo zero while ensuring y (j ω0) The vibration reduction system can be enabled to realize complete isolation of vibration transmission without increasing, and meanwhile, the isolation system is guaranteed not to strengthen the vibration transmission. This is still a fairly good design.
If now 1 is selected which satisfies the condition (13), the condition (12) results in:
0＜m2≤0.37. (14)
that is to say when m2When the range (14) is taken, the following are provided:
|g-1|≤1, (15)
get m20.37 gave:
-g＝-0.3344-0.7408j. (16)
at this time, the schematic view of the simultaneous damping feasibility of the damping system and the isolation system becomes fig. 4. It can be seen from the figure that there are a large number of crossing regions, y (j ω), between the two circles0) And z (j ω)0) Attenuation can be obtained simultaneously; and the compensator can be designed such that z (j ω) is0) Decays to zero and holds y (j ω)0) Not increased.
The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.

Claims (1)

1. A method for designing a shock absorber is characterized in that: the method comprises the following steps:
step 1: establishing a shock absorber model;
the specific method of the step 1 comprises the following steps: the model of any vibration damper can be simplified into a two-degree-of-freedom mass block-spring-damper system, wherein d represents external vibration, u represents design input, and x represents1And x2Displacement of the two masses relative to the equilibrium position, m1,k1And c1Parameter representing the damping system, m1Representing the mass of the damping system, k1Spring rate of the damping system, c1Representing a damper coefficient of the damping system; m is2,k2And c2Representing an isolated system parameter to be designed, m2Denotes the quality of the insulation system, k2Showing the spring rate of the isolation system, c2Representing a damper coefficient of the isolation system; it is through m that the shock absorber passes2,k2And c2The design of (3) makes the vibration transmission of the external vibration d to the base reach the minimum even isolation;
the system equation of motion is:
Figure FDA0002518779360000011
wherein M represents a mass matrix, C is a damping coefficient matrix, K is a spring stiffness wash matrix, F is an external force vector, and x represents a state variable and is defined asAccordingly, the number of the first and second electrodes,
Figure FDA0002518779360000013
and
Figure FDA0002518779360000014
the derivative and the second derivative of the state variable are respectively;
at any frequency ω, equation of motion (1) is written as an expression of frequency:
x(jω)＝(K-Mω2+jCω)-1F(jω) (2)
and define
Figure FDA0002518779360000015
Is composed of
Figure FDA0002518779360000016
Writing equation (2) in the form of:
Figure FDA0002518779360000017
wherein:
Figure FDA0002518779360000018
Figure FDA0002518779360000019
Figure FDA00025187793600000110
step 2: evaluating the feasibility of damping the vibration of the vibration damping system and the isolation system simultaneously;
defining:
Figure FDA0002518779360000021
g is expressed by the right formula, and the feasibility of damping the vibration of the vibration damping system and the isolation system simultaneously is evaluated according to the formula (4):
a. at frequency ω ═ ω0，ω0Is a constant, physical parameter m1,m2,k1,k2,c1And c2Satisfy when Im (g (j omega)0) Re (g (j ω)) when equal to 00) 0, i.e. when g (j ω)0) When the imaginary part is zero and the real part is greater than zero, a compensator k (j omega) exists0) So that u (j ω)0)＝k(jω0)y(jω0) After compensation, the vibration amount y (j ω0) And z (j ω)0) Simultaneously attenuating;
b. at frequency ω ═ ω0，ω0Is constant, let z (j ω)0) Decays to zero while ensuring y (j ω)0) Without increasing, then the physical parameter m1,m2,k1,k2,c1And c2Must satisfy g (j ω0) Is true and (-g (j omega)0) Is located within a unit circle of (-1, 0);
at frequency ω ═ ω0，ω0Is constant, let y (j ω0) Decays to zero while ensuring z (j ω)0) Without increasing, then the physical parameter m1,m2,k1,k2,c1And c2Re (g (j ω) must be satisfied0) 0.5, i.e., g (j ω)0) The real part of (a) is greater than 0.5;
and step 3: designing isolation system parameters;
using a domain method, taking the minimum positive real number to make:
|g-1|≤ (5)
converting the design problem of the isolation system into a search parameter m2，c2And k2Satisfaction formula (5)
And 4, step 4: evaluation of vibration damping performance of the vibration damper:
the specific method comprises the following steps: redesigning the Mass m of the isolation System according to equation (5)2Then, there are:
Figure FDA0002518779360000022
will be the parameter m1＝1,k1＝4,k2＝2,c11 and c2Substituting equation (6) for 0.5 can calculate m2The value range of (A):
Figure FDA0002518779360000023
due to the existence of square roots, it is necessary to satisfy: > 0.78.
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CN104361158A (en) * 2014-10-31 2015-02-18 沈阳化工大学 Simulation analysis method for influence of forge hammer vibration system parameters on vibration response
CN106122605A (en) * 2016-08-19 2016-11-16 中国船舶重工集团公司第七〇九研究所 A kind of pipeline dynamic vibration absorber

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