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Why Bassreflex is not Suitable for a Subwoofer

Robert-H Munnig Schmidt 罗伯特-芒尼格-施密特

(C) 2017 The author, RMS Acoustics & Mechatronics and Grimm Audio. All rights reserved.
(C) 2017 作者、RMS Acoustics & Mechatronics 和 Grimm Audio。保留所有权利。
Copying of the complete document is allowed for personal use only.
The author/publisher is not responsible for any problems that might arise by using the contents of this paper.
Published by RMS Acoustics & Mechatronics
由 RMS Acoustics & Mechatronics 出版
The Netherlands 荷兰

Contents 目录

1 Introduction ..... 3 1 简介 .....3
2 Helmholz Resonator ..... 4
2 Helmholz 谐振器 .....4

3 Bassreflex Systems for Very Low Frequencies ..... 5
3 用于极低频的低音反射系统 .....5

4 Bassreflex Dynamics ..... 6
4 低音反射动态 .....6

5 Bassreflex with Diaphragm Resonator ..... 6
5 带振膜谐振器的低音反射式 .....6

5.1 Frequency Response of Diaphragm Resonator ..... 7
5.1 膜片谐振器的频率响应 .....7

5.2 Time Domain Response of Sine Signal . ..... 12
5.2 正弦信号的时域响应 . .....12

5.3 Modal analysis of Diaphragm Resonator. ..... 13
5.3 膜片谐振器的模态分析。.....13

6 Bassreflex with Air-Port Resonator ..... 18
6 带空气端口谐振器的低音反射式 .....18

6.1 Frequency Response of Air-Port Resonator ..... 19
6.1 空气端口谐振器的频率响应 .....19

6.2 Stepresponse of Air-Port Resonator ..... 21
6.2 空气端口谐振器的阶跃响应 .....21

6.3 Modal Analysis of Air-Port Resonator ..... 24
6.3 空气端口谐振器的模态分析 .....24

7 Conclusions on Bassreflex for Very Low Frequencies ..... 29
7 关于极低频低音反射的结论 .....29

1 Introduction 1 引言

Subwoofers have to deliver a high sound power level at low frequencies. this automatically implies that they need to be large because of the large volume of air that needs to be moved.

Generally such large loudspeakers do not get much sympathy from people who strive for home esthetics rather than sound quality due to constraints in space and the dominant techno-style shape of a loudspeaker.

In principle small and inconspicuous loudspeakers can be realised by using smaller drivers, however a small radiating surface requires a large diaphragm excursion to compensate the small radiating surface and generate sufficient sound pressure.

Unfortunately, a large mechanical excursion will automatically account for an increased distortion level by non-linearity of the actuator and the suspension.
To alleviate this issue, loudspeaker manufacturers have applied both impedance matching devices (horns) and acoustic dynamic phenomena to increase the acoustic output of a loudspeaker system without increasing the diaphragm excursion.

This paper deals with the second option, in particular with the bass reflex system, which is still the most widely used example of applying dynamics phenomena to enhance the performance at low frequencies.

In a bassreflex system a passive resonating mass is added, which is elastically coupled to the driver diaphragm and partly takes over the low frequency sound generation below a certain frequency while simultaneously reducing the excursion of the main loudspeaker.
The original purpose of this paper was to give students at the university a real life example of how dynamic eigenmodes determine the vibrational properties of mechanical structures. For that reason the dynamic analysis of a bassreflex system is dominant in this paper.
The second (and my personal) goal was, however, to determine the real value of such an approach for high quality loudspeakers.

And, unfortunately for the majority of loudspeaker manufacturers, the conclusion is that a bass reflex system is at best a compromise but mostly utterly useless when a transparent reproduction of music is strived for.
The paper starts with the Helmholtz Resonator, which is the basis of the bass-reflex principle.

Then, after some first thoughts about the applicability of a resonator at low frequencies, a demonstrator is introduced where the principle was shown in the classroom for a bassreflex system with passive diaphragm radiator.

This example is analysed for its dynamic properties followed by the same analysis for an air-port resonator. Finally conclusions are drawn from the analysis.
Figure 1: A Helmholtz resonator consists of an enclosed volume acting as an air-spring with a tube shaped opening, the port. The mass of the air in the port resonates with the stiffness of the air-spring.
图 1:亥姆霍兹谐振器由一个作为空气弹簧的封闭容积和一个管状开口(端口)组成。端口中的空气质量与空气弹簧的刚度产生共振。

2 Helmholz Resonator 2 Helmholz 谐振器

In most cases a so called "Helmholtz Resonator" is used of which the principle is shown in Figure 1. A Helmholtz resonator consists of a combination of an enclosed volume of air (the cabinet) acting as a spring with a moving amount of air (the mass) in a tube, the bassreflex-port.
在大多数情况下,使用的是所谓的 "亥姆霍兹谐振器",其原理如图 1 所示。亥姆霍兹谐振器由作为弹簧的封闭空气体积(箱体)与管内移动的空气量(质量)(低音反射端口)组合而成。

A well known example of a standalone Helmholtz resonator is the generation of sound by blowing air over the opening in a bottle. The resonance frequency of a Helmholtz resonator is calculated as follows starting with the stiffness of the air in the enclosure:
with being a correction factor for the expansion of air (1.4), the cross section of the air-port, the atmospheric pressure and the volume of the enclosed air. The mass of the air in the port is equal to , where equals the density of air and equals the effective length of the moving air. The effective length is somewhat larger than the physical length of the port alone as the air immediately near the openings of the port also moves rapidly, decreasing with the distance to the opening.
是空气膨胀的修正系数 (1.4), 是气口的横截面, 是大气压力, 是封闭空气的体积。端口中空气的质量等于 ,其中 等于空气密度, 等于移动空气的有效长度。有效长度略大于端口的物理长度,因为端口开口附近的空气也在快速流动,并随着与开口距离的增加而减少。

As an empirically found ballpark figure, the effective length equals the length of the port plus approximately 0.73 times the diameter of the port .
根据经验得出的大致数字,有效长度等于端口长度加上约 0.73 倍的端口直径
With these values the Helmholz resonance frequency can be calculated:
With the expression for the speed of sound the eigenfrequency equals:
根据声速表达式 ,特征频率等于:
When combining a Helmholtz resonator with a loudspeaker in one enclosure, the bassreflex port takes over the sound generation from the loudspeaker at the resonance frequency of the Helmholtz resonator.
当把 Helmholtz 谐振器和扬声器组合在一个箱体中时,低音反射端口会接管扬声器在 Helmholtz 谐振器共振频率处产生的声音。

As a result the diaphragm of the driven loudspeaker hardly moves at that frequency, allowing more electrical power to be supplied to the system, thereby increasing the maximum acoustical output of the system.

3 Bassreflex Systems for Very Low Frequencies
3 用于极低频的低音反射系统

A disadvantage of using a resonator is that it collects energy which is delivered back with some delay after the input signal is terminated, causing coloration of the sound by a delayed resonance at the resonance frequency of the Helmholtz resonator.

The delayed resonance can be limited by damping which dissipates the vibration energy into heat. With a standard bassreflex system damping is determined by the loudspeaker-actuator in combination with the amplifier.

A second damping factor is the dissipated energy of the port which is difficult to tune as it is influenced by the shape of the port and the amount of damping material used inside the cabinet near the port.

This situation almost guarantees differences between each manufactured loudspeaker and to reduce these deviations a special version of the bassreflex principle has been introduced where the moving air is replaced by an additional loudspeaker diaphragm, named a passive radiator with well defined dynamic properties.

Still, as will be shown in Section 5.2 , it is impossible to create a response without any delay unless the benefits of the bassreflex principle are fully sacrificed.
不过,正如第 5.2 节所述,除非完全牺牲低音反射原理的优点,否则不可能产生没有任何延迟的响应。
Another and even more important drawback of the bassreflex system is the higherorder drop-off below the Helmholz resonance frequency, which makes it virtually impossible to boost the sound power by filter corrections below this frequency.
低音反射系统的另一个更重要的缺点是,在 Helmholz 谐振频率以下会出现高阶 音量下降,因此几乎不可能通过滤波器修正来提高该频率以下的音量。

This higher-order drop-off is best understood when realising that at very low frequencies the bassreflex port is just an acoustic short-circuit between the front and the back side of the loudspeaker, which cancels the total sound pressure, just like with a loudspeaker without an enclosure.
When trying to achieve a response until with an acceptable dynamic response without too much delay, one would need to bring the Helmholz resonance also at 20 and this can only be achieved with extremely large enclosures. The alternative to use a very thin pipe will not work as then the velocity in the pipe becomes too large and turbulence will increase the flow resistance.
如果要在 之前获得可接受的动态响应,同时又没有过多的延迟,就需要将赫尔姆霍兹共振也设置在 20 ,而这只能通过超大的箱体来实现。使用极细管道的替代方案也行不通,因为管道中的流速会变得过大,湍流会增加流动阻力。

In fact one can already conclude from mere qualitative reasoning that a bassreflex system does not solve anything for real high end subwoofers. For these reasons the principle is not used for the subwoofers of RMS Acoustics and Mechatronics.
事实上,仅从定性推理就可以得出结论,低音反射系统对真正的高端超低音没有任何作用。由于这些原因,RMS Acoustics 和 Mechatronics 的超重低音扬声器并未采用这一原理。
In the following section the dynamic analysis of bass reflex systems is presented, further underlining these conclusive statements on the low-frequency limitations of the principle.

4 Bassreflex Dynamics 4 低音反射动态

As part of the university classroom lectures on dynamics of motion systems, I have often used a demonstrator with two coupled loudspeakers working according to the bassreflex principle.

The charm of the system is the easy observability of the dynamic effects and the mental connection to real life systems as most of the students have loudspeakers where the bassreflex principle is used.

The demonstrator is based on the same enclosure design that will be presented in the paper on "Sensorless Velocity Feedback Subwoofer", which also was developed for as a classroom demonstrator, using two large loudspeakers and motional feedback.
该演示器基于 "无传感器速度反馈低音炮 "论文中介绍的相同箱体设计,该论文也是作为课堂演示器开发的,使用了两个大型扬声器和运动反馈。
The main difference of the approach in this section when compared with the well known Thiele-Small analysis and many other related methods is found in the more mechanical oriented approach.
与众所周知的 Thiele-Small 分析法和许多其他相关方法相比,本节所采用的方法的主要区别在于它更偏向于机械分析。

Regular analysis translates the lumped mechanical elements like the rigid body the spring and the damper into their electronic equivalents like inductor, capacitor and resistor.

Depending on the method used, the actuator is replaced by a gyrator or transformer and the analysis is further done as if it was an electronic circuit.

The usefulness of this electronic equivalent method is proven over the years with many easily applicable computer programs of which the free version of Scan-Speak which can be downloaded at their website is a good example.
多年来,许多易于使用的计算机程序证明了这种电子等效方法的实用性,可在其网站上下载的免费版 Scan-Speak 就是一个很好的例子。
While this electronic equivalent method has its advantage in the possibility to use dedicated software from the electronic domain, it is not capable of utilising the knowledge on dynamics which has been gained over the years in the mechanical domain with for instance vibration modal analysis which can model effectively breakup phenomena and decoupling of compliant bodies.

The propagation of sound in any medium is a physical phenomenon with a clear relation to the mechanical domain" which is a strong argument to remain in the mechanical domain when searching improvements in reproducing music by loudspeakers.

In this respect this section can be seen is a starting point for learning the mechanical dynamics approach on sound reproduction.

5 Bassreflex with Diaphragm Resonator
5 带振膜谐振器的低音反射式耳机

The enclosure as shown in Figure 2 was originally designed to be used as an active controlled closed-box system as described in a separate paper on velocity feedback.
图 2 所示的外壳最初是设计用作主动受控闭箱系统的,这在另一篇关于速度反馈的论文中有所描述。

Due to the two loudspeakers that share the same enclosure it allows to experiment with the passively radiating diaphragm principle by using one of the loudspeakers as the active driven loudspeaker and the other as the passive radiator.

The damping of each loudspeaker can be controlled by the impedance between the external connections, either from the amplifier for the driven loudspeaker or by a series resistance for the passive radiator.
Figure 2: The enclosure of the subwoofer with two loudspeakers. With the bassreflex principle one of the loudspeakers is driven by an amplifier while the other is passively coupled to the driven loudspeaker by the stiffness of the air spring enclosed by the cabinet.
图 2:带有两个扬声器的低音炮箱体。根据低音反射原理,其中一个扬声器由放大器驱动,而另一个扬声器则通过箱体中空气弹簧的刚度与驱动扬声器被动耦合。

5.1 Frequency Response of Diaphragm Resonator
5.1 膜片谐振器的频率响应

The bass-reflex principle is directly related to the theory about the dynamic response of two elastically coupled bodies to a force on one of the bodies. This theory shows that the motion amplitude of driven loudspeaker should become zero at the antiresonance frequency determined by the moving mass of the passive radiator and the stiffness of the coupling spring between the diaphragms, which is determined by the enclosed air.
低音反射原理与两个弹性耦合体对其中一个体上的力的动态响应理论直接相关。该理论 表明,驱动扬声器的运动振幅应在由无源辐射器的运动质量和膜片之间耦合弹簧的刚度(由封闭空气决定)决定的反谐振频率处变为零。
The stiffness of the air between the two diaphragms can be calculated with the same equation as used with the Helmholtz resonator. With the surface area of the diaphragm , an air pressure of , a volume of the enclosure and due to the fibre filling:
两个膜片之间空气的刚度可以用与亥姆霍兹谐振器相同的公式计算。膜片的表面积为 ,气压为 ,外壳的体积为 ,纤维填充物的体积为
For the total stiffness that the passive radiator experiences the stiffness of the suspension needs to be added, which equals the inverse of the compliance and leads to a total stiffness of:
被动散热器的总刚度需要加上悬挂装置的刚度,这等于顺应性 的倒数,从而得出总刚度为:
Electrical Data 电气数据 Power handling 功率处理
Nominal impedance 标称阻抗 4 ohm 100h RMS noise test (IEC)
100h 有效值噪声测试(IEC)
-- W
Minimum impedance 最小阻抗 3 ohm Long-term Max System Power
-- W
Maximum impedance 最大阻抗 Zo 65.7 ohm
DC resistance 直流电阻 2.6 ohm Max linear SPL (rms) @ power
最大线性声压级(均方根值)@ 功率
Voice coil inductance 音圈电感 Le 1.6 Short Term Max power
-- W
Capacitor in series with x ohm
与 x 欧姆串联的电容器
Cc -- Voice Coil and Magnet Parameters
T-S Parameters T-S 参数 Voice coil diameter 音圈直径 51
Resonance Frequency 共振频率 fs 19.1 Voice coil height 音圈高度 32.6
Mechanical Q factor 机械 Q 因子 Qms 9.29 Voice coil layers 音圈层 4
Electrical Q factor 电气 Q 因子 Qes 0.38 Height of the gap
Total Q factor 总 Q 因子 Qts 0.37 Linear excursion +/- 线性偏移 +/- 13
Ratio fs/Qts -- Max mech. excursion +/-
最大机械偏移 +/-
Force factor 10.3 Flux density of gap
Mechanical resistance 机械阻力 Rms 1.69 Total useful flux 总有用通量 2.3
Moving mass Mms 130.6 Diameter of magnet 磁铁直径 147
Suspension compliance 暂停遵守规定 0.53 Height of magnet 磁铁高度 35
Effective cone diameter 有效锥直径 24.4 Weight of magnet 磁铁重量 2.2
Effective piston area 有效活塞面积 Sd 466
Equivalent volume 等效体积 Vas 159 Itrs
Sensitivity 91.2
Ratio BL/ 6.4
Figure 3: Characteristics of the applied loudspeaker, the Peerless XXLS 12.
图 3:所用扬声器 Peerless XXLS 12 的特性。
With the moving mass of this results in a natural frequency of the passive radiator with this spring equal to:
由于移动质量为 ,因此使用该弹簧的无源辐射器的固有频率等于:
Measurement of this natural frequency showed a slightly lower frequency of which might indicate that the filling works better in achieving an isothermal compression/expansion or that the stiffness of the suspension is lower.
对这一固有频率的测量显示, ,频率略低,这可能表明填充物在实现等温压缩/膨胀 ,或者悬浮液的刚度较低。

This small deviation is acceptable within the accuracy of the approximated values and the model is sufficiently correct to take the estimated values for the moving mass and spring stiffness of the air in the enclosure and calculate the response of the two loudspeakers, taking into account all springs and dampers.

Figure 4 shows the lumped-element model used to derive the frequency response functions where equals the mass of the driven loudspeaker and the mass of the passive radiator. and are the springs and dampers of each element to the enclosure caused by the guiding diaphragm (surround, spider and the electromagnetic damping.
图 4 显示了用于推导频率响应函数的叠加元件模型,其中 等于驱动扬声器的质量, 等于被动辐射器的质量。 是每个元件与箱体之间的弹簧和阻尼,由导向振膜(环绕、蜘蛛和电磁阻尼)引起。

In the model the previously found resemblance between radiated power and acceleration is used and both sound-pressure responses are added together for the total sound pressure.

This corresponds with the earlier found conclusion that two loudspeakers that generate the same sound pressure by a certain movement of the diaphragm will together generate a sound power that is four times the sound power of one loudspeaker ( ).
这与早先发现的结论相吻合,即两个扬声器通过一定的振膜运动产生相同的声压,共同产生的声功率是一个扬声器声功率的四倍 ( )。
Starting with :
Figure 4: The lumped-element model of the bass reflex system with passive radiator is used to derive the frequency response functions.
图 4:带有无源辐射器的低音反射系统的叠加元件模型用于推导频率响应函数。
From this follows: 由此可见
The motion equation for mass equals
质量 的运动方程等于
and the displacement can be written as function from :
而位移 可以写成来自 的函数:
Filling this in Equation (8) and careful applying some algebra leads to the following equations:
将其填入公式 (8),并仔细应用一些代数方法,可以得出以下公式:
And: 还有
With: 有了
Replacing s with and multiplying with ultimately leads to the following radial frequency response functions for the sound pressure. Note that this is only a proportionality relation to the sound pressure as only the acceleration is calculated.
将 s 替换为 并与 相乘,最终得到以下声压的径向频率响应函数。请注意,这只是声压的比例关系,因为只计算了加速度。
To calculate the real soundpressure it must be multiplied with the radiating efficiency of the diaphragm at a certain distance:
and for the passive radiator:
The total sound pressure is than equal to the difference of these equations as being caused by the motion difference between the two diaphragms.
With the help of MATLAB the responses for different levels of damping are calculated using the above equations. By subtracting both responses the sound response is obtained because the difference of movement creates the acoustic pressure/power.
在 MATLAB 的帮助下,利用上述公式计算出不同阻尼水平的响应。将两个响应相减,就得到了声音响应,因为运动的差异产生了声压/声功率。

The first Bode-plot of Figure 5 shows the effect of the situation when the damping is very low as would be the case when the amplifier has a high output impedance, like a current source. At very low frequencies both masses move in phase until a clear resonance at around . This resonance is caused by the surround diaphragm and spider of both loudspeakers and corresponds with the given resonance frequency characteristics of the used loudspeaker when not mounted in an enclosure.
图 5 的第一张 Bode-plot 显示了阻尼非常低时的效果,就像放大器具有高输出阻抗(如电流源)时的情况一样。在很低的频率下,两个质量块相向移动,直到在 左右出现明显的共振。这种共振是由两个扬声器的环绕振膜和蜘蛛引起的,并与未安装在箱体内时所用扬声器的给定共振频率特性相对应。

They move both in the same direction so the air volume in enclosure does not change by this movement, causing no additional stiffness.
At a higher frequency the passive radiator will dynamically decouple from the driven loudspeaker because the spring can not supply enough force to accelerate the passive radiator.

Eventually this causes a negative peak, the anti-resonance in the response of the driven loudspeaker at the predicted . At the second resonance frequency both masses will move in counter phase.
最终,这将导致一个负峰值,即在预测的 处驱动扬声器响应的反谐振。在第二个共振频率,两个质量块将反相移动。

Now each loudspeaker works on half the volume of the enclosure which means that the gas spring of the enclosure is equally divided over each loudspeaker so they both get twice the stiffness of the total enclosed air between both loudspeakers:
Adding the stiffness of the diaphragm suspension results in the total stiffness per loudspeaker:
This results in a resonance frequency of:
As the diaphragms now move in the opposite direction of each other they will create a sound pressure and as a result the summed response shows a very strong resonance.

Figure 5: Bode plot of the undamped and damped responses from both the driven diaphragm (blue), the passive radiator (red) and the combined responses (black). Below the first resonance at both diaphragms move in the same direction and give no sound pressure. The damping matches the situation when the driven loudspeaker is connected to a voltage source amplifier. The damped step response is still quite nervous.
图 5:驱动振膜(蓝色)、无源辐射器(红色)和综合响应(黑色)的无阻尼和有阻尼响应的 Bode 图。在 第一个共振点以下,两个振膜向同一方向移动,不产生声压。阻尼与驱动扬声器连接到电压源放大器时的情况相符。阻尼阶跃响应仍然相当紧张。

This can be improved by also damping the passive radiator but then the beneficial effect of the resonator in the low frequency response is also reduced. Note the fourth order octave slope below the maximum value at in the combined response
这可以通过对无源辐射器进行阻尼来改善,但这样一来,谐振器对低频响应的有利影响也会减弱。请注意,在 的综合响应中,四阶 倍频程斜率低于最大值
Figure 6: The response of the driven and the resonator diaphragm on a starting sinusoidal signal with a frequency equal to the Helmholtz frequency,shows clearly that first the driven diaphragm will create the sound pressure while after a few periods the resonator takes over.
图 6:从动膜片和共振膜片对频率等于亥姆霍兹频率的起始正弦信号的响应可以清楚地看出,从动膜片首先产生声压,而共振膜片在几个周期后开始产生声压。

This is most clearly seen with low damping but then also the total response shows overshoot.

When both diaphragms have some amount of damping the system can be made to act without overshoot, however, in that case the total response becomes almost equal to the response of the driven diaphragm.
In order to reduce this resonance peak, damping is applied on the driven loudspeaker by using a voltage source amplifier. The effect is shown in the second Bode-plot of Figure 5 and also in the stepresponse.
为了降低共振峰值,使用电压源放大器对驱动扬声器施加阻尼。其效果如图 5 的第二幅节点图和阶跃响应所示。

The added damping clearly reduces the high peak in the frequency response but the sound contribution of the second passive diaphragm is also reduced. Still the summed output shows an acceptable resonance with less than increase in magnitude at and a bandwidth @ 30 . The stepresponse is still not very well damped with almost two full periods of ) which would clearly cause an over emphasis of at low frequency transients, creating a "boombox" sound. More damping could be applied at the passive radiator to reduce the resonance but this also reduces the beneficial effect of the reduction of the loudspeaker excursion as demonstrated in Figure 5, when comparing a: and b:.
增加的阻尼明显降低了频率响应的峰值,但第二个被动振膜的声音贡献也减少了。总和输出仍然显示出可以接受的共振,在 时幅度增加不到 ,在 时带宽为 30 。阶跃响应的阻尼仍然不是很好,几乎有两个整周期的 ),这显然会导致 过分强调低频瞬态,从而产生 "boombox "音效。可在无源辐射器上施加更多阻尼,以减少共振,但这也会降低减少扬声器偏移的有利效果,如图 5 中 a:和 b:的比较所示。

Furthermore it is quite expensive to use a full loudspeaker to only contribute some damping at this very limited frequency area.

For this reason normally the electromagnetic actuator is omitted with the passive radiator and only the mass is tuned while the surround is made from damping rubber to reduce the resonance to an acceptable level.

5.2 Time Domain Response of Sine Signal
5.2 正弦信号的时域响应

The stepresponse of Figure 5 showed a clear periodic reaction with an undamped resonator. Musical signals are, however, never like a step function but rather like a discontinuous series of sine functions and it is interesting to see the behaviour of
图 5 的阶跃响应显示了无阻尼谐振器的明显周期性反应。然而,音乐信号绝不是阶跃函数,而是一系列不连续的正弦函数。

both diaphragms on such signals.
Figure 6 shows the calculated time-domain response of a bassreflex system with a passive resonator for two situations, where in both cases a sinusoidal signal with a frequency, equal to the Helmholtz resonance frequency of the passive resonator diaphragm, is started at .
图 6 显示了带有无源谐振器的低音反射系统在两种情况下的时域响应计算结果,在这两种情况下,都是在 处启动频率与无源谐振器膜片的亥姆霍兹共振频率相同的正弦信号。

In the first situation both the driven diaphragm and the resonating diaphragm have a moderate level of damping and it clearly confirms that the dip in the frequency response of the driven diaphragm from Figure 5 only occurs after some time, because the resonator needs to build up its energy.
在第一种情况下,驱动膜片和共振膜片都具有中等程度的阻尼,这清楚地证实了图 5 中驱动膜片频率响应的下降只是在一段时间后才出现,因为共振器需要积累能量。

Furthermore, at a higher level of damping of both diaphragms, as shown in the right graph of Figure 6, the contribution of the resonating diaphragm to the total sound pressure is almost gone. This also corresponds to the frequency response curves from Figure 5.
此外,如图 6 右图所示,当两个振膜的阻尼水平较高时,共振振膜对总声压的贡献几乎消失。这也与图 5 中的频率响应曲线相吻合。
Two important conclusions can be drawn from these graphs.
  • The often assumed benefit that a bassreflex system could allow the use of a smaller driven loudspeaker than with a closed box enclosure for the same maximum low frequency sound pressure is only true for continuous signals and a low damping resonating diaphragm.

    With varying and sudden bass, like with a base drum, this benefit is non-existing.
  • A low level of damping will always create overshoot in the response but a higher level of damping will reduce the benefit of the bassreflex principle.

    For this reason small subwoofers for computers and cheap home-movie surround systems are always equipped with undamped resonators, resulting in an exaggerated boom bass, which is sometimes nice whan watching a war movie but more often very tiring, while cause a headache.
Like most things in real life, there is no such thing as a free lunch. Mostly benefits on one aspect are counteracted by drawbacks on other aspects.

5.3 Modal analysis of Diaphragm Resonator.
5.3 膜片谐振器的模态分析

The analytical expression of the frequency response becomes quickly quite complicated when describing higher order dynamic structures with several lumped bodies, springs and dampers. For that reason a dynamic system is often analysed by means of its vibration eigenmodes.

This is allowed when the system dynamics are essentially linear as then the total dynamic behaviour can be modelled as the superposition of the behaviour of the system in its separate eigenmodes.

The theory of eigenmodes is based on the property that a non-rigid dynamic system, described as a series of bodies connected by springs and dampers, shows several characteristic resonance frequencies.

Excitation at these frequencies will cause a synchronous periodic movement of all bodies of the system.

The characteristic periodic movement is called an "eigenmode" where the German and Dutch word "eigen" means "own", reflecting the fact that it is a characteristic system property. The corresponding
这种特征性的周期运动被称为 "特征模式",在德语和荷兰语中,"特征 "的意思是 "自己的",这反映了它是一种特征性的系统属性。相应的
a: Eigenmode 1 a: 特征模式 1
b: Eigenmode 1 equivalent (combined mass and stiffness)
b:特征模式 1 等效(质量和刚度的组合)
c: Eigenmode 2 c:特征模式 2
: Eigenmode 2 equivalent ( mirrored) and simplified (added elements)
:特征模式 2 等效( 镜像)和简化(增加元素)
Figure 7: Splitting of the fourth-order dynamic system in two second-order mass-spring systems according to the eigenmodes of the system. The first eigenmode is the rigid-body mode where both diaphragms and move in the same direction as if they were one body with modal mass . The mass and suspension stiffness of both diaphragms can then be added to determine the dynamic response of the first eigenmode. The second eigenmode is a bit more complicated to comprehend.
图 7:根据系统的特征模态,将四阶动力系统拆分为两个二阶质量弹簧系统。第一个特征模态是刚体模态,即 两个膜片沿同一方向运动,就好像它们是一个具有模态质量的体 。将两个隔膜的质量和悬挂刚度相加,即可确定第一特征模态的动态响应。第二个特征模态的理解要复杂一些。

It is the mode where both masses move opposite to each other with the same amplitude as if driven by a mechanism. The symmetry allows a mirroring of the second body with its related springs and like with the first eigenmode the modal mass . Special attention is needed for the connecting air spring which is a factor four larger in the equivalent simple mass-spring system.
在该模态中,两个质量块以相同的振幅相对运动,就像由机械装置驱动一样。对称性使得第二个物体与相关弹簧形成镜像,与第一个特征模态一样,模态质量为 。需要特别注意的是连接空气弹簧,它比等效的简单质量弹簧系统大四倍。

resonance frequency is called the eigenfrequency of that mode, while the movement amplitude as function of the bodies is called the "mode-shape" described by the shape function, a vector notation with terms for each body, where the sign of the value represents the phase at that point relative to the reference body.
共振频率被称为该模态的特征频率,而运动振幅与各机构的函数关系被称为 "模态形状",由形状函数描述。形状函数是一种矢量符号,包含每个机构的项,值的符号代表该点相对于参考机构的相位。
As an example the undamped response of Figure 5 shows two clearly distinguishable eigenfrequencies, one at and one at . The eigenmode that corresponds to has a mode shape that is uniform and equal for both loudspeaker diaphragms (Shape function [ ). The second eigenmode at has a mode shape where both diaphragms move opposite to each other with an equal amplitude (Shape function ).
例如,图 5 的无阻尼响应显示了两个明显不同的特征频率,一个位于 ,另一个位于 。与 相对应的特征模态的模态形状是一致的,两个扬声器振膜的模态形状相等(形状函数 [ )。 处的第二个特征模态的模态形状是两个振膜相对运动,振幅相等(形状函数