x(t) = A * cos(ωt + φ)
These include:
X (t) is the displacement of the object at time t;
-A is the amplitude, indicating the maximum displacement of the object;
-ω is the angular frequency, which is related to the period t of vibration: ω = 2π/t; ;
-t is time;
-φ is a phase constant, which determines the initial phase of vibration.
The characteristic of simple harmonic vibration is that its displacement is a sinusoidal function with time, which is periodic and symmetrical. Simple harmonic vibration is widely used in physics, engineering and other scientific fields, such as spring vibrator, pendulum clock and so on.
Simple harmonic vibration has the following characteristics
1. Periodic
Simple harmonic vibration vibrates with a fixed period t, that is, the same motion is repeated at equal time intervals. The period t is the time required to vibrate once.
2. Symmetry
The displacement-time curve of simple harmonic vibration is a sine function or cosine function with symmetry. When the object is in an equilibrium position, the displacement is zero; When the object reaches the maximum displacement, the velocity is zero.
3. The restoring force is proportional to the displacement.
The restoring force of simple harmonic vibration is proportional to the displacement of the object, and the direction of restoring force is opposite to the direction of displacement. This conforms to Hooke's law, that is, F = -kx, where f is restoring force, k is restoring force constant and x is displacement.
4. Maximum displacement and amplitude
The maximum displacement of simple harmonic vibration is defined as amplitude (a), which indicates the distance from the equilibrium position to the maximum displacement. The amplitude depends on the initial conditions and the energy of the system.
5. Angular frequency and angular velocity
The angular frequency (ω) of simple harmonic vibration is defined as the reciprocal of the vibration period, that is, ω = 2π/t, and the angular velocity (ω) represents the rate of change of the phase angle through which the vibration passes in unit time.
6. Law of Conservation of Energy
In simple harmonic vibration, mechanical energy (kinetic energy+potential energy) is conserved. When an object is at its maximum displacement, its kinetic energy is the largest, while at its equilibrium position, its potential energy is the largest.
These characteristics describe the basic characteristics of simple harmonic vibration, making simple harmonic vibration an important concept and model in physics and engineering.
Application of simple harmonic vibration formula
1. physics
The simple harmonic vibration formula is used to describe the vibration of physical systems such as spring oscillator, simple harmonic pendulum and sound wave. Their motion can be modeled and analyzed by simple harmonic vibration formula to study their frequency, amplitude and phase characteristics.
2. Engineering
Simple harmonic vibration formula is also widely used in engineering. For example, for bridges, buildings or mechanical systems in structural engineering, the simple harmonic vibration formula can be used to study their vibration response under external excitation, so as to evaluate the stability and safety of the structure.
3. Circuit science
Analysis of simple harmonic vibration formula applied to AC circuit in circuit science. For example, in oscillation circuits, such as LC oscillation circuit, RC oscillation circuit and resonance circuit, the simple harmonic vibration formula can be used to describe the vibration behavior of voltage and current.
4. optics
In the field of optics, the simple harmonic vibration formula can be used to describe the propagation and vibration of light waves. For example, the interference, diffraction and polarization of light can be better understood by representing the electromagnetic field in the form of simple harmonic vibration.
5. musicology
The sound in music can also be described by simple harmonic vibration formula. The tone and timbre produced by musical instruments can be analyzed and explained by simple harmonic vibration formula.
Examples of simple harmonic vibration formula
Question: A particle vibrates in a simple harmonic way near the equilibrium position, with an amplitude of 0. 1 m and an angular frequency of 5 rad/s. Q:
A) the displacement function of particles;
B) particle displacement and velocity at b)t = 2s.
Answer:
A) The general form of the displacement function is x(t) = A * cos(ωt+φ), where a is the amplitude, ω is the angular frequency, t is the time, and φ is the phase difference.
According to the information given in the topic, the amplitude A = 0. 1 m and the angular frequency ω = 5 rad/s, so the displacement function of the particle is x(t) = 0. 1 * cos(5t+φ).
B) When t = 2 s, the displacement and velocity of particles can be calculated by substituting the value of t. ..
Displacement: x(2) = 0. 1 * cos(5 * 2+φ)
Speed: v (2) = dx/dt =-0.1* 5 * sin (5 * 2+φ)
Note: Due to the lack of specific initial velocity or phase information, specific displacement and velocity values cannot be obtained. You can only get their faces.