Harmonic oscillator spring pendulum. Ideal harmonic oscillator

F, proportional to the displacement x :

If a F- the only force acting on the system, then the system is called simple or conservative harmonic oscillator. Free oscillations of such a system represent a periodic movement around the equilibrium position (harmonic oscillations). The frequency and amplitude are constant, and the frequency does not depend on the amplitude.

Mechanical examples of a harmonic oscillator are the mathematical pendulum (with small angles of deflection), a weight on a spring, a torsion pendulum, and acoustic systems. Among the non-mechanical analogues of a harmonic oscillator, one can single out an electric harmonic oscillator (see LC circuit).

Let x- displacement of a material point relative to its equilibrium position, and F- acting on a point restoring force of any nature of the form

where k= const. Then, using Newton's second law, one can write the acceleration as

The amplitude is reduced. This means that it can have any value (including zero - this means that the material point is at rest in the equilibrium position). The sine can also be reduced, since the equality must hold at any time t. Thus, the condition for the oscillation frequency remains:

Simple harmonic motion is the basis of some ways of analyzing more complex types of motion. One of these methods is based on the Fourier transform, the essence of which is to decompose a more complex type of motion into a series of simple harmonic motions.

Any system in which simple harmonic motion occurs has two key properties:

A typical example of a system in which simple harmonic motion occurs is an idealized mass-spring system in which a mass is attached to a spring and is placed on a horizontal surface. If the spring is not compressed and not stretched, then no variable forces act on the load and it is in a state of mechanical equilibrium. However, if the load is removed from the equilibrium position, the spring is deformed and a force will act from its side, tending to return the load to the equilibrium position. In the case of a load-spring system, such a force is the elastic force of the spring, which obeys Hooke's law:

where k has a very specific meaning - this is the coefficient of spring stiffness.

Once the displaced load is subjected to the action of a restoring force, accelerating it and tending to return it to the starting point, that is, to the equilibrium position. As the load approaches the equilibrium position, the restoring force decreases and tends to zero. However, in position x = 0 the load has a certain amount of motion (momentum), acquired due to the action of the restoring force. Therefore, the load skips the equilibrium position, starting to deform the spring again (but in the opposite direction). The restoring force will tend to slow it down until the speed is zero; and the force will again seek to return the load to its equilibrium position.

If there is no energy loss, the load will oscillate as described above; this movement is periodic.

Simple harmonic motion shown simultaneously in real space and phase space. Real Space - real space; Phase Space - phase space; velocity - speed; position - position (position).

In the case of a load vertically suspended on a spring, along with the elastic force, gravity acts, that is, the total force will be

Measurements of the frequency (or period) of oscillations of a load on a spring are used in devices for determining body mass - the so-called mass meters, used on space stations when the balance cannot function due to weightlessness.

Simple harmonic motion can in some cases be considered as a one-dimensional projection of universal circular motion.

If an object moves with a constant angular velocity ω along a circle of radius r, whose center is the origin of the plane x − y, then such motion along each of the coordinate axes is simple harmonic with amplitude r and circular frequency ω .

In the approximation of small angles, the motion of a simple pendulum is close to simple harmonic. The period of oscillation of such a pendulum attached to a rod of length ℓ , is given by the formula

where g- acceleration of gravity. This shows that the period of oscillation does not depend on the amplitude and mass of the pendulum, but depends on g, therefore, with the same length of the pendulum, on the Moon it will swing more slowly, since gravity is weaker there and the value of the free fall acceleration is lower.

The specified approximation is correct only at small deflection angles, since the expression for the angular acceleration is proportional to the sine of the coordinate:

where I- moment of inertia ; in this case I = mℓ 2. Small angles are realized under conditions when the oscillation amplitude is much less than the length of the rod.

which makes the angular acceleration directly proportional to the angle θ, and this satisfies the definition of simple harmonic motion.

When considering a damped oscillator, the model of a conservative oscillator is taken as a basis, to which the viscous friction force is added. The force of viscous friction is directed against the speed of the load relative to the medium and is directly proportional to this speed. Then the total force acting on the load is written as follows:

Using Newton's second law, we get differential equation describing the damped oscillator:

Therefore, in pointer indicators (for example, in ammeters), they usually try to introduce precisely critical attenuation so that the arrow calms down as quickly as possible to read its readings.

An oscillator with critical damping has a quality factor of 0.5. Accordingly, the quality factor indicates the nature of the behavior of the oscillator. If the quality factor is greater than 0.5, then the free movement of the oscillator is an oscillation; theoretically, over time, it will cross the equilibrium position an unlimited number of times. A quality factor less than or equal to 0.5 corresponds to the non-oscillatory movement of the oscillator; in free motion, it will cross the equilibrium position at most once.

In the case of oscillatory motion, attenuation is also characterized by such parameters as:

This time is considered as the time required for damping (cessation) of oscillations (although, formally, free oscillations continue indefinitely).

The oscillations of an oscillator are called forced when some additional external influence is made on it. This influence can be produced by various means and according to various laws. For example, force excitation is the effect on the load by a force that depends only on time according to a certain law. Kinematic excitation is the action on the oscillator by the movement of the spring fixing point according to a given law. The effect of friction is also possible, when, for example, the medium with which the load experiences friction moves according to a given law.

Consider oscillations of a weight m on a spring with stiffness coefficient k, which lies on a flat horizontal table, assuming that there is no friction of the weight on the table surface. If the weight is removed from the equilibrium position, it will oscillate about this position. We will describe these oscillations by a time-dependent function, assuming that it determines the deviation of the weight from its equilibrium position at time t.

In the horizontal direction, only one force acts on the weight - the elastic force of the spring, determined by the well-known Hooke's law

The deformation of the spring is a function of time, which is why it is also a variable.

From Newton's second law we have

because the acceleration is the second derivative of the displacement: .

Equation (9) can be rewritten in the form

where. This equation is called the harmonic oscillator equation.

Comment. In the mathematical literature, when writing a differential equation, one usually does not indicate the argument (t) near all functions that depend on it. This dependency is assumed by default. When using the mathematical package Maple in (10), it is necessary to indicate the explicit dependence of the function.

In contrast to the previous example of body motion under the action of a constant force, in our case the force changes over time, and equation (10) can no longer be solved using the usual integration procedure. Let's try to guess the solution of this equation, knowing that it describes some oscillatory process. As one of the possible solutions to equation (10), we can choose the following function:

Differentiating function (11), we have

Substituting expression (12) into equation (10), we make sure that it is satisfied identically for any value of t.

However, function (11) is not the only solution to the harmonic oscillator equation. For example, one can choose a function as another solution, which is also easy to check in a similar way. Moreover, one can check that any linear combination of these two randomly named solutions

with constant coefficients A and B is also a solution to the harmonic oscillator equation.

It can be proved that the two-constant solution (13) is the general solution of the harmonic oscillator equation (10). This means that formula (13) exhausts all possible solutions to this equation. In other words, the harmonic oscillator equation has no other particular solutions, except for those obtained from formula (13) by fixing arbitrary constants A and B.

Note that in physics it is most often necessary to look for just some particular solutions of individual ODEs or their systems. Let's consider this question in more detail.

It is possible to excite oscillations in the system of weight on a spring we are considering different ways. Let us set the following initial conditions

This means that at the initial moment of time, the weight was removed from the equilibrium position by a value a and freely released (ie, it starts its movement with zero initial speed). One can imagine many other ways of excitation, for example, a weight in the equilibrium position is given some initial speed by a “click”, etc. [ general case, ].

We consider the initial conditions (14) as some additional conditions for separating from the general solution (13) some particular solution corresponding to our method of excitation of the weight oscillations.

Assuming t=0 in expression (13), we have, whence it follows that B=a. Thus, we have found one of the previously arbitrary constants in solution (13). Further, differentiating in formula (13), we have

Assuming t=0 in this expression and taking into account the second initial condition from (14), we obtain, hence it follows that A=0 and, thus, the initial particular solution has the form

It describes the oscillatory mode of the considered mechanical system, which is determined by the conditions of the initial excitation (14).

It is known from the school physics course that in formula (16) a is the amplitude of the oscillations (it sets the maximum deviation of the weight from its equilibrium position), is the cyclic frequency, and is the phase of the oscillations (the initial phase turns out to be equal to zero).

The harmonic oscillator equation (10) is an example of a linear ODE. This means that the unknown function and all its derivatives are included in each term of the equation to the first degree. Linear differential equations have an extremely important distinctive property: they satisfy the principle of superposition. This means that any linear combination of any two solutions of a linear ODE is also its solution.

In the example of the harmonic oscillator equation we are considering, an arbitrary linear combination of two particular solutions is not just some new solution, but a general solution to this equation (it exhausts all its possible solutions).

In general, this is not the case. For example, if we were dealing with a third-order linear differential equation (i.e., if the equation included a third derivative), then a linear combination of any two of its particular solutions would also be a solution to this equation, but would not represent him common decision.

In the course of differential equations, a theorem is proved that the general solution of an ODE of the Nth order (linear or non-linear) depends on N arbitrary constants. In the case of a nonlinear equation, these arbitrary constants can enter the general solution (in contrast to (13)), in a non-linear manner.

The superposition principle plays an extremely important role in the theory of ODEs, since it can be used to construct a general solution of a differential equation in the form of a superposition of its particular solutions. For example, for the case of linear ODEs with constant coefficients and their systems (the harmonic oscillator equation belongs precisely to this type of equations), a general solution method has been developed in the theory of differential equations. Its essence is as follows. We are looking for a particular solution in the form As a result of its substitution into the original equation, all time-dependent factors cancel and we arrive at some characteristic equation, which for the Nth order ODE is algebraic equation Nth degree. Solving it, we find, thereby, all possible particular solutions, an arbitrary linear combination of which gives the general solution of the original ODE. We will not dwell on this issue further, referring the reader to the appropriate textbooks on the theory of differential equations, where one can find further details, in particular, the case when the characteristic equation contains multiple roots.

If a linear ODE with variable coefficients is considered (its coefficients depend on time), then the superposition principle is also valid, but it is no longer possible to construct a general solution to this equation in an explicit form by any standard method. We will return to this issue later, discussing the phenomenon of parametric resonance and the Mathieu equation related to its study.

VASCULATION. WAVES. OPTICS

VASCULATION

Lecture 1

HARMONIC OSCILLATIONS

Ideal harmonic oscillator. Ideal oscillator equation and its solution. Amplitude, frequency and phase of oscillations

Oscillation is one of the most common processes in nature and technology. Fluctuations are processes that repeat over time. High-rise buildings and high-voltage wires oscillate under the influence of the wind, the pendulum of a wound clock and a car on springs during movement, the level of the river during the year and the temperature of the human body during illness. Sound is fluctuations in air pressure, radio waves are periodic changes in the strength of the electrical and magnetic field, light is also electromagnetic oscillations. Earthquakes - ground vibrations, ebbs and flows - changes in the levels of the seas and oceans caused by the attraction of the moon, etc.

Oscillations are mechanical, electromagnetic, chemical, thermodynamic, etc. Despite such a variety, all oscillations are described by the same differential equations.

The first scientists to study vibrations were Galileo Galilei and Christian Huygens. Galileo established the independence of the period of oscillations from the amplitude. Huygens invented the pendulum clock.

Any system that, when slightly out of balance, oscillates steadily is called a harmonic oscillator. In classical physics, such systems are a mathematical pendulum within small deflection angles, a load within small oscillation amplitudes, an electrical circuit consisting of linear capacitance and inductance elements.

A harmonic oscillator can be considered linear if the displacement from the equilibrium position is directly proportional to the perturbing force. The oscillation frequency of a harmonic oscillator does not depend on the amplitude. For the oscillator, the principle of superposition is fulfilled - if several disturbing forces act, then the effect of their total action can be obtained as a result of adding the effects from active forces separately.

Harmonic oscillations are described by the equation (Fig. 1.1.1)

(1.1.1)

where X- displacement of the oscillating value from the equilibrium position, BUT– amplitude of oscillations equal to the value of the maximum displacement, - phase of oscillations, which determines the displacement at time , - initial phase, which determines the magnitude of the displacement at the initial moment of time, - cyclic frequency of oscillations.

The time of one complete oscillation is called the period, where is the number of oscillations completed during the time.

The oscillation frequency determines the number of oscillations per unit time, it is related to the cyclic frequency by the ratio, then the period.

The speed of an oscillating material point

acceleration

Thus, the speed and acceleration of the harmonic oscillator also change according to harmonic law with amplitudes and respectively. In this case, the speed is ahead of the phase displacement by , and acceleration - by (Fig. 1.1.2).

From a comparison of the equations of motion of a harmonic oscillator (1.1.1) and (1.1.2) it follows that , or

This second-order differential equation is called the harmonic oscillator equation. His solution contains two constants a and , which are determined by the task initial conditions

If a periodically repeating process is described by equations that do not coincide with (1.1.1), it is called anharmonic. A system that performs anharmonic oscillations is called an anharmonic oscillator.

1.1.2 . Free oscillations of systems with one degree of freedom. complex form representations of harmonic vibrations

In nature, small oscillations that a system makes near its equilibrium position are very common. If a system taken out of equilibrium is left to itself, that is, external forces do not act on it, then such a system will perform free undamped oscillations. Consider a system with one degree of freedom.

A stable equilibrium corresponds to a position of the system in which its potential energy has a minimum ( q is the generalized coordinate of the system). The deviation of the system from the equilibrium position leads to the emergence of a force that tends to bring the system back. We denote the value of the generalized coordinate corresponding to the equilibrium position, then the deviation from the equilibrium position

We will count the potential energy from the minimum value . Let's take the resulting function, expand it in a Maclaurin series and leave the first term of the expansion, we have: o

where . Then, taking into account the introduced notation:

, (1.1.4)

Taking into account the expression (1.1.4) for the force acting on the system, we obtain:

According to Newton's second law, the equation of motion of the system has the form:

Expression (1.1.5) coincides with the equation (1.1.3) of free harmonic oscillations, provided that

and has two independent solutions: and , so the general solution is:

From formula (1.1.6) it follows that the frequency is determined only by the intrinsic properties of the mechanical system and does not depend on the amplitude and on the initial conditions of motion.

The dependence of the coordinate of the oscillating system on time can be determined as the real part of the complex expression , where A=Xe-iα is a complex amplitude, its modulus coincides with the usual amplitude, and its argument coincides with the initial phase.

1.1.3 . Examples of oscillatory motions of various physical nature

Fluctuations of the load on the spring

Consider the oscillations of a load on a spring, provided that the spring is not deformed beyond the limits of elasticity. We will show that such a load will perform harmonic oscillations relative to the equilibrium position (Fig. 1.1.3). Indeed, according to Hooke's law, a compressed or stretched spring creates a harmonic force:

where - coefficient of spring stiffness, is the coordinate of the equilibrium position, X is the coordinate of the load (material point) at the moment of time , is the displacement from the equilibrium position.

Let us place the origin of the coordinate in the equilibrium position of the system. In this case .

If the spring is stretched by X, then release at time t=0, then the equation of motion of the load according to Newton's second law will take the form -kx=ma, or , and

(1.1.6)

This equation coincides in form with the equation of motion (1.1.3) of a system performing harmonic oscillations, we will look for its solution in the form:

. (1.1.7)

We substitute (1.17) into (1.1.6), we have: that is, expression (1.1.7) is a solution to equation (1.1.6) provided that

If at the initial moment of time the position of the load was arbitrary, then the equation of motion will take the form:

Let's consider how the energy of the load changes, making harmonic oscillations in the absence of external forces (Fig. 1.14). If at the time t=0 send offset to cargo x=A, then its total energy will become equal to the potential energy of the deformed spring, kinetic energy equals zero (point 1).

Force acting on the load F= -kx, seeking to return it to the equilibrium position, so the load moves with acceleration and increases its speed, and, consequently, its kinetic energy. This force reduces the displacement of the load X, the potential energy of the load decreases, turning into kinetic. The "load - spring" system is closed, so its total energy is conserved, that is:

. (1.1.8)

At the moment of time, the load is in equilibrium (point 2), its potential energy is zero, and its kinetic energy is maximum. We find the maximum speed of the load from the law of conservation of energy (1.1.8):

Due to the stock of kinetic energy, the load does work against the elastic force – and passes through the equilibrium position. Kinetic energy gradually turns into potential. When the load has a maximum negative displacement - BUT, kinetic energy wk=0, the load stops and starts moving to the equilibrium position under the action of an elastic force F= -kx. Further movement is similar.

Pendulums

A pendulum is a rigid body that oscillates around a fixed point or axis under the action of gravity. There are physical and mathematical pendulums.

A mathematical pendulum is an idealized system consisting of a weightless inextensible thread on which a mass concentrated at one material point is suspended.

A mathematical pendulum, for example, is a ball on a long thin thread.

The deviation of the pendulum from the equilibrium position is characterized by the angle φ , which forms a thread with a vertical (Fig. 1.15). When the pendulum deviates from the equilibrium position, a moment of external forces (gravity) arises: , where m- weight, - pendulum length

This moment tends to return the pendulum to the equilibrium position (similar to the quasi-elastic force) and is directed opposite to the displacement φ , so there is a minus sign in the formula.

Equation of dynamics rotary motion for a pendulum has the form: Iε=,

We will consider the case of small fluctuations, therefore sin φ ≈φ, denote ,

we have: , or , and finally

This is the equation of harmonic oscillations, its solution:

The oscillation frequency of a mathematical pendulum is determined only by its length and the acceleration of gravity, and does not depend on the mass of the pendulum. The period is:

If the oscillating body cannot be represented as a material point, then the pendulum is called physical (Fig. 1.1.6). We write the equation of its motion in the form:

In the case of small fluctuations , or =0 , where . This is the equation of motion of a body that performs harmonic oscillations. The oscillation frequency of a physical pendulum depends on its mass, length and moment of inertia about the axis passing through the suspension point.

Let's denote . Value is called the reduced length of the physical pendulum. This is the length of a mathematical pendulum whose period of oscillation coincides with the period of a given physical pendulum. A point on a straight line connecting the suspension point with the center of mass, lying at a distance of the reduced length from the axis of rotation, is called the swing center of a physical pendulum ( O'). If the pendulum is suspended in the center of the swing, then the reduced length and period of oscillation will be the same as at the point O. Thus, the suspension point and the swing center have the properties of reciprocity: when the suspension point is transferred to the swing center, the old suspension point becomes the new swing center.

A mathematical pendulum that swings with the same period as the physical one under consideration is called isochronous to the given physical pendulum.

1.1.4. Addition of vibrations (beats, Lissajous figures). Vector description of oscillation addition

The addition of equally directed oscillations can be performed using the method of vector diagrams. Any harmonic oscillation can be represented as a vector as follows. Let's choose an axis X with origin at point O(fig.1.1.7)

From a point O construct a vector that makes up the angle with axle X. Let this vector rotate with angular velocity . Projection of a vector onto an axis X is equal to:

that is, it performs harmonic oscillations with an amplitude a.

Consider two harmonic oscillations of the same direction and the same cyclic small , given by the vectors and . Offsets along the axis X are equal:

the resulting vector has a projection and represents the resulting oscillation (Fig. 1.1.8), according to the cosine theorem Thus, the addition of harmonic oscillations is carried out by adding the vectors.

Let us carry out the addition of mutually perpendicular oscillations. Let the material point make two mutually perpendicular oscillations with a frequency:

The material point itself will then move along some curvilinear trajectory.

From the equation of motion follows: ,

. (1.1.9)

From equation (1.1.9) you can get the ellipse equation (Fig.1.1.9):

Consider special cases of this equation:

1. Oscillation phase difference α= 0. At the same time those. or This is the equation of a straight line, and the resulting oscillation occurs along this straight line with amplitude (Fig. 1.1.10). a.

its acceleration is equal to the second derivative of the displacement with respect to time then the force acting on the oscillating point, according to Newton's second law, is equal to

That is, the force is proportional to the displacement X and is directed against the displacement to the equilibrium position. This force is called the restoring force. In the case of a load on a spring, the restoring force is the elastic force, in the case of a mathematical pendulum, it is the component of gravity.

The restoring force by nature obeys Hooke's law F=-kx, where

is the coefficient of the restoring force. Then the potential energy of the oscillating point is:

(the integration constant is chosen equal to zero, so that when X).

Anharmonic Oscillator

HARMONIC OSCILLATIONS

Lecture 1

VASCULATION

VASCULATION. WAVES. OPTICS

Oscillation is one of the most common processes in nature and technology. Fluctuations are processes that repeat over time. High-rise buildings and high-voltage wires oscillate under the influence of the wind, the pendulum of a wound clock and a car on springs during movement, the level of the river during the year and the temperature of the human body during illness. Sound is fluctuations in air pressure, radio waves are periodic changes in the strength of the electric and magnetic fields, light is also electromagnetic vibrations. Earthquakes - ground vibrations, ebbs and flows - changes in the levels of the seas and oceans caused by the attraction of the moon, etc.

Oscillations are mechanical, electromagnetic, chemical, thermodynamic, etc. Despite such a variety, all oscillations are described by the same differential equations.

A harmonic oscillator can be considered linear if the displacement from the equilibrium position is directly proportional to the perturbing force. The oscillation frequency of a harmonic oscillator does not depend on the amplitude. For the oscillator, the principle of superposition is fulfilled - if several perturbing forces act, then the effect of their total action can be obtained as a result of adding the effects of the acting forces separately.

Harmonic oscillations are described by the equation (Fig. 1.1.1)

(1.1.1)

The time of one complete oscillation is called the period, where is the number of oscillations completed during the time.

The oscillation frequency determines the number of oscillations per unit time, it is related to the cyclic frequency by the ratio, then the period.

The speed of an oscillating material point

acceleration

Thus, the speed and acceleration of the harmonic oscillator also change according to the harmonic law with amplitudes and respectively. In this case, the speed is ahead of the phase displacement by , and acceleration - by (Fig. 1.1.2).

From a comparison of the equations of motion of a harmonic oscillator (1.1.1) and (1.1.2) it follows that , or

This second-order differential equation is called the harmonic oscillator equation. His solution contains two constants a and , which are determined by setting the initial conditions

1.1.2 . Free oscillations of systems with one degree of freedom. Complex form of representation of harmonic oscillations

We will count the potential energy from the minimum value . Let's take the resulting function, expand it in a Maclaurin series and leave the first term of the expansion, we have: o

Harmonic oscillator(in classical mechanics) - a system that, when displaced from an equilibrium position, experiences the action of a restoring force F, proportional to the displacement x(according to Hooke's law):

F = − k x (\displaystyle F=-kx)

where k- coefficient rigidity of the system.

Mechanical examples of a harmonic oscillator are a mathematical pendulum (with small deflection angles), a torsion pendulum, and acoustic systems. Among other analogues of the harmonic oscillator, it is worth highlighting the electric harmonic oscillator (see LC circuit).

Encyclopedic YouTube

1 / 5

Elementary particles | quantum field theory | study number 6 | quantum oscillator

Forced oscillations of a linear oscillator | General physics. Mechanics | Evgeniy Butikov

Elementary particles | quantum field theory | study number 5 | classic oscillator

Oscillators: what are they and how to use them? Education for traders from I-TT.RU

Sytrus 01 of 16 Working with the oscillator shape

Subtitles

Free vibrations

Conservative harmonic oscillator

As a model of a conservative harmonic oscillator, we take the mass load m, fixed on a spring with rigidity k .

Let x- displacement of the load relative to the equilibrium position. Then, according to Hooke's law, the restoring force will act on it:

F = − k x . (\displaystyle F=-kx.)

We substitute into the differential equation.

x ¨ (t) = − A ω 2 sin ⁡ (ω t + φ) , (\displaystyle (\ddot (x))(t)=-A\omega ^(2)\sin(\omega t+\varphi) ,) − A ω 2 sin ⁡ (ω t + φ) + ω 0 2 A sin ⁡ (ω t + φ) = 0. (\displaystyle -A\omega ^(2)\sin(\omega t+\varphi)+\ omega _(0)^(2)A\sin(\omega t+\varphi)=0.)

The amplitude is reduced. This means that it can have any value (including zero - this means that the load is at rest in the equilibrium position). The sine can also be reduced, since the equality must hold at any time t. Thus, the condition for the oscillation frequency remains:

− ω 2 + ω 0 2 = 0 , (\displaystyle -\omega ^(2)+\omega _(0)^(2)=0,) ω = ± ω 0 . (\displaystyle \omega =\pm \omega _(0).) U = 1 2 k x 2 = 1 2 k A 2 sin 2 ⁡ (ω 0 t + φ) , (\displaystyle U=(\frac (1)(2))kx^(2)=(\frac (1) (2))kA^(2)\sin ^(2)(\omega _(0)t+\varphi),)

then the total energy is constant

E = 1 2 k A 2 . (\displaystyle E=(\frac (1)(2))kA^(2).)

Simple harmonic movement is a simple movement harmonic oscillator, a periodic motion that is neither forced nor damped. A body in simple harmonic motion is subjected to a single variable force that is directly proportional in absolute value to the displacement x from the equilibrium position and is directed in the opposite direction.

This movement is periodic: the body oscillates around the equilibrium position according to a sinusoidal law. Each subsequent oscillation is the same as the previous one, and the period, frequency and amplitude of the oscillations remain constant. If we assume that the equilibrium position is at a point with a coordinate equal to zero, then the displacement x body from the equilibrium position at any time is given by the formula:

x (t) = A cos ⁡ (2 π f t + φ) , (\displaystyle x(t)=A\cos \left(2\pi \!ft+\varphi \right),)

where A- oscillation amplitude, f- frequency, φ - initial phase.

The frequency of movement is determined by the characteristic properties of the system (for example, the mass of the moving body), while the amplitude and initial phase are determined by the initial conditions - the movement and speed of the body at the moment the oscillations begin. The kinetic and potential energies of the system also depend on these properties and conditions.

Simple harmonic motion can be viewed as a mathematical model various kinds movement, such as the oscillation of a spring. Other cases that can be roughly considered as simple harmonic motion are the motion of a pendulum and the vibrations of molecules.

A typical example of a system in which simple harmonic motion occurs is an idealized mass-spring system in which a mass is attached to a spring. If the spring is not compressed and not stretched, then no variable forces act on the load, and the load is in a state of mechanical equilibrium. However, if the load is removed from the equilibrium position, the spring is deformed, and a force will act on the load from its side, which will tend to return the load to the equilibrium position. In the case of a load-spring system, such a force is the elastic force of the spring, which obeys Hooke's law:

F = − k x , (\displaystyle F=-kx,) F- restoring force x- movement of the load (spring deformation), k- coefficient of stiffness of the spring.

Any system in which simple harmonic motion occurs has two key properties:

When a system is taken out of equilibrium, there must be a restoring force tending to bring the system back into equilibrium.
The restoring force must be exactly or approximately proportional to the displacement.

The weight-spring system satisfies both of these conditions.

Once the displaced load is subjected to the action of a restoring force, accelerating it, and tending to return to the starting point, that is, to the equilibrium position. As the load approaches the equilibrium position, the restoring force decreases and tends to zero. However, in position x = 0 the load has a certain amount of motion (momentum), acquired due to the action of the restoring force. Therefore, the load skips the equilibrium position, starting to deform the spring again (but in the opposite direction). The restoring force will tend to slow it down until the speed is zero; and the force will again seek to return the load to its equilibrium position.

As long as there is no energy loss in the system, the load will oscillate as described above; such a movement is called periodic.

Further analysis will show that in the case of a mass-spring system, the motion is simple harmonic.

Dynamics of simple harmonic motion

For an oscillation in one-dimensional space, taking into account Newton's Second Law ( F= m d² x/d t² ) and Hooke's law ( F = −kx, as described above), we have a second-order linear differential equation:

m d 2 x d t 2 = − k x , (\displaystyle m(\frac (\mathrm (d) ^(2)x)(\mathrm (d) t^(2)))=-kx,) m- body mass, x- its displacement relative to the equilibrium position, k- constant (spring stiffness factor).

The solution to this differential equation is sinusoidal; one solution is this:

x (t) = A cos ⁡ (ω t + φ) , (\displaystyle x(t)=A\cos(\omega t+\varphi),)

where A, ω and φ - constants, and the equilibrium position is taken as the initial one. Each of these constants is an important physical property movements: A is the amplitude, ω = 2π f is the circular frequency, and φ is the initial phase.

U (t) = 1 2 k x (t) 2 = 1 2 k A 2 cos 2 ⁡ (ω t + φ) . (\displaystyle U(t)=(\frac (1)(2))kx(t)^(2)=(\frac (1)(2))kA^(2)\cos ^(2)(\ omega t+\varphi).)

Universal circular motion

Simple harmonic motion can in some cases be considered as a one-dimensional projection of universal circular motion.

If an object moves with a constant angular velocity ω along a circle of radius r, whose center is the origin of the coordinates of the plane x − y, then such motion along each of the coordinate axes is simple harmonic with amplitude r and circular frequency ω .

Weight as a simple pendulum

In the approximation of small angles, the motion of a simple pendulum is close to simple harmonic. The period of oscillation of such a pendulum attached to a rod of length ℓ with free fall acceleration g is given by the formula

T = 2πℓg. (\displaystyle T=2\pi (\sqrt (\frac (\ell )(g))).)

This shows that the period of oscillation does not depend on the amplitude and mass of the pendulum, but depends on the free fall acceleration g, therefore, with the same length of the pendulum, on the Moon it will swing more slowly, since gravity is weaker there and the value of the free fall acceleration is lower.

This approximation is correct only for small deflection angles, since the expression for angular acceleration is proportional to the sine of the coordinate:

ℓ m g sin ⁡ θ = I α , (\displaystyle \ell mg\sin \theta =I\alpha ,)

where I- moment of inertia ; in this case I = mℓ 2 .

ℓ m g θ = I α (\displaystyle \ell mg\theta =I\alpha ),

which makes the angular acceleration directly proportional to the angle θ, and this satisfies the definition of simple harmonic motion.

Damped Harmonic Oscillator

Taking the same model as a basis, we add the force of viscous friction to it. The force of viscous friction is directed against the speed of the load relative to the medium and is directly proportional to this speed. Then the total force acting on the load is written as follows:

F = − k x − α v (\displaystyle F=-kx-\alpha v)

Carrying out similar actions, we obtain a differential equation describing a damped oscillator:

x ¨ + 2 γ x ˙ + ω 0 2 x = 0 (\displaystyle (\ddot (x))+2\gamma (\dot (x))+\omega _(0)^(2)x=0)

Here is the notation: 2 γ = α m (\displaystyle 2\gamma =(\frac (\alpha )(m))). Coefficient γ (\displaystyle \gamma ) is called the damping constant. It also has the dimension of frequency.

The solution falls into three cases.

x (t) = A e − γ t s i n (ω f t + φ) (\displaystyle x(t)=Ae^(-\gamma t)sin(\omega _(f)t+\varphi)),

where ω f = ω 0 2 − γ 2 (\displaystyle \omega _(f)=(\sqrt (\omega _(0)^(2)-\gamma ^(2))))- frequency of free oscillations.

x (t) = (A + B t) e − γ t (\displaystyle \ x(t)=(A+Bt)e^(-\gamma t)) x (t) = A e − β 1 t + B e − β 2 t (\displaystyle x(t)=Ae^(-\beta _(1)t)+Be^(-\beta _(2)t )),

where β 1 , 2 = γ ± γ 2 − ω 0 2 (\displaystyle \beta _(1,2)=\gamma \pm (\sqrt (\gamma ^(2)-\omega _(0)^(2) ))).

Critical damping is notable for the fact that it is during critical damping that the oscillator tends most rapidly to the equilibrium position. If the friction is less than critical, it will reach the equilibrium position faster, however, it will “slip” it by inertia, and will oscillate. If the friction is greater than critical, then the oscillator will exponentially tend to the equilibrium position, but the slower, the greater the friction.

Therefore, in pointer indicators (for example, in ammeters), they usually try to introduce precisely critical attenuation so that the arrow calms down as quickly as possible to read its readings.

The damping of an oscillator is also often characterized by a dimensionless parameter called the quality factor. Quality factor is usually denoted by the letter Q (\displaystyle Q). By definition, the quality factor is:

Q = ω 0 2 γ (\displaystyle Q=(\frac (\omega _(0))(2\gamma )))

The greater the quality factor, the slower the oscillations of the oscillator decay.

The quality factor is sometimes called the gain of the oscillator, since with some methods of excitation, when the excitation frequency coincides with the resonant frequency of oscillations, their amplitude is set to approximately Q (\displaystyle Q) times greater than when excited with the same intensity at a low frequency.

Also, the quality factor is approximately equal to the number of oscillatory cycles, during which the oscillation amplitude decreases in e (\displaystyle e) times multiplied by π (\displaystyle \pi ).

In the case of oscillatory motion, attenuation is also characterized by such parameters as:

Lifetime fluctuations (aka decay time, it is relaxation time) τ is the time during which the oscillation amplitude will decrease in e once.

τ = 1 / γ . (\displaystyle \tau =1/\gamma .) This time is considered as the time required for the damping (cessation) of oscillations (although formally free oscillations continue indefinitely).