多个简谐振动的合成

Synthesis of Multiple Simple Harmonic Vibrations

关键词:  简谐振动  合成  MATLAB  波形  轨迹

Key words:  simple harmonic vibration  synthesis  MATLAB  wave form  locus

摘要:

在一维、二维及三维坐标中合成多个同频率、不同频率简谐振动,并基于MATLAB软件绘出不同频率简谐振动合成的波形及轨迹.结果显示,多个一维同频率简谐振动可合成为一个同频率的简谐振动;n个一维同振幅、同相位频率相差不大简谐振动的合成结果是形成(n-1)个拍;多个二维同频率简谐振动的合振动是两个相互垂直同频率简谐振动的叠加,合振动的轨迹为椭圆;多个三维同频率简谐振动的合振动是3个相互垂直同频率简谐振动的叠加,合振动的轨迹为椭圆.多个一维频率比为有理数简谐振动的合振动具有周期性,而多个一维频率比为无理数简谐振动的合振动则无周期性;多个二维、三维频率比为有理数简谐振动合成的轨迹是稳定的闭合曲线,而多个二维、三维频率比为无理数简谐振动合成的轨迹则是复杂的非闭合曲线.

Abstract:

This paper analyzes the synthesis of multiple simple harmonic vibration with same or different frequency in one-dimensional,2-d,3-d coordinate. Based on Matlab, descriptions on wave from and locus of the synthesis with different frequency are made. The results showed that the multiple simple harmonic vibrations with same frequency in 1-d can be synthesized as one vibration. N-vibrations of simple harmonic with same amplitude, same phase and little frequency difference can be synthesized as (n-1) beats. The synthesis of multiple simple harmonic vibrations in 2-d or 3-d with same frequency is superposition, which is two or three vibrations of mutually perpendicular, and their synthesis locus is an ellipse. When the frequency ratio is a rational number, the synthesis of multiple simple harmonic vibrations in 1-d is periodicity, which on the contrary is not periodicity. Under the same conditions, the synthesis locus of multiple simple harmonic vibrations in 2-d, 3-d is a closed curve, which on the contrary is a non-closed curve.