小提琴振动机理及声学品质研究
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摘要
本文在分析国内外小提琴研究现状的基础上，通过理论研究、有限元数值模拟及实验手段，研究小提琴振动以及力学和声学的一些关键问题，揭示小提琴的振动机理和力学特性以及对声学品质的影响，探索从客观的物理角度改善和评判小提琴音质的方法，为小提琴的制造和音质调整提供科学的指导。
小提琴弓弦之间的相互作用机理非常复杂，粘滑摩擦作用形成一个复杂的振动系统。本文在分析琴弦亥姆霍兹运动、传统的弓弦粘滑摩擦振动模型以及现代弓弦振动理论的基础上，提出了单摆摩擦振动模型。该模型考虑了振动系统的能量转换，并对振动周期给出限制，从而较好体现了小提琴琴弦的振动行为。
为了深入研究小提琴弦的振动机理，本文使用高速摄像技术，设计和构建了一种非接触式光学测量系统，将高速摄影的二维平面拍摄扩展为三维检测，测量拉弦和拨弦时琴弦上标定点的三维振动特性。开发了基于圆形霍夫变换的图像处理算法，对记录的数以万计的序列图像进了分析处理，精确地提取了琴弦上标定点的位移曲线和三维空间轨迹，并达到亚像素分辨率。利用开发的实验技术对拨弦和拉弦两种不同的弦振动机理进行了深入实验研究。通过测量弦上不同位置点的位移曲线研究了拉弦时振动锯齿波的正、逆程时间变化和振动包络线的形成过程，探究和验证了拉弦时琴弦的“亥姆霍兹运动”特征，为建立弦振动的精确模型提供了实验支持。
小提琴的琴码，特别是在琴码频率响应中2-3kHz频率范围内的宽带共振峰对小提琴的动态频率响应和声学品质有很大影响。本文首先通过四个理论模型，包括确定性统计分割模型、卷积模型、模态模型和动态接触振动模型，分析了琴码频率响应对整个小提琴动态响应的影响。然后开发了基于动态接触振动理论的仿真模型，利用有限元方法研究了琴码的振动机理，特别是琴弦－琴码和琴码脚－面板这两个动态接触界面的动态接触刚度、激励条件以及琴码的孔洞结构对琴码振动模态与动态频率响应的影响。
为了验证琴码的理论与数值分析结果，本文设计了基于压电式三向测力仪的实验装置，测量拨弦和拉奏时琴码作用到面板上的横向、纵向和垂直方向动态作用力。提出了基于滑拉弦的琴码频率响应测试技术。该技术为实验研究真实小提琴琴码频率响应提供了一个更为实用和精确的测试手段。实验结果验证了有限元仿真结果。
小提琴共鸣箱作为小提琴发声系统最为核心的部分，其振动状态对于小提琴的声学特性起着重要作用。本文通过几何建模和有限元仿真，分析了小提琴共鸣箱的振动模态以及频率响应，研究了小提琴共鸣箱面板和背板的琴板厚度、拱高、拱的形状等几何参数，以及低音梁和音柱等对共鸣箱振动特性的影响。设计了基于加速度传感器的振动实验系统，对共鸣箱的频率响应进行测量和分析，并与有限元仿真的结果进行比较。实验结果验证了共鸣箱振动有限元仿真和数值分析的结论。
本文最后根据上述小提琴振动机理和力学特性的研究结果，进一步分析了小提琴的各结构部件的结构、材料、振动形态以及各结构部件之间的相互作用对小提琴声学品质的影响。提出了改善小提琴声学品质的方法和途径。
After reviewing the current status of violin study, a number of key issues with regard tothe dynamic vibration, mechanics and acoustics of violin were investigated in this thesis.
The interaction between violin bow and strings is very complicated. Through the analysisof the Helmholtz motion of strings, the traditional stick-slip friction string vibration modeland modern string vibration theory, a pendulum frictional vibration model was developed inthis thesis. By taking account of the energy conversion and the vibration period in thebow-string vibration system, the vibration behavior of violin string can be modelled moreaccurately in the proposed model.
In order to understand the physics of a bowed violin string and develop an accurate model,experimental measurement of the bowed string motion is crucial. In this thesis, a high-speedphotography based non-contact optical measurement system was designed to measure thestring motion without interference for movement when plucking or bowing a string fitted in aviolin. Through a novel optical system design and setting up an artificial marker on the string,the instantaneous motions of the marker in the X-Y and X-Z planes were simultaneouslyrecorded in every single image. The string vibration was videoed using a single high-speedvideo camera, generating more than10,000sequential images. An image processing algorithmbased on the Hough transform were developed to extract the marker position from a recordedimage at sub-pixel resolution. All the recorded sequential images were processedautomatically using the developed image processing algorithm, to track the three-dimensionalmotion of the marker. Experimental results showed that the proposed measurement systemcan accurately track the violin string motion and trajectory. The tracked motion verified thepredicted Helmholtz motion of the bowed string. The designed measurement system andimage processing algorithm provide a potential experimental tool for studying the mechanismof violin string vibration.
The dynamic response of a violin is often divided into a deterministic region and astatistic region. However, the statistic region is not well understood. The bridge role in thedynamic response was firstly analyzed through four theoretical models: thedeterministic-statistical division model, the convolution model, the modal model, and thedynamic contact vibration model. The bridge mobility under in-plane and out-of-planeexcitations was then explored based on the dynamic contact vibration model for a real bridgeand a solid plate bridge. Theoretical analysis and numerical results showed that the bridgemobility is not only affected by the bridge geometrical structure and material, but the interaction among the strings, bridge and the front plate also plays a vital role. Especially, thecontact vibration boundaries in the two contact interfaces: strings-bridge, and bridge feet-topplate, have a great impact on the bridge mobility and as a result on the violin dynamicresponse，helping us gain an further understanding to the statistic region in the dynamicresponse of a violin.
In order to verify the theoretical and numerical findings on bridge mobility, a novelexperimental system was designed on the basis of a piezoelectric dynamometer for bridgemobility analysis in this thesis. The dynamic forces in three directions acted on the front plateby the bridge when bowing a string with finger slurring on the finger board were recordedthrough the experiment system, and frequency responses were further analyzed. Experimentalresults confirmed the impact of contact vibration boundaries on the bridge mobility, whichwere discovered in the finite element simulation.
Violin corpus plays a key role in the acoustic characteristics of violin. In this thesis,vibration mode analysis and frequency response analysis of the violin corpus were carried outby means of finite element simulation. The impact of the material and geometrical structure ofthe violin front and back plates, the bass bar and the sound-post on the violin vibrationcharacteristics were investigated. An experiment system based on acceleration sensors wasthen designed to measure the frequency response of violin. The experimental results verifiedthe simulated results.
Based on the obtained theoretical and experimental findings on the vibration mechanismand mechanics of violin, the influence of the violin structures, materials, and interactionamong different parts on the sound quality were further analyzed, and the ways and methodsto improve the violin sound quality were proposed.
小提琴弓弦之间的相互作用机理非常复杂，粘滑摩擦作用形成一个复杂的振动系统。本文在分析琴弦亥姆霍兹运动、传统的弓弦粘滑摩擦振动模型以及现代弓弦振动理论的基础上，提出了单摆摩擦振动模型。该模型考虑了振动系统的能量转换，并对振动周期给出限制，从而较好体现了小提琴琴弦的振动行为。
为了深入研究小提琴弦的振动机理，本文使用高速摄像技术，设计和构建了一种非接触式光学测量系统，将高速摄影的二维平面拍摄扩展为三维检测，测量拉弦和拨弦时琴弦上标定点的三维振动特性。开发了基于圆形霍夫变换的图像处理算法，对记录的数以万计的序列图像进了分析处理，精确地提取了琴弦上标定点的位移曲线和三维空间轨迹，并达到亚像素分辨率。利用开发的实验技术对拨弦和拉弦两种不同的弦振动机理进行了深入实验研究。通过测量弦上不同位置点的位移曲线研究了拉弦时振动锯齿波的正、逆程时间变化和振动包络线的形成过程，探究和验证了拉弦时琴弦的“亥姆霍兹运动”特征，为建立弦振动的精确模型提供了实验支持。
小提琴的琴码，特别是在琴码频率响应中2-3kHz频率范围内的宽带共振峰对小提琴的动态频率响应和声学品质有很大影响。本文首先通过四个理论模型，包括确定性统计分割模型、卷积模型、模态模型和动态接触振动模型，分析了琴码频率响应对整个小提琴动态响应的影响。然后开发了基于动态接触振动理论的仿真模型，利用有限元方法研究了琴码的振动机理，特别是琴弦－琴码和琴码脚－面板这两个动态接触界面的动态接触刚度、激励条件以及琴码的孔洞结构对琴码振动模态与动态频率响应的影响。
为了验证琴码的理论与数值分析结果，本文设计了基于压电式三向测力仪的实验装置，测量拨弦和拉奏时琴码作用到面板上的横向、纵向和垂直方向动态作用力。提出了基于滑拉弦的琴码频率响应测试技术。该技术为实验研究真实小提琴琴码频率响应提供了一个更为实用和精确的测试手段。实验结果验证了有限元仿真结果。
小提琴共鸣箱作为小提琴发声系统最为核心的部分，其振动状态对于小提琴的声学特性起着重要作用。本文通过几何建模和有限元仿真，分析了小提琴共鸣箱的振动模态以及频率响应，研究了小提琴共鸣箱面板和背板的琴板厚度、拱高、拱的形状等几何参数，以及低音梁和音柱等对共鸣箱振动特性的影响。设计了基于加速度传感器的振动实验系统，对共鸣箱的频率响应进行测量和分析，并与有限元仿真的结果进行比较。实验结果验证了共鸣箱振动有限元仿真和数值分析的结论。
本文最后根据上述小提琴振动机理和力学特性的研究结果，进一步分析了小提琴的各结构部件的结构、材料、振动形态以及各结构部件之间的相互作用对小提琴声学品质的影响。提出了改善小提琴声学品质的方法和途径。
After reviewing the current status of violin study, a number of key issues with regard tothe dynamic vibration, mechanics and acoustics of violin were investigated in this thesis.
The interaction between violin bow and strings is very complicated. Through the analysisof the Helmholtz motion of strings, the traditional stick-slip friction string vibration modeland modern string vibration theory, a pendulum frictional vibration model was developed inthis thesis. By taking account of the energy conversion and the vibration period in thebow-string vibration system, the vibration behavior of violin string can be modelled moreaccurately in the proposed model.
In order to understand the physics of a bowed violin string and develop an accurate model,experimental measurement of the bowed string motion is crucial. In this thesis, a high-speedphotography based non-contact optical measurement system was designed to measure thestring motion without interference for movement when plucking or bowing a string fitted in aviolin. Through a novel optical system design and setting up an artificial marker on the string,the instantaneous motions of the marker in the X-Y and X-Z planes were simultaneouslyrecorded in every single image. The string vibration was videoed using a single high-speedvideo camera, generating more than10,000sequential images. An image processing algorithmbased on the Hough transform were developed to extract the marker position from a recordedimage at sub-pixel resolution. All the recorded sequential images were processedautomatically using the developed image processing algorithm, to track the three-dimensionalmotion of the marker. Experimental results showed that the proposed measurement systemcan accurately track the violin string motion and trajectory. The tracked motion verified thepredicted Helmholtz motion of the bowed string. The designed measurement system andimage processing algorithm provide a potential experimental tool for studying the mechanismof violin string vibration.
The dynamic response of a violin is often divided into a deterministic region and astatistic region. However, the statistic region is not well understood. The bridge role in thedynamic response was firstly analyzed through four theoretical models: thedeterministic-statistical division model, the convolution model, the modal model, and thedynamic contact vibration model. The bridge mobility under in-plane and out-of-planeexcitations was then explored based on the dynamic contact vibration model for a real bridgeand a solid plate bridge. Theoretical analysis and numerical results showed that the bridgemobility is not only affected by the bridge geometrical structure and material, but the interaction among the strings, bridge and the front plate also plays a vital role. Especially, thecontact vibration boundaries in the two contact interfaces: strings-bridge, and bridge feet-topplate, have a great impact on the bridge mobility and as a result on the violin dynamicresponse，helping us gain an further understanding to the statistic region in the dynamicresponse of a violin.
In order to verify the theoretical and numerical findings on bridge mobility, a novelexperimental system was designed on the basis of a piezoelectric dynamometer for bridgemobility analysis in this thesis. The dynamic forces in three directions acted on the front plateby the bridge when bowing a string with finger slurring on the finger board were recordedthrough the experiment system, and frequency responses were further analyzed. Experimentalresults confirmed the impact of contact vibration boundaries on the bridge mobility, whichwere discovered in the finite element simulation.
Violin corpus plays a key role in the acoustic characteristics of violin. In this thesis,vibration mode analysis and frequency response analysis of the violin corpus were carried outby means of finite element simulation. The impact of the material and geometrical structure ofthe violin front and back plates, the bass bar and the sound-post on the violin vibrationcharacteristics were investigated. An experiment system based on acceleration sensors wasthen designed to measure the frequency response of violin. The experimental results verifiedthe simulated results.
Based on the obtained theoretical and experimental findings on the vibration mechanismand mechanics of violin, the influence of the violin structures, materials, and interactionamong different parts on the sound quality were further analyzed, and the ways and methodsto improve the violin sound quality were proposed.
引文
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