table of Contents
- Vibration Fundamentals
Vibration Fundamentals
11.1 & description simple harmonic motion
The definition of simple harmonic motion
- Concept: displacement away from the equilibrium position (or angular displacement) by changes in the law cosine (or sine function) movement
- Kinetic defined: particle motion in its linear restoring force proportional to the displacement of the equilibrium position is the reverse of the simple harmonic motion
- Kinematics Characteristics: simple harmonic motion acceleration is proportional to the displacement from the reverse
Simple harmonic motion expression
- official:
\[ x=A\cos(\omega t+\varphi) \]
- The meaning of each physical quantity
- Amplitude _A: indicates the maximum displacement of simple harmonic motion of the object away from the equilibrium position
- Angular frequency _ \ (\ Omega \) : represents the object \ (2 \ pi \) times the reciprocal vibration of time, also known as the angular frequency
- Period _T: reciprocation time of a vibration experienced by
- Vibration frequency _v: reciprocating vibration frequency per unit time
- First phase _ $ \ varphi $: motion state (t = 0) is the initial time vibration system
- Transforming the relationship between physical quantities period:
\[ T=\frac{2\pi}{\omega}\\v=\frac{1}{T}=\frac{\omega}{2\pi}\\\omega=2\pi v \]
Speed and acceleration simple harmonic motion
- Speed formula:
\[ v=v_m\cos(\omega t+\varphi+\frac{\pi}{2})\\【v_m=\omega A】 \]
- Acceleration:
\[ a=a_m\cos(\omega t+\varphi+\pi)=-\omega^2x\\【a_m=\omega^2A】 \]
Phase simple harmonic motion
- Phase _ ( \ (\ Omega X + \ varphi \) ): simple harmonic motion state quantity determined
- Retardation _ \ (\ of Delta \ varphi = \ varphi_2- \ varphi_1 \)
- Retardation characteristics:
- In the same direction: \ (\ of Delta \ varphi = 0 2K + \ PI, \ Quad K \ in the Z \)
- Reverse: \ (\ of Delta \ varphi = \ + 2K PI \ PI, \ Quad K \ in the Z \)
Rotation vector
- Angle: current time phase
- Radius: amplitude
- Reference circle: corresponding circumferential
11.2 & simple harmonic motion of Kinetics
Dynamics defined
- Definition: vibration process, the object suffered combined force proportional to the displacement with respect to its equilibrium position and the reverse
- Linear restoring force: combined force suffered
- official:
\ [F = -kx \ quad is the stiffness coefficients [k] \]
Physical quantities in power
- Angular frequency / natural frequency _ $ \ omega $:
\[ \omega=\sqrt{\frac{k}{m}} \]
- Natural period _T:
\[ T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{k}} \]
- Amplitude _A:
\[ A=\sqrt{x_0^2+\left(\frac{v_0}{\omega}\right)^2} \]
- Initial phase _ \ (\ varphi \) :
\[ \varphi=\arctan\left( -\frac{v_0}{\omega x_0} \right) \]
Examples of simple harmonic motion
- In the case of a small angle of swing, the pendulum is a simple harmonic vibration
\[ \omega =\sqrt{\frac{g}{l}}\qquad T=2\pi\sqrt{\frac{l}{g}} \]
Simple harmonic motion energy
\ [E + = E_k E_p = \ frac {1} {2} 2 kA \]
11.3 & synthetic simple harmonic motion
The same frequency and the direction of movement of the simple harmonic synthesis
Combined vibration amplitude:
\ [A = \ sqrt {A_1 ^ 2 + 2 ^ + A_2 2A_1A_2 \ COS (\ varphi_2- \ varphi_1)} \]
- If the vibration of the two phases, together maximum amplitude, \ (A + A_2 A_1 = \)
- If two opposite-phase vibration, together smallest amplitude, \ (A = | A_1-A_2 | \)
First combined vibration phase:
\ [\ Tan \ varphi = \ {A_1 FRAC \ SiN \ varphi_1 A_2 + \ SiN \ varphi_2 A_1} {\ COS \ varphi_1 A_2 + \ COS \ varphi_2} \]
Different frequency harmonic synthesis with the direction of movement
The concept shot
Combined vibration amplitude periodic variation phenomenon
Derivation
- \(x_1=Acos\omega_1t=Acos2\pi \upsilon_1t\)
- \(x_2=Acos\omega_2t=Acos2\pi \upsilon_2t\)
- Use and the difference of the plot formula conversion:
\[ x=x_1+x_2=2Acos2\pi\frac{\upsilon_2-\upsilon_1}{2}t·cos2\pi\frac{\upsilon_1+\upsilon_2}{2}t \]
- The amplitude term: \ (2Acos2 \ PI \ FRAC {\ upsilon_2- \ upsilon_1 2} {} T \)
- Item vibration (resonance factor): \ (cos2 \ PI \ FRAC {\ upsilon_1 + \ upsilon_2 2} {} T \)
Related parameters
- Modulation frequency : \ (\ FRAC {\ omega_2- \ omega_1 {2}} \)
- Carrier : \ (\ FRAC {\ omega_1 + \ omega_2 {2}} \)
Beat
- Concept : co-frequency amplitude variations
- Formula : difference between two points of vibration, i.e. \ (v = v_2-v_1 \ )
Beat note
Two acoustic frequency difference small tuning-fork vibration simultaneously, there will be a low buzz of high
Different frequency harmonic motion perpendicular to the Synthesis of
Lissajous figure
Graphic shapes
Application: unknown wave frequency and phase measurements
11.4 & vibration damping
Type of oscillation
Amplitude vibration : simple harmonic motion
Vibration damping (vibration reduction): frictional damping, radiation damping
Viscosity: fluid molecules constantly performed with an irregular thermal motion , whether the fluid is a static state or flowing state. Such irregular thermal motion causes the gas mass flow of the different layers of the exchange, and the fluid velocity different layers, then two adjacent layers of fluid molecules momentum different. Existing mass exchange between adjacent layers, there must be momentum exchange. Fast fluid layer due to thermal motion of the molecular layer of the fluid molecules ran slow, fast layer begins to flow away a momentum flow in the slow layer, thereby speeding up the slow flow of the fluid layer; conversely, slow layer due to thermal motion of fluid molecules went fast formation fluids in the molecule, layer begins to flow away a momentum slow to fast flow in the layer, thereby slowing the fluid flow fast layer. Therefore, the viscosity depends only on the molecular thermal motion speed, fluid temperature is the thermal motion of molecules is a direct sign of kinetic energy, the same fluid viscosity and therefore depends only on the temperature of the fluid, and the pressure has nothing to do. Liquid and gas dynamic viscosity coefficient varies with temperature opposite trend, as they produce different physical reasons viscosity, the former mainly between liquid molecules from the cohesion , the viscosity is proportional to temperature; primary gas molecules from the latter to the heat motion , viscosity inversely proportional to temperature.
Viscous substances object motion equations
Derivation:
- Resistance to media object are:
\[ F_r=-\gamma v=-\gamma\frac{dx}{dt} \]
- Points have to object equations of motion :
\[ m\frac{d^2x}{dt^2}=-kx-\gamma \frac{dx}{dt} \]
- 令\(\omega^2=\frac{k}{m}\)，\(2\beta=\frac{\gamma}{m}\)：
\[ \frac{d^2x}{dt^2}+2\beta\frac{dx}{dt}+\omega_0^2x=0 \]
- Solutions of differential equations are:
\[ x=A_0e^{-\beta t}cos(\sqrt{\omega_0^2-\beta^2}t+\varphi_0) \]
Related parameters
- \ (\ omega O \) : When undamped oscillator natural angular frequency
- \ (\ Beta \) : damping
Type vibration damping
- Quasi-periodic oscillation (underdamped)
- Characteristics: \ (\ omega O> \ Beta \)
- Cycle: \ (T = \ {2 FRAC \ PI} {\ Omega} = \ {2 FRAC \ PI} {\ sqrt {\ 2- omega O ^ \ ^ Beta 2}} \)
- Critical damping :
- Features: from a moving object to a stationary equilibrium position in a minimum time
- Tokusei: \ (\ Omega_0 = \ Beta \)
- Overdamped :
- Characteristics: \ (\ omega O <\ Beta \)
Damping curve
11.5 & forced vibration resonance
Forced vibration
Definition: one cycle of an external force applied to the vibration system, the vibration which it occurs
Driving force: this force periodicity, deduced the following simply referred to as \ (F = F_0 \ cos \ omega t \)
Equations of motion:
\ [^ {X = A_0e - \ Beta} T \ COS (\ sqrt {\ omega O ^ 2- \ Beta} ^ 2 + T \ varphi_0) + A \ COS (\ Omega T + \ varphi) \]
- Transient key: \ (A_0e ^ {- \ Beta} T \ COS (\ sqrt {\ omega O ^ 2- \ Beta} ^ 2 + T \ varphi_0) \)
- Term stability: \ (A \ COS (\ Omega T + \ varphi) \)
Vibration parameter equation:
\ [A = \ {FRAC f_0} {\ sqrt {(\ omega O ^ 2- \ Omega ^ 2) ^ 2 +. 4 \ 2 Beta ^ \ ^ 2}} Omega \\\ varphi = \ arctan \ frac {-2 \ beta \ omega} {\ omega_0 ^ 2- \ omega ^ 2} \]Features:
Angular frequency, but the driving force by the natural angular frequency is not forced vibration angular frequency oscillator; A (Steady forced vibration amplitude) and (forced vibration and the steady-state driving force retardation) are independent of the initial conditions and the frequency of the driving force, magnitude \ (F_0 \) and damping factors
Resonance
- Amplitude reaches a maximum value
- Angular frequency: \ (\ Omega the _r = \ sqrt {\ omega O ^ 2-2 \ Beta} 2 ^ \)
- Amplitude: \ (A_r = \ FRAC f_0 {2} {\ Beta \ sqrt {\ 2- omega O ^ \ ^ Beta 2}} \)
- Shift in the resonance: maximum displacement amplitude
- Speed resonance: the driving force exactly equal to the frequency of the natural frequency
- The driving force behind the phase of the vibration displacement \ (\ frac {\ pi} {2} \)
- Speed of the vibration in phase with the driving force, the maximum energy of the system to obtain supplementary
11.6 & electromagnetic oscillations
Electromagnetic oscillations: periodically changing voltage and current of the circuit
- LC circuit: simple oscillating circuit composed of a capacitor and coil inductance
Free undamped oscillations: radiation damping resistors and ignore
Features: In the LC resonant circuit, and the current charge amount are as simple harmonic motion, oscillation period and oscillation frequency depends on the nature of the circuit itself
Oscillating circuit total energy:
\ [W is = W_e + w_m = \ FRAC {Q_0 ^ 2} {2C} [\ COS ^ 2 (\ Omega T + \ varphi) + \ SiN ^ 2 (\ Omega T + \ varphi)] = \ frac {q_0 ^ 2} {2C } \]
[Expand] our ability to emit visible light by means of LC oscillator?
Author: Nikola Tesla
Theoretically possible, but it is impossible to realize, just from the circuit in terms of knowledge, L the impedance of the inductor is proportional to the frequency of high-frequency, inductive reactance on the large, to a certain extent it is basically not the energy output . In addition, all communication circuit has a response time, if the frequency is too high, the whole circuit will not work substantially.
General oscillator frequency up to MHz order of magnitude have only a small amount can achieve GHz order of magnitude, but the frequency of visible light is necessary to achieve THz orders of magnitude! !
Deep from another perspective, the first so-called photon: it has a specific triggering mechanism, so there is a specific phase structure, such as the intrinsic field: Rainbow frequency of the photon frequency, pulse modulation harmonic spectrum fine structure superposition state, i.e. : a first photon channel system is an amplitude modulated wave phase modulation +, Common LC tank circuit to produce it?
Second: large photon particular center frequency, of course, the base LC tank circuit frequency is generated by a crystal oscillator, the LC tank circuit at least various multiplication / division circuit, a photon can be seen by human eyes: This is due to photons can achieve atomic orbital electron secondary excitation with photon resonance effect, in order to have the greatest effect on brain nerve stimulation, resulting in brain biochemistry feelings, ordinary atomic orbital electron waves can achieve secondary challenge it? unable!
Photon excitation mechanisms: level transitions atomic orbital electrons generated high dimensional quantum tunneling mechanism, it is possible to generate photon pulses. Electromagnetic waves generated LC circuit: generating source is effectively the crystal oscillation circuit provided in the electronic circuit, a so-called LC electromagnetic radiation, is a circuit for oscillator frequency signal times divided signal, and then the divide the signal into electromagnetic radiation in the LC oscillator circuit high antenna radiation impedance generated current, which generates electromagnetic wave radiation antenna via a high frequency impedance of the radiation, so that electromagnetic radiation mechanism forced oscillation due to the electron acceptor, such as: electrical coupling exciton radiation, it is impossible to generate an optical pulse as structure!
There is, if the higher current frequency, there will be a series of effects, whether or not an LC circuit, as long as the high frequency current to a certain value, the electromagnetic waves are radiated outward. Or upside down, figuratively speaking: "Current" is a very slow frequency "electromagnetic waves", we finally put the power supply, wire and so on various types of electrical components to "wave" of the frequency becomes lower, applies to us all kinds of electrical appliances needs, why should we turn a way to increase the current frequency of light it? It does not make sense, it is difficult to achieve.
To emit light, we have a lot of other ways, such as atomic energy level transitions, thermal radiation, etc., these methods better than the oscillation circuit! !
The second problem relates to the wave-particle duality, de Broglie relation: E = hv, p = h / λ. Description of particle parameters E, p and volatility parameter v, the relationship of λ, i.e. the relationship between the transition energy of the electromagnetic wave. As for heat radiation, in general, it refers to the very long wavelength infrared, but most of the energy it passes by way of thermal motion is absorbed, so we call it thermal radiation.
From a higher perspective, as long as you learn more, you will find that the emission of electromagnetic waves oscillating circuit and atomic energy level transition emission of electromagnetic waves, heat waves, and so on radiative transfer of various forms of nature have similar properties, or simply with a mechanism! ! In other words, the transition between the atomic level is a high frequency of the LC oscillator circuit. (You can interpret it this way, though still a little different)
Ultimately: a structure specific photon excitation due to a particular mechanism, it can be human eye-brain response optic maximum resonance of any radio channel can only be seen through various times / divider, AM / PM, approximate simulation of photons but to be able to obtain a field effect has been seen in the human eye is impossible!
So you say: Can we help LC oscillator circuit to emit visible light? Of course, not be achieved