How to judge the displacement and velocity direction of simple harmonic vibration, urgent!

1.When the displacement of the particle from the equilibrium position x varies with time t, the law of the empty band follows the cosine function.

or sinusoidal function bucket reed: x=acos(2* *t t+ ) This linear vibration is the simple harmonic vibration. where a is the absolute value of the maximum displacement (x=0) when the particle leaves the equilibrium position.

Called "vibrating radiation", t is the period of the simple harmonic vibration, and the angle of (2* *t t+ ) is called the circumferential phase angle or phase of the simple harmonic vibration.

2.The motion of an object under the action of an external force that is proportional to the displacement and is in the opposite direction is called simple harmonic vibration.

3.The positive or negative displacement is the opposite of the positive or negative recovery force.

4.There is no direct relationship between the positive or negative velocity and the positive or negative displacement, and the positive or negative recovery force.

To determine that the motion of an object is simple harmonic motionThe first thing to determine is that the motion of the object is a mechanical vibration, that is, to see whether the object is in reciprocating motion, to see if the object has a balanced position in the process of motion. See if the resistance of the object in motion is small enough when it leaves the equilibrium position, whether it will be affected by the restoring force directed towards the equilibrium position.

Then find the equilibrium position and establish the coordinate system with the equilibrium position as the origin, and then let the object deviate from the equilibrium position along the positive direction of the x-axis, and find the magnitude of the restoring force on the object, if the restoring force is f=-kx, then the motion of the object is simple harmonic motion.

Applications of simple harmonic motion.

Simple harmonic vibration is the simplest and most basic vibration, and any complex vibration can be regarded as the synthesis of several simple harmonic motions. The basic laws of vibration and fluctuation are the basis of acoustics, science, electrical engineering, electronics, optics, etc.

1. Electrical engineering.

In electrical engineering, there is a sinusoidal AC circuit, which is a linear circuit when the excitation (voltage source or current source) changes according to a sinusoidal law, and the response (voltage or current) is also a sinusoidal quantity of the same frequency, and the working state of the circuit is called sinusoidal steady state. The circuit at this time is called a sinusoidal steady state circuit, or sinusoidal AC circuit.

2. Structural dynamics.

The force of the building structure is divided into static load and dynamic load, in which if the load changes greatly with time, the dynamic load design needs to be carried out, such as the first load. In the calculation of dynamic loads, it is necessary to use the free vibration of the simplest single-degree-of-freedom system as the basis, as shown in the figure below, the cantilever column structure can be simplified into a spring oscillator model.

The sufficient and necessary condition for judging simple harmonic motion is that the force is proportional to the displacement from the equilibrium point, i.e.

f=-kx；A negative sign indicates that this force is directed towards the equilibrium position, hindering the relative motion.

So analyze it:

1.There is a segment that is f=mg, so it is not;

So it's not. And because v is proportional to h, it is a simple harmonic motion.

So yes; 5.This is an approximate simple harmonic motion, because the angle is relatively small, so mgsin is approximately equal to mg, this.

It is the radian, because the angle is relatively small, and the radius is relatively large, so it can be regarded as a straight line, so the force is proportional to the displacement from the equilibrium position, so it is a simple harmonic motion.

The direction of velocity is the direction of the object's motion, and the direction of acceleration is determined by the direction of the recovery force. Displacement is also a vector quantity, which is positive and negative, and is relative, and this needs to specify a positive direction. When an object is in simple harmonic motion, the force exerted on the object is proportional to the displacement and always points to the equilibrium position.

It is a periodic motion (e.g., single pendulum motion and spring oscillator motion) that is determined by the nature of its own system. In fact, simple harmonic vibration is sinusoidal vibration. The mathematical model of simple harmonic motion is an ordinary differential equation of linear constant coefficients, and such a vibrational system is called a linear system.

Linear systems are the simplest and most common mathematical model of vibrational systems.

Hello! In simple harmonic motion, the direction of acceleration du always points to the equilibrium position, which is always opposite to the direction of the displacement. Edition.

If you know the direction of displacement weight, you can know the direction of acceleration. (For example, to the left (negative), then the acceleration must be to the right (positive)) Conversely, if you know the direction of acceleration, you can also know the direction of displacement.

Judgment of the direction of speed.

If the object is moving in a negative direction along the x-axis, the direction of velocity is negative (to the left).

If the object is moving in the positive direction of the x-axis, then the direction of velocity is positive (to the right).

The direction of acceleration is opposite to the direction of displacement, and the magnitude is directly proportional to the displacement. There is no necessary relationship between the direction of velocity and the direction of displacement, and the velocity at any point may be in two different directions, but the smaller the displacement, the greater the velocity.

(o equilibrium position, a square bai maximum distance position, dua' negative direction maximum zhi distance position, "dao-"" indicates direction) o-a a a-o o o-a' a' a‘-o o

The displacement is increased within + max + decreased.

Capacity- 0 Increase- Maximum- Decrease- 0Speed Decrease+ 0 Increase- Maximum- Decrease 0 Increase+ Maximum+

Acceleration Increase - Min - Decrease - 0 Increase + Max + Decrease + 0

The equation of motion for simple harmonic motion is x=acos( t+ ).where a is the amplitude of the simple harmonic motion, called the angular frequency (sometimes called the circular frequency) is the initial phase.

The first derivative of displacement is velocity, and the second derivative is acceleration.

Let the simple harmonic equations of motion find the first and second derivatives of time to yield:

v=dx/dt=-asin(ωt+φ）

a=d2x/dt2=-aω2(ωt+φ）

Features:

1) Force characteristics: the restoring force f=-kx$, the magnitude of f$ (or a$) is proportional to the magnitude of x$, and the direction is opposite.

2) Motion characteristics: when close to the equilibrium position, a, f, and x$ all decrease, and v$ increases; When moving away from the equilibrium position, a, f, and x$ all increase, and v$ decreases.

3) Energy characteristics: the greater the amplitude, the greater the energy. During the motion, kinetic energy and potential energy are converted into each other, and the mechanical energy of the system is conserved.

The above content refers to: Encyclopedia - Simple Harmonic Vibration.

The kinematic equation of simple harmonic vibration is x = acos ( 0t+ ) The circular frequency, frequency, and period are determined by the vibration system itself, 0=2 t=2 V; Amplitude a and initial phase are determined by the initial conditions.

where a is the maximum value of the displacement lead x, which is called the amplitude, which indicates the intensity of the vibration; n represents the amplitude increment of the vibration per second, which is called the angular frequency, also known as the circular frequency; The FAI is called the initial phase. The number of cycles of vibration in each second is expressed in f= n2 and is called frequency; Its reciprocal, t=1 f, indicates the time it takes to vibrate for one week, known as the period. The amplitude a, frequency f (or angular frequency n), and initial phase are called the three elements of simple harmonic vibration.

Definition of simple harmonic vibration: The round-trip motion of an object that is only "proportional to the magnitude of the mismatched position that deviates from the equilibrium judgment and points to the restoring force of the equilibrium position" is called simple harmonic vibration.

I feel that the subject wants to ask how to understand the simple harmonic vibration equation, especially how to understand it from the level of physical essence. Personal experience is that all kinds of vibration equations established by mechanical analysis can be understood from the level of force balance, for example, the general formula of the vibration equation mx''+cx'+kx=f(t), you can move the three items on the left to the right, then they also have a negative sign, at this time we can find that these three items correspond to the inertial force, damping force and elastic force of the system, and f(t) is the external force received, the sum of these four forces must be zero, that is, in a state of mechanical equilibrium, This equilibrium state does not refer to static equilibrium, but dynamic equilibrium, which is actually the expression of D'Alembert's principle: in any mechanical system, as long as the inertial force is regarded as an external force, then the sum of all the forces experienced by the system in any case (static or dynamic casual) must be zero.