1 Introduction ..... 3 1 简介 ...... 3
2 Helmholz Resonator ..... 4
3 Bassreflex Systems for Very Low Frequencies ..... 5 3 适用于极低频的 Bassreflex 系统 ..... 5
4 Bassreflex Dynamics ..... 6 4 低音反射动态..... 6
5 Bassreflex with Diaphragm Resonator ..... 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..... 18
6.1 Frequency Response of Air-Port Resonator ..... 19 6.1 气口谐振器的频率响应 ...... 19
6.2 Stepresponse of Air-Port Resonator ..... 21
6.3 Modal Analysis of Air-Port Resonator ..... 24
7 Conclusions on Bassreflex for Very Low Frequencies ..... 29 关于极低频低音反射的 7 个结论 ..... 29
1 Introduction 1 简介
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. 低音炮必须在低频下提供高声功率级。这自然意味着它们需要很大，因为需要移动大量空气。 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. 一般来说，由于空间的限制和扬声器的主导技术风格形状，这种大型扬声器不会得到那些追求家居美观而不是音质的人们的同情。 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. 原则上，小型且不显眼的扬声器可以通过使用较小的驱动器来实现，但是较小的辐射表面需要较大的振膜偏移来补偿较小的辐射表面并产生足够的声压。 Unfortunately, a large mechanical excursion will automatically account for an increased distortion level by non-linearity of the actuator and the suspension. 不幸的是，大的机械偏移将自动导致致动器和悬架的非线性导致的失真水平增加。
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. 为了缓解这个问题，扬声器制造商应用了阻抗匹配设备（喇叭）和声学动态现象，以在不增加振膜偏移的情况下增加扬声器系统的声输出。 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. 本文讨论第二种选择，特别是低音反射系统，它仍然是应用动态现象来增强低频性能的最广泛使用的示例。 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. 在低音反射系统中添加了一个无源谐振质量，该质量弹性耦合到驱动器振膜，并部分接管低于特定频率的低频声音生成，同时减少主扬声器的偏移。
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. 本文的最初目的是为大学学生提供一个现实生活中的例子，说明动态特征模态如何确定机械结构的振动特性。因此，本文以低音反射系统的动态分析为主。
The second (and my personal) goal was, however, to determine the real value of such an approach for high quality loudspeakers. 然而，第二个（也是我个人的）目标是确定这种方法对于高质量扬声器的真正价值。 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. 不幸的是，对于大多数扬声器制造商来说，结论是低音反射系统充其量只是一种妥协，但在追求透明的音乐再现时几乎毫无用处。
The paper starts with the Helmholtz Resonator, which is the basis of the bass-reflex principle. 本文从亥姆霍兹谐振器开始，它是低音反射原理的基础。 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. 然后，在对谐振器在低频下的适用性进行一些初步思考后，介绍了一个演示器，其中在教室中展示了带有无源振膜辐射器的低音反射系统的原理。 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. 分析了该示例的动态特性，然后对机场谐振器进行了相同的分析。最后通过分析得出结论。
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封闭的体积组成，该体积充当空气弹簧，具有管状开口（端口）。端口中的空气质量与空气弹簧的刚度产生共振。
2 Helmholz Resonator
In most cases a so called "Helmholtz Resonator" is used of which the principle is shown in Figure 1. 在大多数情况下，使用所谓的“亥姆霍兹谐振器”，其原理如图 1 所示。 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. 亥姆霍兹共鸣器由充当弹簧的封闭空气（箱体）与管中移动的空气（质量）（低音反射端口）的组合组成。 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: 独立亥姆霍兹谐振器的一个众所周知的例子是通过将空气吹过瓶子的开口来产生声音。亥姆霍兹谐振器的谐振频率从刚度开始计算如下 外壳内的空气：
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. 和 是空气膨胀的修正系数 (1.4)， 机场的横截面， 大气压和 封闭空气的体积。端口中空气的质量等于 ， 在哪里 等于空气的密度并且 等于移动空气的有效长度。有效长度比端口单独的物理长度稍大，因为紧邻端口开口的空气也快速移动，并随着到开口的距离而减小。 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 . 根据经验得出的大概数字，有效长度等于端口长度加上大约 0.73 倍直径 港口的 。
With these values the Helmholz resonance frequency can be calculated: 利用这些值可以计算亥姆霍兹共振频率：
With the expression for the speed of sound the eigenfrequency equals: 用声速的表达式 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. 当将亥姆霍兹谐振器与扬声器组合在一个外壳中时，低音反射端口以亥姆霍兹谐振器的谐振频率接管扬声器的声音生成。 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. 因此，驱动扬声器的振膜在该频率下几乎不移动，从而允许向系统提供更多电力，从而增加系统的最大声学输出。
3 Bassreflex Systems for Very Low Frequencies 3 个适用于极低频的 Bassreflex 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. 使用谐振器的一个缺点是，它会收集能量，并在输入信号终止后以一定的延迟传回，从而导致亥姆霍兹谐振器谐振频率处的延迟谐振导致声音染色。 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. 延迟共振可以通过阻尼来限制，阻尼将振动能量耗散成热量。对于标准的低音反射系统，阻尼由扬声器执行器与放大器的组合决定。 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. 第二个阻尼因素是端口的耗散能量，它很难调节，因为它受到端口形状和端口附近机柜内使用的阻尼材料量的影响。 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. 这种情况几乎保证了每个制造的扬声器之间存在差异，为了减少这些偏差，引入了特殊版本的低音反射原理，其中移动的空气被附加的扬声器振膜取代，称为具有明确动态特性的无源辐射器。 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. 尽管如此，正如第 5.2 节所示，除非完全牺牲低音反射原理的好处，否则不可能毫无延迟地创建响应。
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甚至更重要的缺点是高阶 衰减低于亥姆霍兹共振频率，这使得几乎不可能通过低于该频率的滤波器校正来提高声功率。 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. 当认识到在非常低的频率下，低音反射端口只是扬声器前侧和后侧之间的声学短路时，可以最好地理解这种高阶衰减，这会抵消总声压，就像扬声器一样没有外壳。
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. 当试图获得响应时，直到 如果具有可接受的动态响应且没有太多延迟，则需要将亥姆霍兹共振也设置为 20 而这只有通过非常大的外壳才能实现。使用非常细的管道的替代方案将不起作用，因为管道中的速度变得太大并且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，人们已经可以从纯粹的定性推理中得出结论，低音反射系统并不能解决真正的高端低音炮的任何问题。由于这些原因，该原理不用于 RMS Acoustics and Mechatronics 的低音炮。
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. 在下一节中，将介绍低音反射系统的动态分析，进一步强调有关该原理的低频限制的结论性陈述。
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. 作为关于运动系统动力学的大学课堂讲座的一部分，我经常使用带有两个根据低音反射原理工作的耦合扬声器的演示器。 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. 该系统的魅力在于易于观察动态效果以及与现实生活系统的心理联系，因为大多数学生都有使用低音反射原理的扬声器。 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”论文中介绍的相同外壳设计，该设计也是为教室演示器开发的，使用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 分析和许多其他相关方法相比，本节中的方法的主要区别在于更面向机械的方法。 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. 定期分析将刚体、弹簧和阻尼器等集总机械元件转换为电感器、电容器和电阻器等电子等效元件。 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. 根据所使用的方法，执行器被回转器或变压器取代，并且进一步进行分析，就好像它是电子电路一样。 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，这种电子等效方法的实用性已经通过许多易于应用的计算机程序得到了证明，其中可以在其网站上下载的免费版本的 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. 虽然这种电子等效方法的优点在于可以使用电子领域的专用软件，但它无法利用多年来在机械领域获得的动力学知识，例如可以建模的振动模态分析。有效地分解现象和顺应体的脱钩。 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. 声音在任何介质中的传播都是一种与机械领域有明确关系的物理现象”，这是在寻求扬声器再现音乐的改进时留在机械领域的有力论据。 In this respect this section can be seen is a starting point for learning the mechanical dynamics approach on sound reproduction. 在这方面，可以看出本节是学习声音再现的机械动力学方法的起点。
5 Bassreflex with Diaphragm Resonator
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 所示的外壳最初设计用作主动控制的封闭箱系统，如另一篇关于速度反馈的论文所述。 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. 由于两个扬声器共享同一外壳，因此可以通过使用其中一个扬声器作为有源驱动扬声器而另一个作为无源辐射器来试验无源辐射振膜原理。 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. 每个扬声器的阻尼可以通过外部连接之间的阻抗来控制，外部连接可以来自驱动扬声器的放大器，也可以通过无源辐射器的串联电阻来控制。
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外壳。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封闭的空气弹簧的刚度被动耦合到驱动扬声器。
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物体对其中一个物体上的力的动态响应的理论直接相关。这个理论 表明驱动扬声器的运动幅度应在反谐振频率处变为零，该反谐振频率由无源辐射器的移动质量和膜片之间的耦合弹簧的刚度确定，而膜片之间的耦合弹簧的刚度由封闭的空气确定。
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: 两个隔膜之间的空气刚度可以使用与亥姆霍兹谐振器相同的方程来计算。与隔膜的表面积 ，气压为 ，外壳的体积 和 由于纤维填充：
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: 对于无源辐射器所经历的总刚度，需要添加悬架的刚度，这等于柔量的倒数 并导致总刚度为：
Specs:
Electrical Data 电气数据
Power handling 功率处理
Nominal impedance 标称阻抗
4
ohm 欧姆
100h RMS noise test (IEC)
--
W
Minimum impedance
3
ohm 欧姆
Long-term Max System Power
--
W
Maximum impedance
Zo
65.7
ohm 欧姆
DC resistance
2.6
ohm 欧姆
Max linear SPL (rms) @ power
--
Voice coil inductance 音圈电感
Le
1.6
Short Term Max power 短期最大功率
--
W
Capacitor in series with x ohm 串联x欧姆电容
Cc
--
Voice Coil and Magnet Parameters 音圈和磁铁参数
T-S Parameters 传输参数
Voice coil diameter
51
Resonance Frequency 共振频率
fs
19.1
Voice coil height 音圈高度
32.6
Mechanical Q factor 机械品质因数
Qms
9.29
Voice coil layers 音圈层数
4
Electrical Q factor
Qes
0.38
Height of the gap 间隙高度
8
Total Q factor
Qts
0.37
Linear excursion +/- 线性偏移+/-
13
Ratio fs/Qts
--
Max mech. excursion +/- 最大机甲。偏移+/-
--
Force factor
10.3
Flux density of gap
--
Mechanical resistance
Rms
1.69
Total useful flux
2.3
Moving mass 移动质量
Mms 多发性硬化症
130.6
Diameter of magnet
147
Suspension compliance 暂停合规性
0.53
Height of magnet 磁铁高度
35
Effective cone diameter
24.4
Weight of magnet
2.2
Effective piston area 活塞有效面积
Sd 英石
466
Equivalent volume
Vas
159
Itrs 伊特斯
Sensitivity 灵敏度
91.2
Ratio BL/
6.4
Figure 3: Characteristics of the applied loudspeaker, the Peerless XXLS 12. 图 3：所应用扬声器 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: 随着移动质量 这导致带有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. 该固有频率的测量显示出略低的频率 这可能表明填充在实现等温压缩/膨胀方面效果更好 或者悬架的刚度较低。 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. 这种小偏差在近似值的精度范围内是可以接受的，并且模型足够正确，可以获取外壳中空气的移动质量和弹簧刚度的估计值，并计算两个扬声器的响应，同时考虑所有弹簧和阻尼器。 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 显示了用于导出频率响应函数的集总元件模型，其中 等于驱动扬声器的质量 无源辐射器的质量。 和 是由导向膜片（环绕、星形和电磁阻尼）引起的每个元件对外壳的弹簧和阻尼器。 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. 在该模型中，使用了先前发现的辐射功率和加速度之间的相似性，并将两个声压响应加在一起以获得总声压。 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 ( ). 这与之前发现的结论相对应，即通过振膜的一定运动产生相同声压的两个扬声器将共同产生的声功率是一个扬声器声功率的四倍（ ）。
Starting with : 从...开始 :
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集总元件模型用于推导频率响应函数。
From this follows: 由此可知：
The motion equation for mass equals 质量的运动方程 等于
and the displacement can be written as function from : 和位移 可以写成函数 :
Filling this in Equation (8) and careful applying some algebra leads to the following equations: 将其填入方程 (8) 并仔细应用一些代数可得出以下方程：
And: 和：
With: 和：
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 并乘以 最终得出以下声压的径向频率响应函数。请注意，这只是与声压的比例关系，因为仅计算加速度。
To calculate the real soundpressure it must be multiplied with the radiating efficiency of the diaphragm at a certain distance: 要计算真实声压，必须将其乘以一定距离处振膜的辐射效率：
and for the passive radiator: 对于无源辐射器：
The total sound pressure is than equal to the difference of these equations as being caused by the motion difference between the two diaphragms. 总声压不等于由两个膜片之间的运动差异引起的这些方程的差。
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，使用上述方程计算不同阻尼级别的响应。通过减去这两个响应，可以获得声音响应，因为运动的差异产生了声压/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 的第一个波特图显示了当阻尼非常低时情况的影响，就像放大器具有高输出阻抗（如电流源）时的情况一样。在非常低的频率下，两个质量块同相移动，直到在大约 。这种谐振是由两个扬声器的环绕振膜和星形轮引起的，并且与所用扬声器未安装在外壳中时的给定谐振频率特性相对应。 They move both in the same direction so the air volume in enclosure does not change by this movement, causing no additional stiffness. 它们沿相同方向移动，因此外壳中的空气量不会因该移动而改变，不会导致额外的刚度。
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. 在较高频率下，无源辐射器将与驱动扬声器动态解耦，因为弹簧无法提供足够的力来加速无源辐射器。 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. 最终这会导致负峰值，即驱动扬声器在预测的响应中的反谐振 。在第二共振频率下，两个质量块将以反相移动。 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: 现在，每个扬声器在外壳体积的一半上工作，这意味着外壳的气弹簧在每个扬声器上均分，因此它们的刚度都是两个扬声器之间总封闭空气的两倍：
Adding the stiffness of the diaphragm suspension results in the total stiffness per loudspeaker: 添加振膜悬架的刚度可得出每个扬声器的总刚度：
This results in a resonance frequency of: 这导致谐振频率为：
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. 当隔膜现在沿彼此相反的方向移动时，它们将产生声压，因此总响应显示出非常强的共振。
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：驱动膜片（蓝色）、无源辐射器（红色）和组合响应（黑色）的无阻尼和阻尼响应的波特图。低于第一共振点 两个隔膜沿相同方向移动并且不产生声压。阻尼与驱动扬声器连接到电压源放大器时的情况相匹配。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 这也可以通过阻尼无源辐射器来改善，但谐振器在低频响应中的有益效果也会降低。注意第四个顺序 八度斜率低于最大值 在综合响应中
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：从动隔膜和谐振器隔膜对频率等于亥姆霍兹频率的起始正弦信号的响应清楚地表明，首先从动隔膜将产生声压，而在几个周期后，谐振器接管。 This is most clearly seen with low damping but then also the total response shows overshoot. 这在低阻尼时最为明显，但总响应也显示出过冲。 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. 当两个膜片都具有一定量的阻尼时，系统可以在没有超调的情况下起作用，但是，在这种情况下，总响应几乎等于从动膜片的响应。
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. 为了减少该谐振峰值，通过使用电压源放大器对驱动扬声器施加阻尼。图 5 的第二个波特图以及阶跃响应中显示了该效果。 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，但第二个无源振膜的声音贡献也降低了。总输出仍然显示出可接受的谐振，小于 增加幅度为 和一个 带宽@30 。阶跃响应仍然没有得到很好的抑制，几乎有两个完整的周期 ）这显然会导致过度强调 在低频瞬变时，产生“音箱”声音。可以在无源辐射器处应用更多阻尼以减少谐振，但这也会降低减少扬声器偏移的有益效果，如图 5 所示（比较 a: 和 b: 时）。 Furthermore it is quite expensive to use a full loudspeaker to only contribute some damping at this very limited frequency area. 此外，使用全扬声器仅在这个非常有限的频率区域提供一些阻尼是相当昂贵的。 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. 因此，通常情况下，无源辐射器会省略电磁执行器，仅调整质量，而周围则由阻尼橡胶制成，以将共振降低到可接受的水平。
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清晰周期性反应。然而，音乐信号从来不像阶跃函数，而是像一系列不连续的正弦函数，并且有趣的是看到
both diaphragms on such signals. 两个隔膜都在此类信号上。
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 显示了在两种情况下计算出的具有无源谐振器的低音反射系统的时域响应，在这两种情况下，正弦信号的频率等于无源谐振器膜片的亥姆霍兹谐振频率，起始位置为 。 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，从动膜片和谐振膜片都具有中等程度的阻尼，并且它清楚地证实了图 5 中的从动膜片频率响应的下降仅在一段时间后发生，因为谐振器需要建立其活力。 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，在两个膜片的阻尼水平较高时，如图 6 右图所示，谐振膜片对总声压的贡献几乎消失。这也对应于图 5 中的频率响应曲线。
Two important conclusions can be drawn from these graphs. 从这些图中可以得出两个重要的结论。
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. 通常认为，低音反射系统可以允许使用比封闭箱外壳更小的驱动扬声器来获得相同的最大低频声压，这一好处仅适用于连续信号和低阻尼谐振膜。 With varying and sudden bass, like with a base drum, this benefit is non-existing. 对于变化和突然的低音，例如基鼓，这种好处是不存在的。
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. 低水平的阻尼总是会导致响应过冲，但高水平的阻尼会降低低音反射原理的优势。 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. 因此，用于计算机的小型低音炮和廉价的家庭电影环绕系统总是配备无阻尼谐振器，导致夸张的低音，有时在观看战争电影时很不错，但更多时候很累，同时会引起头痛。
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. 就像现实生活中的大多数事情一样，世界上没有免费的午餐。大多数情况下，一方面的好处会被其他方面的缺点所抵消。
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. 当描述具有多个集总体、弹簧和阻尼器的高阶动态结构时，频率响应的解析表达式很快就会变得非常复杂。因此，动态系统通常通过其振动本征模态进行分析。 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. 当系统动力学本质上是线性的时，这是允许的，因为这样总的动态行为可以被建模为系统在其单独的本征模态下的行为的叠加。 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. 本征模态理论基于以下性质：非刚性动力系统（描述为一系列由弹簧和阻尼器连接的物体）显示出多个特征共振频率。 Excitation at these frequencies will cause a synchronous periodic movement of all bodies of the system. 这些频率的激励将引起系统所有物体的同步周期性运动。 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 特征周期运动被称为“本征模式”，其中德语和荷兰语“eigen”的意思是“自己的”，反映了它是特征系统属性的事实。相应的
a: Eigenmode 1 a：本征模 1
b: Eigenmode 1 equivalent (combined mass and stiffness) b：本征模 1 等效值（质量和刚度组合）
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：根据系统本征模态将四阶动态系统分裂为两个二阶质量弹簧系统。第一个本征模态是刚体模态，其中两个膜片 和 朝同一方向移动，就好像它们是具有模态质量的一个物体一样 。然后可以将两个隔膜的质量和悬架刚度相加，以确定第一本征模式的动态响应。第二本征模理解起来有点复杂。 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. 在这种模式下，两个质量块以相同的幅度彼此相反地移动，就像由机构驱动一样。对称性允许第二物体及其相关弹簧的镜像，并且与第一本征模态一样，模态质量 。需要特别注意连接空气弹簧，它是等效简单质量弹簧系统的四倍。
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. 共振频率称为该模式的本征频率，而作为物体函数的运动幅度称为由形状函数描述的“模式形状”，形状函数是每个物体的矢量符号，其中值的符号表示该点相对于参考体的相位。
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 ). 作为示例，图 5 的无阻尼响应显示了两个明显可区分的特征频率，其中一个为 和一个在 。对应的本征模态 两个扬声器振膜具有均匀且相等的模态振型（形状函数 [ ）。第二本征模态为 具有模态形状，其中两个膜片以相等的幅度彼此相反地移动（形状函数 ）。
From this example one could conclude that the amount of eigenmodes is equal to the square root of the order of the system, which is correct. 从这个例子可以得出结论，本征模的数量等于系统阶数的平方根，这是正确的。 In principle the exact model of a real system should consist of an infinite amount of springs, dampers and bodies with a corresponding large amount of eigenmodes. 原则上，真实系统的精确模型应由无限数量的弹簧、阻尼器和具有相应大量特征模态的物体组成。 In a loudspeaker these are most visible at the higher frequencies where diaphragm-breakup, edge diffraction and other dynamic phenomena represent each their own eigenmodes. 在扬声器中，这些在较高频率下最为明显，其中隔膜破裂、边缘衍射和其他动态现象代表各自的本征模式。 In practice the infinite amount of eigenmodes can be reduced to a smaller set by neglecting eigenmodes with a very high eigenfrequency, outside the frequency range of interest. With the example of the passive radiator bassreflex system this set can be restricted to the two mentioned eigenmodes, because this analysis focuses on low frequency sound reproduction. 实际上，通过忽略感兴趣频率范围之外的具有非常高本征频率的本征模，可以将无限数量的本征模减少到较小的集合。以无源辐射器低音反射系统为例，该组可以限制为提到的两种本征模式，因为该分析侧重于低频声音再现。
The reason why this modal analysis is introduced here with this symplified system is its usefulness to explain the anti-resonance of the driven loudspeaker as not being a resonance at all. For that reason it also does not correspond to an eigenmode. 之所以在此引入该模态分析与该简化系统，是因为它有助于将驱动扬声器的反谐振解释为根本不是谐振。因此，它也不对应于本征模。
Figure 7 shows the mode-shapes that belong to the two eigenmodes of this fourthorder system while re-arranging the lumped-elements such that their modal behaviour can be directly determined. 图 7 显示了属于该四阶系统的两个本征模态的振型，同时重新排列集总元件，以便可以直接确定它们的模态行为。 One should be aware that this simplification is only valid for this specific symmetric situation with equal mass and stiffness values. 人们应该意识到，这种简化仅适用于质量和刚度值相等的特定对称情况。 In the next section it will be shown that an asymmetric system like the bassreflex system with air-port needs some additional adaptations to enable the analysis. 在下一节中，我们将展示像带空气端口的低音反射系统这样的非对称系统需要一些额外的调整才能进行分析。
The first eigenmode is the low frequency mode where both diaphragms move in the same direction with equal amplitude and phase, supported by the suspension rubber and spider. 第一个本征模式是低频模式，其中两个膜片在悬架橡胶和星形轮的支撑下以相同的幅度和相位沿相同方向移动。 With this mode the connecting air-spring is not deforming and its influence can thus be neglected from the analysis. 在这种模式下，连接空气弹簧不会变形，因此在分析中可以忽略其影响。 This first eigenmode will have an eigenfrequency which is equal to the eigenfrequency of the unmounted loudspeaker, because the combined masses of the two diaphragms work together with a total, so called "modal mass" on the combined stiffness of the two suspensions: 该第一本征模态的本征频率等于未安装的扬声器的本征频率，因为两个振膜的组合质量一起工作，称为“模态质量” 两种悬架的组合刚度：
when and . 什么时候 和 。
This corresponds with the first resonance peak in Figure 8. 这对应于图 8 中的第一个共振峰。
The second eigenmode is, as mentioned, related to the opposite movement of the 如前所述，第二本征模态与
two bodies, elastically coupled with the connecting air-spring and an eigenfrequency as calculated in Equation (17). 两个物体，与连接的空气弹簧弹性耦合，特征频率如方程（17）所示。 The movement amplitude is defined by the masses of the two bodies where a larger mass shows a lower amplitude to correlate the acceleration with the equal force in the connecting spring . 运动幅度由两个物体的质量定义，其中较大的质量显示较低的幅度，以将加速度与连接弹簧中的相等力相关联 。
To imagine both eigenmodes is not extremely difficult. The first eigenmode is the most easy to imagine with the two loudspeakers moving in the same direction, even when only one of them is driven. Like a car with a caravan. 想象这两种本征模态并不是极其困难。第一个本征模式是最容易想象的，两个扬声器朝同一方向移动，即使只驱动其中一个。就像一辆带有大篷车的汽车。
To imagine the second eigenmode one should think of two balls connected with a spring hanging in outer space. An astronaut grabs these balls, stretches the spring and lets the balls loose. 要想象第二本征模，我们应该想象两个与悬挂在外层空间的弹簧相连的球。宇航员抓住这些球，拉伸弹簧并使球松开。 Now it is easy to see that the balls first approach each other until the spring is compressed, then they separate again, etc. When the balls have a different mass the smaller ball will move more quickly. 现在很容易看出，球首先相互接近，直到弹簧被压缩，然后它们再次分开，等等。当球具有不同的质量时，较小的球会移动得更快。 In extremis, like with a car as the first body connected via its suspension to the earth as second body, a bumping car will hardly move the earth due to the immense mass difference. 在极端情况下，就像汽车作为第一物体通过其悬架连接到地球作为第二物体一样，由于巨大的质量差异，碰撞的汽车几乎不会移动地球。
In the complete system the combination of both modes determines the total behaviour and as a result a force in the positive -direction will also result in a movement of the second body in the positive -direction but less than with the first eigenmode only. For the first body the response of the second eigenmode is positively added to the first eigenmode while the second eigenmode is subtracted from the first eigenmode for the second body. 在完整的系统中，两种模式的组合决定了总体行为，从而产生积极的力量 - 方向也会导致第二个物体朝正方向运动 -方向，但小于仅第一本征模。对于第一物体，第二本征模式的响应被正加到第一本征模式，而对于第二物体，从第一本征模式减去第二本征模式。
It is interesting to see how the elements can be rearranged in ones imagination for the analytical understanding in this case where both moving diaphragms have the same mass. Especially the impact of the air-spring will prove to be significant. 有趣的是，在两个移动隔膜具有相同质量的情况下，如何在想象中重新排列元素以进行分析理解。尤其是空气弹簧的影响将被证明是显着的。 In this specific situation with two equal masses the magnitude of the movement of both masses is equal. As a consequence the middle of the air-spring does not move. 在这种具有两个相同质量的特定情况下，两个质量的运动幅度是相等的。因此，空气弹簧的中部不会移动。 One might call it a "node" where only force is transferred and for that reason this middle point could be connected to the stationary world like a wall without effect on the eigenmode from a dynamic point of view. 人们可能将其称为仅传递力的“节点”，因此该中间点可以像墙一样连接到静止世界，而从动态角度来看不会对本征模产生影响。 This means that each body works on half the spring with double the stiffness as the full spring. 这意味着每个主体在一半弹簧上工作，其刚度是整个弹簧的两倍。
The presence of a imaginary wall in the middle allows to imagine the second body mirrored to the other side of the wall, while it also could be directly connected to the first body. This is allowed for the analysis as the movements are equal for this eigenmode. 中间存在一堵假想的墙，可以想象第二个身体镜像到墙的另一侧，同时它也可以直接连接到第一个身体。这是允许进行分析的，因为该特征模态的运动是相等的。 As a last step all mass and stiffness values can be added like with the first eigenmode. This means that also for this second eigenmode the modal mass becomes equal to and the air-spring stiffness appears with a factor four times in the equivalent simplified mass-spring system. For the example this stiffness equals 作为最后一步，可以像第一个特征模态一样添加所有质量和刚度值。这意味着对于第二本征模态，模态质量 变得等于 空气弹簧的刚度增加了四倍 在等效简化的质量弹簧系统中。例如，该刚度等于
The combined frequency-response transfer function can be derived from the two responses as shown in Figure 7. 组合频率响应传递函数可以从两个响应中得出，如图 7 所示。
The frequency responses start at low frequencies on a different level because of the difference in stiffness of both modes due to the air-spring. At the common 由于空气弹簧导致两种模式的刚度存在差异，因此频率响应从不同级别的低频开始。在 共同点
Figure 8: Bode plots of the diaphragms as combined response from the two undamped eigenmodes. The difference in phase of the driven loudspeaker and the passive radiator in the second eigenmode causes an "anti-resonance" at approximately because at that frequency the contributions of both eigenmodes are equal but with an opposite sign. With the passive radiator the phases are equal hence the values simply add to the double value . Note the combined sound response which is simply double ( ) the response of the second eigenmode for each loudspeaker. 图 8：隔膜的伯德图，作为两个无阻尼本征模的组合响应。驱动扬声器的相位差 和无源辐射器 在第二本征模中引起大约“反谐振” 因为在该频率下，两个本征模态的贡献相等，但符号相反。对于无源辐射器，相位相等，因此这些值只需添加到双倍值 。请注意组合声音响应，它是简单的两倍（ ）每个扬声器的第二本征模式的响应。
eigenfrequency of the first eigenmode is visible in the response of both diaphragms. At the magnitude of both modes is equal and to determine the combined movement it is important to look at the phase of both modes. For the driven loudspeaker the first eigenmode has almost phase at while the second eigenmode has phase. This means both contributions to the movement of the first diaphragm will cancel each other out and cause the anti-resonance which appears to be no resonance at all but just the combination of two equal opposite movements. 第一本征模式的本征频率在两个隔膜的响应中可见。在 两种模式的幅度相等，为了确定组合运动，重要的是查看两种模式的相位。对于驱动扬声器来说，第一本征模态几乎 相位在 而第二本征模有 阶段。这意味着对第一隔膜运动的两种贡献将相互抵消并导致反共振，该反共振看起来根本没有共振，而只是两个相等的相反运动的组合。 For the passive radiator the situation is different because here the second eigenmode moves in the opposite direction of the driven loudspeaker. An opposite movement means phase and as a consequence the first and second eigenmode have the same phase at for the passive radiator. As a consequence the movements add to a factor two (+6 dB). At higher frequencies first the eigenfrequency of the second eigenmode at shows its characteristic resonance peak. Above the driven loudspeaker follows a flat response corresponding by a constant acceleration, like with a closed box loudspeaker. The passive radiator however shows a -2 slope at increasing frequencies which is caused by the fact that both eigenmodes approach 对于无源辐射器，情况有所不同，因为这里第二本征模沿与驱动扬声器相反的方向移动。相反的运动意味着 相位，因此第一和第二本征模具有相同的 相位在 对于无源辐射器。因此，运动增加了两倍 (+6 dB)。在较高频率处，首先是第二本征模的本征频率 显示其特征共振峰。多于 驱动扬声器遵循与恒定加速度相对应的平坦响应，就像封闭式箱体扬声器一样。然而，无源辐射器在频率增加时显示出 -2 斜率，这是由于两个本征模接近的事实造成的
the same mass-determined response corresponding with with a phase difference of which means that they cancel each other out more at higher frequencies. 相同的质量决定响应对应于 相位差为 这意味着它们在较高频率下相互抵消得更多。
The effect of damping is equal as shown with the analytical calculations. It should be noted that damping of the driven loudspeaker acts on both eigenmodes but with different levels. The quality factor is higher with the high stiffness of the second eigenmode which means that more damping is needed to suppress this second eigenmode. This is not always sufficiently possible leading to "boombass". 如分析计算所示，阻尼效果是相等的。应该注意的是，驱动扬声器的阻尼作用于两种本征模，但程度不同。品质因数 第二本征模态的刚度较高，这意味着需要更多的阻尼来抑制第二本征模态。这并不总是有足够的可能导致“轰动”。 A loudspeaker must be designed according to the application by tuning the electromagnetic properties of the actuator to the mass and the enclosure otherwise the result will not be acceptable. 扬声器必须根据应用进行设计，根据质量和外壳调整执行器的电磁特性，否则结果将不可接受。
Further it is good to be aware that the sound is only produced by the second eigenmode which matches the green line in the figure which is a factor above the movement of each diaphragm apart for this second eigenmode only. This factor 2 is due to the fact that the calculation is made for both eigenmodes separately, when driven with a unit force. 此外，值得注意的是，声音仅由第二本征模产生，该本征模与图中的绿线相匹配，这是一个因素 仅针对第二本征模的每个隔膜的运动上方。这个因子 2 是因为当用单位力驱动时，计算是分别针对两个本征模态进行的。 When combined this force would be divided by two over the two eigenmodes, thereby cancelling the factor 2 in reality. 当合并时，该力将在两个本征模上除以二，从而在现实中消除因子 2。
From this observation one can conclude that it is better to only drive the system in its second eigenmode with sufficient damping. This requires that the first eigenmode is not excited and that is only possible when the passive radiator is also driven. 从这一观察可以得出结论，最好仅在具有足够阻尼的第二本征模式下驱动系统。这要求第一本征模式不被激励，并且只有当无源辐射器也被驱动时才可能实现。 When doing so the system becomes identical to a closed box with two drivers, which do deliver when compared to one driver because of the twice supplied electrical power. 这样做时，系统变得与带有两个驱动器的封闭盒子相同，它们确实提供 与一名驾驶员相比，因为提供了两倍的电力。
6 Bassreflex with Air-Port Resonator 6 低音反射带空气端口谐振器
A passive radiator is always more expensive than an air-port made by means of a plastic tube. This low cost level is the main reason that the latter is mostly applied even though it is more difficult to achieve a well defined low frequency behaviour. 无源散热器总是比由塑料管制成的空气端口更昂贵。这种低成本水平是大多数应用后者的主要原因，尽管实现明确定义的低频行为更加困难。 Figure 9,a: shows a schematic drawing of the principle where the passive radiator is replaced by a volume of air contained within a tube that is open both to the outside and to the inside of the enclosure. 图 9,a：显示了原理示意图，其中无源散热器被包含在向外壳外部和内部开放的管内的一定体积的空气取代。 The air-volume in the bassreflex-port will act as the second passively radiating body and determine a resonating eigenfrequency with the spring of the enclosure and the mass of the driven loudspeaker diaphragm. 低音反射端口中的风量将充当第二个无源辐射体，并通过外壳的弹簧和驱动扬声器振膜的质量确定共振本征频率。 Unfortunately the modal analysis of this system is less simple as with the previously described symmetrical system. This is caused by the difference in diameter of the bassreflex port and the loudspeaker diaphragm and the different mass values. 不幸的是，该系统的模态分析不如前面描述的对称系统那么简单。这是由于低音反射端口和扬声器振膜的直径差异以及质量值不同造成的。 Still it will be shown that the same "anti-resonance" effect occurs on the driven loudspeaker as with the passive radiator. 尽管如此，仍将显示出与无源辐射器相同的“反谐振”效应发生在驱动扬声器上。 This corresponds approximately with the eigenfrequency of a Helmholtz resonator determined by the air-mass in the port and the enclosed volume of air as explained in Figure 1. 这与亥姆霍兹谐振器的特征频率大致对应，该特征频率由端口中的空气质量和封闭的空气体积决定，如图 1 所示。 It is also important to note that the same dynamic limitations that were described in Section 5.2 are valid 同样重要的是要注意第 5.2 节中描述的相同动态限制是有效的
a: Schematic cross-section
b: Lumped element model
Figure 9: A normal bassreflex loudspeaker enclosure (a:) applies a volume of air as passive radiator. This volume of air is enclosed by a pipe or port that is open at both sides, connecting the enclosure volume to the environment. 图 9：普通的低音反射扬声器外壳 (a:) 使用一定量的空气作为无源辐射器。该体积的空气被两侧开放的管道或端口封闭，将外壳体积与环境连接起来。 The mass of the volume of air inside the air-port or bassreflex-port port will resonate with the stiffness of the air spring by the enclosure volume causing a reduction of the excursion amplitude of the driven loudspeaker diaphragm at that frequency. 空气端口或低音反射端口端口内的空气体积的质量将通过外壳体积与空气弹簧的刚度共振，导致驱动扬声器振膜在该频率下的偏移幅度减小。 The lumped element equivalent scheme (b:) is used to derive the frequency transfer functions. 集总元件等效方案 (b:) 用于导出频率传递函数。
for bassreflex systems with an air-port resonator, because, as will be shown, the dynamics are essentially equal. 对于带有空气端口谐振器的低音反射系统，因为正如将要显示的，动态基本上是相等的。
6.1 Frequency Response of Air-Port Resonator 6.1 气口谐振器的频率响应
The frequency response for both moving elements is determined with the help of the lumped-element model of Figure 9,b:. New terms in this model are the radiating surfaces for the driven loudspeaker and for the opening of the bassreflex port, where stands for the diameter. The mass of the passive radiator is determined by the volume of the port and the density of air . As mentioned with the Helmholtz resonator, the effective length equals the length of the port plus approximately 0.73 times the diameter of the port . The factor for the adiabatic compression/expansion of the air in the enclosure can be accounted with as a factor reducing the volume of air in the enclosure . 两个移动元件的频率响应是借助图 9,b: 的集总元件模型来确定的。该模型中的新术语是辐射面 对于驱动扬声器和 用于打开低音反射端口，其中 代表直径。无源辐射器的质量由端口的体积决定 和空气的密度 。正如亥姆霍兹谐振器中提到的，有效长度 等于端口长度加上大约 0.73 倍端口直径 。因素 对于外壳中空气的绝热压缩/膨胀，可以将其视为减少外壳中空气体积的一个因素 。
Starting with the driven loudspeaker with mass : 从具有质量的驱动扬声器开始 :
where equals the stiffness of the surround suspension of the loudspeaker and equals the air pressure of the environment . Both fractions in the above equation represent the relative volume change by a movement of the respective elements which, after multiplication with the average environmental pressure and the radiating surface of the loudspeaker, gives the force on that surface due to the movement of each element. 在哪里 等于扬声器环绕悬架的刚度，并且 等于环境气压 。上式中的两个分数都表示各个元件运动引起的相对体积变化，与平均环境压力和辐射表面相乘后 扬声器的，给出了由于每个元件的运动而作用在该表面上的力。
Written as force in terms of and this equation is written as: 写成力的形式为 和 该方程写为：
Doing the same steps as with the loudspeaker the motion equation for the passive radiating air-mass can be derived, resulting in the following relation between and : 执行与扬声器相同的步骤，计算无源辐射气团的运动方程 可以推导，得到以下关系 和 :
To simplify further calculations three stiffness terms are defined: 为了进一步简化计算，定义了三个刚度项：
With these terms Equation (20) is simplified into: 利用这些项，方程 (20) 简化为：
and with Equation (21) the displacement can be written as function from : 并用方程（21）计算位移 可以写成函数 :
Filling this in Equation (23) and careful applying some algebra leads to the following transfer functions from force to motion: 将其填入方程（23）并仔细应用一些代数可得出以下从力到运动的传递函数：
and: 和：
with: 和：
Replacing s with and multiplying the numerator with to get the acceleration response, ultimately leads to the following proportional radial frequency response functions for the soundpressure of the loudspeaker diaphragm: 将 s 替换为 并将分子乘以 为了获得加速度响应，最终得出扬声器振膜声压的以下比例径向频率响应函数：
To retain the same proportionality for the soundpressure of the passive radiator it is necessary to correct for the much smaller radiating surface for which reason the transfer function is multiplied with the ratio : 为了保持无源辐射器的声压相同的比例，有必要针对更小的辐射表面进行校正，因此传递函数要乘以该比率 :
The total sound pressure is than equal to the difference of these equations as being caused by the motion difference between the driven loudspeaker and the moving air in the bassreflex port. 总声压等于这些方程的差，这是由驱动扬声器和低音反射端口中的移动空气之间的运动差引起的。
As an example the applied loudspeaker of the previous part is used in the same enclosure cabinet while the passive radiating diaphragm is replaced by a tube with a diameter of and a length . The air volume is then approximately because, as mentioned before, also the air just outside the port has to be taken into account, giving an effective length . The resulting moving mass of the passive radiator is then approximately . This is more then a factor 100 below the moving mass of the active driven loudspeaker diaphragm and one would expect little effect. 举例来说，前一部分所应用的扬声器被用在同一个箱体中，而无源辐射振膜则被直径为 和一个长度 。那么风量大约为 因为，如前所述，还必须考虑端口外部的空气，从而给出有效长度 。由此产生的无源辐射器的移动质量大约为 。这比有源驱动扬声器振膜的移动质量低 100 倍以上，预计影响不大。 The frequency response functions for the little damped and optimally damped situation are calculated in MATLAB and shown in Figure 10, unexpectedly indicating a comparable dynamic characteristic as with the passive radiating diaphragm. 小阻尼和最佳阻尼情况下的频率响应函数在 MATLAB 中计算，如图 10 所示，出乎意料地表明了与无源辐射膜片相当的动态特性。 As will be showed with the modal analysis this is caused by the ratios between the active surfaces of the air in the tube and the loudspeaker diaphragm. This is very prominently shown with the first eigenfrequency which is significantly below the resonance frequency of the unmounted loudspeaker. When looking at the modal mass analysis in the next section it is showed that the small mass of the air is perceived as a large mass on the driven loudspeaker diaphragm to the ratio of the radiating surfaces squared. 正如模态分析所示，这是由管内空气的活性表面与扬声器振膜之间的比率引起的。这在第一特征频率中表现得非常明显，该特征频率明显低于 未安装扬声器的谐振频率。当查看下一节中的模态质量分析时，可以发现，与辐射表面平方之比相比，空气的小质量在驱动扬声器振膜上被感知为大质量。
6.2 Stepresponse of Air-Port Resonator
The stepresponse from Figure 10 is calculated for four different settings of the mainly resistive amplifier output impedance. 图 10 中的阶跃响应是针对主要电阻放大器输出阻抗的四种不同设置进行计算的。 A true voltage source amplifier which almost all modern amplifiers are, shows a better damped stepresponse than the passive radiator with only one period of delayed response. 几乎所有现代放大器都是真正的电压源放大器，它比仅具有一个延迟响应周期的无源辐射器表现出更好的阻尼阶跃响应。 This difference in dynamic behaviour is related to a somewhat higher damping of the air-port by the high velocity 这种动态行为的差异与高速运动对空气端口的较高阻尼有关。
Figure 10: The Bode plots with little (a:) and optimal (b:) damping and step response with different levels of damping of the bassreflex system designed with the same enclosure as the passive loudspeaker diaphragm system of the previous section, where the passive radiating diaphragm is exchanged by an air volume in a pipe of diameter and length. When provided with the damping caused by a voltage source amplifier the frequency responses look almost equal as with the passive radiating diaphragm. 图 10：使用相同设计的低音反射系统的小阻尼 (a:) 和最佳阻尼 (b:) 的伯德图以及不同阻尼级别的阶跃响应 外壳作为上一节的无源扬声器振膜系统，其中无源辐射振膜与管道中的空气体积进行交换 直径和 长度。当提供由电压源放大器引起的阻尼时，频率响应看起来几乎与无源辐射膜片相同。 The step response is improved although it is clearly seen that an amplifier with a non-zero output impedance rapidly results in a deterioration with a strong delayed resonance after a transient. 尽管可以清楚地看到，具有非零输出阻抗的放大器在瞬态后会因强烈的延迟谐振而迅速导致性能恶化，但阶跃响应得到了改善。
Figure 11: The impact of the damping in by the port on two settings of amplifier impedance. With a voltage source amplifier the level of port-damping influences the magnitude rather than the periodicity of the response while a high output impedance amplifier benefits more of a stronger damped port. 图 11：阻尼的影响 通过放大器阻抗的两种设置的端口。带电压源放大器 端口阻尼水平影响响应的幅度而不是周期性，而高输出阻抗放大器 更强的阻尼端口更有好处。
of the air than was the case with the modelled undamped passive diaphragm of the previous case. 空气比前一个案例的建模无阻尼无源隔膜的情况要大。 An increasing amplifier impedance shows a significant effect on the delayed resonance with almost three periods when the impedance of the amplifier becomes equal to the resistive value of the loudspeaker impedance. 当放大器的阻抗变得等于扬声器阻抗的电阻值时，增加的放大器阻抗显示出对几乎三个周期的延迟谐振的显着影响。 Even though this is a high value especially tube amplifiers often show an output impedance in the Ohmic range due to lack of feedback and these amplifiers require a higher level of internal damping by for instance the port. 尽管这是一个很高的值，但由于缺乏反馈，电子管放大器通常会显示出欧姆范围内的输出阻抗，并且这些放大器需要通过端口等更高水平的内部阻尼。 This is demonstrated in Figure 11 where four different levels of port damping are calculated for two settings of amplifier impedance. Both situations show a beneficial effect of increased port damping but the effect is most prominent with the high output impedance amplifier. 图 11 对此进行了演示，其中针对两种放大器阻抗设置计算了四种不同级别的端口阻尼。这两种情况都显示了增加端口阻尼的有益效果，但这种效果在高输出阻抗放大器中最为突出。 This indicates that a less fortunate amplifier loudspeaker combination can be improved by increasing the port damping. A reason why several people prefer to combine their tube amplifier with a bassreflex loudspeaker over a closed box loudspeaker. 这表明可以通过增加端口阻尼来改善不太幸运的放大器扬声器组合。这就是为什么有些人更喜欢将电子管放大器与低音反射扬声器而不是封闭箱式扬声器结合起来的原因。
The beneficial effect of the port damping on the dynamics is however at a sacrifice of noise as most of the energy is dissipated in turbulence around the edges of the port. 然而，端口阻尼对动力学的有益影响是以牺牲噪声为代价的，因为大部分能量在端口边缘周围的湍流中消散。 This can be improved by rounding the edges but then the damping is decreased and one might insert a piece of fibre padding or rubber foam with open cells inside the tube to increase the damping. 这可以通过将边缘倒圆来改善，但随后阻尼会降低，并且可以在管内插入一块具有开孔的纤维垫或橡胶泡沫以增加阻尼。 This again is quite unpredictable, resulting in a larger spread in performance of different loudspeakers with the same design. 这也是相当不可预测的，导致具有相同设计的不同扬声器的性能差异更大。 On the other hand it gives the possibility to tune a loudspeaker to the amplifier and with suitable measuring microphones one can even optimise the system for the listening room to a limited extent. 另一方面，它提供了将扬声器调谐到放大器的可能性，并且使用合适的测量麦克风甚至可以在有限的范围内优化听音室的系统。
6.3 Modal Analysis of Air-Port Resonator
It was previously explained that the modal analysis of a "normal" mechanical system consisting of lumped bodies, springs and dampers is based on the principle that each eigenmode can exist independent of the others and will show a resonance when excited in its eigenfrequency. 之前解释过，由集总体、弹簧和阻尼器组成的“正常”机械系统的模态分析基于这样的原理：每个本征模态可以独立于其他模态存在，并且在以其本征频率激发时会表现出共振。 It also implies that no other external forces act on the system other than the excitation force by the actuator and the forces in the springs and dampers that connect the bodies to each other and to the stationary world. 它还意味着，除了执行器的激振力以及将物体彼此连接以及连接到静止世界的弹簧和阻尼器中的力之外，没有其他外力作用在系统上。 The passive radiator had the same diameter as the driven loudspeaker for which reason the connecting air could be modelled straightforward as a mechanical spring acting equally on both diaphragms. The system with a bassreflex port is however quite different as the driven loudspeaker will experience another stiffness value by the enclosed air volume as the air-mass in the port, due to the diameter difference that comes squared in the equation for the stiffness value. 无源辐射器的直径与从动扬声器的直径相同，因此可以将连接空气直接建模为作用在两个振膜上的机械弹簧。然而，具有低音反射端口的系统则完全不同，因为驱动扬声器将通过封闭的空气体积（作为端口中的空气质量）经历另一个刚度值，这是由于刚度值方程中的直径差的平方。 Furthermore the volume of air acts like a compressible medium creating forces to all surfaces inside the enclosure. 此外，空气体积就像可压缩介质一样，对外壳内的所有表面产生力。
The easy part is the fact that also in this case it is allowed to limit the relevant eigenmodes to just two as the non-modal analysis shows two clearly distinguishable eigenfrequencies, corresponding to a first eigenmode where both bodies (diaphragm and air-column in the bassreflex port) move in the same direction and a second eigenmode where they move opposite to each other. 简单的部分是，在这种情况下，也允许将相关本征模态限制为仅两个，因为非模态分析显示了两个明显可区分的本征频率，对应于第一个本征模态，其中两个物体（膜片和空气柱）低音反射端口）沿相同方向移动，而第二本征模则彼此相反移动。
The first eigenmode will have a mode-shape where the air in the bassreflex port will show a higher amplitude than the loudspeaker diaphragm in the ratio of the cross-section of the loudspeaker and the bassreflex port. 第一本征模将具有模态形状，其中低音反射端口中的空气将在扬声器和低音反射端口的横截面的比率方面显示出比扬声器振膜更高的振幅。 When assuming the air to be incompressible, the corresponding shape function would equal ). The assumption of incompressibility at the low frequency is based on the understanding that at this frequency the air in the port will not yet receive much motion resistance, hence not exert large forces. 当假设空气不可压缩时，相应的形函数将等于 ）。低频不可压缩性的假设是基于这样的理解：在该频率下，端口中的空气还不会受到太多运动阻力，因此不会施加大的力。 It will be shown that this assumption is only allowed for a very rough approximation and in any case the mass of the air is accelerated with a large factor higher than the first body and this has a very interesting effect on the equivalent modal mass as observed at the point of excitation, which is the first moving mass. 将会表明，这种假设只允许非常粗略的近似，并且在任何情况下，空气质量都会以比第一物体高得多的系数加速，这对等效模态质量有非常有趣的影响，如在激发点，即第一个移动质量。 This is best explained with mathematics, starting with Newton's second law on inertia with the variables as defined in Figure 9: 这最好用数学来解释，从牛顿第二惯性定律开始 变量如图 9 中定义：
where equals the acceleration of the driven loudspeaker with mass and equals the reactive force by the mass of the air in the bassreflex port. This reactive force is equal to the pressure that is created by the force that accelerates the air in the port. 在哪里 等于驱动扬声器的质量加速度 和 等于反作用力除以质量 低音反射端口中的空气。该反作用力等于压力 这是由力量创造的 加速港口内的空气流动。
Note that in the relation between the acceleration levels the condition of incompressibility is assumed. Both equations combined give the following value for the modal 请注意，在加速度水平之间的关系中，假定了不可压缩条件。两个方程组合起来给出以下模态值
mass: 重量：
With for the driven loudspeaker, for the port and , the modal mass for the first eigenmode becomes equal to . This means that the mass of the air is even more dominant than the mass of the driven loudspeaker for this eigenmode. In reality it is necessary to take into account the real finite stiffness for the connecting air spring. 和 对于驱动扬声器， 对于港口和 ，第一本征模态的模态质量变得等于 。这意味着对于该本征模，空气的质量甚至比驱动扬声器的质量更占主导地位。实际上，有必要考虑连接空气弹簧的实际有限刚度。 It is well imaginable that a lower stiffness value than infinite will create a smaller movement of the mass of the air in the bassreflex port, thus reducing the reactive force. 可以想象，比无穷大更低的刚度值将在低音反射端口中产生更小的空气质量运动，从而减小反作用力。 Most probably the modal mass component of the air in the bassreflex port is approximately equal to the mass of the loudspeaker diaphragm like is the case for the second eigenmode as will be shown further on. 最有可能的是，低音反射端口中的空气的模态质量分量大约等于扬声器振膜的质量，就像第二本征模态的情况一样，如将进一步示出的。
The stiffness of the connection of the first eigenmode to the stationary enclosure is equal to the stiffness of the suspension of the driven loudspeaker. With this stiffness the more than doubled mass will result in a significantly lower eigenfrequency at times the eigenfrequency of the unmounted loudspeaker which was . The resulting resonance at around corresponds with the value shown in Figure 10.a:. Unfortunately this lower frequency does not mean that the loudspeaker will reproduce sound at this frequency as the sound pressure is not produced by the first eigenmode, which was also the case with the passive diaphragm version. 第一本征模式与固定外壳的连接的刚度等于从动扬声器的悬架的刚度。有了这种刚度，超过一倍的质量将导致显着降低的特征频率 乘以未安装的扬声器的特征频率，即 。产生的共振大约为 对应于图 10.a: 中所示的值。不幸的是，这个较低的频率并不意味着扬声器将在此频率再现声音，因为声压不是由第一本征模产生的，无源振膜版本也是如此。 Even though the air in the port moves faster in the ratio of the radiating surfaces, the same ratio compensates the effect on sound pressure as it is linear proportional to both surface and excursion . 尽管端口中的空气在辐射表面的比率上移动得更快，但相同的比率可以补偿对声压的影响，因为它与表面和偏移成线性比例 。
For the sound radiation the second eigenmode is the determining factor and this analysis is even more complicated because now the compressibility of the air must be taken into account. 对于声辐射，第二本征模态是决定因素，并且这种分析更加复杂，因为现在必须考虑空气的可压缩性。 The starting point for this modal analysis is the assumption that the enclosure is small in respect to the wavelength of the sound at the eigenfrequency of the second eigenmode. From Figure 10.a: this eigenfrequency is expected around where the second resonance is shown. This corresponds to a wavelength of several metres so the condition is met and as a consequence the air pressure can be assumed homogeneous inside the enclosure. 该模态分析的出发点是假设外壳相对于第二本征模的本征频率处的声音波长而言较小。从图 10.a 可以看出：该特征频率预计约为 其中显示了第二个共振。这对应于几米的波长，因此满足条件，因此可以假设外壳内的气压是均匀的。 This means that the forces acting on both moving masses will relate to the radiating surfaces and . At the eigenfrequency of the second eigenmode the system is in full equilibrium and assuming no energy is dissipated it will keep resonating at this frequency. 这意味着作用在两个移动质量上的力将与辐射表面相关 和 。在第二本征模的本征频率处，系统处于完全平衡，并且假设没有能量耗散，它将在此频率下保持谐振。 In that case the relative periodic accelerations and the directly proportional relative periodic displacements of both bodies can then be calculated as follows using Newton's second law with the necessary equilibrium in pressure inside the enclosure: 在这种情况下，两个物体的相对周期加速度和成正比的相对周期位移可以使用牛顿第二定律以及外壳内压力的必要平衡计算如下：
This enables to write down the following ratio between and : 这使得能够写下以下比率 和 :
The sound pressure is a function of the excursion and the radiating surfaces giving: 声压是偏移和辐射表面的函数 给予：
With the numbers from the example this ratio equals approximately 1.4, meaning both radiating surfaces act almost equal on the sound pressure at the eigenfrequency of the second eigenmode. 根据示例中的数字，该比率大约等于 1.4，这意味着两个辐射表面对第二本征模式的本征频率处的声压的作用几乎相等。 The determination of the eigenfrequency is based on the fact that there should be a neutral zone in the air spring as both bodies move opposite to each other. This means that there is a dividing plane inside the enclosure where the molecules of air stand still. 特征频率的确定基于这样的事实：当两个物体彼此相反地运动时，空气弹簧中应该存在一个中性区。这意味着外壳内部有一个分隔面，空气分子在其中保持静止。 This plane is determined by the volume change which is related to a displacement of both diaphragms of which the relation is given by Equation (33). 该平面由与两个隔膜的位移相关的体积变化确定，其关系由方程（33）给出。
which is not unexpectedly the same relation as between the sound pressures with a value of 1.4 for the practical example, meaning that about of the enclosure volume is used by the loudspeaker and by the bassreflex-port. These findings all point stronger and stronger to the previously found similarity between the passive radiator and the port-loaded bassreflex system and indeed this is a true finding. 这与实际示例中值为 1.4 的声压之间的关系并不意外，这意味着大约 扬声器使用了外壳体积，并且 通过低音反射端口。这些发现都越来越有力地证明了之前发现的无源辐射器和端口加载的低音反射系统之间的相似性，这确实是一个真实的发现。 If the ratio value was equal to one the systems would be exactly the same and this can be arranged in this example by increasing the air mass with a longer bassreflex port. 如果比率值等于 1，则系统将完全相同，并且在此示例中可以通过使用更长的低音反射端口增加空气质量来进行安排。 The eigenfrequency of the second eigenmode is then equal to the value found with the passive radiator with but even with the given dimensions the system acts almost the same. As a check whether this reasoning is true the eigenfrequency of the second eigenmode can be calculated on both the loudspeaker mass and bassreflex port mass each with the stiffness of their own part of the enclosure volume using Equation (30) of the paper on "Low Frequency Sound Generation by Loudspeaker Drivers". Using the values for the loudspeaker with and , and for the moving air with and and taking for both stiffness values and , the following is obtained: 第二本征模的本征频率等于无源辐射器的值 但即使给定尺寸，系统的行为也几乎相同。为了检查这个推理是否正确，可以根据扬声器质量计算第二本征模式的本征频率 和低音反射端口质量 每个都使用“扬声器驱动器产生的低频声音”论文中的方程（30）计算其自身外壳体积部分的刚度。使用扬声器的值 和 ，对于流动的空气 和 并取两个刚度值 和 ，得到如下：
Adding the stiffness of the diaphragm suspension results in the total stiffness for the loudspeaker of . With the moving mass values 增加隔膜悬架的刚度 结果扬声器的总刚度为 。随着移动质量值
of and this results respectively in a natural frequency of: 的 和 这分别导致固有频率为：
The difference in these calculated values, which should have been equal, is small enough to prove the assumption, while it is easily caused by the approximation of the factor 1.4 between the volume parts of the enclosure, having a large impact on the stiffness. 这些计算值的差异本应相等，但足够小，足以证明该假设，但很容易因外壳各体积部分之间系数1.4的近似而引起，对刚度影响较大。 A 10% larger part of the enclosure volume for the loudspeaker and a corresponding smaller part for the moving air in the bassreflex port would result in a different frequency thus equalising the values. 扬声器外壳体积的 10% 较大部分和低音反射端口中移动空气的相应较小部分将导致 不同的频率从而使值相等。
Based on the found similarity with the passive radiator it is not without logic to expect that the modal mass, effective on the point where the actuator drives the system is most probably approximately equal to the modal mass of the first eigenmode. 基于所发现的与无源辐射器的相似性，不无逻辑地预期在致动器驱动系统的点上有效的模态质量很可能近似等于第一本征模态的模态质量。 More exactly it can be determined starting with the first part of Equation (30): 更准确地说，可以从方程（30）的第一部分开始确定：
Using Equation (33) the following relation between the accelerations is obtained 使用方程（33）可以获得加速度之间的以下关系
which leads to a simple expression for the reactive force: 由此得出反作用力的简单表达式：
This means that the reactive force by the coupled mass on the point of insertion of the driving force is equal to force needed for the acceleration of the driven loudspeaker diaphragm alone and the coupled mass just doubles the perceived mass of the driven loudspeaker for the second eigenmode. 这意味着驱动力插入点上耦合质量的反作用力等于驱动扬声器振膜单独加速所需的力，并且耦合质量恰好是第二本征模式的驱动扬声器感知质量的两倍。 This fully complies with the "gutfeel" that the second eigenmode should be in mass balance. It is also logical to state then that the modal stiffness is twice the perceived stiffness of the enclosure by the driven loudspeaker. 这完全符合第二本征模应该处于质量平衡的“直觉”。也合乎逻辑的是，模态刚度是驱动扬声器感知到的外壳刚度的两倍。 With these findings the frequency response can be derived using Matlab with the following steps. 根据这些发现，可以使用 Matlab 通过以下步骤导出频率响应。
The first step is to model the first eigenmode for the driven loudspeaker which can be based on the modal mass with the stiffness of the suspension of the driven loudspeaker. 第一步是对驱动扬声器的第一本征模式进行建模，该模型可以基于模态质量 与驱动扬声器的悬架的刚度。
The acoustic radiation by the first eigenmode of the air-mass in the bassreflex tube is equal and with opposite sign to the acoustic radiation by the first eigenmode of the driven loudspeaker. 低音反射管中空气团的第一本征模的声辐射与驱动扬声器的第一本征模的声辐射相等且符号相反。
Figure 12: The construction of the frequency response by means of eigenmodes with a bassreflex system with air port gives comparable results as with the analytical solution. The deviations are larger due to the many assumptions on the system. 图 12：通过带有空气端口的低音反射系统的本征模构建频率响应，得到与解析解相当的结果。由于系统的假设较多，偏差较大。
The second eigenmode is calculated first on the driven loudspeaker with the modal mass equal to the double mass of the moving diaphragm acting on the double stiffness value of the air-spring defined by the volume part of the enclosure that is compressed/expanded by the driven loudspeaker. 首先在驱动扬声器上计算第二本征模式，其模态质量等于运动膜片的双倍质量，作用于空气弹簧的双刚度值，空气弹簧的双刚度值由驱动扬声器压缩/膨胀的箱体体积部分定义。喇叭。
Finally the responses are combined to give the result. 最后将响应组合起来给出结果。
Figure 12 shows the result of this exercise and when comparing with Figure 10 several deviations are visible. First of all the änti-resonance is not at the same frequency. This is due to the large impact of relative gains of both eigenmodes on the point where they intersect. 图 12 显示了此练习的结果，与图 10 相比，可以看到一些偏差。首先，反共振的频率不同。这是由于两个本征模态的相对增益对其相交点的影响很大。 Secondly the small remaining resonance in the sound output at the first eigenfrequency is not seen. But for the remainder the results are qualitatively comparable. 其次，在第一特征频率处的声音输出中没有看到小的剩余共振。但对于其余部分，结果在质量上具有可比性。 Certainly this all points out that these simplified calculations have to be seen as not better than rough approximations to obtain qualitative indications of the phenomena that can be expected in reality as a large series of assumptions are made which are all not completely true: 当然，这一切都表明，这些简化的计算必须被视为不比粗略的近似更好，以获得现实中可以预期的现象的定性指示，因为所做的一系列假设并不完全正确：
The air is assumed to flow frictionless for both modes without turbulence. 假设空气在两种模式下都无摩擦流动，没有湍流。
All is modelled linear. 一切都是线性建模的。
The moving mass in the bassreflex port is estimated with a "rule of thumb" correction factor for the air just outside the port. 低音反射端口中的移动质量是根据端口外部空气的“经验法则”校正因子来估计的。
The mass of the air in the enclosure is not taken into account. 不考虑外壳中空气的质量。
The mass of air that is driven to produce sound is neglected for reason of the low coupling between a diaphragm and the surrounding air but it is not zero. 由于振膜与周围空气之间的低耦合，被驱动产生声音的空气质量被忽略，但它不为零。
The wavelength and speed of the sound can be neglected in the enclosure. 声音的波长和速度在外壳中可以忽略不计。
etc.
Nevertheless the shown dynamic responses are sufficiently representative for real bassreflex systems to be conclusive. 尽管如此，所显示的动态响应足以代表真实的低音反射系统，从而得出结论。
7 Conclusions on Bassreflex for Very Low Frequencies 关于极低频 Bassreflex 的 7 个结论
When observing the resulting responses from Figure 5 and Figure 10 it is clear that even with a good sized enclosure of 60 litres and a large loudspeaker the achieved lowest frequency is around , with a very steep octave slope below this frequency depending on the applied damping. One can in principle extend this range and reduce the dynamic effect of the resonance by a compensation filter at the input signal. 当观察图 5 和图 10 的响应时，可以清楚地看出，即使使用 60 升的大尺寸外壳和大型扬声器，也能达到最低的 频率大约是 ，具有非常陡峭的 低于该频率的倍频斜率取决于所应用的阻尼。原则上可以通过输入信号处的补偿滤波器来扩展该范围并减少谐振的动态影响。 This will however drive the driven loudspeaker diaphragm in extreme excursions, while the entire purpose of bass reflex was to prevent this. 然而，这将在极端偏移中驱动驱动扬声器振膜，而低音反射的全部目的就是防止这种情况发生。 Furthermore, increasing the movement of large volumes of air below the frequency, where the port takes over the main part of the sound reproduction, will increase flow noise. 此外，在端口接管声音再现的主要部分的频率以下增加大量空气的运动将增加流动噪声。
The only way to really extend the response to is to decrease all resonance frequencies with a factor two. Due to the square root relation with mass or stiffness this means a factor four less stiffness or higher mass or a factor two in both. 真正将响应扩展到 是将所有共振频率降低两倍。由于与质量或刚度的平方根关系，这意味着刚度要小四倍，质量要高，或者两者都要两倍。 While a higher mass will further decrease efficiency, only a four times lower stiffness would work as long as all stiffness values are decreased that much, so including the surround, spider and the volume of the enclosure. 虽然更高的质量会进一步降低效率，但只要所有刚度值都降低那么多，包括环绕件、星形轮和外壳的体积，只有四倍的刚度才有效。 The last one can also be decreased by means of a smaller loudspeaker but then the moving mass is also decreased. 最后一个也可以通过较小的扬声器来减少，但移动质量也会减少。
From this reasoning it can be concluded that only extremely large bass-reflex systems can produce frequencies around . And even then the stepresponse will always show a delayed reaction, giving the impression of uncontrolled "woolly" bass. 从这个推理可以得出结论，只有非常大的低音反射系统才能产生大约 。即使如此，阶跃响应始终会表现出延迟反应，给人一种不受控制的“羊毛状”低音的印象。
Finally there have been times that people believed and seriously stated that a bassreflex system has a higher efficiency. They used as argument that the pipe is open and transfers the sound from the back like in a delay-line. 最后，有时人们相信并认真地声明低音反射系统具有更高的效率。他们用管道打开并像延迟线一样从后面传输声音作为论据。 The fact is, however, that a higher output for the same input power only occurs, when the bassreflex resonance on the second eigenmode ( in the examples) is insufficiently damped and then only around that frequency. Below the roll-off is 4th order, so the power output and efficiency is lower than with the second order roll-off of a closed-box enclosure with bandwidth. Above the air in the pipe or the 然而，事实是，只有当第二本征模上的低音反射谐振（ 在示例中）阻尼不足，并且仅在该频率附近。以下 滚降是四阶，因此功率输出和效率低于封闭式外壳的二阶滚降 带宽。多于 管道或管道中的空气
Figure 13: By closing off the pipe or passive membrane the effect of the bassreflex principle on both frequency and time response is made clear. In a closed-box enclosure the sound output matches the diaphragm motion of the driven loudspeaker. 图 13：通过关闭管道或无源膜，低音反射原理对频率和时间响应的影响变得清晰。在封闭箱体中，声音输出与驱动扬声器的振膜运动相匹配。
passive diaphragm hardly moves and does not transfer anything. This means that it effectively closes off the enclosure at those frequencies and consequently the sound output is then equal to the situation with a closed-box enclosure. 被动隔膜几乎不移动并且不传输任何东西。这意味着它在这些频率下有效地封闭了外壳，因此声音输出等于封闭箱外壳的情况。
Figure 13 is derived from the optimally damped graphs of Figure 5 where the passive membrane part is omitted and the response of the system is added when closing off the passive membrane. 图 13 源自图 5 的最佳阻尼图，其中省略了无源膜部分，并在关闭无源膜时添加了系统响应。 It clearly shows the benefit of a smaller diaphragm excursion, however as was shown before in Figure 6 only after the transient periods are over!. It is also clear that there is hardly an increase in efficiency, while the steeper slope with less output below is also evident. Finally the stepresponse of the closed-box situation is better controlled. When increasing the damping of the bassreflex system this can be improved, but then the benefit on diaphragm excursion will also decrease. 它清楚地显示了较小的隔膜偏移的好处，但是如之前图 6 所示，只有在瞬态周期结束之后！同样明显的是，效率几乎没有增加，而下面的斜率更陡，输出更少 也是显而易见的。最后，闭箱情况的阶跃响应得到了更好的控制。当增加低音反射系统的阻尼时，这可以得到改善，但对振膜偏移的好处也会减少。 This all underlines the conclusion from Section 3 that for high quality well-controlled low-frequency sound reproduction one should use a closed-box enclosure. 这一切都强调了第 3 节的结论，即为了获得高质量、控制良好的低频声音再现，应该使用封闭式外壳。 In two other papers it is shown that the time response can even be further improved by active velocity or acceleration feedback. 另外两篇论文表明，甚至可以通过主动速度或加速度反馈进一步改善时间响应。
