JP2008101782A - Dynamic vibration absorber - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a dynamic vibration absorber having consistently stable vibration damping effects without being influenced by a parameter fluctuation of a vibration damped object or the dynamic vibration absorber itself. <P>SOLUTION: The dynamic vibration absorber 1 comprises a plurality of spring members 6a, 6b extending from a common supporting member 2, weights 7 mounted at tips of the spring members 6a, 6b, respectively, so as to be cantilevered and supported in a rockable manner, and a damping member 12 mounted for damping the rocking of the weights 7. <P>COPYRIGHT: (C)2008,JPO&INPIT

Description

  The present invention relates to a dynamic vibration absorber for the purpose of vibration and noise prevention of various structures using metal materials such as steel materials as structural members in bridges, iron bridges, ships, buildings, automobiles, and various industrial machines. is there.

  Since steel materials are excellent as structural members, they are used in many structures. On the other hand, since the internal damping is extremely small, there is also a problem that causes vibration and noise. For example, railway railway bridges generate noise when vehicles pass through, and the magnitude of the noise increases as the vehicle speed increases, so speed regulation is unavoidable to keep the noise level below the standard. . In the recent Kobe earthquake, concrete piers were broken over 500 meters and became a big topic. However, concrete was used to lower the noise level in the city area, and as a result, this was caused. Conceivable. In order to quiet the interior of an automobile, it is necessary to prevent engine vibration and vibration from the road surface from being amplified by the vehicle body. For this reason, the more expensive a car is, the more putty is put on the wall of the car body. This problem goes against trying to save energy by reducing the weight of the car body.

  Bridge vibrations caused by the passing of trucks, etc. are transmitted to the ground and invade the habitability of local residents, creating a social problem. In order to solve this problem, an attempt to suppress vibration of the bridge by attaching a dynamic vibration absorber is also being studied. However, the vehicle weight on the bridge is constantly changing, and the natural frequency of the bridge changes accordingly. A dynamic vibration absorber is an effective means for a vibration control target having a certain natural frequency, but its effectiveness is greatly reduced for a variable object. In addition, the temperature environment where the bridge is placed varies greatly between summer and winter, daytime and nighttime, but dynamic vibration absorbers have the problem that the vibration control performance is greatly affected by temperature changes. For various industrial machines, vibration dampers, such as dynamic vibration absorbers, have been used to eliminate vibration problems, but they cannot respond to changes in the natural frequency or environmental temperature of structures. Cause similar problems.

  As described above, the conventional techniques are summarized and the methods for reducing vibration and noise are listed as follows.

  (1) Use structural materials with large internal damping such as damping materials and concrete.

  (2) A vibration-absorbing material such as rubber or putty is stretched on the surface of the structural member.

  (3) A vibration damper such as a dynamic vibration absorber is attached to provide damping of the structural member from the outside.

  However, the method of item (1) has a problem in strength as a structural member, and the internal damping is larger than that of steel materials, but it is not possible to obtain such a large damping that the vibration problem can be solved. Item (2) is an extremely empirical and aggressive method that is effective at high frequencies but is known to have little effect on damping low frequencies such as low-order mode vibrations. .

  For item (3), the optimum design method of the vibration damper is well known. If a dynamic vibration absorber or the like is designed and adjusted based on this, vibration is well suppressed from the low-order vibration mode to the high-order vibration mode. In particular, a dynamic vibration absorber is often used because it represents the vibration damper and exhibits the most excellent vibration damping performance. However, it has the following problems, which has become a major obstacle to practical use.

  (A) The damping effect is drastically impaired by parameter fluctuations of the damping target such as a change in the natural frequency of the damping target.

  (B) The vibration damping effect is impaired by fluctuations in parameters of the dynamic vibration absorber such as a change in the damping coefficient of the dynamic vibration absorber.

  (C) Although the dynamic vibration absorber can be made small, it is still difficult to attach to the outside so as not to stand out.

  (D) A dynamic vibration absorber designed to suppress low-frequency vibrations does not contribute much to suppression of high-frequency noise.

  The dynamic vibration absorber according to the present invention extends a plurality of spring members from a common support member, attaches weights to the tips of the spring members, supports the weights in a swingable manner, and swings the weights. A damping member for damping the movement is attached.

  In other words, by dividing a single dynamic vibration absorber into multiple parts and applying an optimal design to each of them, the influence of parameter fluctuations on the vibration suppression target and the dynamic vibration absorber itself can be reduced, and the structure can be made compact. In addition, it is also effective for high-order mode noise control.

  The present invention exhibits the following excellent effects.

  (1) A stable vibration control effect can always be exhibited without being affected by parameter variations of the vibration control target or the dynamic vibration absorber itself.

  (2) It is also effective in suppressing high frequency noise.

  (3) Miniaturization can be achieved.

  Preferred embodiments of the present invention will be described below in detail with reference to the accompanying drawings.

  FIG. 1 shows an example of the structure of a double dynamic vibration absorber that forms the minimum configuration of the dynamic vibration absorber according to the present invention.

As shown in the figure, the double dynamic vibration absorber 1 has a support member 2 fixed to the structure S to be controlled by an appropriate means (bolt, adhesive, etc.), and the support member 2 can be divided. The lower block 3 and the upper block 4 are formed. The lower block 3 and the upper block 4 are fastened with two bolts 5. At the time of fastening, the leaf springs 6 are sandwiched between the blocks 3 and 4 and between the upper block 4 and the bolts 5. . The upper and lower leaf springs 6 are parallel springs, and both ends thereof are extended to both sides of the support member 2, and a block member 7 as a weight is disposed between the upper and lower ends. Two long holes 8 are provided at both ends of the leaf spring 6, and bolts 9 are inserted into the long holes 8, and the bolts 9 fix the block member 7 at an appropriate position. In this way, the block member 7 is movably attached along the longitudinal direction of the leaf spring 6, and its position can be adjusted. In particular, the left extension 6a and the right extension 6b of the leaf spring 6 extending from the support member 2 have different lengths L ₁ and L ₂ , and here the left extension 6a is on the right. It is longer than the extending portion 6b (L ₁ > L ₂ ).

  Thus, the two spring members formed by the substantially half portions of the parallel spring (upper and lower sets of the left extension portion 6a and the right extension portion 6b) are extended from the common support member 2, and are respectively attached to the tips of these spring members. Weights are attached, and these weights are supported in a swingable manner.

  From both side surfaces of the upper block 4, a threaded shaft 10 penetrating and fixed to the upper block 4 is extended, and an adjustment nut member 11 is screwed and attached to the tip of the threaded shaft 10. The adjustment nut member 11 integrally has a damping member 12 at its tip. The damping member 12 is made of a damping material such as a polymer material such as silicon gel. Then, by adjusting the position of the adjusting nut member 11 along the shaft 10 or along the extending direction of the leaf spring 6, the damping member 12 gives a variable pressing force to the block member 7, and the block member 7 is appropriately swung. Attenuates.

  As will be described in detail later, the two spring members are defined as spring constants k1 and k2, the two weights are defined as m1 and m2, and the two damping materials are defined as c1 and c2. Design to take. In this structure, the fine tuning can be performed so that the optimum tuning can be obtained by adjusting the weight and the optimum damping can be obtained by adjusting the position of the adjusting nut member 11.

  FIG. 2 shows a configuration example of a quadruple dynamic vibration absorber that is another embodiment of the dynamic vibration absorber according to the present invention. In the quadruple vibration absorber 21, a leaf spring 25 formed in a cross shape is fixed between a lower block 23 and an upper block 24 constituting a support member 22 by a bolt 26. The four extending portions 25a of the leaf spring 25 form spring members, and block members 27 as weights are placed and fixed to the distal ends of these extending portions 25a by bolts 28a and nuts 28b, respectively. A long hole 28c is provided at the distal end of the extending portion 25a, whereby the position of the block member 26 can be adjusted. The lengths of the four extending portions 25a are slightly different from each other.

  A damping member 29 is compressed and inserted into a gap defined by the outer surface of the upper block 24, the upper surface of the extending portion 25a, and the inner surface of the block member 26. The attenuation member 29 fills up the gap and is flush with the upper surfaces of the upper block 24 and the block member 26.

  As described in detail later, a thin and compact dynamic vibration absorber can be realized by dividing the weight equivalent to the single dynamic vibration absorber into four equal parts and distributing them. If the weights are divided into four and m1 to m4 are determined, the spring constants k1 to k4 and the damping coefficients c1 to c4 are determined by equations (22) and (29) described later.

  FIG. 3 shows an eight-fold dynamic vibration absorber that is another embodiment of the dynamic vibration absorber according to the present invention. In the eight-fold vibration damper 31, the support member 32 is integrally formed by a rectangular parallelepiped upper block 33 and a lower block 34 that are long in the front-rear direction. Four leaf springs 35 are arranged in parallel and extend from both side surfaces of the upper block 33. The lengths of the leaf springs 35 are different from each other, and a block member 36 serving as a weight is integrally attached to each tip. Both ends of the leaf spring 35 are connected to the height center positions of the upper block 33 and the block member 36, and a damping member 37 is integrally fixed to the upper and lower gaps of the leaf spring 35 unlike the above. .

  In particular, this eight-fold vibration absorber 31 has four of these attached as a basic unit. If the number is two, a quadruple dynamic vibration absorber is formed, and if the number is three, a six dynamic vibration absorber is formed.

  FIG. 4 shows a twelfth dynamic vibration absorber constituting another form of the dynamic vibration absorber according to the present invention. In the twelve double vibration damper 41, a support member 42 is integrally formed by a short columnar upper block 43 and a lower block 44. From the side surface of the upper block 43, twelve sheets having the same or different length are provided. The leaf springs 45 extend radially at equal intervals (in the radial direction with the upper block 43 as the center). A block member 46 as a weight is integrally attached to the tip of each leaf spring 45. Further, the damping member 47 is integrally fixed to the upper and lower gaps of the leaf spring 45 as described above.

  Thus, the dynamic vibration absorber according to the present invention is a multi-dynamic vibration absorber (combined dynamic vibration absorber) having a plurality of one unit of dynamic vibration absorbers including a spring member, a weight, and a damping member.

  Now, the basic configuration of the multi-vibration absorber of the present invention represented by the above embodiment and an analysis method for optimally designing the same will be described.

  (A) Analysis of a multi-vibration absorber and a dynamic model in which a multi-vibration absorber is installed on a displacement amplitude ratio damping object is shown in FIG. Here, M and K are the mass and spring constant of the one-degree-of-freedom model of the object to be controlled, and mi, ki, and ci are the mass, spring constant, and damping coefficient of the i-th dynamic vibration absorber. In this model, the damping of the structure is considered to be negligible and negligible. The equation of motion derived from this dynamic model is shown below. Where x is the displacement of the vibration suppression object, xi and fi are the displacement of the i th dynamic vibration absorber and the force acting on the vibration suppression object, and fin is the external force acting on the vibration suppression object. (I = 1,2,3, ....., n)

  However,

  And

  Next, in this vibration system,

  far. However, X, Xi, and F represent the complex amplitude of x, xi, and fi. Substituting these into equation (3) gives

  Also,

  From the relationship

  Therefore,

  It becomes. Substituting this into equation (1) gives

If the ratio fin / K between the external force fin and the spring constant K of the main vibration system is defined as the static deflection Xst, the displacement amplitude ratio X / Xst can be expressed as follows.

  Here, the dimensionless numbers shown below are introduced.

  λ is a forced frequency ratio, μ is a mass ratio, γ is a natural frequency ratio, and ζ is a damping rate. Therefore, the displacement amplitude ratio is

  However,

The magnitude of the displacement amplitude ratio is

  It becomes.

  (B) Optimum adjustment condition for multi-vibration absorber The purpose of optimum adjustment of the compound dynamic absorber is to reduce the maximum displacement amplitude ratio | X / Xst | max of the main vibration system. As the method, when the number of installed dynamic vibration absorbers is one or two, the fixed point theory can be applied. Fixed point theory is as follows.

  When the spring constant of the dynamic vibration absorber is set to a constant value and the attenuation coefficient is changed, there are a plurality of fixed points that are not related to the value of the attenuation coefficient on the displacement amplitude curve. Therefore, the following method is used.

(1) Align the heights of these fixed points (optimal tuning)
(2) Make local maximum values near these fixed points (optimum attenuation)
A fixed point theory is one in which optimum adjustment is performed by these two methods.

  However, if the number of installed dynamic vibration absorbers is 3 or more, the fixed points as described above do not exist, and optimal adjustment based on the fixed point theory cannot be expected.

Therefore, when the displacement amplitude curve when n dynamic vibration absorbers are attached to the main vibration system, μ ₁ ... Μn is a constant value and γ ₁ ... Γn and ζ ₁ . I understood. (1) There are at most (n + 1) maximum values and n minimum values.

(2) The location (x coordinate) of n local minimum values is greatly affected by n 1 / γ ₁ .

(3) The difference in height (y coordinate) between a certain minimum value and the maximum value located on both sides thereof is greatly affected by n ζ ₁ .

Here, the vicinity of the maximum value is called a peak, and the vicinity of the minimum value is called a notch, and how (1), (2), and (3) are used to lead to optimum tuning and optimum attenuation is described below. Show.

  The optimum tuning condition is defined as “aligning heights of (n + 1) peaks”. As an evaluation function for that purpose, a value obtained by squaring and adding the difference between the average height of (n + 1) peaks and the height of each peak was considered. This is I.

That is, the optimum tuning is to change 1 / γ ₁ ... 1 / γn and minimize I.

  The optimum attenuation condition is defined as “minimizing the average height of (n + 1) peaks”. As an evaluation function for that purpose, the average height of (n + 1) peaks was considered according to the conditions. This is J.

That is, the optimum attenuation is obtained by changing ζ ₁ .

  Calculation is performed by a program that creates a numerical solution that satisfies the above two equations (20) and (21).

(C) As an example of the optimum design chart and the practical approximation formula, FIG. 6 shows an optimum design chart of the quadruple vibration damper obtained by performing numerical analysis by calculation. 4 and the mass ratio mu ₄ parameters of ponderomotive vibration absorber, the optimum natural frequency ratio of the dynamic vibration reducer _{1 / γ 1, 1 / γ} 2, 1 / γ 3, 1 / γ 4, damping factor zeta _1, zeta ₂ , ζ ₃ , ζ ₄ and the maximum displacement amplitude ratio | X / Xst | max at that time are shown.

  When designing a dynamic vibration absorber, it is only necessary to set the mass ratio and read the natural frequency ratio and damping rate from this chart. However, it is inconvenient in actual use. Therefore, the following practical approximation formula is proposed.

  The plot points in FIG. 6 are values obtained by calculation, and it can be seen that each approximate expression well approximates the characteristics of the chart.

  Next, the damping effect of the multi-dynamic vibration absorber optimally designed by the above method will be described.

  (A) Damping effect of multi-vibration absorber The damping effect is shown when the total mass ratio of the dynamic absorber is 0.1.

  In addition, in order to compare with a single dynamic vibration absorber, the mass ratio of each dynamic vibration absorber was determined as follows, and the total mass ratio of the double dynamic vibration absorber and the combined dynamic vibration absorber was all the same.

Single dynamic vibration absorber (SDA) μ ₁ = 0.1
2 ponderomotive vibration absorber _{(DDA) μ 2 = μ 1} /2 (= 0.05)
4 ponderomotive vibration absorber _{(4.DA) μ 4 = μ 1} /4 (= 0.025)
6 ponderomotive vibration absorber _{(6.DA) μ 6 = μ 1} /6 (= 0.0167)
8 ponderomotive vibration absorber _{(8.DA) μ 8 = μ 1} /8 (= 0.0125)
FIG. 7 shows a displacement amplitude curve obtained by the optimum adjustment method for each heavy vibration absorber. Moreover, the impulse response under the same conditions is shown in FIG.

  As a result, the maximum displacement amplitude ratios of the quadruple, sixfold, and eightfold vibration absorbers are suppressed by about 9.3%, 12.5%, and 13.7%, respectively, compared with the double motion vibration absorber. Further, since the dynamic vibration absorber is excited and vibrated by the vibration to be controlled, the effect of the composite dynamic vibration absorber does not appear in the first period wave in FIG. Shows the effect of the combined dynamic vibration absorber. Obviously, the amplitude of the waveform in the second period is reduced by using a quadruple or quintuple dynamic vibration absorber, and it can be seen that vibration is quickly suppressed. Thereby, it turns out that the multi dynamic vibration absorber of this invention exhibits the high damping effect.

  (B) Effect on fluctuation of damping coefficient Among the components of the dynamic vibration absorber, the damping element (damper) is most susceptible to the use environment. Currently, there are materials using oil, magnetism and gel materials as damping materials.

  The fluctuation of the damping coefficient c of the dynamic vibration absorber is shown as a change in the damping rate ζ. Compare the displacement amplitude curves when the optimally adjusted attenuation rate is varied from 0.2 times to 3.0 times. FIG. 9 shows a displacement amplitude curve when each dynamic vibration absorber that is optimally adjusted is installed in the main vibration system, and the damping rate ζ is changed by taking a quadruple vibration vibration absorber as an example.

  FIG. 10 shows the relationship between the change in the maximum displacement amplitude ratio of each curve and the change rate of the damping rate ζ of the dynamic vibration absorber.

  As a result, it has been clarified that a composite dynamic vibration absorber having a plurality of dynamic vibration absorbers can be less affected by fluctuations in the damping coefficient c than single and double dynamic vibration absorbers. Particularly noteworthy is that the multi-vibration absorber can be made extremely insensitive to fluctuations in attenuation if the damping rate is set to a value larger than the optimum value in advance. I understood.

  (C) Effect on natural frequency fluctuation The fluctuation of the natural frequency Ω (= √ (K / M)) is a change of the spring constant K of the main vibration system. Compare the displacement amplitude curves when the spring constant K is changed from 0.7 times to 1.4 times. FIG. 11 shows an example of a quadruple vibration damper. Further, FIG. 12 shows the change of the maximum displacement amplitude ratio with respect to the fluctuation of the natural frequency of the double, quadruple, six-fold, and eight-fold dynamic vibration absorbers as compared with the single dynamic vibration absorber. In this case, the region where the multi-vibration absorber is good has a spring constant of 0.9 or more and 1.1 or less, and beyond that, the single-vibration absorber gives better results. But this issue is not important. This can be easily solved by a design method in which the natural frequency of each dynamic vibration absorber is shifted up and down in anticipation of the fluctuation when the spring constant is expected to fluctuate.

  FIG. 13 shows changes in the maximum displacement amplitude ratio with respect to fluctuations in the natural frequency of each heavy vibration absorber when the spring constant is shifted up and down by 5%. Although the maximum amplitude ratio near the optimum value increases, it can be seen that the maximum amplitude ratio is kept low over a wide range of fluctuation.

  From the above, it is understood that the multi-dynamic vibration absorber of the present invention including the result of (b) can always exhibit a stable vibration-damping effect without being affected by parameter variations of the vibration-damping target or the dynamic vibration absorber itself. The

  (D) Noise control effect Finally, a prototype quadruple vibration damper is mounted on a flat plate and a comparison of the sound emitted from the flat plate is shown in FIG. In the figure, the blackened portion is where the peak is reduced due to the installation of the dynamic vibration absorber. In particular, it can be seen that the resonance peak is greatly reduced from 1 KHz to 4 KHz, which is sensitive to the human ear. This is due to the hood damper effect brought about by the quadruple vibration absorber, and it was confirmed from this result that the multi-vibration absorber of the present invention is also effective in a noise boundary.

  As described above, the multi-dynamic vibration absorber according to the present invention optimizes each dynamic vibration absorber and always produces a stable vibration damping effect. If space restrictions allow, the number of members can be increased to further increase the number of members. In particular, according to such a form, a flat, thin and compact dynamic vibration absorber can be realized, and it can be attached to a structure so as to be inconspicuous. It is optional to accommodate the dynamic vibration absorber in the casing.

  In particular, when the equivalent mass to be damped is extremely large, a general dynamic vibration absorber becomes large due to a large weight mass, and is easily limited by the installation location. The dynamic vibration absorber of the present invention is obtained by dividing a single dynamic vibration absorber into a plurality of parts, thereby reducing the size of the spring member, weight, and damping member as constituent elements.

  In addition, the conventional single dynamic vibration absorber is designed to be damped based on the natural frequency of the structure, but if this occurs, the natural frequency changes, for example, when there is a mass change due to vehicle passage on a bridge. The damping effect will be significantly worsened. Although the multi-vibration absorber of the present invention is designed to be damped based on the natural frequency of the structure as a whole, each dynamic vibration absorber has a different natural frequency, and the natural vibration of the structure. One of the dynamic vibration absorbers acts in response to the number change, absorbs the natural frequency change, and exhibits a high damping effect in a wide frequency range.

  As a configuration for this, in such a multi-vibration absorber, the length of the leaf spring is made unequal, or the position of the weight can be changed, so that the spring constant and the natural frequency are individually given different values. I am doing so. In the double vibration damper 1, the pressing force of the damping member 12 is variable, and in the four vibration damper 21, the compression state of the attenuation member 29 is variable by adjusting the position of the weight. In the dynamic vibration absorbers 31 and 41, the damping coefficients 37 and 47 are changed so that the damping coefficients are individually set to different values. In particular, in the double and quadruple dynamic vibration absorbers 1 and 21, the damping coefficient also changes in accordance with the change in the weight position.

  And the dynamic vibration absorber of this invention exhibits an effect also in the reduction of the noise accompanying vibration. Furthermore, the vibration state of the plate material or wall material varies depending on the location. If the dynamic vibration absorber of the present invention is applied to this, the variation can be absorbed and vibration and noise can be effectively reduced.

  In addition, the dynamic vibration absorber of the present invention can be manufactured lighter and cheaper than a conventional dynamic vibration absorber using a magnetic damper or the like, and the values of the spring constant, weight mass, damping coefficient, etc. can be easily changed and adjusted. There are benefits.

Embodiment of the dynamic vibration damper which concerns on this invention is shown, (a) is a top view, (b) is a front view. The other form of the dynamic vibration damper which concerns on this invention is shown, (a) is a top view, (b) is a front view. The other form of the dynamic vibration damper which concerns on this invention is shown, (a) is a top view, (b) is a front view. The other form of the dynamic vibration damper which concerns on this invention is shown, (a) is a top view, (b) is AA sectional drawing. It is a figure which shows the dynamic model of the dynamic vibration damper which concerns on this invention. It is an optimal design chart of a quadruple dynamic vibration absorber. It is a graph which shows the comparison of each dynamic vibration damper in an optimal design value. It is a graph which shows the impulse response in an optimal design value. It is a graph which shows the displacement amplitude curve at the time of changing an attenuation factor. It is a graph which shows the relationship between the change of the maximum displacement amplitude ratio of each curve of FIG. 9, and the change rate of the attenuation factor of a dynamic vibration absorber. It is a graph which shows the displacement amplitude curve at the time of changing a spring constant. It is a graph which shows the change of the maximum displacement amplitude ratio with respect to the fluctuation | variation of the natural frequency of a double, quadruple, sixfold, and eightfold dynamic vibration absorber compared with a single dynamic vibration absorber. It is a graph which shows the change of the maximum displacement amplitude ratio with respect to the fluctuation | variation of the natural frequency of each heavy dynamic vibration absorber when a spring constant is shifted only 5% up and down. It is a graph which shows the comparison of the radiated sound at the time of attaching a quadruple vibration damper to a flat plate.

Explanation of symbols

1, 2, double vibration absorber 2, 22, 32, 42 Support member 6, 25, 35, 45 Leaf spring (spring member)
6a Left extension (spring member)
6b Right-hand extension (spring member)
7, 27, 36, 46 Block member (weight)
12, 29, 37, 47 Damping member 21 Quadruple vibration absorber 25a Extension part (spring member)
31 8 double vibration absorber 41 12 double vibration absorber

Claims (7)

   A plurality of spring members are extended from a common support member, weights are attached to the tips of the spring members, and the weights are cantilevered in a swingable manner, and a damping member for attenuating the swing of the weights is provided. A dynamic vibration absorber characterized by being attached.   The dynamic vibration absorber according to claim 1, wherein the weight is attached so as to be movable along the spring member.   The dynamic vibration absorber according to claim 1 or 2, wherein the damping member is attached so as to be movable along the extending direction of the spring member to give a variable pressing force to the weight.   The dynamic vibration absorber according to claim 1, wherein the damping member is integrally fixed to the support member, the spring member, and the weight.   The dynamic vibration absorber according to claim 1, wherein the spring member extends in a radial direction centering on the support member.   The dynamic vibration absorber according to any one of claims 1 to 4, wherein the spring members are arranged in parallel on both sides of the support member.   The dynamic vibration absorber according to any one of claims 1 to 6, wherein a natural frequency for at least one weight or a damping coefficient by the damping member for at least one weight is different from that for other weights.

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