“Simple Harmonic Motion”的概念、定义、习题解题方法及翻译-数学辞典

单词

Simple Harmonic Motion

释义

Simple Harmonic Motion

Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion isexecuted by any quantity obeying the Differential Equation

(1)

where

denotes the second Derivative of

with respect to

, and

is the angular frequency of oscillation. This Ordinary Differential Equation has an irregular Singularity at

. The general solutionis

			(2)
			(3)

where the two constants

and

(or

and

) are determined from the initial conditions.

Many physical systems undergoing small displacements, including any objects obeying Hooke's law, exhibit simple harmonic motion. This equation arises, for example, in the analysis of the flow ofcurrent in an electronic CL circuit (which contains acapacitor and an inductor ). If a damping force such asFriction is present, an additional term must be added to the Differential Equation andmotion dies out over time.

Adding a damping force proportional to , the first derivative of withrespect to time, the equation of motion for damped simple harmonic motion is

(4)

where

is the damping constant. This equation arises, for example, in the analysis of the flow ofcurrent

in an electronic CLR circuit,

(which contains acapacitor,

an inductor,

and aresistor

). This Ordinary Differential Equation can be solved by looking for trial solutionsof the form

. Plugging this into (4) gives

(5)

(6)

This is a Quadratic Equation with solutions

(7)

There are therefore three solution regimes depending on the Sign of the quantity inside the Square Root,

(8)

The three regimes are

: 1. is Positive: overdamped,
: 2. is Zero: critically damped,
: 3. is Negative: underdamped.

If a periodic (sinusoidal) forcing term is added at angular frequency , the same three solution regimes are againobtained. Surprisingly, the resulting motion is still periodic (after an initial transient response, corresponding to thesolution to the unforced case, has died out), but it has an amplitude different from the forcing amplitude.

The ``particular'' solution to the forced second-order nonhomogeneous Ordinary Differential Equation

(9)

due to forcing is given by the equation

(10)

where

and

are the homogeneous solutions to the unforced equation

(11)

and

is the Wronskian of these two functions. Once the sinusoidal case of forcing is solved, it canbe generalized to any periodic function by expressing the periodic function in a Fourier Series.

Critical damping is a special case of damped simple harmonic motion in which

(12)

(13)

The above plot shows an underdamped simple harmonic oscillator with

. The solid curve is for

, the dot-dashed for (0, 1), and the dotted for (1/2, 1/2). In this case,

so the solutions of theform

satisfy

(14)

One of the solutions is therefore

(15)

In order to find the other linearly independent solution, we can make use of the identity

(16)

Since we have

simplifies to

. Equation (16) thereforebecomes

(17)

The general solution is therefore

(18)

In terms of the constants

and

, the initial values are

			(19)
			(20)

			(21)
			(22)

For sinusoidally forced simple harmonic motion with critical damping, the equation of motion is

(23)

and the Wronskian is



			(24)

Plugging this into the equation for the particular solution gives






			(25)

In order to put this in the desired form, note that we want to equate


			(26)

This means

			(27)
			(28)

			(29)
			(30)

Plugging in,


			(31)
			(32)

The solution in the requested form is therefore


			(33)

where

is defined by (32).

Overdamped simple harmonic motion occurs when

(34)

(35)

The above plot shows an overdamped simple harmonic oscillator with

. The solid curveis for

, the dot-dashed for (0, 1), and the dotted for (1/2, 1/2). The solutions are

			(36)
			(37)

where

(38)

The general solution is therefore

(39)

where

and

are constants. The initial values are

			(40)
			(41)

			(42)
			(43)

For a cosinusoidally forced overdamped oscillator with forcing function , the particular solutions are

			(44)
			(45)

where

			(46)
			(47)

These give the identities

			(48)
			(49)

and


			(50)

The Wronskian is


			(51)

The particular solution is

(52)

where

			(53)

			(54)

Therefore,




			(55)

where

(56)

Underdamped simple harmonic motion occurs when

(57)

(58)

The above plot shows an underdamped simple harmonic oscillator with

. The solid curveis for

, the dot-dashed for (0, 1), and the dotted for (1/2, 1/2). Define

(59)

then solutions satisfy

(60)

where

(61)

and are of the form

(62)

Using the Euler Formula

(63)

this can be rewritten

(64)

We are interested in the real solutions. Since we are dealing here with a linear homogeneous ODE, linearsums of Linearly Independent solutions are also solutions. Since we have a sum of such solutions in (64), itfollows that the Imaginary and Real Parts separately satisfy the ODE and are therefore the solutions we seek. Theconstant in front of the sine term is arbitrary, so we can identify the solutions as