“normal mode”的概念、定义、翻译、参考文献-科学参考

normal mode

Mathematics

An oscillating mechanical system with n degrees of freedom has normal coordinates q₁, q₂, …, q_n about an equilibrium q₁ = q₂ = ⋯ = q_n = 0 if

$\begin{array}{l} T & = & \frac{1}{2} ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + \dots + {\dot{q}}_{n}^{2}), \\ V & = & \frac{1}{2} (ω_{1}^{2} q_{1}^{2} + ω_{2}^{2} q_{2}^{2} + \dots + ω_{n}^{2} q_{n}^{2}), \end{array}$

where T denotes kinetic energy and V denotes potential energy. Here ω‎₁, ω‎₂, …, ω‎_n are the natural frequencies of the system. The normal modes of the system are given by q_k(t) = A_kcos(ω‎_kt + ε‎_k) with all other q_j being 0.

For a simple light pendulum of length l with a mass m at the end, performing small oscillations about the downward vertical θ‎ = 0, then

$T = \frac{1}{2} m l^{2} {\dot{θ}}^{2}, V = m g l (1 - \cos θ) \approx \frac{1}{2} m g l θ^{2} .$

Thus, the normal coordinate is $\sqrt{m} l θ$ and the natural frequency is $\sqrt{g / l} .$ For a string y(x,t) fixed at y(0,t) = y(l,t) = 0 and performing small transverse oscillations, there are infinitely many degrees of freedom and the normal modes are

$A_{k} sin (\frac{k π x}{l}) cos (\frac{k π c t}{l} + ε_{k}) k = 1, 2, 3, \dots$

and the corresponding natural frequency is kπ‎c/l. Here c² = T/ρ‎ where T is the tension in the string and ρ‎ is its density.

Electronics and Electrical Engineering

The mode of an oscillating system in which every part is in sinusoidal motion at the same frequency and maintains the same phase relationship with every other part throughout. The frequencies at which this can occur for a given system are known as its resonant frequencies. The general motion of such a system can be expressed as a superposition of its normal modes over different frequencies.

单词	normal mode
释义	normal mode Mathematics An oscillating mechanical system with n degrees of freedom has normal coordinates q₁, q₂, …, q_n about an equilibrium q₁ = q₂ = ⋯ = q_n = 0 if $\begin{array}{l} T & = & \frac{1}{2} ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + \dots + {\dot{q}}_{n}^{2}), \\ V & = & \frac{1}{2} (ω_{1}^{2} q_{1}^{2} + ω_{2}^{2} q_{2}^{2} + \dots + ω_{n}^{2} q_{n}^{2}), \end{array}$ where T denotes kinetic energy and V denotes potential energy. Here ω‎₁, ω‎₂, …, ω‎_n are the natural frequencies of the system. The normal modes of the system are given by q_k(t) = A_kcos(ω‎_kt + ε‎_k) with all other q_j being 0. For a simple light pendulum of length l with a mass m at the end, performing small oscillations about the downward vertical θ‎ = 0, then $T = \frac{1}{2} m l^{2} {\dot{θ}}^{2}, V = m g l (1 - \cos θ) \approx \frac{1}{2} m g l θ^{2} .$ Thus, the normal coordinate is $\sqrt{m} l θ$ and the natural frequency is $\sqrt{g / l} .$ For a string y(x,t) fixed at y(0,t) = y(l,t) = 0 and performing small transverse oscillations, there are infinitely many degrees of freedom and the normal modes are $A_{k} sin (\frac{k π x}{l}) cos (\frac{k π c t}{l} + ε_{k}) k = 1, 2, 3, \dots$ and the corresponding natural frequency is kπ‎c/l. Here c² = T/ρ‎ where T is the tension in the string and ρ‎ is its density. Electronics and Electrical Engineering The mode of an oscillating system in which every part is in sinusoidal motion at the same frequency and maintains the same phase relationship with every other part throughout. The frequencies at which this can occur for a given system are known as its resonant frequencies. The general motion of such a system can be expressed as a superposition of its normal modes over different frequencies.
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