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57-428: Assume that the molecule consists of harmonic oscillators , which are connected and can exchange energy with each other. Assume that A is an excited molecule: where P stands for product, and A for the critical atomic configuration with the maximum energy E 0 along the reaction coordinate . The unimolecular rate constant k u n i {\displaystyle k_{\mathrm {uni} }}

114-414: A k , … , a 1 , a 0 {\displaystyle a_{k},\dotsc ,a_{1},a_{0}} are the coefficients. This includes the possibility that some terms have coefficient 0; for example, in x 3 − 2 x + 1 {\displaystyle x^{3}-2x+1} , the coefficient of x 2 {\displaystyle x^{2}}

171-410: A x 2 + b x + c {\displaystyle ax^{2}+bx+c} have coefficient parameters a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} . A constant coefficient , also known as constant term or simply constant , is a quantity either implicitly attached to the zeroth power of

228-409: A coefficient is a multiplicative factor involved in some term of a polynomial , a series , or any other type of expression . It may be a number without units , in which case it is known as a numerical factor . It may also be a constant with units of measurement , in which it is known as a constant multiplier . In general, coefficients may be any expression (including variables such as

285-404: A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x : F → = − k x → , {\displaystyle {\vec {F}}=-k{\vec {x}},} where k is a positive constant . If F is the only force acting on the system,

342-458: A steady state that is independent of initial conditions and depends only on the driving amplitude F 0 {\displaystyle F_{0}} , driving frequency ω {\displaystyle \omega } , undamped angular frequency ω 0 {\displaystyle \omega _{0}} , and the damping ratio ζ {\displaystyle \zeta } . The steady-state solution

399-403: A , b and c ). When the combination of variables and constants is not necessarily involved in a product , it may be called a parameter . For example, the polynomial 2 x 2 − x + 3 {\displaystyle 2x^{2}-x+3} has coefficients 2, −1, and 3, and the powers of the variable x {\displaystyle x} in the polynomial

456-574: A mass m , which experiences a single force F , which pulls the mass in the direction of the point x = 0 and depends only on the position x of the mass and a constant k . Balance of forces ( Newton's second law ) for the system is F = m a = m d 2 x d t 2 = m x ¨ = − k x . {\displaystyle F=ma=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=m{\ddot {x}}=-kx.} Solving this differential equation , we find that

513-655: A mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing, since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits the instability phenomenon. The equation d 2 q d τ 2 + 2 ζ d q d τ + q = 0 {\displaystyle {\frac {\mathrm {d} ^{2}q}{\mathrm {d} \tau ^{2}}}+2\zeta {\frac {\mathrm {d} q}{\mathrm {d} \tau }}+q=0}

570-399: A variable or not attached to other variables in an expression; for example, the constant coefficients of the expressions above are the number 3 and the parameter c , involved in 3= c  ⋅  x . The coefficient attached to the highest degree of the variable in a polynomial of one variable is referred to as the leading coefficient ; for example, in the example expressions above,

627-416: Is 0, and the term 0 x 2 {\displaystyle 0x^{2}} does not appear explicitly. For the largest i {\displaystyle i} such that a i ≠ 0 {\displaystyle a_{i}\neq 0} (if any), a i {\displaystyle a_{i}} is called the leading coefficient of the polynomial. For example,

SECTION 10

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684-409: Is a constant function . For avoiding confusion, in this context a coefficient that is not attached to unknown functions or their derivatives is generally called a constant term rather than a constant coefficient. In particular, in a linear differential equation with constant coefficient , the constant coefficient term is generally not assumed to be a constant function. In mathematics, a coefficient

741-442: Is a multiplicative factor in some term of a polynomial , a series , or any expression . For example, in the polynomial 7 x 2 − 3 x y + 1.5 + y , {\displaystyle 7x^{2}-3xy+1.5+y,} with variables x {\displaystyle x} and y {\displaystyle y} , the first two terms have the coefficients 7 and −3. The third term 1.5

798-476: Is called the viscous damping coefficient . The balance of forces ( Newton's second law ) for damped harmonic oscillators is then F = − k x − c d x d t = m d 2 x d t 2 , {\displaystyle F=-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}}=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}},} which can be rewritten into

855-483: Is generally assumed that x is the only variable, and that a , b and c are parameters; thus the constant coefficient is c in this case. Any polynomial in a single variable x can be written as a k x k + ⋯ + a 1 x 1 + a 0 {\displaystyle a_{k}x^{k}+\dotsb +a_{1}x^{1}+a_{0}} for some nonnegative integer k {\displaystyle k} , where

912-723: Is independent of the forcing function. Apply the " complex variables method" by solving the auxiliary equation below and then finding the real part of its solution: d 2 q d τ 2 + 2 ζ d q d τ + q = cos ⁡ ( ω τ ) + i sin ⁡ ( ω τ ) = e i ω τ . {\displaystyle {\frac {\mathrm {d} ^{2}q}{\mathrm {d} \tau ^{2}}}+2\zeta {\frac {\mathrm {d} q}{\mathrm {d} \tau }}+q=\cos(\omega \tau )+i\sin(\omega \tau )=e^{i\omega \tau }.} Supposing

969-760: Is known as the universal oscillator equation , since all second-order linear oscillatory systems can be reduced to this form. This is done through nondimensionalization . If the forcing function is f ( t ) = cos( ωt ) = cos( ωt c τ ) = cos( ωτ ) , where ω = ωt c , the equation becomes d 2 q d τ 2 + 2 ζ d q d τ + q = cos ⁡ ( ω τ ) . {\displaystyle {\frac {\mathrm {d} ^{2}q}{\mathrm {d} \tau ^{2}}}+2\zeta {\frac {\mathrm {d} q}{\mathrm {d} \tau }}+q=\cos(\omega \tau ).} The solution to this differential equation contains two parts:

1026-486: Is minimal, since a reactance (not a resistance) is varied. Another common use is frequency conversion, e.g., conversion from audio to radio frequencies. For example, the Optical parametric oscillator converts an input laser wave into two output waves of lower frequency ( ω s , ω i {\displaystyle \omega _{s},\omega _{i}} ). Parametric resonance occurs in

1083-468: Is obtained as follows: where k ( E , J ) {\displaystyle k(E,J)} is the microcanonical transition state theory rate constant, G ‡ {\displaystyle G^{\ddagger }} is the sum of states for the active degrees of freedom in the transition state, J {\displaystyle J} is the quantum number of angular momentum, ω {\displaystyle \omega }

1140-430: Is of the order τ = 1/( ζω 0 ) . In physics, the adaptation is called relaxation , and τ is called the relaxation time. In electrical engineering, a multiple of τ is called the settling time , i.e. the time necessary to ensure the signal is within a fixed departure from final value, typically within 10%. The term overshoot refers to the extent the response maximum exceeds final value, and undershoot refers to

1197-782: Is proportional to the driving force with an induced phase change φ {\displaystyle \varphi } : x ( t ) = F 0 m Z m ω sin ⁡ ( ω t + φ ) , {\displaystyle x(t)={\frac {F_{0}}{mZ_{m}\omega }}\sin(\omega t+\varphi ),} where Z m = ( 2 ω 0 ζ ) 2 + 1 ω 2 ( ω 0 2 − ω 2 ) 2 {\displaystyle Z_{m}={\sqrt {\left(2\omega _{0}\zeta \right)^{2}+{\frac {1}{\omega ^{2}}}(\omega _{0}^{2}-\omega ^{2})^{2}}}}

SECTION 20

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1254-504: Is the phase of the oscillation relative to the driving force. The phase value is usually taken to be between −180° and 0 (that is, it represents a phase lag, for both positive and negative values of the arctan argument). For a particular driving frequency called the resonance , or resonant frequency ω r = ω 0 1 − 2 ζ 2 {\textstyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}} ,

1311-431: Is the absolute value of the impedance or linear response function , and φ = arctan ⁡ ( 2 ω ω 0 ζ ω 2 − ω 0 2 ) + n π {\displaystyle \varphi =\arctan \left({\frac {2\omega \omega _{0}\zeta }{\omega ^{2}-\omega _{0}^{2}}}\right)+n\pi }

1368-401: Is the collision frequency between A ∗ {\displaystyle A^{*}} molecule and bath molecules, Q r {\displaystyle Q_{r}} and Q v {\displaystyle Q_{v}} are the molecular vibrational and external rotational partition functions. Harmonic oscillators In classical mechanics ,

1425-438: Is the constant coefficient. In the final term, the coefficient is 1 and is not explicitly written. In many scenarios, coefficients are numbers (as is the case for each term of the previous example), although they could be parameters of the problem—or any expression in these parameters. In such a case, one must clearly distinguish between symbols representing variables and symbols representing parameters. Following René Descartes ,

1482-485: Is the driving amplitude, and ω {\displaystyle \omega } is the driving frequency for a sinusoidal driving mechanism. This type of system appears in AC -driven RLC circuits ( resistor – inductor – capacitor ) and driven spring systems having internal mechanical resistance or external air resistance . The general solution is a sum of a transient solution that depends on initial conditions, and

1539-419: Is varied periodically. The circuit that varies the diode's capacitance is called the "pump" or "driver". In microwave electronics, waveguide / YAG based parametric oscillators operate in the same fashion. The designer varies a parameter periodically to induce oscillations. Parametric oscillators have been developed as low-noise amplifiers, especially in the radio and microwave frequency range. Thermal noise

1596-435: Is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves. A simple harmonic oscillator is an oscillator that is neither driven nor damped . It consists of

1653-1769: The "transient" and the "steady-state". The solution based on solving the ordinary differential equation is for arbitrary constants c 1 and c 2 q t ( τ ) = { e − ζ τ ( c 1 e τ ζ 2 − 1 + c 2 e − τ ζ 2 − 1 ) ζ > 1  (overdamping) e − ζ τ ( c 1 + c 2 τ ) = e − τ ( c 1 + c 2 τ ) ζ = 1  (critical damping) e − ζ τ [ c 1 cos ⁡ ( 1 − ζ 2 τ ) + c 2 sin ⁡ ( 1 − ζ 2 τ ) ] ζ < 1  (underdamping) {\displaystyle q_{t}(\tau )={\begin{cases}e^{-\zeta \tau }\left(c_{1}e^{\tau {\sqrt {\zeta ^{2}-1}}}+c_{2}e^{-\tau {\sqrt {\zeta ^{2}-1}}}\right)&\zeta >1{\text{ (overdamping)}}\\e^{-\zeta \tau }(c_{1}+c_{2}\tau )=e^{-\tau }(c_{1}+c_{2}\tau )&\zeta =1{\text{ (critical damping)}}\\e^{-\zeta \tau }\left[c_{1}\cos \left({\sqrt {1-\zeta ^{2}}}\tau \right)+c_{2}\sin \left({\sqrt {1-\zeta ^{2}}}\tau \right)\right]&\zeta <1{\text{ (underdamping)}}\end{cases}}} The transient solution

1710-415: The acceleration is in the direction opposite to the displacement. The potential energy stored in a simple harmonic oscillator at position x is U = 1 2 k x 2 . {\displaystyle U={\tfrac {1}{2}}kx^{2}.} In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases in proportion to

1767-415: The acting frictional force. While in a simple undriven harmonic oscillator the only force acting on the mass is the restoring force, in a damped harmonic oscillator there is in addition a frictional force which is always in a direction to oppose the motion. In many vibrating systems the frictional force F f can be modeled as being proportional to the velocity v of the object: F f = − cv , where c

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1824-439: The amplitude (for a given F 0 {\displaystyle F_{0}} ) is maximal. This resonance effect only occurs when ζ < 1 / 2 {\displaystyle \zeta <1/{\sqrt {2}}} , i.e. for significantly underdamped systems. For strongly underdamped systems the value of the amplitude can become quite large near the resonant frequency. The transient solutions are

1881-477: The case where ζ ≤ 1 . The amplitude A and phase φ determine the behavior needed to match the initial conditions. In the case ζ < 1 and a unit step input with  x (0) = 0 : F ( t ) m = { ω 0 2 t ≥ 0 0 t < 0 {\displaystyle {\frac {F(t)}{m}}={\begin{cases}\omega _{0}^{2}&t\geq 0\\0&t<0\end{cases}}}

1938-665: The damping ratio ζ critically determines the behavior of the system. A damped harmonic oscillator can be: The Q factor of a damped oscillator is defined as Q = 2 π × energy stored energy lost per cycle . {\displaystyle Q=2\pi \times {\frac {\text{energy stored}}{\text{energy lost per cycle}}}.} Q is related to the damping ratio by Q = 1 2 ζ . {\textstyle Q={\frac {1}{2\zeta }}.} Driven harmonic oscillators are damped oscillators further affected by an externally applied force F ( t ). Newton's second law takes

1995-839: The differential equation gives − ω 2 A e i ( ω τ + φ ) + 2 ζ i ω A e i ( ω τ + φ ) + A e i ( ω τ + φ ) = ( − ω 2 A + 2 ζ i ω A + A ) e i ( ω τ + φ ) = e i ω τ . {\displaystyle -\omega ^{2}Ae^{i(\omega \tau +\varphi )}+2\zeta i\omega Ae^{i(\omega \tau +\varphi )}+Ae^{i(\omega \tau +\varphi )}=(-\omega ^{2}A+2\zeta i\omega A+A)e^{i(\omega \tau +\varphi )}=e^{i\omega \tau }.} Dividing by

2052-413: The exponential term on the left results in − ω 2 A + 2 ζ i ω A + A = e − i φ = cos ⁡ φ − i sin ⁡ φ . {\displaystyle -\omega ^{2}A+2\zeta i\omega A+A=e^{-i\varphi }=\cos \varphi -i\sin \varphi .} Equating

2109-691: The extent the response falls below final value for times following the response maximum. In the case of a sinusoidal driving force: d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x = 1 m F 0 sin ⁡ ( ω t ) , {\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {1}{m}}F_{0}\sin(\omega t),} where F 0 {\displaystyle F_{0}}

2166-414: The form d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x = 0 , {\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x=0,} where The value of

2223-873: The form F ( t ) − k x − c d x d t = m d 2 x d t 2 . {\displaystyle F(t)-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}}=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}.} It is usually rewritten into the form d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x = F ( t ) m . {\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {F(t)}{m}}.} This equation can be solved exactly for any driving force, using

2280-592: The friction coefficient, the system can: The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped . If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator . Mechanical examples include pendulums (with small angles of displacement ), masses connected to springs , and acoustical systems . Other analogous systems include electrical harmonic oscillators such as RLC circuits . The harmonic oscillator model

2337-496: The leading coefficient of the first row is 1; that of the second row is 2; that of the third row is 4, while the last row does not have a leading coefficient. Though coefficients are frequently viewed as constants in elementary algebra, they can also be viewed as variables as the context broadens. For example, the coordinates ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\dotsc ,x_{n})} of

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2394-457: The leading coefficient of the polynomial 4 x 5 + x 3 + 2 x 2 {\displaystyle 4x^{5}+x^{3}+2x^{2}} is 4. This can be generalised to multivariate polynomials with respect to a monomial order , see Gröbner basis § Leading term, coefficient and monomial . In linear algebra , a system of linear equations is frequently represented by its coefficient matrix . For example,

2451-409: The leading coefficients are 2 and a , respectively. In the context of differential equations , these equations can often be written in terms of polynomials in one or more unknown functions and their derivatives. In such cases, the coefficients of the differential equation are the coefficients of this polynomial, and these may be non-constant functions. A coefficient is a constant coefficient when it

2508-466: The motion is described by the function x ( t ) = A sin ⁡ ( ω t + φ ) , {\displaystyle x(t)=A\sin(\omega t+\varphi ),} where ω = k m . {\displaystyle \omega ={\sqrt {\frac {k}{m}}}.} The motion is periodic , repeating itself in a sinusoidal fashion with constant amplitude A . In addition to its amplitude,

2565-426: The motion of a simple harmonic oscillator is characterized by its period T = 2 π / ω {\displaystyle T=2\pi /\omega } , the time for a single oscillation or its frequency f = 1 / T {\displaystyle f=1/T} , the number of cycles per unit time. The position at a given time t also depends on the phase φ , which determines

2622-419: The oscillator, such as the damping or restoring force. A familiar example of parametric oscillation is "pumping" on a playground swing . A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing and squatting, in rhythm with

2679-1181: The real and imaginary parts results in two independent equations A ( 1 − ω 2 ) = cos ⁡ φ , 2 ζ ω A = − sin ⁡ φ . {\displaystyle A(1-\omega ^{2})=\cos \varphi ,\quad 2\zeta \omega A=-\sin \varphi .} Squaring both equations and adding them together gives A 2 ( 1 − ω 2 ) 2 = cos 2 ⁡ φ ( 2 ζ ω A ) 2 = sin 2 ⁡ φ } ⇒ A 2 [ ( 1 − ω 2 ) 2 + ( 2 ζ ω ) 2 ] = 1. {\displaystyle \left.{\begin{aligned}A^{2}(1-\omega ^{2})^{2}&=\cos ^{2}\varphi \\(2\zeta \omega A)^{2}&=\sin ^{2}\varphi \end{aligned}}\right\}\Rightarrow A^{2}[(1-\omega ^{2})^{2}+(2\zeta \omega )^{2}]=1.} Coefficient In mathematics ,

2736-416: The same as the unforced ( F 0 = 0 {\displaystyle F_{0}=0} ) damped harmonic oscillator and represent the systems response to other events that occurred previously. The transient solutions typically die out rapidly enough that they can be ignored. A parametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of

2793-704: The solution is x ( t ) = 1 − e − ζ ω 0 t sin ⁡ ( 1 − ζ 2 ω 0 t + φ ) sin ⁡ ( φ ) , {\displaystyle x(t)=1-e^{-\zeta \omega _{0}t}{\frac {\sin \left({\sqrt {1-\zeta ^{2}}}\omega _{0}t+\varphi \right)}{\sin(\varphi )}},} with phase φ given by cos ⁡ φ = ζ . {\displaystyle \cos \varphi =\zeta .} The time an oscillator needs to adapt to changed external conditions

2850-1089: The solution is of the form q s ( τ ) = A e i ( ω τ + φ ) . {\displaystyle q_{s}(\tau )=Ae^{i(\omega \tau +\varphi )}.} Its derivatives from zeroth to second order are q s = A e i ( ω τ + φ ) , d q s d τ = i ω A e i ( ω τ + φ ) , d 2 q s d τ 2 = − ω 2 A e i ( ω τ + φ ) . {\displaystyle q_{s}=Ae^{i(\omega \tau +\varphi )},\quad {\frac {\mathrm {d} q_{s}}{\mathrm {d} \tau }}=i\omega Ae^{i(\omega \tau +\varphi )},\quad {\frac {\mathrm {d} ^{2}q_{s}}{\mathrm {d} \tau ^{2}}}=-\omega ^{2}Ae^{i(\omega \tau +\varphi )}.} Substituting these quantities into

2907-883: The solutions z ( t ) that satisfy the unforced equation d 2 z d t 2 + 2 ζ ω 0 d z d t + ω 0 2 z = 0 , {\displaystyle {\frac {\mathrm {d} ^{2}z}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} z}{\mathrm {d} t}}+\omega _{0}^{2}z=0,} and which can be expressed as damped sinusoidal oscillations: z ( t ) = A e − ζ ω 0 t sin ⁡ ( 1 − ζ 2 ω 0 t + φ ) , {\displaystyle z(t)=Ae^{-\zeta \omega _{0}t}\sin \left({\sqrt {1-\zeta ^{2}}}\omega _{0}t+\varphi \right),} in

SECTION 50

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2964-414: The starting point on the sine wave. The period and frequency are determined by the size of the mass m and the force constant k , while the amplitude and phase are determined by the starting position and velocity . The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases. The velocity is maximal for zero displacement, while

3021-422: The swing's oscillations. The varying of the parameters drives the system. Examples of parameters that may be varied are its resonance frequency ω {\displaystyle \omega } and damping β {\displaystyle \beta } . Parametric oscillators are used in many applications. The classical varactor parametric oscillator oscillates when the diode's capacitance

3078-409: The system is called a simple harmonic oscillator , and it undergoes simple harmonic motion : sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude). If a frictional force ( damping ) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator . Depending on

3135-551: The system of equations { 2 x + 3 y = 0 5 x − 4 y = 0 , {\displaystyle {\begin{cases}2x+3y=0\\5x-4y=0\end{cases}},} the associated coefficient matrix is ( 2 3 5 − 4 ) . {\displaystyle {\begin{pmatrix}2&3\\5&-4\end{pmatrix}}.} Coefficient matrices are used in algorithms such as Gaussian elimination and Cramer's rule to find solutions to

3192-491: The system. The leading entry (sometimes leading coefficient ) of a row in a matrix is the first nonzero entry in that row. So, for example, in the matrix ( 1 2 0 6 0 2 9 4 0 0 0 4 0 0 0 0 ) , {\displaystyle {\begin{pmatrix}1&2&0&6\\0&2&9&4\\0&0&0&4\\0&0&0&0\end{pmatrix}},}

3249-440: The variables are often denoted by x , y , ..., and the parameters by a , b , c , ..., but this is not always the case. For example, if y is considered a parameter in the above expression, then the coefficient of x would be −3 y , and the constant coefficient (with respect to x ) would be 1.5 + y . When one writes a x 2 + b x + c , {\displaystyle ax^{2}+bx+c,} it

#901098