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. 超重低音扬声器必须在低频时提供高功率，这自然意味着它们需要很大，因为需要移动的空气量很大。 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技术风格外形占主导地位，这种大型扬声器并不会得到追求家居美学而非音质的人们的同情。 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，小型和不显眼的扬声器可以通过使用较小的驱动器来实现，但小的辐射面需要大的振膜偏移来补偿小的辐射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. 遗憾的是，由于致动器和悬挂装置的非线性，较大的机械偏移会自动导致失真度增加。
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. 为了缓解这一问题，扬声器制造商采用了阻抗匹配装置（号角）和声学动态现象，在不增加振膜偏移的情况下提高扬声器系统的声学输出。 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. 本文讨论的是第二种选择，特别是低音反射系统，它仍然是应用动力学现象来提高低频性能的最广泛的例子。 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，会添加一个被动共振块，它与驱动器振膜弹性耦合，在降低主扬声器偏移的同时，部分接收一定频率以下的低频声音。
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. 本文的初衷是为大学学生提供一个真实的例子，让他们了解动态特征模态如何决定机械结构的振动特性。因此，本文主要对低音反射系统进行动态分析。
The second (and my personal) goal was, however, to determine the real value of such an approach for high quality loudspeakers. 不过，第二个目标（也是我个人的目标）是确定这种方法对高品质扬声器的真正价值。 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折衷方案，但在追求音乐的通透再现时，低音反射系统大多毫无用处。
The paper starts with the Helmholtz Resonator, which is the basis of the bass-reflex principle. 本文从低音反射原理的基础--亥姆霍兹谐振器说起。 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初步思考后，介绍了一个演示器，在课堂上展示了带有无源膜片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. 在分析了这个例子的动态特性后，我们又对气口谐振器进行了同样的分析。最后从分析中得出结论。
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: 一个众所周知的独立亥姆霍兹谐振器的例子是通过向瓶口吹气来产生声音。亥姆霍兹谐振器共振频率的计算方法如下，首先计算外壳中空气的刚度 ：
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: 根据这些值可以计算出亥姆霍兹共振频率：
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. 因此，被驱动扬声器的振膜在该频率下几乎不会移动，从而可以向系统提供更多电力，从而提高系统的最大声学输出。
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. 使用谐振器的一个缺点是，它收集的能量在输入信号终止后会延迟传递回来，从而在亥姆霍兹谐振器的共振频率上产生延迟共振，导致声音变色。 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. 延迟共振可通过阻尼来限制，阻尼可将振动能量消耗为热量。标准低音反射系统的阻尼由扬声器驱动器和放大器共同决定。 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. 第二个阻尼系数是端口的耗散能量，由于它受到端口形状和端口附近机柜内阻尼材料用量的影响，因此很难调整。 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. 为了减少这些偏差，我们引入了低音反射原理的一个特殊版本，即用一个额外的扬声器振膜（被称为具有明确动态特性的无源辐射器）来替代流动的空气。 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. 当我们意识到在极低频时，低音反射端口只是扬声器正面和背面之间的一个声学短路，它会抵消总声压，就像没有外壳的扬声器一样，我们就能很好地理解这种高阶衰减。
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. 下一节将介绍低音反射系统的动态分析，进一步强调低音反射原理在低频方面的局限性。
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. 在大学课堂上讲授运动系统动力学时，我经常使用一个根据低音反射原理工作的两个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. 该系统的魅力在于动态效果的易观测性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. 常规分析将刚体、弹簧和阻尼器等块状机械元件转化为电感器、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，致动器被一个回旋器或变压器所取代，然后像分析电子电路一样进行分析。 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. 虽然这种电子等效方法的优势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，例如振动模态分析，它可以有效地模拟顺应体的断裂现象和解耦。 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物理现象，与机械领域有着明确的关系"，这就有力地证明了在寻找扬声器重现音乐的改进方法时，应将其保留在机械领域。 In this respect this section can be seen is a starting point for learning the mechanical dynamics approach on sound reproduction. 因此，本节可视为学习声音重现机械动力学方法的起点。
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. 由于两个扬声器共用一个箱体，因此可以使用其中一个扬声器作为有源驱动扬声器，另一个作为无源辐射器，从而尝试无源辐射振膜原理。 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. 每个扬声器的阻尼都可以通过外部连接之间的阻抗来控制，对于驱动型扬声器，阻抗可以来自放大器，对于无源辐射器，阻抗可以来自串联电阻。
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：
Specs:
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 长期最大系统功率
--
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 短期最大功率
--
W
Capacitor in series with x ohm 与 x 欧姆串联的电容器
Cc
--
Voice Coil and Magnet Parameters 音圈和磁铁参数
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 间隙高度
8
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 间隙的通量密度
--
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. 这种微小的偏差在近似值的精确度范围内是可以接受的，而且该模型的正确性足以在考虑到所有弹簧和阻尼器的情况下，采用箱体内空气的运动质量和弹簧刚度的估计值，并计算出两个扬声器的响应。 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。 和 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. 在该模型中，使用了之前发现的辐射功率与加速度之间的相似性，并将两种声压响应相加，得出总声压。 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 : 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: 要计算实际声压，必须将其与一定距离内隔膜的辐射效率相乘：
and for the passive radiator: 和被动散热器：
The total sound pressure is than equal to the difference of these equations as being caused by the motion difference between the two diaphragms. 总声压等于这些等式的差值，因为它是由两个膜片之间的运动差引起的。
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. 它们的运动方向相同，因此外壳中的空气量不会因这一运动而发生变化，也就不会产生额外的刚度。
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. 在频率较高时，无源辐射器将与驱动扬声器动态脱钩，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: 现在，每个扬声器在一半体积的箱体上工作，这意味着箱体的空气弹簧平均分配给每个扬声器，因此它们都能获得两个扬声器之间总封闭空气刚度的两倍：
Adding the stiffness of the diaphragm suspension results in the total stiffness per loudspeaker: 加上振膜悬挂装置的刚度，就是每个扬声器的总刚度：
This results in a resonance frequency of: 因此，共振频率为
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. 由于膜片现在朝相反的方向运动，它们会产生声压，因此总和响应显示出非常强烈的共振。
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. 这种情况在低阻尼时最为明显，但总响应也会出现过冲。 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的阻尼时，系统可以在没有过冲的情况下工作，但在这种情况下，总响应几乎等于驱动膜片的响应。
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. 此外，使用一个完整的扬声器仅在这一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，无源辐射器省去了电磁致动器，只对质量进行调整，而四周则由阻尼橡胶制成，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. 在这种信号下，两个隔膜都会产生振动。
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. 从这些图表中可以得出两个重要结论。
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的好处是，在相同的最大低频声压下，可以使用比封闭箱体箱体更小的驱动扬声器，但这只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. 如果低音忽高忽低，就像底鼓一样，这种优势就不存在了。
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. 阻尼水平过低会导致响应过冲，但阻尼水平过高则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，用于电脑和廉价家庭影院环绕声系统的小型低音炮总是配备无阻尼谐振器，从而产生夸张的轰鸣低音，这种低音有时在观看战争电影时很好听，但更多时候会让人感到非常疲惫，同时还会引起头痛。
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. 与现实生活中的大多数事情一样，天下没有免费的午餐。大多数情况下，某一方面的好处会被其他方面的弊端所抵消。
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. 在描述具有多个块体、弹簧和阻尼器的高阶动态结构时，频率响应的分析表达很快就会变得相当复杂。因此，通常采用振动特征模态来分析动态系统。 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候，就可以这样做，因为此时总的动力学行为可以被模拟为系统在不同特征模式下行为的叠加。 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. 特征模态理论基于这样一个特性，即一个非刚性动态系统（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. 这些频率的激励将使系统中的所有机构产生同步周期性运动。 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 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）
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 和 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 ）。
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. 从这个例子中，我们可以得出结论：特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，实际系统的精确模型应由无限量的弹簧、阻尼器和机构组成，并具有相应的大量特征模态。 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，这些现象在较高频率时最为明显，因为振膜破裂、边缘衍射和其他动态现象代表了各自的特征模式。 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. 在实践中，可以通过忽略频率范围之外特征频率非常高的特征模型，将无限量的特征模型减小到更小的集合。 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. 以无源辐射器低音反射系统为例，这套系统可以仅限于上述两种特征模式，因为本分析侧重于低频声音重现。
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. 之所以在此引入这种模态分析，是因为它有助于解释驱动扬声器的反谐振并非共振。因此，它也不属于特征模态。
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两个特征模态的模态振型，同时重新排列了组合元件，从而可以直接确定其模态行为。 One should be aware that this simplification is only valid for this specific symmetric situation with equal mass and stiffness values. 需要注意的是，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. 下一节将说明，像带气孔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. 第一个特征模态是低频模态，在该模态下，两个膜片在悬挂橡胶和蜘蛛的支撑下，以相等的振幅和相位沿同一方向运动。 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: 第一特征模态的特征频率等于未安装扬声器的特征频率，这是因为两个振膜的总质量（即所谓的 "模态质量 " ）与两个悬挂架的总刚度共同作用：
when and . 当 和 .
This corresponds with the first resonance peak in Figure 8. 这与图 8 中的第一个共振峰相对应。
The second eigenmode is, as mentioned, related to the opposite movement of the 如前所述，第二特征模式与 "彗星 "的反向运动有关。
two bodies, elastically coupled with the connecting air-spring and an eigenfrequency as calculated in Equation (17). 两个主体，与连接空气弹簧弹性耦合，特征频率按公式 (17) 计算。 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 . 运动振幅由两个物体的质量决定，质量越大，振幅越小，从而使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. 要想象这两种特征模式并非难事。第一种特征模式最容易想象，两个扬声器朝同一方向移动，即使只驱动其中一个扬声器。就像一辆大篷车。
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. 要想象第二种特征模式，我们应该想到两个用弹簧连接的球悬挂在外太空。宇航员抓住这两个球，拉伸弹簧，然后让球松开。 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. 现在我们不难看出，小球先是相互靠近，直到弹簧被压缩，然后又分开，如此反复。当小球的质量不同时，小球移动得更快。 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，比如汽车作为第一体，通过悬挂装置与作为第二体的地球相连，由于巨大的质量差，汽车的撞击几乎不会撼动地球。
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. 在整个系统中，两种模式的组合决定了整个系统的行为，因此，正 - 方向上的力也会导致第二个物体在正 - 方向上的移动，但移动幅度小于仅使用第一特征模式时的移动幅度。对于第一个物体，第二个特征模态的响应与第一个特征模态的响应相加，而对于第二个物体，第二个特征模态的响应与第一个特征模态的响应相减。
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，在两个运动膜片质量相同的情况下，如何根据自己的想象重新排列这些元素，以便进行分析理解。特别是空气弹簧的影响将被证明是非常重要的。 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. 在两个质量相等的特定情况下，两个质量的运动幅度相等。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. 我们可以称它为 "节点"，在这里只有力的传递，因此这个中间点可以像一堵墙一样与静止的世界相连，从动态的角度来看不会对特征模式产生影响。 This means that each body works on half the spring with double the stiffness as the full spring. 这意味着，每个弹簧体在一半的弹簧上工作，其刚度是全弹簧的两倍。
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. 由于中间有一堵假想的墙，因此可以想象第二个身体镜像到墙的另一侧，同时也可以直接连接到第一个身体。由于这种特征模式下的运动是相等的，因此可以进行分析。 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 最后一步是将所有质量和刚度值相加，就像第一个特征模态一样。这意味着对于第二个特征模态，模态质量 也等于 ，而空气弹簧刚度在等效简化质量弹簧系统中的系数为 的四倍。在本例中，该刚度等于
The combined frequency-response transfer function can be derived from the two responses as shown in Figure 7. 如图 7 所示，从这两个响应可以得出频率-响应组合传递函数。
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不同，因此频率响应从低频开始就不同。在 ，普通
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：振膜的 Bode 图为两个无阻尼特征模式的综合响应。在第二个特征模式中，驱动扬声器 和无源辐射器 的相位差会在大约 处产生 "反谐振"，因为在该频率下，两个特征模式的贡献相等，但符号相反。而在无源辐射器中，相位相等，因此数值相加即可得到双倍值 。请注意综合声音响应，即每个扬声器的第二个特征模态响应的两倍（ ）。
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. 第一个特征模态的特征频率在两个膜片的响应中都可以看到。在 ，两个模式的幅度相等，要确定组合运动，必须查看两个模式的相位。对于驱动扬声器，第一个特征模态在 时的相位几乎为 ，而第二个特征模态的相位为 。这意味着对第一振膜运动的两个贡献将相互抵消，并导致反共振，这似乎根本不是共振，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 对于无源辐射器而言，情况则有所不同，因为这里的第二特征模式与驱动扬声器的运动方向相反。相反的运动意味着 相位，因此对于无源辐射器而言，第一和第二特征模态在 时具有相同的 相位。因此，两个特征模式的相位相差 2 倍（+6 dB）。在较高频率下，位于 的第二特征模式的特征频率首先显示出其特有的共振峰值。在 以上，驱动扬声器的响应平缓，与恒定加速度相对应，就像封闭箱式扬声器一样。然而，无源辐射器在频率增加时显示出-2 斜坡，这是由于两个特征模态都接近于
the same mass-determined response corresponding with with a phase difference of which means that they cancel each other out more at higher frequencies. 与 相对应的相同质量决定响应的相位差为 ，这意味着它们在较高频率下会相互抵消。
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". 阻尼的影响与分析计算结果相同。值得注意的是，驱动扬声器的阻尼作用于两个特征模态，但程度不同。当第二个特征模式的刚度较高时，品质因数 会更高，这意味着需要更多的阻尼来抑制第二个特征模式。但这并不总能实现，从而导致 "boombass "现象。 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. 扬声器的设计必须符合应用要求，根据质量和箱体调整推杆的电磁特性，否则结果将无法接受。
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，最好注意声音仅由第二种特征模式产生，这与图中的绿线相吻合，绿线比第二种特征模式下每个膜片的运动高出一个系数 。系数 2 的原因是，在以单位力驱动时，两个特征模态是分开计算的。 When combined this force would be divided by two over the two eigenmodes, thereby cancelling the factor 2 in reality. 在两个特征模态之间，这个力将被二除以二，从而抵消了现实中的系数 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. 从这一观察结果可以得出结论，最好只在具有足够阻尼的第二特征模式下驱动系统。这就要求第一特征模式不被激发，而这只有在被动辐射器也被驱动的情况下才有可能实现。 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. 这样一来，系统就等同于一个带有两个驱动器的封闭箱，与一个驱动器相比，由于供电功率增加了一倍， 。
6 Bassreflex with Air-Port Resonator 6 带空气端口谐振器的低音反射式耳机
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. 无源辐射器总是比用塑料管制成的气口昂贵。这种低成本水平是后者被广泛应用的主要原因，尽管它更难获得明确的低频特性。 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 9a：显示的是原理示意图，其中被动散热器被一个管子中的空气量所取代，该管子既向机壳外部开放，也向机壳内部开放。 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. 低音反射端口中的空气体积将充当第二个被动辐射体，并与箱体弹簧和驱动扬声器振膜的质量共同决定共振特征频率。 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. 遗憾的是，该系统的模态分析不如之前描述的对称系统简单。这是由于低音反射端口和扬声器振膜的直径不同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，我们仍然可以看到，驱动扬声器与无源辐射器产生了相同的 "反谐振 "效应。 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. It is also important to note that the same dynamic limitations that were described in Section 5.2 are valid 如图 1 所示，这与亥姆霍兹谐振器的特征频率大致吻合，该谐振器的特征频率由端口中的空气质量和封闭的空气体积决定。还需要注意的是，第 5.2 节中所述的动态限制同样有效。
a: Schematic cross-section a:横截面示意图
b: Lumped element model b:块状元素模型
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 9：一个普通的低音反射式扬声器箱体（a：）使用一个空气体积作为被动辐射器。该空气体积由两侧开放的管道或端口封闭，将箱体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. 空气导孔或低音反射导孔内空气体积的质量将与箱体体积的空气弹簧刚度产生共振，导致该频率下驱动扬声器振膜的偏移振幅减小。 The lumped element equivalent scheme (b:) is used to derive the frequency transfer functions. 采用叠加元件等效方案 (b:) 来推导频率传递函数。
for bassreflex systems with an air-port resonator, because, as will be shown, the dynamics are essentially equal. 对于带有气孔谐振器的低音反射系统而言，这是因为，如图所示，动态效果基本相同。
6.1 Frequency Response of Air-Port Resonator 6.1 空气端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均借助图 9.b:中的块状元件模型确定。该模型中的新术语是驱动扬声器的辐射表面 和低音反射端口的开口 ，其中 代表直径。无源辐射器的质量由端口的体积 和空气密度 决定。如亥姆霍兹谐振器所述，有效长度 等于端口长度加上约 0.73 倍的端口直径 。 中空气的绝热压缩/膨胀因子可作为减少外壳中空气体积的因子来计算 。
Starting with the driven loudspeaker with mass : 从质量为 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. 其中 等于扬声器环绕悬挂装置的刚度， 等于环境的气压 。上式中的两个分数均表示各元件运动时的相对体积变化，与平均环境压力和扬声器的辐射表面 相乘后，得出各元件运动对该表面的作用力。
Written as force in terms of and this equation is written as: 用 和 表示力，这个方程可以写成
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 : 按照与扬声器相同的步骤，可推导出无源辐射空气质量 的运动方程，从而得出 和 之间的关系如下：
To simplify further calculations three stiffness terms are defined: 为了进一步简化计算，我们定义了三个刚度项：
With these terms Equation (20) is simplified into: 有了这些项，方程 (20) 简化为
and with Equation (21) the displacement can be written as function from : 根据公式 (21)，位移 可以写成来自 的函数：
Filling this in Equation (23) and careful applying some algebra leads to the following transfer functions from force to motion: 将其填入公式 (23)，并仔细应用一些代数，就可以得到以下从力到运动的传递函数：
and: 和
with: 用：
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 以获得加速度响应，最终得出以下扬声器振膜声压的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声压的比例不变，有必要对小得多的辐射表面进行修正，因此传递函数乘以比率 ：
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. 总声压等于这些等式的差值，这是由驱动扬声器和低音反射端口中移动空气之间的运动差造成的。
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. 例如，在同一箱体中使用前一部分的扬声器，而无源辐射振膜则由直径为 、长度为 的管子取代。这样，空气体积约为 ，因为如前所述，端口外的空气也必须考虑在内，因此有效长度为 。因此，无源散热器的运动质量约为 。这比有源驱动扬声器振膜的运动质量低 100 多倍，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. 在 MATLAB 中计算了小阻尼和最佳阻尼情况下的频率响应函数，如图 10 所示，出乎意料地显示了与被动辐射膜片相似的动态特性。 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. 正如模态分析所显示的那样，这是由导管中空气的有效表面与扬声器振膜之间的比率造成的。这在第一个特征频率上表现得非常明显，该频率明显低于未安装扬声器的 共振频率。下一节中的模态质量分析表明，空气的小质量在驱动扬声器振膜上被认为是与辐射表面平方之比相当的大质量。
6.2 Stepresponse of Air-Port Resonator 6.2 空气端口谐振器的阶跃响应
The stepresponse from Figure 10 is calculated for four different settings of the mainly resistive amplifier output impedance. 图 10 中的阶跃响应是根据主要是电阻放大器输出阻抗的四种不同设置计算得出的。 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. 真正的电压源放大器（几乎所有现代放大器都是如此）比无源辐射器显示出更好的阻尼阶跃响应，只有一段延迟响应。 This difference in dynamic behaviour is related to a somewhat higher damping of the air-port by the high velocity 这种动态性能上的差异与气口的高速阻尼较高有关。
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：低音反射系统（a:）和最佳（b:）阻尼的 Bode 图，以及不同阻尼水平下的阶跃响应，低音反射系统的设计与上一节中的无源扬声器振膜系统相同 ，其中无源辐射振膜由直径为 、长度为 的管道中的空气量交换。通过电压源放大器产生的阻尼，频率响应与无源辐射振膜几乎相同。 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. 虽然输出阻抗不为零的放大器在瞬态后会迅速产生强烈的延迟共振，导致性能下降，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： 中端口阻尼对放大器阻抗两种设置的影响。对于电压源放大器 ，端口阻尼水平影响的是响应的幅度而不是周期，而高输出阻抗放大器 则更受益于较强的端口阻尼。
of the air than was the case with the modelled undamped passive diaphragm of the previous case. 与上一个案例中模拟的无阻尼被动隔膜相比，空气中的阻尼值更小。 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. 当放大器的阻抗等于扬声器阻抗的电阻值时，放大器阻抗的增加对延迟共振有显著影响，几乎有三个周期。 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数值，特别是电子管放大器，由于缺乏反馈，其输出阻抗通常在欧姆范围内，这些放大器需要更高水平的内部阻尼，例如端口阻尼。 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 演示了这一点，图中计算了两种放大器阻抗设置下的四种不同水平的端口阻尼。两种情况都显示出增加端口阻尼的有益效果，但高输出阻抗放大器的效果最为突出。 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理想的放大器和扬声器组合。这也是为什么很多人喜欢将电子管放大器与低音反射式扬声器组合在一起，而不是封闭箱式扬声器的原因。
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利影响是以牺牲噪音为代价的，因为大部分能量都在端口边缘的湍流中耗散了。 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. 可以通过将边缘修圆来改善这种情况，但这样一来阻尼就会减小，人们可能会在管内插入一块纤维衬垫或带有开口孔的橡胶泡沫来增加阻尼。 This again is quite unpredictable, resulting in a larger spread in performance of different loudspeakers with the same design. 这也是相当难以预测的，导致具有相同设计的不同扬声器的性能差异较大。 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，它还可以将扬声器调谐到放大器上，使用合适的测量麦克风甚至可以在一定程度上优化聆听室的系统。
6.3 Modal Analysis of Air-Port Resonator 6.3 空气端口谐振器的模态分析
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. 前面已经解释过，对由块体、弹簧和阻尼器组成的 "正常 "机械系统进行模态分析所依据的原则是，每个特征模态都可以独立于其他模态而存在，并且在其特征频率下受到激励时会产生共振。 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. 这也意味着，除了致动器的激振力以及连接各机构和静止世界的弹簧和阻尼器的力之外，没有其他外力作用于系统。 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驱动扬声器的直径相同，因此可将连接空气直接模拟为机械弹簧，对两个振膜产生相同的作用。然而，带有低音反射端口的系统则完全不同，由于在刚度值方程中直径差的平方，驱动扬声器将通过端口中的空气质量所包围的空气体积体验到另一个刚度值。 Furthermore the volume of air acts like a compressible medium creating forces to all surfaces inside the enclosure. 此外，空气体积就像一种可压缩介质，会对外壳内的所有表面产生作用力。
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. 最简单的是，在这种情况下也可以将相关的特征模态限制为两个，因为非模态分析显示出两个明显不同的特征频率，分别对应于两个主体（膜片和低音反射端口中的气柱）同向运动的第一特征模态和两个主体相对运动的第二特征模态。
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. 第一特征模态的模态形状是，在扬声器和低音反射端口的横截面比例中，低音反射端口中的空气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. 假设空气不可压缩，则相应的形状函数等于 ）。低频不可压缩的假设是基于这样一种理解，即在此频率下，端口中的空气不会受到太大的运动阻力，因此不会产生很大的力。 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. 这将表明，这一假设只适用于非常粗略的近似，而且在任何情况下，空气质量的加速度都比第一个物体的加速度高出一个很大的系数，这对在激振点（即第一个运动质量）观察到的等效模态质量产生了非常有趣的影响。 This is best explained with mathematics, starting with Newton's second law on inertia with the variables as defined in Figure 9: 最好用数学来解释这一点，从牛顿惯性第二定律开始 ，变量如图 9 所示：
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. 其中， 等于质量为 的驱动扬声器的加速度， 等于低音反射端口中空气质量 所产生的反作用力。该反作用力等于 加速端口内空气的力所产生的压力 。
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，在加速度等级之间的关系中，假定了不可压缩的条件。这两个方程的组合得出以下模态值
mass: 质量
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. 表示驱动扬声器， 表示端口， ，则第一特征模式的模态质量等于 。这意味着在该特征模式下，空气质量比驱动扬声器的质量更为重要。实际上，有必要考虑到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，如果刚度值低于无穷大，低音反射端口中空气质量的移动就会变小，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. 低音反射端口中空气的模态质量分量很可能与扬声器振膜的质量近似相等，就像第二特征模态的情况一样，这一点将进一步说明。
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. 第一特征模态与固定外壳连接的刚度等于驱动扬声器悬架的刚度。在此刚度下，超过一倍的质量将导致 的特征频率大大降低，是未安装扬声器特征频率的 倍。由此产生的共振频率约为 ，与图 10.a:中显示的数值一致。遗憾的是，较低的频率并不意味着扬声器能在此频率重现声音，因为声压不是由第一特征模式产生的，这也是无源振膜版本的情况。 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 . 即使端口中的空气以辐射表面的比例移动得更快，同样的比例也能补偿对声压的影响，因为声压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. 对于声辐射来说，第二特征模态是决定因素，而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. 这种模态分析的出发点是假定在第二特征模态的特征频率处，外壳相对于声音波长较小。从图 10.a：该特征频率预计在 附近，此处显示了第二共振。这相当于几米的波长，因此符合条件，因此可以假定箱体内的气压是均匀的。 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运动质量上的力将与辐射表面 和 有关。在第二特征模式的特征频率下，系统处于完全平衡状态，假设没有能量耗散，系统将在此频率下保持共振。 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通过牛顿第二定律计算如下，并在外壳内实现必要的压力平衡：
This enables to write down the following ratio between and : 这样就可以写出 和 之间的比率：
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：
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，这一比率约等于 1.4，这意味着两个辐射表面对第二特征模特征频率处的声压作用几乎相等。 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一个事实，即空气弹簧中应该有一个中性区域，因为两个物体的运动方向相反。这意味着在外壳内有一个分界面，空气分子在此静止不动。 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). 该平面由体积变化决定，体积变化与两块膜片的位移有关，其关系式为公式 (33)。
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. 这与实际例子中声压值为 1.4 的关系并不意外，这意味着扬声器使用了约 的箱体容积，而低音反射端口使用了 的箱体容积。这些发现都愈发证明了之前发现的无源辐射器与端口加载低音反射系统之间的相似性，而这也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. 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: 如果比率值等于 1，系统将完全相同，在本例中，可以通过加长低音反射端口来增加空气质量。这样，第二个特征模式的特征频率就等于使用 的无源辐射器时的值，但即使在给定的尺寸下，系统的作用也几乎相同。为了验证这一推理是否正确，可使用 "扬声器驱动器产生的低频声音 "论文中的公式 (30) 计算扬声器质量 和低音反射端口质量 的第二个特征模式的特征频率，每个特征频率都与箱体体积中各自部分的刚度有关。使用 和 得出扬声器的数值，使用 和 得出移动空气的数值，并将这两个刚度值取为 和 ，得出以下结果：
Adding the stiffness of the diaphragm suspension results in the total stiffness for the loudspeaker of . With the moving mass values 加上振膜悬挂装置的刚度 ，扬声器的总刚度为 。运动质量值
of and this results respectively in a natural frequency of: 和 ，自然频率分别为
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. 这些本应相等的计算值之间的差异很小，足以证明这一假设，而这很容易是由于外壳体积部分之间的系数 1.4 近似造成的，对刚度有很大影响。 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. 扬声器的箱体容积增大 10%，低音反射孔中的流动空气容积相应减小 10%，就会产生 不同的频率，从而使数值相等。
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. 根据所发现的与被动散热器的相似性，我们不难理解，在致动器驱动系统的点上有效的模态质量很可能与第一特征模态的模态质量大致相等。 More exactly it can be determined starting with the first part of Equation (30): 更确切地说，可以从公式 (30) 的第一部分开始计算：
Using Equation (33) the following relation between the accelerations is obtained 利用公式 (33) 可以得出加速度之间的关系如下
which leads to a simple expression for the reactive force: 由此可以得到一个简单的反作用力表达式：
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. 这意味着耦合质量对驱动力插入点的反作用力等于单独驱动扬声器振膜加速所需的力，而耦合质量只是第二特征模式下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. 这完全符合第二特征模态应达到质量平衡的 "直觉"。同样符合逻辑的是，模态刚度是驱动扬声器的箱体感知刚度的两倍。 With these findings the frequency response can be derived using Matlab with the following steps. 有了这些结论，就可以使用 Matlab 按以下步骤得出频率响应。
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. 第一步是建立驱动扬声器的第一特征模态模型，该模型可基于模态质量 和驱动扬声器悬挂装置的刚度。
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. 低音反射管中空气质量的第一特征模式的声辐射与驱动扬声器的第一特征模式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：通过带有气孔的低音反射系统的特征模态来构建频率响应，结果与分析解法相当。由于对系统做了许多假设，因此偏差较大。
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. 第二特征模态首先在驱动扬声器上进行计算，其模态质量等于移动振膜的双倍质量，作用于由驱动扬声器压缩/膨胀的箱体体积部分所定义的空气弹簧的双倍刚度值。
Finally the responses are combined to give the result. 最后，将答复合并得出结果。
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 显示了这一练习的结果，与图 10 相比，可以看到一些偏差。首先，反共振不在同一频率。这是因为两个特征模态的相对增益对它们的交点有很大影响。 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，在第一个特征频率的声音输出中，没有看到剩余的少量共振。但就其余部分而言，结果在性质上具有可比性。 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，这些简化计算只能被看作是粗略的近似值，无法定性地反映现实中可能出现的现象，因为所做的一系列假设都不完全正确：
The air is assumed to flow frictionless for both modes without turbulence. 假定两种模式下的空气流动均无摩擦，没有湍流。
All is modelled linear. 所有模型都是线性的。
The moving mass in the bassreflex port is estimated with a "rule of thumb" correction factor for the air just outside the port. 低音反射端口中的运动质量是根据端口外空气的 "经验法则 "修正系数估算出来的。
The mass of the air in the enclosure is not taken into account. 外壳中的空气质量未考虑在内。
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. 由于膜片与周围空气之间的耦合度较低，因此产生声音的空气质量被忽略，但它并不是零。
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. 等等
Nevertheless the shown dynamic responses are sufficiently representative for real bassreflex systems to be conclusive. 不过，所显示的动态响应足以代表真实的低音反射系统，因此可以得出结论。
7 Conclusions on Bassreflex for Very Low Frequencies 7 关于极低频低音反射的结论
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. 通过观察图 5 和图 10 得出的响应，我们可以清楚地看到，即使使用 60 升的超大箱体和大型扬声器，所实现的最低 频率也在 左右，根据所应用的阻尼，该频率以下的 倍频程斜率非常陡峭。原则上，我们可以通过在输入信号处使用补偿滤波器来扩大这一范围并降低共振的动态效果。 This will however drive the driven loudspeaker diaphragm in extreme excursions, while the entire purpose of bass reflex was to prevent this. 然而，这会使驱动的扬声器振膜产生极端偏移，而低音反射的整个目的就是为了防止这种情况发生。 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，在端口占据声音重现主要部分的频率以下，增加大量空气的流动会增加流动噪音。
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. 只有将所有共振频率降低 2 倍，才能真正扩大 的响应范围。由于质量或刚度的平方根关系，这意味着刚度降低 4 倍或质量增加 2 倍，或者两者都降低 2 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. 虽然较高的质量会进一步降低效率，但只要所有的刚度值都降低这么多，包括环绕、蛛网和箱体的体积，降低四倍的刚度也是可行的。 The last one can also be decreased by means of a smaller loudspeaker but then the moving mass is also decreased. 最后一项也可以通过使用较小的扬声器来减少，但移动质量也会随之减少。
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. 由此可以得出结论，只有超大型低音反射系统才能产生 左右的频率。即便如此，阶跃响应也总是会出现延迟反应，给人一种不受控制的 "毛糙 "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，人们曾一度相信并严肃地指出低音反射系统具有更高的效率。他们的理由是，管道是开放式的，可以像延迟线一样从后面传递声音。 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 但事实上，只有在第二特征模态的低音反射共振（示例中为 ）阻尼不足时，才会出现相同输入功率下较高的输出，而且仅在该频率附近出现。在 以下，滚降为四阶，因此功率输出和效率低于带宽为 的封闭式箱体的二阶滚降。高于 时，管道中的空气或
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便一目了然。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. 被动式振膜几乎不会移动，也不会传递任何声音。这意味着在这些频率下，它能有效地封闭箱体，因此声音输出与封闭箱体的情况相同。
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 由图 5 的最佳阻尼图推导而来，其中省略了被动膜部分，增加了关闭被动膜时的系统响应。 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. 它清楚地显示了振膜偏移较小的好处，但正如图 6 所示，只有在瞬态周期结束后才会出现这种情况！此外，效率几乎没有提高。同样明显的是，效率几乎没有提高，而在 以下输出较少的陡峭斜率也很明显。最后，封闭箱体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，这一点可以得到改善，但对振膜偏移的好处也会随之减少。 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 节中的结论，即要重现高质量、控制良好的低频声音，就应该使用封闭箱体。 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，主动速度或加速度反馈甚至可以进一步改善时间响应。
For more detailed background information see the book "The Design of High Performance Mechatronics" as presented on the website http://rmsacoustics.nl/education.html 有关更详细的背景信息，请参阅网站 http://rmsacoustics.nl/education.html 上介绍的《高性能机电一体化设计》一书。
