ATIAM Master’s degree
阿蒂亚姆硕士学位
Synthesis of electric motors sounds based
onphysical models
基于物理模型的电动机声音合成
Author:Supervised by:Joshua Reiss
作者: 督导: Joshua Reiss
Leo Sebal
Leo Sebal 里奥 · 西巴尔
August 29, 2016
August 29,20162016 年 8 月 29 日
Contents
目录
1 Operation and decomposition of an electric motor4
1 电动机的操作和分解 4
1.1 Operation of the DC motor..............................4
1.1 直流电动机的操作 4
1.2 Decomposition of an electric motor..........................4
1.2 电动机的分解... 4
1.2.1Sources.....................................5
1.2.1 来源... ... ... ... ... ... ... ... ... ... ... ... 5
1.2.2Resonators...................................6
1.2.2 谐振器... ... ... ... ... ... ... ... ... ... ... ... 6
2 Methods10
2 种方法 10
2.1 Sources......................................... 10
2.1 来源... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 10
2.2 Resonators....................................... 10
2.2 谐振器... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 10
2.2.1Finite differences method for a beam..................... 10
2.2.1 梁的有限差分法
2.2.2Finite differences method for a plate..................... 11
2.2.2 板的有限差分法 11
2.3 Stator magnet...................................... 11
2.3 定子磁铁 11
3 Results12
3 个结果 12
3.1 Details of the reimplementation............................ 12
3.1 重新执行的细节... ... ... ... ... ... ... ... ... 12
3.2 Analysis of the results................................. 12
3.2 结果分析 12
3.2.1Decomposition................................. 12
3.2.1 分解... 12
3.3 Comparisons...................................... 12
3.3 比较... ... ... ... ... ... ... ... ... ... ... ... ... ... 12
Bibliography20
参考书目 20
Abstract
摘要
Most widespread sound
synthesis methods im-ply the
extraction of sound
parameters through analysis,
then synthesis according to
those parameters, or
combination of short sound
chunks. They mostly consist in
analyzing the sound signal.
Physical modeling sound
synthe-sis, on another hand,
consists in studying of the
physics phenomena behind the
production of sound, and
reproduction of those phenomena. It therefore requires a
specific study of the
instrument or body producing
the sound,and the application
of physical methods and
equations, which aim to
reproduce the way the sound is
physically produced.
大多数广泛使用的声音合成方法
是通过分析提取声音参数，然后
根据这些参数进行合成，或者将
短的声块组合起来。它们主要包
括分析声音信号。另一方面，物
理建模声音合成包括对声音产生
背后的物理现象的研究，以及对
这些现象的再现。因此，它需要
对产生声音的仪器或物体进行具
体研究，并应用物理方法和公式，
目的是再现物理产生声音的方式。
The objects we will be
interested in here are
electrical
motors.
An
electrical motor is nat-urally
quite silent: as opposed to
combustion engines, sound is
not produced by successive
explosions,
but
by
the
different vibrating parts of
the object. The study of each
vibrating part, their coupling
and the description of those
physical
phenomena
constitute therefore the basis
of the work.
我们在这里感兴趣的对象是电动
机。电动机本质上是相当安静的:
与内燃机相反，声音不是由连续
的爆炸产生的，而是由物体的不
同振动部分产生的。对每一个振
动部件的研究，它们的耦合以及
对这些物理现象的描述构成了这
项工作的基础。
This report presents the work
made during an internship in
the Audio Engineering team of
the Center for Digital Music, in
the Queen Mary University of
London. The objective of this
in-ternship is the study of a
sound synthesis based on
physical modeling working in
real time.
本报告介绍了在伦敦玛丽女王大
学数字音乐中心音频工程团队实
习期间所做的工作。实习的目的
是研究基于实时物理模型的声音
合成。
Tandis que les techniques de
synth`ese sonore les plus r
´epandues impliquent
l’extraction de param`etres du
son existant par analyse, puis
la synth`ese suivant ces
param`etres, par synth`ese
additive ou soustractive, ou
encore par com-binaison
d’´echantillons sonores de diff
´erentes longueurs.La
synth`ese par mod´elisation
physique consiste, quant`a
elle,`a ´etudier les ph
´enom`enes physiques
conduisant`a la cr´eation du
son, et la reproduction de ces
ph´enom`enes.Elle n´ecessite
donc l’´etude sp´ecifique de
l’instrument ou corps sonnant
et l’application de diverses m
´ethodes physiques,selon la
fac¸on dont le son est cr´e´e.
同时，合成这些声纳的技术越来
越广泛，这意味着要通过分析从
存在的其他声纳中提取其他声纳，
然后再通过合成这些声纳，再通
过合成这些加法或加法的声纳，
或者再结合不同长耳猴的声纳来
合成其他声纳。因此，它不需要
专门研究乐器或声乐器，也不需
要多种物理方法的应用，因为事
实上它是可信的。
Les objets qui nous int
´eressent ici sont les mo-teurs
´electriques.Ceux-ci ont la sp
´ecificit´e
d’ˆetre, par nature, assez
silencieux: contraire-ment`a
un
moteur`a
explosion,
l’essentiel du son n’est pas cr
´e´e par une succession
d’explosions,mais par les diff
´erentes parties vibrantes de
l’objet.L’´etude de chaque
partie vibrante,leur couplage
et la description des ph
´enom`enes physiques les r
´egissant est donc l’essentiel
de nos pr´eoccupations.
我们在这里看到的物体是电子仪
器。它本质上是非常安静的: 与发
动机“爆炸”相反，它的本质并
不是一连串的爆炸，而是由物体
的不同震动部分引起的。研究每
一个部位的振动，他们的联系和
描述的物理现象，这些都是我们
关注的重点。
Ce rapport pr´esente le travail
effectu´e au cours d’un stage
effectu´e au sein de l’´equipe
Au-dio Engineering du Center
for Digital Music (C4DM) de
l’Universit´e de Queen Mary de
Londres. L’objectif de ce stage
est l’´etude d’un mod`ele de
synth`ese sonore sur mod`ele
physique fonctionnant en
temps r´eel.
这份报告介绍了伦敦大学数字音乐中心(C4DM)和伦敦玛丽女王学院的工程师团队的工作效果和一个阶段的效
果。这一阶段的目标是研究一种模型，这种模型的声学合成器是基于一种能随时间变化而运转的身体模型。
Introduction
引言
This report is the temporary version of this internship report. It will actually end on the 31st of
August, and numerous points need some exploration and adjustments, including in sections 2and 3.
The entirety of the theory and most of the methods already have, however, been developed;that
allows us to present this report, which will be completed. The final version, containing all
adjustments and discoveries made at that stage, will be given on the 17th August.
这份报告是这份实习报告的临时版本。它实际上将在 8 月 31 日结束，许多要点需要一些探索和调整，包括
第 2 和第 3 节。然而，整个理论和大多数方法已经得到发展，这使我们能够提出这份报告，该报告将完成。
最终版本，包括在这个阶段所做的所有调整和发现，将在 8 月 17 日发布。
Physical synthesis consists in generating the sound by reproducing the phenomena generating it,
instead of reproducing the parameters of a recorded sound. Sound sources can be of different kinds:
vibrations of plates, strings, membranes, etc. To reproduce them, one needs to know the physical
properties and the equations behind the studied object: for the membranes and strings, the wave
equation, respectively in one and two dimensions, for a thin plate, Love-Kirchoff equations, and for
beams, Euler-Bernoulli equations.
物理合成是通过再现产生声音的现象来产生声音，而不是再现记录声音的参数。声源可以是不同种类的: 板、
弦、膜等的振动。为了再现它们，我们需要知道研究对象背后的物理性质和方程: 对于膜和弦，分别在一维
和二维中的波动方程，对于薄板，Love-Kirchoff 方程，对于梁，Euler-Bernoulli 方程。
Advantages of that kind of synthesis are multiple; the main one is to be able to create a sound
entirely based on the physical properties of the studied object. This allows, for example,to anticipate
the sound of a virtual object, for a physical modelization. Insofar, as the model works in real time, it is
also possible to conceive the incorporation of such a model into a video game engine, or a total and
immediate control of the sound, based on the physical parameters of the object.
这种合成方法的优点是多方面的，主要的优点是能够完全基于所研究对象的物理特性来制造声音。这允许，
例如，预测一个虚拟物体的声音，为一个物理模型。由于该模型是实时工作的，因此也可以设想将该模型整
合到视频游戏引擎中，或者根据对象的物理参数对声音进行全面和立即的控制。
Physical synthesis was first conceived in the 1960s, with Kelly and Lochbaum’s works on voice
synthesis[13], which models the trachea with a succession of accoustic pipes. In 1971, Hiller and Ruiz
use finite differences models in order to solve the wave equation[12]. However, the discipline grew
most notably in the 1980s, with the works of the CCRMA on guided wavebands [13] and Jean-Marie
Adrien’s, in the Ircam, with modal synthesis[1] and Modalys.
物理合成最早是在 20 世纪 60 年代构想出来的，当时 Kelly 和 Lochbaum 在声音合成方面的工作[13] ，他们
用一系列的声学管道来模拟气管。1971 年，Hiller 和 Ruiz 使用有限差分模型来解决波动方程[12]。然而，
这一学科在 20 世纪 80 年代发展得最为显著，CCRMA 在导波带上的工作[13]和 Jean-Marie Adrien 在 Ircam
上的工作，模态综合[1]和 Modalys。
Motor sounds, as a whole, have been studied in different forms. Among the notable synthesis works,
we can cite Andy Farnell’s synthesis of combustion engines and electric motors[11],inspired by
physics and using additive and substractive synthesis, or various works of granular synthesis, as
AudioGaming’s plugins. A heavily physics-inspired sound synthesis model has also been studied by
Joshua Reiss and Simon Hendry[19], which uses a number of notions initially explored by Andy
Farnell, and adding to it a number of parameters and physical controls.
电机声音作为一个整体，已经以不同的形式进行了研究。在著名的合成作品中，我们可以引用 Andy Farnell
的内燃机和电动机的合成[11] ，灵感来自物理学，使用加法和减法合成，或各种粒状合成作品，作为
AudioGaming 的插件。Joshua Reiss 和 Simon Hendry [19]也研究了一个受物理启发的声音合成模型，这
个模型使用了 Andy Farnell 最初探索的一些概念，并增加了一些参数和物理控制。
The objective of this internship is to create a synthesis model of electric motors, based as much as
possible on the physical phenomena governing them. Another goal is to be able to make that model
work in real time. This report will first address the question of operation of electric motors, will
explore why and how the sounds of an electric motor is created, before presenting the equations
used in order to describe those sounds. Then, it will talk about how to apply those equations, before
commenting on the results.
这次实习的目的是建立一个综合模型的电动机，尽可能多地基于物理现象支配他们。另一个目标是使模型能
够实时工作。本报告将首先讨论电动马达的运行问题，将探讨为什么和如何产生电动马达的声音，然后介绍
用于描述这些声音的方程式。然后，在评论结果之前，将讨论如何应用这些方程式。
Chapter 1
第一章
Operation and decomposition of an
electric motor
电动机的操作和分解
An electric motor is a machine that converts electric energy into mechanical energy. We will be
interested in electric motors powered by a direct current, in short DC motors. It is interesting to note
that a dynamo also works on the same principle, but converts mechanical energy into electric
energy. Therefore, all methods described here can also apply to a dynamo; we will, however,talk
about DC motors to simplify explanations. We will start by quickly explaining how a DC motor works,
before lingering on sounds it can produce and how to put those phenomena in equations.
电动机是一种将电能转化为机械能的机器。我们将对直流电动机感兴趣，简称直流电动机。有趣的是，发电
机也是基于同样的原理工作的，但是它将机械能转化为电能。因此，这里描述的所有方法也适用于发电机;
然而，我们将讨论直流电机来简化解释。我们将从快速解释直流电动机的工作原理开始，然后继续讨论它可
以产生的声音，以及如何将这些现象写入方程式。
1.1Operation of the DC motor
1.1 直流电动机的操作
Most of the information in this paragraph is from[2]. All information from other source is explicitly
mentioned.
本段中的大部分信息来自[2]。所有来自其他来源的信息都被明确提及。
Electric motors all have an electric current going through a coil; the rotor, fixed on a shaft, has a
given number of protrusions and, on each of those protrusions, is rolled up a coil. That current is
transmitted by a constant contact between the brushes, thanks to the commutator, a set of
conductive plates, one per coil. When it goes through the coils, the current generates a magnetic
field. This magnetic field then reacts with the stator, a given number of permanent magnets,usually 2
or 4, fixed on the casing of the motor; this creates an initial rotating movement.When the rotor has
rotated enough, the polarization of the magnetic field in which each coil is immersed changes. The
commutator has then turned with the motor, and the brushes are not in contact with the same
commutator plate anymore, which reverses the polarity of the coils:the magnetic field of the coil,
called rotoric field, and the one of the magnets, called statoric,therefore keep their orthogonality,
which maintains the rotating movement.
电动机都有一个电流通过线圈; 转子，固定在轴上，有一定数量的突出物，在每个突出物上，卷起一个线圈。
多亏了换向器，电流通过电刷之间的持续接触传递，换向器是一组导电板，每个线圈一个。当电流通过线圈
时，会产生一个磁场。这个磁场然后与定子反应，一定数量的永久磁铁，通常是 2 或 4，固定在电机的外壳
上，这创建了一个初始的旋转运动。当转子旋转足够大时，每个线圈所处磁场的极化就会发生变化。换向器
随着电机转动，电刷不再与同一换向盘接触，这就使线圈的极性相反: 线圈的磁场称为旋转磁场，而其中一
个磁体称为静止磁场，因此保持它们的正交性，维持旋转运动。
Another kind of motor, called brushless, works by replacing the brushes and commutator by a
system of electronic commands. It ensures the conservation of orthogonality of rotoric and statoric
fields. However, brushes and commutator are the main source of noise, and we will therefore be
interested here mostly in brushed motors, leaving aside brushless motors.
另一种电机叫做无刷电机，它的工作原理是用电子指令系统代替电刷和换向器。它确保了旋转和静电场的正
交性守恒。然而，电刷和换向器是噪声的主要来源，因此我们在这里将主要对刷电机感兴趣，撇开无刷电机
不谈。
1.2Decomposition of an electric motor
1.2 电动机的分解
As any source of sound, those coming from an electric motor can be separated into two
kinds:exciters and resonators. The first ones generate the sound, and the second ones will make it
resound or, when the sources create a sound outside of the audible range, can make it audible.In the
case of the motor, it is essential to understand how the sound is created and what makes it actually
heard.
作为任何声源，来自电动机的声源可以分为两种: 激励器和谐振器。第一种声音产生声音，第二种声音会使
声音产生回响，或者，当声源产生超出可听范围的声音时，可以使声音变得可听到。就马达而言，必须了解
声音是如何产生的，以及是什么使声音真正被听到。
Brushes
刷子
Figure 1.1 – Lateral section of a DC motor[11]
图 1.1-直流电动机的横截面[11]
Before going on and identifying excitors and resonators, it is important to determine which parts of
the motor actually make noise, namely what parts vibrate. The study of each component allows to
eliminate from the start the stator as a source or resonator, too thick to vibrate in the audible range,
and is therefore considered as rigid. The sources of sound are, by order of importance:
在继续和识别激励器和谐振器之前，重要的是要确定哪些部分的电机实际上制造噪音，即哪些部分振动。对
每个元件的研究可以从一开始就消除定子作为一个源或谐振器，太厚而不能在可听范围内振动，因此被认为
是刚性的。声源按重要性排列如下:
• The periodic contact between the brushes and the commutator. Each contacts
results in a slight displacement of one of the non-fixed extremities of the brush, •
The displacement of the stator. The electromagnetic force it is subjected to displaces
it slightly. It therefore acts in a similar way as the membrane of a highspeaker at a
low frequency, • If the shaft is not perfectly balanced, a periodic contact between it
and the metallic casing,
刷子和换向器之间的周期性接触。每次接触都会导致电刷的一个非固定端的轻微位移。•
定子的位移。它所承受的电磁力使它轻微移位。因此，它的作用方式类似于低频率的高音
喇叭的膜，如果轴不完全平衡，它与金属外壳之间的周期性接触,
• Electronical noises.
‧电子噪音。
And the resonant parts are:
共振部分是:
• The shaft, periodically excited, either by its contact with the casing, or
through the com-mutator, which is fixed,
•轴，周期性地受到激励，或者通过与套管接触，或者通过固定的换向器,
• The brushes,
刷子,
• The casing
‧外壳
1.2.1Sources
1.2.1 资料来源
Contact between the brushes and the commutator This contact induces a small dis-placement of the
brush. To a given number Nrpm of rotations per minute, and for a given number R of coils on the
rotor, which is also equal to the number of plates on the commutator,this equals a frequency of
displacements of RNrpm60Hz
电刷和换向器之间的接触这种接触会引起电刷的轻微位移。对于每分钟转动的给定数目的 Nrpm，以及对于
转子上线圈的给定数目的 r，也等于换向器上板的数目，这等于 RNrpm60Hz 的位移频率
Moreoever, the torque on the moving shaft can be described by the following equation[19]:
此外，移动轴上的扭矩可以用以下公式来描述[19] :
where T is the torque mentioned, Ts the stall torque, as the torque when the
angular fre-quency ω is zero, and ωf is the final angular frequency: ωf= 2πrpm60.
That torque can be expressed as a function of the moment of inertia J, thanks to the
relation T= J ω˙. The equation above then becomes a first order differential equation
on ω:
其中 t 是扭矩，t 是失速扭矩，作为角频率 ω 为零时的扭矩，ωf 是最终角频率 ωf =
2πrpm60。这个转矩可以表示为转动惯量 j 的函数，由于关系式 t = jω。上面的方程就变
成了 ω 上的一阶微分方程:
The frequency of the motor ramping up and down can therefore be described thanks to the solution
of that equation:
因此，由于这个方程的解，可以描述电机上下运动的频率:
We can see in the equation above that the ramping of the motor directly depends on the moment of
inertia on the shaft, J. On a motor, it is directly related to the load on the motor:
我们可以在上面的方程式中看到，电机的转动惯量直接取决于轴上的转动惯量。在电机上，转动惯量与电机
的负载直接相关:
where d(x,∆) is the distance between the central axis of the shaft and the given point, ρ the density
and dV an element of volume.
其中 d
(x，δδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδδ
δδδdv。
In simpler words, the heavier the load, the more time the motor will take to ramp up to its maximum
frequency ωf. This can be synthesized by just varying the frequency of the pulses.
简单地说，负载越重，电机需要更多的时间来达到它的最大频率 ωf。这可以通过改变脉冲的频率来合成。
Electromagnetic resonance The electromagnetic resonances implies a very fast displace-ment of the
stator magnets. The electrostatic forced exerted by one of the magnets on one of the rotor coils can
be written as(fig. 1.2):
电磁共振电磁共振意味着定子磁铁的快速位移。其中一个磁铁施加在一个转子线圈上的静电力可以写成(图
1.2) :
where Fel is the electrostatic force, B the magnetic field generated by the named magnet, R
the number of coils, dr the distance between the magnetic poles, approximated by the
center of the magnet and the center of the rth coil. The distance dr can be written as
其中 Fel 是静电力，b 是指定磁体产生的磁场，r 是线圈的数目，r 是磁极之间的距离，近似于磁体
的中心和 rth 线圈的中心。距离 dr 可以写成
where θr(t) is the angle at instant t between the rth coil, and rt the distance between the
center of the shaft and the center of the magnetic plate(fig. 1.2),
其中 θr (t)是 r 线圈与 r 线圈之间瞬时 t 的角度，rt 是轴心与磁盘中心之间的距离(图 1.2) ,
The overall force is the sum of the R individual forces of the equation(1.2).
总的力是方程(1.2)的 r 个体力的总和。
Imbalance of the shaft If the shaft is imbalanced, it can bring a periodic contact between itself and
the casing. The frequency of this contact heavily depends on the imbalance, but it has to be of a
fundamental frequency 60Nrpm, and can be approximated in the same manner as the contact
between the brush and the commutator, namely a very short pulse.
轴的不平衡如果轴是不平衡的，它可以带来周期性的接触之间的自己和套管。这种接触的频率在很大程度上
取决于不平衡，但它必须是一个基本频率 60nrpm，并且可以用与电刷和换向器之间的接触相同的方式进行
近似，即非常短的脉冲。
Electronic noises This part of the sound is created in an entirely other manner, since it is not due to
vibrations. It has been considered in a fashion similar to the methods used by Andy Farnell and
Simon Hendry, knowingly substractive synthesis.
电子噪音这部分声音是以完全不同的方式产生的，因为它不是由振动引起的。它被认为是一种类似于 Andy
Farnell 和 Simon Hendry 使用的方法，即有意识的减法合成。
1.2.2Resonators
1.2.2 共振器
A large part, and non negligible part of the sound, also comes from the resonators. We can count
mostly the brushes, the shaft and the casing.
声音的很大一部分，不可忽略的一部分，也来自共振器。我们可以数出大部分的刷子，轴和外壳。
Figure 1.2 – Front section of a 3-phase electric motor. d1, d2 and d3 are the approximations
of the distances between the magnet and each of the coils. θr is therefore the angle
between the rth protrusion of the rotor and the center of the stator..
图 1.2-三相电动机的前部。D1，d2 和 d3 是磁铁和每个线圈之间距离的近似值。因此，θr 是转子
的 rth 突起与定子中心之间的夹角。.
Figure 1.3 – Lateral section of an electric motor. L1, L2,... L7 are the landmarks on the axis x,
alongside the shaft.
图 1.3-电动机的横截面。L1，L2，... l7 是 x 轴上的地标，沿着轴。
Brushes The brushes are two very thin strips of conductive metal, displaced with each rotation by the
contact with the commutator. The movement of the brushes can be described by Euler-Bernoulli’s
beam theory, simplified by its projection on the main axis. Euler-Bernoulli’s equation describes the
displacement due to a force, applied normally to the main axis of the beam[24].Projected on this
axis, is can be expressed as:
刷子刷子是两条非常细的导电金属条，每转一圈都会被换向器的接触所取代。电刷的运动可以用 EulerBernoulli 的光束理论来描述，通过它在主轴上的投影来简化。Euler-Bernoulli 的方程描述了通常应用于梁
的主轴的力引起的位移[24]。投影在这个轴上，可以表示为:
where u(x, y, z, t) is the displacement of the beam regarding its position at rest, I(x) its second
moment of inertia, E(x) the Young modulus of the said material, ρ(x) its lineic mass,and f(x, t) the
force applied to the beam. It’s that force that makes the brush vibrate. The motors are mostly made
of metals, isotropic materials, and the section of the brushes is constant,therefore E, I and ρ do not
depend on x, and the above equation can be simplified:
其中 u (x，y，z，t)表示梁在静止位置上的位移，i (x)表示第二转动惯量，e (x)表示材料的弹性模量，ρ (x)表
示线性质量，f (x，t)表示作用在梁上的力。正是这个力使得刷子发生了振动。电机主要由金属、各向同性材
料制成，电刷的截面是恒定的，因此 e、 i 和 ρ 不依赖于 x，上述方程可以简化:
Boundary conditions described above are, fixed on one extremity, and the force f(x, t) applied to the
other one. Without that force, the equation(1.3) could be solved analytically; however, its presence
compels the use of a numerical resolution method.
上面描述的边界条件是，固定在一个边上，力 f (x，t)施加到另一个边上。如果没有这个力，方程(1.3)可以用
解析方法求解; 然而，它的存在迫使我们使用数值解析方法。
For a beam with a rectangular section, the second moment area can be expressed as:
对于矩形截面的梁，第二弯矩面积可表示为:
where Ib is the second moment area, b the width of the section and h its height.
其中 Ib 是第二个弯矩区域，b 是截面的宽度和 h 的高度。
It is also important to determine the minimal frequency for which the brushes can be con-sidered as
flexible, as for which they are effectively considered as vibrating bodies. Considering that each brush
is fixed at one of its extremities, we can express the modal frequencies as([24],sec. 5.2):
同样重要的是要确定最小频率，在这个频率上，刷子可以被认为是灵活的，因为它们实际上被认为是振动体。
考虑到每个刷子固定在其一个末端，我们可以将模态频率表示为([24] ，第 5.2 节) :
where fn is the frequency of the nth mode, Lb the length of the shaft. Through the second
moment area I, we find that all dimensions of the shaft are included in that equation. The
frequency of the first mode, which will determine if the brush has to be seen as stiff or
not,therefore is:
其中 fn 是第 n 模式的频率，Lb 是轴的长度。通过第二个力矩区域 i，我们发现轴的所有尺寸都包
含在这个方程中。第一种模式的频率，将决定画笔是否被视为僵硬，因此是:
Shaft The shaft can be described as a finite cylinder, of length L7 a lot longer than the dimensions of
its cross section. Euler-Bernoulli’s beam equation(1.3) can describe its movement as well. However,
the boundary conditions are more complex and numerous than for the brushes (fig. 1.3):
轴可以描述为一个有限的圆柱体，长度为 L7，比其横截面的尺寸长得多。Euler-Bernoulli 的梁方程(1.3)也
可以描述它的运动。然而，边界条件比笔刷更加复杂和多样(图 1.3) :
• free in x= L0 and x= L7,
在 x = l0 和 x = l7 中自由,
• rigid between x= L2 and x= L3,
刚性介于 x = l2 和 x = l3 之间,
• coupled with the casing in x= L1 and x= L6, • coupled with the
movements of the commutator in x= L4 and x= L5.
结合 x = l1 和 x = l6 中的外壳，结合 x = l4 和 x = l5 中换向器的运动。
There are more boundary conditions than previously, which favors some modes of vibration and
eliminates others.
有更多的边界条件比以前，这有利于某些模式的振动和消除其他。
The second moment area of a cylindrical section can be written as:
圆柱截面的第二个弯矩区域可以写成:
where Is is the second moment area and r the radius of the cylinder.
其中是第二个矩面积和圆柱体的半径。
As previously, we can study the dimensions of the shaft for when it has to be seen as stiff. For an
indicative value, we can simplify the boundary conditions to their simplest form, considering both
points where the shaft is fixed to the casing. The boundary conditions are slightly different from
previously, and the modal frequencies are now([24], sec. 5.2)
和以前一样，我们可以研究轴的尺寸，以便在它看起来很坚硬的时候。对于一个指示值，我们可以将边界条
件简化为最简单的形式，考虑轴固定在套管上的两个点。边界条件与以前略有不同，模态频率现在是([24] ，
第 5.2 节)
,
where n is the nth mode. The lowest frequency is then
N 是第 n 个模式，那么最低的频率是
Casing The cylindrical shape of the casing poses some issues with its description. As it is,describing
the vibrations of a thin cylinder is not as simple as describing those of a beam. To simplify this
probem, it has been chosen to use the description of the vibrations of a plate,with a displacement
continuity at both ends of the plate. Vibrations of plates can be described with an extension of EulerBernoulli’s beam theory, Kirchoff-Love’s theory. For an isotropic,quasi-static material, for which
deformations in the main plan are negligible, the displacements of a plate of width h on the normal
axis to it can be written as[6]
套管外壳的圆柱形状给它的描述带来了一些问题。事实上，描述薄圆柱体的振动并不像描述梁的振动那么简
单。为了简化这个问题，选用了描述板的振动，在板的两端具有位移连续性。板的振动可以用 EulerBernoulli 梁理论和 Kirchoff-Love 理论的延伸来描述。对于主平面变形可以忽略的各向同性准静态材料，宽
度为 h 的板在法线上的位移可以写为[6]
where u(x, y, t) is the displacement previously mentioned, E the Young modulus of the material and ν
its Poisson coefficient. Unsurprisingly, this equation is extremely similar to 1.3.The boundary
conditions now are:
其中 u (x，y，t)是前面提到的位移，e 是材料的杨氏模量，ν 是泊松系数。不出所料，这个方程非常类似于
1.3。现在的边界条件是:
• Continuity of displacements at opposed ends, to approximate the plate as a
cylinder, • Coupling with the shaft in x= L2 and x= L3,
两端相对位移的连续性，将板近似为一个圆柱体，在 x = l2 和 x = l3 中与轴连接,
• Coupled to the brushes in x= L0.
•耦合到 x = l0 中的笔刷。
Chapter 2Methods
第二章方法
The sound synthesis of the sound of the DC motor can now be obtain by solving the equa-tions
described in section 1. We can regroup all the phenomena creating sounds in three large categories:
直流电机的声音合成现在可以通过求解第 1 节中描述的方程来获得。我们可以将所有产生声音的现象分为三
大类:
• Sources, namely the displacement of the brushes(see 1.2.1) and, sometimes,
the imbalance of the shaft(see 1.2.1), • Resonators, that is vibrations of the
brushes and of the shaft(eq. 1.3) on one side, and of the casing(eq. 1.8) on
the other side, • Displacements of the stator magnets, as described by the eq.
1.2, which are seen in a different manner.
•源，即刷子的位移(见 1.2.1) ，有时，轴的不平衡(见 1.2.1)。•共振器，即刷子和
轴的振动(等式)。1.3)在一边，和外壳(公式。1.8)在另一边，•定子磁铁的位移，如
公式所述。1.2，这是以不同的方式看到。
All those sounds are still mostly governed by the speed of the motor, described with the equa-tion
1.1. They are therefore all harmonic, or pseudo-harmonic, with a fundamental frequency given by
this equation.
所有这些声音仍然主要由马达的速度控制，用公式 1.1 来描述。因此，它们都是谐波，或伪谐波，由这个方
程给出一个基本频率。
2.1Sources
2.1 来源
As described previously, sources are essentially displacements. Each displacement is accounted for a
pulse, approximated with a Dirac of a short Hann window, of the order of 100 ms.Those
displacements let us define a number of coupling and boundary conditions.
如前所述，源本质上是位移。每个位移都是一个脉冲，用一个短 Hann 窗口的 Dirac 近似，大约 100 毫秒。
这些位移让我们定义了一些耦合和边界条件。
2.2Resonators
2.2 谐振器
Vibrations of the resonators are obtained by solving each partial differential equation represent-ing
them.
通过求解表示谐振子的各个偏微分方程，得到了谐振子的振动。
Several methods of numerical resolution exist to solve partial differential equations. The simplest
ones are, however, the fastest, and were therefore preferred, for a real-time oriented resolution.
Therefore, it has been chosen to use a finite differences methods. It adapts Euler’s method for the
resolution of differential equations to partial differential equations. It offers the multiple advantages
of being easy to manipulate and apply. Moreover, it is possible to vectorize it, which is a certain
advantage as soon as an interpreted language is used.
有几种数值分辨方法可以解偏微分方程。然而，最简单的方法是最快的，因此在实时分辨率方面更受欢迎。
因此，它被选择使用有限差分方法。它采用欧拉的方法解决微分方程的偏微分方程。它提供了易于操作和应
用的多重优势。此外，它可以向量化，这是一定的优势，一旦使用直译语言。
2.2.1Finite differences method for a beam
2.2.1 梁的有限差分法
Euler’s method informs us of the approximation of partial derivates of order n on any function x −→
f(x). Specifically:
Euler 方法告诉我们任何函数 x-→ f (x)上 n 阶偏导数的近似。具体来说:
where fi= f(xi), and xi= i∆x.
其中 fi = f (xi) ，xi = i δx。
Let’s now approximate any displacement u(x, t) with its discretization: u(xi,
tk)= uki, with xi= i∆x and tk= k∆t. According to the equations above, the
equation 1.3 becomes
现在让我们用它的离散化来近似任何位移 u (x，t) : u (xi，tk) = uki，其中 xi = i
δδx，tk = k δt。根据上面的等式，等式 1.3 变成
where fi k= f(xi, tk), the force given at a point x alongside the axis, at the
instant t, also describing the couplings.
其中 fi k = f (xi，tk) ，在瞬间 t 时，沿轴向 x 点施加的力，也描述了耦合。
This equation allows us to define the movement at instant t+2∆t, knowing the displacement in t and
t+∆t:
这个等式允许我们定义瞬间 t + 2 δt 时的运动，知道 t 和 t + δt 中的位移:
Boundary conditions are defined by fixing the value at those points.
边界条件是通过固定这些点的值来定义的。
2.2.2Finite differences method for a plate
2.2.2 板的有限差分法
Above equations stay valid for a plate. We can now approximate the displacement u(x, y,
t)with uki,j, so that u(xi, yi, tk)= uki,k. We also have
上述方程对于一个板块仍然有效。现在我们可以用 uki，j 逼近位移 u (x，y，t) ，使 u (xi，yi，tk)
= uki，k。我们还有
.
Equation 1.8 then becomes
方程 1.8 变成了
where we have to replace the remaining partial derivatives with their discretization.
As previously, fi k,k allows us to integrate the couplings, the boundary conditions
are defined by with values of uki,j, and knowing the displacement at instants tk and
tk+1 allows us to know it at instant tk+2.
我们必须用它们的离散化来代替剩余的偏导数。如前所述，fi k，k 允许我们对耦合进行积
分，边界条件由 uki，j 的值定义，知道瞬时 tk 和 tk + 1 的位移可以让我们知道瞬时 tk + 2
的位移。
2.3Stator magnet
2.3 定子磁铁
Displacement of the stator magnets gives us directly the appropriate waveform. Using the frequency
given by the equation 1.1 in the equation 1.2, we get
定子磁铁的位移直接给我们合适的波形。利用方程 1.1 给出的频率，我们可以得到
which is straight the equation that needs to be implemented.
这就是需要实现的方程。
Chapter 3Results
第三章结果
The modelization described in sections 1 and 2 didn’t work. Therefore, it has been chosen to use
Simon Hendry’s model[19], itself almost entirely signal-based, with an implementation of results
obtained from the studies shown previously. We can also note that most of Hendry’s model is based
on Andy Farnell’s, thoroughly explained in[11].
第一节和第二节描述的模型化没有起作用。因此，它被选择使用西蒙亨德利的模型[19] ，本身几乎完全基于
信号，实现了从前面显示的研究结果。我们还可以注意到，Hendry 的大部分模型是基于 Andy Farnell 的，
在[11]中彻底解释。
3.1Details of the reimplementation
3.1 重新执行的详情
Hendry’s patch is decomposed in subpatches as follows:
Hendry 的补丁在子补丁中分解如下:
• The main one is the central waveform. It represents of the normla force applied to
the shaft, and takes the form of a phasor. This signal is then sent to three
subpatches.
•主要是中央波形。它代表施加在轴上的法向力，形成一个相量。然后这个信号被发送到三
个子区域。
• One synthesizes the sound of the rotor. It takes as inputs the aforementioned signal, the
frequency of the rotor, the amplitude of the signal of the rotor, and the amplitude of the
signal of the brush.
一个合成转子的声音。它以上述信号、转子频率、转子信号幅值和电刷信号幅值作为输入。
• One for the signal of the stator(a rougher version of what is described in eq. 1.2),
which takes as an input the main waveform and the amplitude of the signal.
•一个是定子的信号(方程式中描述的粗略版本)。1.2) ，它接受作为输入的主要波形和信号
的振幅。
• One generates the sound of the casing. It takes as an input the main waveform,
the amplitude of this signal and the resonating frequency of the casing.
一个产生外壳的声音。它把主要的波形、信号的振幅和壳体的共振频率作为输入。
The signals generated by the rotor, stator and casing are then summed, and filtered. Those filters
were originally integrated by Andy Farnell.
然后对转子、定子和机壳产生的信号进行求和和滤波。这些滤波器最初是由 Andy Farnell 集成的。
Many of the parameters used in this patch can be calculated thanks to the physics: most of the input
frequencies, the speed of the build-up and build-down, the different amplitudes...Our study also
allows to improve the functions with the physical parameters. Therefore, all frequencies, the ramp-up
speed, etc., have been calculated thanks to equations shown in chapter 1.
由于物理原理，这个补丁中使用的许多参数都可以计算出来: 大多数的输入频率，增加和减少的速度，不同
的振幅... 我们的研究也允许用物理参数来改善函数。因此，所有的频率，增长速度等，都是根据第一章中的
方程式计算出来的。
3.2Analysis of the results
3.2 结果分析
3.2.1Decomposition
3.2.1 分解
All the sections are excited thanks to a similar force. They therefore have very similar time envelopes,
but the frequency content is quite different.
由于相似的力，所有的部分都很兴奋。因此它们有非常相似的时间信封，但频率内容却大不相同。
3.3Comparisons
3.3 比较
Figure 3.1 – Spectrogram of the rotor signal. The rotor shows only some noise. In fact,the resonating
frequencies of the brushes are too high to be considered as a hearing frequency,and only noise
remains.
图 3.1-转子信号的光谱图。转子只显示出一些噪音。事实上，电刷的共振频率太高，不能被认为是听觉频率，
只剩下噪音。
(a) The frequency is evenly distributed from 0 to 20 kHz.
(a)频率均匀分布在 0 到 20 千赫之间。
(b) The stator output shows a fundamental frequency around 120 Hz. That fits the main speed of the
motor. A strong harmonic appears around 1200 Hz, related to the number of stator plates. The rampup and ramp-down of the motor is very distinct.
(b)定子输出的基频约为 120 赫兹。这符合电机的主速度。强谐波出现在 1200hz 左右，与定子片的数量有关。
电动机的上升和下降是非常明显的。
Figure 3.2 – Spectrograms of the stator signal
图 3.2-定子信号的光谱图
(a) Spectrogram of the casing signal The casing signal shows some high energy in lower
frequencies,quite little on higher frequencies: most of the sound generated there is in the low
frequencies, except during the ramp-up and ramp-down, particularly around 0.5 s.
(a)套管信号的频谱图套管信号在较低频率时显示出一些高能量，在较高频率时则显示出相当少的能量: 在那
里产生的大部分声音都是在低频率中产生的，除了在斜坡上升和斜坡下降期间，特别是在 0.5 秒左右。
(b) Spectrogram of the casing signal(0-5kHz) The motor ramp-up is quite obvious. As expected,we
can see a strong fundamental, but a very strong 4th harmonic is also noticeable. The surprising part
is what seems to be a parallel signal of higher frequency during the ramp-up and ramp-down. It
could be due to some aliasing or beating. Some research about what could cause that result have
allowed to its origin, but not to explain it. It is to note that this anomaly doesn’t appear on Simon
Hendry’s model,although it is in theory similar: this is certainly related to the fact that PureData and
MaxMSP solve by themselves all abnormalities related to aliasing, pointing moreso to that direction.
(b)外壳信号的光谱图(0-5kHz)马达的上升非常明显。正如预期的那样，我们可以看到一个很强的基波，但是
一个非常强的第四次谐波也是显而易见的。令人惊讶的是，在上升和下降过程中，似乎出现了一个频率更高
的平行信号。这可能是由于一些混淆或跳动。一些关于导致这种结果的原因的研究已经揭示了它的起源，但
并没有解释它。值得注意的是，这种异常并没有出现在 Simon Hendry 的模型中，尽管在理论上是相似的:
这肯定与 PureData 和 MaxMSP 自己解决所有与混淆相关的异常有关，并且更多地指向那个方向。
Figure 3.3 – Spectrograms of the casing signal
图 3.3-套管信号的光谱图
(a) Load of 0.2 kg
(a)负荷 0.2 公斤
(b) Load of 0.4 kg
(b)负荷 0.4 公斤
Figure 3.4 – Spectrograms of the full signal for two different loads The time the motor takes to rampup and down is significantly higher for a higher mass. This is in adequation with eq. 1.1, which shows
that the time the motor takes to ramp-up is directly related to the moment of inertia, which includes
the load. Although the load is the double of the initial load, it doesn’t include the weight and load of
the axis; therefore, the second ramp-up time is not twice the first one.
图 3.4-两种不同负载下的全信号谱图对于质量较大的电机，电机上升和下降的时间明显较长。这个结果与公
式一致。1.1，这表明，时间的电机采取斜坡直接相关的转动惯量，其中包括负荷。虽然载荷是初始载荷的两
倍，但是它不包括轴的重量和载荷，因此，第二次加载时间不是第一次加载时间的两倍。
(a) Brushes of 0.5 × 2 × 20 mm
(a)0.5 × 2 × 20 毫米的刷子
(b) Brushes of 1 × 4 × 40 mm
(b)1 × 4 × 40mm 的刷子
Figure 3.5 – Comparison for brushes of two different sizes For the larger brushes, the white noise is
located in lower frequencies: there is far less energy in higher frequencies.
图 3.5-两种不同尺寸的笔刷的比较对于较大的笔刷，白噪声位于较低的频率: 在较高的频率有少得多的能量。
(a) Signal for a stator of radius of 2 cm, length of 2.5 cm and width of 1.5 mm
(a)半径为 2 厘米，长度为 2.5 厘米，宽度为 1.5 毫米的定子信号
(b) Signal for a stator of radius of 4 cm, length of 5 cm and width of 3 mm
(b)半径为 4 厘米、长度为 5 厘米、宽度为 3 毫米的定子信号
Figure 3.6 – Comparison for stators of two differents sizes. Once again, the larger the part, the lower
the energy.
图 3.6-两种不同尺寸定子的比较。同样，部件越大，能量越低。
(a) Sample of a recorded motor This one has been chosen because its dimensions are known.
选择这台电机是因为它的尺寸是已知的。
(b) Synthesized signal The physical parameters used there are similar to the ones of the sample
above.
(b)合成信号那里使用的物理参数与上述样品的相似。
Figure 3.7 – Comparison between a recorded sample and a synthesized sample. The ramp-up and
ramp-downs fit with the sample’s, as does the fundamental. The 2nd to 4th harmonics, however are
stronger than the ones from the sample, and harmonics after the 6th disappear on the synthesized
signal, when they are still present on the sample. Other artifacts,due to the casing signal, also
appear.
图 3.7-记录样本与合成样本的比较。斜坡上升和斜坡下降符合样本的基本情况。然而，第二次到第四次谐波
比样品中的谐波强，第六次之后的谐波在合成信号中消失，当它们仍然存在于样品中时。其他由于外壳信号
而产生的伪影也会出现。
Conclusion
结论
This internship will have allowed me to address physical synthesis in a new and complete
angle.Among the objectives given at the beginning of the internship, several have been fulfilled.
First,a physically based model of sound synthesis, has been entirely created. Unfortunately, its
application doesn’t work in its current state, but the first results seem to indicate that it
can.Nevertheless, the results obtained from this study were used to complete the physical aspects of
Simon Hendry’s model, itself heavily based on Andy Farnell’s. The only remaining issues,for example
the shape of the casing signal, seem solvable, and highly related to the signal-based aspects of the
model. However, the real-time operation of the model has not been completed.
这次实习让我能够从一个全新的角度来研究物理合成。在实习开始时给出的目标中，有几个已经实现了。首
先，一个基于物理的声音合成模型已经完全建立起来。不幸的是，它的应用程序在目前的状态下不能工作，
但是第一个结果似乎表明它可以。尽管如此，从这项研究中得到的结果被用来完成西蒙亨德利模型的物理方
面的研究，这个模型本身很大程度上是基于安迪法内尔的。唯一剩下的问题，例如外壳信号的形状，似乎是
可以解决的，并且与模型的基于信号的方面高度相关。然而，模型的实时操作尚未完成。
In order to continue that project, and considering what has been and not been successful,several
directions can be seen. The main and most interesting one would be to work on making the
theoritical model operational. That would allow to have a fully functional physical model.Such a
model would fill the gaps left by the current one: mainly coupling, but also the use of the solutions
instead of signal-based approximations. It would also allow to find out which parts of the
modelization are relevant and which ones might not be, for example. A next step to this would be to
improve it in order to make it work in real-time.
为了继续这个项目，考虑到什么是成功的，什么是不成功的，可以看到几个方向。最主要也是最有趣的一个
方法是致力于使理论模型可操作。这样就可以建立一个功能完备的物理模型。这样的模型将填补当前模型留
下的空白: 主要是耦合，但也使用解决方案，而不是基于信号的近似。例如，它还可以找出模型中哪些部分
是相关的，哪些部分可能不是。下一步就是改进它，使其能够实时工作。
During those months spent in the C4DM, I will have had the occasion to be integrated to the Audio
Engineering team. The extremely diverse and international aspects of the team, of the C4DM more
generally speaking, and even of London in its entirety, have allowed me to work with different people
from all nationalies and backgrounds, and to bond with a new generation of researchers from all
horizons. Those contacts also allowed me to note the advancement of numerous steps of a PHD, and
to discover research from a new perspective and in far more depth than any time before, during my
previous internships, in the LAM, in the Ircam when it was in public research laboratories, or with
AudioGaming in a company.
在 C4DM 的那几个月里，我将有机会被整合到音频工程团队。C4DM 团队极其多元化和国际化的一面，更广
泛地说，甚至整个伦敦，使我能够与来自不同国家和背景的不同人士一起工作，并与来自不同视野的新一代
研究人员建立联系。这些接触也使我注意到博士学位的许多步骤的进步，并从一个新的视角发现研究，比以
往任何时候都深入得多，在我以前的实习期间，在 LAM，在 Ircam，当它在公共研究实验室，或在一家公司
的音频游戏。
Bibliography
参考书目
[1] Jean-Marie Adrien. The missing link: Modal synthesis. In Representations of musical signals,
pages 269–298. MIT Press, 1991.
[1]让-玛丽 · 阿德里安。缺失的环节: 模态综合。音乐信号表征，269-298 页。麻省理工学院出版社，1991 年。
[2] H Wayne Beaty and James L Kirtley. Electric motor handbook. McGraw-Hill Professional,1998.
[2] h Wayne Beaty 和 James l Kirtley. Electric motor handbook. McGraw-Hill Professional，1998。
[3] Thomas Bertolini and Thomas Fuchs.Schwingungen und Ger¨ausche elektrischer Kleinantriebe:
Messung, Analyse, Interpretation, Optimierung. Su¨ddeutscher Verlag on-pact, 2011.
[3]托马斯 · 贝托里尼和托马斯 · 福克斯。摆动和电动小动力: 测量，分析，解释，优化。Su ddeutscher
Verlag on-pact，20112011 年德国出版公司达成协议。
[4] JF Bird, RW Hart, and FT McClure. Vibrations of thick-walled hollow cylinders: Exact numerical
solutions. The Journal of the Acoustical Society of America, 32(11):1404–1412,1960.
[4] JF Bird，RW Hart，FT McClure.厚壁空心圆柱体的振动: 精确的数值解。美国声学学会杂志，32(11) :
1404-1412,1960。
[5] Niels B¨ottcher and Stefania Serafin. Design and evaluation of physically inspired mod-els of
sound effects in computer games. In Audio Engineering Society Conference: 35th International
Conference: Audio for Games. Audio Engineering Society, 2009.
[5] Niels b otcher 和 Stefania Serafin。计算机游戏中声音效果的物理启发模型的设计和评估。In Audio
Engineering Society Conference: 35th International Conference: Audio for Games 音频工程协会会议:
第 35 届国际会议: 游戏音频。音频工程学会，2009 年。
[6] Antoine Chaigne and Jean Kergomard. Acoustique des instruments de musique. 2008.
[6]安东尼 · 沙伊恩和让 · 克尔戈马德。音乐乐器的声学。2008。
[7] DH Chambers. Acoustically driven vibrations in cylindrical structures. Technical report,Lawrence
Livermore National Laboratory(LLNL), Livermore, CA, 2013.
[7] DH 室。圆柱结构中的声驱动振动。技术报告，劳伦斯利福摩尔国家实验室(LLNL) ，利弗莫尔，加利福尼
亚州，2013。
[8] Robert C Chanaud. Tools for analyzing sound sources, 2010.
[8] Robert c Chanaud。分析声源的工具，2010。
[9] Andrew James Farnell. Designing sound: procedural audio research based on the book by Andy
Farnell. PhD thesis, Universidade Catolica Portuguesa, 2012.
[9]安德鲁 · 詹姆斯 · 法内尔。设计声音: 基于安迪 · 法内尔(Andy Farnell)的书的程序性音频研究。博士论文，
葡萄牙天主教大学，2012 年。
[10] Andy Farnell. An introduction to procedural audio and its application in computer games.In
Audio mostly conference, pages 1–31, 2007.
[10] Andy Farnell.程序音频及其在计算机游戏中的应用介绍。在音频大会上，第 1-31 页，2007。
[11] Andy Farnell. Designing sound. Mit Press Cambridge, 2010.
[11]安迪 · 法内尔。设计声音。麻省理工学院出版社，剑桥，2010。
[12] Lejaren Hiller and Pierre Ruiz. Synthesizing musical sounds by solving the wave equation for
vibrating objects, parts 1 and 2. Journal of the Audio Engineering Society, 19(6,7),1971.
勒哈伦 · 希勒(Lejaren Hiller)和皮埃尔 · 鲁兹(Pierre Ruiz)。通过解振动物体的波动方程合成音乐声音，第 1
部分和第 2 部分。音频工程学会杂志，19(6,7) ，1971。
[13] John L Kelly and Carol C Lochbaum. Speech synthesis. 1962.
[13]约翰 · l · 凯利和卡罗尔 · c · 洛赫鲍姆. 语音合成. 1962。
[14] Javier R Movellan et al. Dc motors. Machine Perception Laboratory, University of Califor-nia, San
Diego, USA, 2010.
[14] Javier r Movellan 等人。直流电机。机器感知实验室，加利福尼亚大学，圣地亚哥，美国，2010。
[15] Sebastiao Lauro Nau and Hugo Gustavo Gomez Mello. Acoustic noise in induction motors:causes
and solutions. In Petroleum and Chemical Industry Conference, 2000. Record of Conference Papers.
Industry Applications Society 47th Annual, pages 253–263. IEEE, 2000.
[15]塞巴斯蒂安 · 劳罗 · 瑙和雨果 · 古斯塔沃 · 戈麦斯 · 梅洛。感应电动机中的声学噪声: 原因和解决方案。在
2000 年石油和化学工业会议上。会议文件记录。工业应用协会第 47 届年会，253-263 页。IEEE，2000.
[16] Sami Oksanen, Julian Parker, and Vesa V¨alim¨aki. Physically informed synthesis of jack-hammer
tool impact sounds. 2013.
[16] Sami Oksanen，Julian Parker 和 Vesa v alim aki。物理信息合成 jack-hammer 工具撞击声。2013。
[17] Utkarsh Pateriya, Ajay Srivastava, Rajiv Singh, and Birendra Vikram Singh. A review on noise
reduction of brushed dc motor. 2015.
[17] Utkarsh Pateriya，Ajay Srivastava，Rajiv Singh 和 Birendra Vikram Singh。刷式直流电动机降噪综
述。二○一五年。
[18] P Pfliegel, F Augusztinovicz, and J Gran´at. Noise analysis and noise reduction of small dc motors.
In Proceedings of the International Seminar on Modal Analysis, volume 1, pages 119–124. KU
Leuven; 1998, 2001.
[18] p Pfliegel，f Augusztinovicz 和 j Gran at。小型直流电动机的噪声分析和降噪。模态分析国际研讨会
会议记录，第 1 卷，第 119-124 页。KU Leuven; 1998,2001.
[19] Joshua Reiss and Simon Hendry. Physical modeling and synthesis of motor noise for repli-cation
of a sound effects library. In Audio Engineering Society Convention 129. Audio Engineering Society,
2010.
[19] Joshua Reiss 和 Simon Hendry。用于声效库复制的电机噪声的物理建模和合成。在音频工程学会会议
129。音频工程学会，2010。
[20] Raul Rojas. Models for dc motors. Unpublished document.
[20]劳尔 · 罗哈斯。直流电动机模型。未发表文件。
[21] Gheorghe Scutaru, Andrei Negoita, and Razvan Mihai Ionescu. Three-phase induction motor
design software. In Automation Quality and Testing Robotics(AQTR), 2010 IEEE International
Conference on, volume 3, pages 1–4. IEEE, 2010.
[21] Gheorghe Scutaru Andrei Negoita 和 Razvan Mihai Ionescu。三相感应电动机设计软件。在自动化
质量和测试机器人(AQTR) ，2010 年 IEEE 国际会议，卷 3，第 1-4 页。IEEE，2010.
[22] Julius O Smith. Physical modeling using digital waveguides. Computer music journal,16(4):74–91,
1992.
[22] Julius o Smith。数字波导物理建模。计算机音乐杂志，16(4) : 74-91,1992。
[23] Gergely Tak´acs et al. Basics of vibration dynamics. In Model Predictive Vibration Control,pages
25–64. Springer London, 2012.
[23] Gergely Tak acs et al.振动动力学的基础。在模型预测振动控制，页 25-64。斯普林格伦敦，2012。
[24] Benson H Tongue. Principles of vibration. Oxford University Press New York, 2002.
[24]本森 h 舌。振动原理。牛津大学出版社，纽约，2002。