Misplaced Pages

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87-419: The cassotto enclosure works as a passive frequency filter, attenuating higher frequencies and reinforcing lower ones ( resonance ). Instruments with cassotto have a sound quality distinctly different from instruments without. Usually only some reed blocks are placed in the cassotto enclosure, typically for low reed sets 16' and/or 8'. The remaining treble reed blocks are mounted straight. The firm Öllerer built

174-565: A Bode plot . For the RLC circuit's capacitor voltage, the gain of the transfer function H ( iω ) is Note the similarity between the gain here and the amplitude in Equation ( 3 ). Once again, the gain is maximized at the resonant frequency ω r = ω 0 1 − 2 ζ 2 . {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.} Here,

261-410: A circuit consisting of a resistor with resistance R , an inductor with inductance L , and a capacitor with capacitance C connected in series with current i ( t ) and driven by a voltage source with voltage v in ( t ). The voltage drop around the circuit is Rather than analyzing a candidate solution to this equation like in the mass on a spring example above, this section will analyze

348-420: A steady state solution that is independent of initial conditions and depends only on the driving amplitude F 0 , driving frequency ω , undamped angular frequency ω 0 , and the damping ratio ζ . The transient solution decays in a relatively short amount of time, so to study resonance it is sufficient to consider the steady state solution. It is possible to write the steady-state solution for x ( t ) as

435-416: A derivation of the resonant frequency for a driven, damped harmonic oscillator is shown. An RLC circuit is used to illustrate connections between resonance and a system's transfer function, frequency response, poles, and zeroes. Building off the RLC circuit example, these connections for higher-order linear systems with multiple inputs and outputs are generalized. Consider a damped mass on a spring driven by

522-612: A diatonic instrument with cassotto without register switches and two 8' (M) reed sets. Making a cassotto for diatonic instruments is quite more difficult due to the different reed plate arrangement. For chromatic accordions, a cassotto does not perform equally in every tone. On piano keyboards, the reed plates for black keys are usually located closer to the filling than those for most white keys. On chromatic button keyboards, two reed sets are typically distributed across 3 or 2 reed blocks (the latter solution often requiring particularly slim reed plates). The cassotto effect tends to be stronger

609-477: A function proportional to the driving force with an induced phase change φ , where 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 .} The phase value

696-540: A harmonic oscillator near equilibrium. An example of this is the Lennard-Jones potential , where the potential is given by: U ( r ) = U 0 [ ( r 0 r ) 12 − ( r 0 r ) 6 ] {\displaystyle U(r)=U_{0}\left[\left({\frac {r_{0}}{r}}\right)^{12}-\left({\frac {r_{0}}{r}}\right)^{6}\right]} The equilibrium points of

783-461: A mechanical oscillation. Oscillation, especially rapid oscillation, may be an undesirable phenomenon in process control and control theory (e.g. in sliding mode control ), where the aim is convergence to stable state . In these cases it is called chattering or flapping, as in valve chatter, and route flapping . The simplest mechanical oscillating system is a weight attached to a linear spring subject to only weight and tension . Such

870-848: A natural frequency and a damping ratio, ω 0 = 1 L C , {\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}},} ζ = R 2 C L . {\displaystyle \zeta ={\frac {R}{2}}{\sqrt {\frac {C}{L}}}.} The ratio of the output voltage to the input voltage becomes H ( s ) ≜ V out ( s ) V in ( s ) = ω 0 2 s 2 + 2 ζ ω 0 s + ω 0 2 {\displaystyle H(s)\triangleq {\frac {V_{\text{out}}(s)}{V_{\text{in}}(s)}}={\frac {\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}} H ( s )

957-458: A natural frequency depending upon their structure; this frequency is known as a resonant frequency or resonance frequency . When an oscillating force, an external vibration, is applied at a resonant frequency of a dynamic system, object, or particle, the outside vibration will cause the system to oscillate at a higher amplitude (with more force) than when the same force is applied at other, non-resonant frequencies. The resonant frequencies of

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1044-418: A new restoring force in the opposite sense. If a constant force such as gravity is added to the system, the point of equilibrium is shifted. The time taken for an oscillation to occur is often referred to as the oscillatory period . The systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the spring-mass system, are described mathematically by

1131-549: A quadratic equation. ( 3 k − m ω 2 ) ( k − m ω 2 ) = 0 ω 1 = k m , ω 2 = 3 k m {\displaystyle {\begin{aligned}&\left(3k-m\omega ^{2}\right)\left(k-m\omega ^{2}\right)=0\\&\omega _{1}={\sqrt {\frac {k}{m}}},\;\;\omega _{2}={\sqrt {\frac {3k}{m}}}\end{aligned}}} Depending on

1218-706: A similar solution, but now there is a different equation for every direction. x ( t ) = A x cos ⁡ ( ω t − δ x ) , y ( t ) = A y cos ⁡ ( ω t − δ y ) , ⋮ {\displaystyle {\begin{aligned}x(t)&=A_{x}\cos(\omega t-\delta _{x}),\\y(t)&=A_{y}\cos(\omega t-\delta _{y}),\\&\;\,\vdots \end{aligned}}} With anisotropic oscillators, different directions have different constants of restoring forces. The solution

1305-483: A sinusoidal position function: x ( t ) = A cos ⁡ ( ω t − δ ) {\displaystyle x(t)=A\cos(\omega t-\delta )} where ω is the frequency of the oscillation, A is the amplitude, and δ is the phase shift of the function. These are determined by the initial conditions of the system. Because cosine oscillates between 1 and −1 infinitely, our spring-mass system would oscillate between

1392-475: A sinusoidal, externally applied force. Newton's second law takes the form where m is the mass, x is the displacement of the mass from the equilibrium point, F 0 is the driving amplitude, ω is the driving angular frequency, k is the spring constant, and c is the viscous damping coefficient. This can be rewritten in the form where Many sources also refer to ω 0 as the resonant frequency . However, as shown below, when analyzing oscillations of

1479-455: A system approaches continuity ; examples include a string or the surface of a body of water . Such systems have (in the classical limit ) an infinite number of normal modes and their oscillations occur in the form of waves that can characteristically propagate. The mathematics of oscillation deals with the quantification of the amount that a sequence or function tends to move between extremes. There are several related notions: oscillation of

1566-402: A system can be identified when the response to an external vibration creates an amplitude that is a relative maximum within the system. Small periodic forces that are near a resonant frequency of the system have the ability to produce large amplitude oscillations in the system due to the storage of vibrational energy . Resonance phenomena occur with all types of vibrations or waves : there

1653-411: A system may be approximated on an air table or ice surface. The system is in an equilibrium state when the spring is static. If the system is displaced from the equilibrium, there is a net restoring force on the mass, tending to bring it back to equilibrium. However, in moving the mass back to the equilibrium position, it has acquired momentum which keeps it moving beyond that position, establishing

1740-418: Is ω r = ω 0 , {\displaystyle \omega _{r}=\omega _{0},} and the gain is one at this frequency, so the voltage across the resistor resonates at the circuit's natural frequency and at this frequency the amplitude of the voltage across the resistor equals the input voltage's amplitude. Some systems exhibit antiresonance that can be analyzed in

1827-522: Is mechanical resonance , orbital resonance , acoustic resonance , electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and resonance of quantum wave functions . Resonant systems can be used to generate vibrations of a specific frequency (e.g., musical instruments ), or pick out specific frequencies from a complex vibration containing many frequencies (e.g., filters). The term resonance (from Latin resonantia , 'echo', from resonare , 'resound') originated from

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1914-474: Is also complex, can be written as a gain and phase, H ( i ω ) = G ( ω ) e i Φ ( ω ) . {\displaystyle H(i\omega )=G(\omega )e^{i\Phi (\omega )}.} A sinusoidal input voltage at frequency ω results in an output voltage at the same frequency that has been scaled by G ( ω ) and has a phase shift Φ ( ω ). The gain and phase can be plotted versus frequency on

2001-1749: Is given by resolving the motion into normal modes . The simplest form of coupled oscillators is a 3 spring, 2 mass system, where masses and spring constants are the same. This problem begins with deriving Newton's second law for both masses. { m 1 x ¨ 1 = − ( k 1 + k 2 ) x 1 + k 2 x 2 m 2 x ¨ 2 = k 2 x 1 − ( k 2 + k 3 ) x 2 {\displaystyle {\begin{cases}m_{1}{\ddot {x}}_{1}=-(k_{1}+k_{2})x_{1}+k_{2}x_{2}\\m_{2}{\ddot {x}}_{2}=k_{2}x_{1}-(k_{2}+k_{3})x_{2}\end{cases}}} The equations are then generalized into matrix form. F = M x ¨ = k x , {\displaystyle F=M{\ddot {x}}=kx,} where M = [ m 1 0 0 m 2 ] {\displaystyle M={\begin{bmatrix}m_{1}&0\\0&m_{2}\end{bmatrix}}} , x = [ x 1 x 2 ] {\displaystyle x={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}} , and k = [ k 1 + k 2 − k 2 − k 2 k 2 + k 3 ] {\displaystyle k={\begin{bmatrix}k_{1}+k_{2}&-k_{2}\\-k_{2}&k_{2}+k_{3}\end{bmatrix}}} The values of k and m can be substituted into

2088-652: Is similar to isotropic oscillators, but there is a different frequency in each direction. Varying the frequencies relative to each other can produce interesting results. For example, if the frequency in one direction is twice that of another, a figure eight pattern is produced. If the ratio of frequencies is irrational, the motion is quasiperiodic . This motion is periodic on each axis, but is not periodic with respect to r, and will never repeat. All real-world oscillator systems are thermodynamically irreversible . This means there are dissipative processes such as friction or electrical resistance which continually convert some of

2175-459: Is small, the resonant frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple and distinct resonant frequencies. A familiar example is a playground swing , which acts as a pendulum . Pushing a person in a swing in time with the natural interval of the swing (its resonant frequency) makes the swing go higher and higher (maximum amplitude), while attempts to push

2262-427: Is some interaction between the types of sound materials and the cassotto. Frequencies between 500 Hz and 1000 Hz will be strengthened in general, but not evenly. Some frequencies will stand out more than others, over 1000 Hz everything will be rather subdued. The sound effects of the cassottos can be described as: Resonance Resonance is a phenomenon that occurs when an object or system

2349-602: Is subjected to an external force or vibration that matches its natural frequency . When this happens, the object or system absorbs energy from the external force and starts vibrating with a larger amplitude . Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is often desirable in certain applications, such as musical instruments or radio receivers. However, resonance can also be detrimental, leading to excessive vibrations or even structural failure in some cases. All systems, including molecular systems and particles, tend to vibrate at

2436-432: Is that reed plates are better accessible for tuning than with a traditional cassotto construction. The effect of a cassotto is less pronounced with various instruments. The impact is therefore also not commonly evaluated. An individual assessment is required for each instrument. Oftentimes the cassotto sounds will be described as warm and chubby. Due to the filter effects, there are few strong, pronounced overtones . There

2523-409: Is the resonant frequency for this system. Again, the resonant frequency does not equal the undamped angular frequency ω 0 of the oscillator. They are proportional, and if the damping ratio goes to zero they are the same, but for non-zero damping they are not the same frequency. As shown in the figure, resonance may also occur at other frequencies near the resonant frequency, including ω 0 , but

2610-468: Is the transfer function between the input voltage and the output voltage. This transfer function has two poles –roots of the polynomial in the transfer function's denominator–at and no zeros–roots of the polynomial in the transfer function's numerator. Moreover, for ζ ≤ 1 , the magnitude of these poles is the natural frequency ω 0 and that for ζ < 1/ 2 {\displaystyle {\sqrt {2}}} , our condition for resonance in

2697-469: Is the transient solution to the differential equation. The transient solution can be found by using the initial conditions of the system. Some systems can be excited by energy transfer from the environment. This transfer typically occurs where systems are embedded in some fluid flow. For example, the phenomenon of flutter in aerodynamics occurs when an arbitrarily small displacement of an aircraft wing (from its equilibrium) results in an increase in

Cassotto - Misplaced Pages Continue

2784-805: Is then found, and used to be the effective potential constant: γ eff = d 2 U d r 2 | r = r 0 = U 0 [ 12 ( 13 ) r 0 12 r − 14 − 6 ( 7 ) r 0 6 r − 8 ] = 114 U 0 r 2 {\displaystyle {\begin{aligned}\gamma _{\text{eff}}&=\left.{\frac {d^{2}U}{dr^{2}}}\right|_{r=r_{0}}=U_{0}\left[12(13)r_{0}^{12}r^{-14}-6(7)r_{0}^{6}r^{-8}\right]\\[1ex]&={\frac {114U_{0}}{r^{2}}}\end{aligned}}} The system will undergo oscillations near

2871-412: Is usually taken to be between −180° and 0 so it represents a phase lag for both positive and negative values of the arctan argument. Resonance occurs when, at certain driving frequencies, the steady-state amplitude of x ( t ) is large compared to its amplitude at other driving frequencies. For the mass on a spring, resonance corresponds physically to the mass's oscillations having large displacements from

2958-578: The Harmona ( Klingenthal ) Cassotta, Supra & Supita models. There is a special construction in the Weltmeister Cassotta (374 + 414), also known as "Klingenthaler Spezialcassotto" or "Füllungscassotto". In those models the reed blocks are arranged in one plane like with ordinary accordions. The effect of cassotto is produced by a small "canopy" ("Vordach") under the treble hood forming the cassotto hollow. An advantage of this construction

3045-419: The angle of attack of the wing on the air flow and a consequential increase in lift coefficient , leading to a still greater displacement. At sufficiently large displacements, the stiffness of the wing dominates to provide the restoring force that enables an oscillation. Resonance occurs in a damped driven oscillator when ω = ω 0 , that is, when the driving frequency is equal to the natural frequency of

3132-405: The simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion . In the spring-mass system, oscillations occur because, at the static equilibrium displacement, the mass has kinetic energy which is converted into potential energy stored in the spring at the extremes of its path. The spring-mass system illustrates some common features of oscillation, namely

3219-923: The Laplace domain the voltage across the inductor is V out ( s ) = s L I ( s ) , {\displaystyle V_{\text{out}}(s)=sLI(s),} V out ( s ) = s 2 s 2 + R L s + 1 L C V in ( s ) , {\displaystyle V_{\text{out}}(s)={\frac {s^{2}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s),} V out ( s ) = s 2 s 2 + 2 ζ ω 0 s + ω 0 2 V in ( s ) , {\displaystyle V_{\text{out}}(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}V_{\text{in}}(s),} using

3306-458: The amplitude of the output's steady-state oscillations to the input's oscillations is called the gain, and the gain can be a function of the frequency of the sinusoidal external input. Peaks in the gain at certain frequencies correspond to resonances, where the amplitude of the measured output's oscillations are disproportionately large. Since many linear and nonlinear systems that oscillate are modeled as harmonic oscillators near their equilibria,

3393-404: The beating of the human heart (for circulation), business cycles in economics , predator–prey population cycles in ecology , geothermal geysers in geology , vibration of strings in guitar and other string instruments , periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy . The term vibration is precisely used to describe

3480-399: The capacitor combined in series. Equation ( 4 ) showed that the sum of the voltages across the three circuit elements sums to the input voltage, so measuring the output voltage as the sum of the inductor and capacitor voltages combined is the same as v in minus the voltage drop across the resistor. The previous example showed that at the natural frequency of the system, the amplitude of

3567-432: The circuit is divided among the three circuit elements, and each element has different dynamics. The capacitor's voltage grows slowly by integrating the current over time and is therefore more sensitive to lower frequencies, whereas the inductor's voltage grows when the current changes rapidly and is therefore more sensitive to higher frequencies. While the circuit as a whole has a natural frequency where it tends to oscillate,

Cassotto - Misplaced Pages Continue

3654-491: The coupled oscillators where energy alternates between two forms of oscillation. Well-known is the Wilberforce pendulum , where the oscillation alternates between the elongation of a vertical spring and the rotation of an object at the end of that spring. Coupled oscillators are a common description of two related, but different phenomena. One case is where both oscillations affect each other mutually, which usually leads to

3741-469: The current and input voltage, respectively, and s is a complex frequency parameter in the Laplace domain. Rearranging terms, I ( s ) = s s 2 L + R s + 1 C V in ( s ) . {\displaystyle I(s)={\frac {s}{s^{2}L+Rs+{\frac {1}{C}}}}V_{\text{in}}(s).} An RLC circuit in series presents several options for where to measure an output voltage. Suppose

3828-737: The different dynamics of each circuit element make each element resonate at a slightly different frequency. Suppose that the output voltage of interest is the voltage across the resistor. In the Laplace domain the voltage across the resistor is V out ( s ) = R I ( s ) , {\displaystyle V_{\text{out}}(s)=RI(s),} V out ( s ) = R s L ( s 2 + R L s + 1 L C ) V in ( s ) , {\displaystyle V_{\text{out}}(s)={\frac {Rs}{L\left(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}\right)}}V_{\text{in}}(s),} and using

3915-427: The displacement x ( t ), the resonant frequency is close to but not the same as ω 0 . In general the resonant frequency is close to but not necessarily the same as the natural frequency. The RLC circuit example in the next section gives examples of different resonant frequencies for the same system. The general solution of Equation ( 2 ) is the sum of a transient solution that depends on initial conditions and

4002-401: The energy stored in the oscillator into heat in the environment. This is called damping. Thus, oscillations tend to decay with time unless there is some net source of energy into the system. The simplest description of this decay process can be illustrated by oscillation decay of the harmonic oscillator. Damped oscillators are created when a resistive force is introduced, which is dependent on

4089-419: The equilibrium point. The force that creates these oscillations is derived from the effective potential constant above: F = − γ eff ( r − r 0 ) = m eff r ¨ {\displaystyle F=-\gamma _{\text{eff}}(r-r_{0})=m_{\text{eff}}{\ddot {r}}} This differential equation can be re-written in

4176-766: The existence of an equilibrium and the presence of a restoring force which grows stronger the further the system deviates from equilibrium. In the case of the spring-mass system, Hooke's law states that the restoring force of a spring is: F = − k x {\displaystyle F=-kx} By using Newton's second law , the differential equation can be derived: x ¨ = − k m x = − ω 2 x , {\displaystyle {\ddot {x}}=-{\frac {k}{m}}x=-\omega ^{2}x,} where ω = k / m {\textstyle \omega ={\sqrt {k/m}}} The solution to this differential equation produces

4263-474: The field of acoustics, particularly the sympathetic resonance observed in musical instruments, e.g., when one string starts to vibrate and produce sound after a different one is struck. Resonance occurs when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a simple pendulum). However, there are some losses from cycle to cycle, called damping . When damping

4350-802: The first derivative of the position, or in this case velocity. The differential equation created by Newton's second law adds in this resistive force with an arbitrary constant b . This example assumes a linear dependence on velocity. m x ¨ + b x ˙ + k x = 0 {\displaystyle m{\ddot {x}}+b{\dot {x}}+kx=0} This equation can be rewritten as before: x ¨ + 2 β x ˙ + ω 0 2 x = 0 , {\displaystyle {\ddot {x}}+2\beta {\dot {x}}+\omega _{0}^{2}x=0,} where 2 β = b m {\textstyle 2\beta ={\frac {b}{m}}} . This produces

4437-1022: The form of a simple harmonic oscillator: r ¨ + γ eff m eff ( r − r 0 ) = 0 {\displaystyle {\ddot {r}}+{\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}(r-r_{0})=0} Thus, the frequency of small oscillations is: ω 0 = γ eff m eff = 114 U 0 r 2 m eff {\displaystyle \omega _{0}={\sqrt {\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}}={\sqrt {\frac {114U_{0}}{r^{2}m_{\text{eff}}}}}} Or, in general form ω 0 = d 2 U d r 2 | r = r 0 {\displaystyle \omega _{0}={\sqrt {\left.{\frac {d^{2}U}{dr^{2}}}\right\vert _{r=r_{0}}}}} This approximation can be better understood by looking at

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4524-455: The frequency response of this circuit. Taking the Laplace transform of Equation ( 4 ), s L I ( s ) + R I ( s ) + 1 s C I ( s ) = V in ( s ) , {\displaystyle sLI(s)+RI(s)+{\frac {1}{sC}}I(s)=V_{\text{in}}(s),} where I ( s ) and V in ( s ) are the Laplace transform of

4611-556: The function are then found: d U d r = 0 = U 0 [ − 12 r 0 12 r − 13 + 6 r 0 6 r − 7 ] ⇒ r ≈ r 0 {\displaystyle {\begin{aligned}{\frac {dU}{dr}}&=0=U_{0}\left[-12r_{0}^{12}r^{-13}+6r_{0}^{6}r^{-7}\right]\\\Rightarrow r&\approx r_{0}\end{aligned}}} The second derivative

4698-595: The further the reed plates are from the filling. Cassotto registers are only available on few (and usually higher quality) accordion models. Especially famous are the Beltuna , Pigini , Bugari and Öllerer models. In traditional German accordion making, the cassotto barely plays a role. One can find only a few cassotto registers in German accordions, but they can be found in the Hohner Morino and Gola models, and

4785-445: The gain in Equation ( 6 ) using the capacitor voltage as the output, this gain has a factor of ω in the numerator and will therefore have a different resonant frequency that maximizes the gain. That frequency is ω r = ω 0 1 − 2 ζ 2 , {\displaystyle \omega _{r}={\frac {\omega _{0}}{\sqrt {1-2\zeta ^{2}}}},} So for

4872-419: The gain, notice that the gain goes to zero at ω = ω 0 , which complements our analysis of the resistor's voltage. This is called antiresonance , which has the opposite effect of resonance. Rather than result in outputs that are disproportionately large at this frequency, this circuit with this choice of output has no response at all at this frequency. The frequency that is filtered out corresponds exactly to

4959-573: The general solution. ( k − M ω 2 ) a = 0 [ 2 k − m ω 2 − k − k 2 k − m ω 2 ] = 0 {\displaystyle {\begin{aligned}\left(k-M\omega ^{2}\right)a&=0\\{\begin{bmatrix}2k-m\omega ^{2}&-k\\-k&2k-m\omega ^{2}\end{bmatrix}}&=0\end{aligned}}} The determinant of this matrix yields

5046-608: The general solution: x ( t ) = e − β t ( C 1 e ω 1 t + C 2 e − ω 1 t ) , {\displaystyle x(t)=e^{-\beta t}\left(C_{1}e^{\omega _{1}t}+C_{2}e^{-\omega _{1}t}\right),} where ω 1 = β 2 − ω 0 2 {\textstyle \omega _{1}={\sqrt {\beta ^{2}-\omega _{0}^{2}}}} . The exponential term outside of

5133-456: The harmonic oscillator example, the poles are closer to the imaginary axis than to the real axis. Evaluating H ( s ) along the imaginary axis s = iω , the transfer function describes the frequency response of this circuit. Equivalently, the frequency response can be analyzed by taking the Fourier transform of Equation ( 4 ) instead of the Laplace transform. The transfer function, which

5220-927: The mass on a spring example, the resonant frequency remains ω r = ω 0 1 − 2 ζ 2 , {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}},} but the definitions of ω 0 and ζ change based on the physics of the system. For a pendulum of length ℓ and small displacement angle θ , Equation ( 1 ) becomes m ℓ d 2 θ d t 2 = F 0 sin ⁡ ( ω t ) − m g θ − c ℓ d θ d t {\displaystyle m\ell {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=F_{0}\sin(\omega t)-mg\theta -c\ell {\frac {\mathrm {d} \theta }{\mathrm {d} t}}} and therefore Consider

5307-628: The matrices. m 1 = m 2 = m , k 1 = k 2 = k 3 = k , M = [ m 0 0 m ] , k = [ 2 k − k − k 2 k ] {\displaystyle {\begin{aligned}m_{1}=m_{2}=m,\;\;k_{1}=k_{2}=k_{3}=k,\\M={\begin{bmatrix}m&0\\0&m\end{bmatrix}},\;\;k={\begin{bmatrix}2k&-k\\-k&2k\end{bmatrix}}\end{aligned}}} These matrices can now be plugged into

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5394-548: The maximum response is at the resonant frequency. Also, ω r is only real and non-zero if ζ < 1 / 2 {\textstyle \zeta <1/{\sqrt {2}}} , so this system can only resonate when the harmonic oscillator is significantly underdamped. For systems with a very small damping ratio and a driving frequency near the resonant frequency, the steady state oscillations can become very large. For other driven, damped harmonic oscillators whose equations of motion do not look exactly like

5481-434: The object. Light and other short wavelength electromagnetic radiation is produced by resonance on an atomic scale , such as electrons in atoms. Other examples of resonance include: Resonance manifests itself in many linear and nonlinear systems as oscillations around an equilibrium point. When the system is driven by a sinusoidal external input, a measured output of the system may oscillate in response. The ratio of

5568-465: The occurrence of a single, entrained oscillation state, where both oscillate with a compromise frequency . Another case is where one external oscillation affects an internal oscillation, but is not affected by this. In this case the regions of synchronization, known as Arnold Tongues , can lead to highly complex phenomena as for instance chaotic dynamics. In physics, a system with a set of conservative forces and an equilibrium point can be approximated as

5655-586: The oscillation is said to be driven . The simplest example of this is a spring-mass system with a sinusoidal driving force. x ¨ + 2 β x ˙ + ω 0 2 x = f ( t ) , {\displaystyle {\ddot {x}}+2\beta {\dot {x}}+\omega _{0}^{2}x=f(t),} where f ( t ) = f 0 cos ⁡ ( ω t + δ ) . {\displaystyle f(t)=f_{0}\cos(\omega t+\delta ).} This gives

5742-445: The others. This leads to a coupling of the oscillations of the individual degrees of freedom. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronise. This phenomenon was first observed by Christiaan Huygens in 1665. The apparent motions of the compound oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description

5829-622: The output voltage of interest is the voltage drop across the capacitor. As shown above, in the Laplace domain this voltage is V out ( s ) = 1 s C I ( s ) {\displaystyle V_{\text{out}}(s)={\frac {1}{sC}}I(s)} or V out = 1 L C ( s 2 + R L s + 1 L C ) V in ( s ) . {\displaystyle V_{\text{out}}={\frac {1}{LC(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}})}}V_{\text{in}}(s).} Define for this circuit

5916-404: The parenthesis is the decay function and β is the damping coefficient. There are 3 categories of damped oscillators: under-damped, where β < ω 0 ; over-damped, where β > ω 0 ; and critically damped, where β = ω 0 . In addition, an oscillating system may be subject to some external force, as when an AC circuit is connected to an outside power source. In this case

6003-500: The positive and negative amplitude forever without friction. In two or three dimensions, harmonic oscillators behave similarly to one dimension. The simplest example of this is an isotropic oscillator, where the restoring force is proportional to the displacement from equilibrium with the same restorative constant in all directions. F → = − k r → {\displaystyle {\vec {F}}=-k{\vec {r}}} This produces

6090-423: The potential curve of the system. By thinking of the potential curve as a hill, in which, if one placed a ball anywhere on the curve, the ball would roll down with the slope of the potential curve. This is true due to the relationship between potential energy and force. d U d t = − F ( r ) {\displaystyle {\frac {dU}{dt}}=-F(r)} By thinking of

6177-432: The potential in this way, one will see that at any local minimum there is a "well" in which the ball would roll back and forth (oscillate) between r min {\displaystyle r_{\text{min}}} and r max {\displaystyle r_{\text{max}}} . This approximation is also useful for thinking of Kepler orbits . As the number of degrees of freedom becomes arbitrarily large,

6264-413: The resonance corresponds physically to having a relatively large amplitude for the steady state oscillations of the voltage across the capacitor compared to its amplitude at other driving frequencies. The resonant frequency need not always take the form given in the examples above. For the RLC circuit, suppose instead that the output voltage of interest is the voltage across the inductor. As shown above, in

6351-400: The same RLC circuit but with the voltage across the inductor as the output, the resonant frequency is now larger than the natural frequency, though it still tends towards the natural frequency as the damping ratio goes to zero. That the same circuit can have different resonant frequencies for different choices of output is not contradictory. As shown in Equation ( 4 ), the voltage drop across

6438-470: The same definitions for ω 0 and ζ as in the previous example. The transfer function between V in ( s ) and this new V out ( s ) across the inductor is H ( s ) = s 2 s 2 + 2 ζ ω 0 s + ω 0 2 . {\displaystyle H(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.} This transfer function has

6525-455: The same natural frequency and damping ratio as in the capacitor example the transfer function is H ( s ) = 2 ζ ω 0 s s 2 + 2 ζ ω 0 s + ω 0 2 . {\displaystyle H(s)={\frac {2\zeta \omega _{0}s}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.} This transfer function also has

6612-451: The same natural frequency and damping ratios as the previous examples, the transfer function is H ( s ) = s 2 + ω 0 2 s 2 + 2 ζ ω 0 s + ω 0 2 . {\displaystyle H(s)={\frac {s^{2}+\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.} This transfer has

6699-669: The same poles as the previous RLC circuit examples, but it only has one zero in the numerator at s = 0. For this transfer function, its gain is G ( ω ) = 2 ζ ω 0 ω ( 2 ω ω 0 ζ ) 2 + ( ω 0 2 − ω 2 ) 2 . {\displaystyle G(\omega )={\frac {2\zeta \omega _{0}\omega }{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.} The resonant frequency that maximizes this gain

6786-671: The same poles as the previous examples but has zeroes at Evaluating the transfer function along the imaginary axis, its gain is G ( ω ) = ω 0 2 − ω 2 ( 2 ω ω 0 ζ ) 2 + ( ω 0 2 − ω 2 ) 2 . {\displaystyle G(\omega )={\frac {\omega _{0}^{2}-\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.} Rather than look for resonance, i.e., peaks of

6873-623: The same poles as the transfer function in the previous example, but it also has two zeroes in the numerator at s = 0 . Evaluating H ( s ) along the imaginary axis, its gain becomes G ( ω ) = ω 2 ( 2 ω ω 0 ζ ) 2 + ( ω 0 2 − ω 2 ) 2 . {\displaystyle G(\omega )={\frac {\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.} Compared to

6960-410: The same way as resonance. For antiresonance, the amplitude of the response of the system at certain frequencies is disproportionately small rather than being disproportionately large. In the RLC circuit example, this phenomenon can be observed by analyzing both the inductor and the capacitor combined. Suppose that the output voltage of interest in the RLC circuit is the voltage across the inductor and

7047-1065: The solution: x ( t ) = A cos ⁡ ( ω t − δ ) + A t r cos ⁡ ( ω 1 t − δ t r ) , {\displaystyle x(t)=A\cos(\omega t-\delta )+A_{tr}\cos(\omega _{1}t-\delta _{tr}),} where A = f 0 2 ( ω 0 2 − ω 2 ) 2 + 4 β 2 ω 2 {\displaystyle A={\sqrt {\frac {f_{0}^{2}}{(\omega _{0}^{2}-\omega ^{2})^{2}+4\beta ^{2}\omega ^{2}}}}} and δ = tan − 1 ⁡ ( 2 β ω ω 0 2 − ω 2 ) {\displaystyle \delta =\tan ^{-1}\left({\frac {2\beta \omega }{\omega _{0}^{2}-\omega ^{2}}}\right)} The second term of x ( t )

7134-413: The spring's equilibrium position at certain driving frequencies. Looking at the amplitude of x ( t ) as a function of the driving frequency ω , the amplitude is maximal at the driving frequency ω r = ω 0 1 − 2 ζ 2 . {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.} ω r

7221-410: The starting point of the masses, this system has 2 possible frequencies (or a combination of the two). If the masses are started with their displacements in the same direction, the frequency is that of a single mass system, because the middle spring is never extended. If the two masses are started in opposite directions, the second, faster frequency is the frequency of the system. More special cases are

7308-453: The swing at a faster or slower tempo produce smaller arcs. This is because the energy the swing absorbs is maximized when the pushes match the swing's natural oscillations. Resonance occurs widely in nature, and is exploited in many devices. It is the mechanism by which virtually all sinusoidal waves and vibrations are generated. For example, when hard objects like metal , glass , or wood are struck, there are brief resonant vibrations in

7395-439: The system. When this occurs, the denominator of the amplitude is minimized, which maximizes the amplitude of the oscillations. The harmonic oscillator and the systems it models have a single degree of freedom . More complicated systems have more degrees of freedom, for example, two masses and three springs (each mass being attached to fixed points and to each other). In such cases, the behavior of each variable influences that of

7482-835: The voltage drop across the resistor equals the amplitude of v in , and therefore the voltage across the inductor and capacitor combined has zero amplitude. We can show this with the transfer function. The sum of the inductor and capacitor voltages is V out ( s ) = ( s L + 1 s C ) I ( s ) , {\displaystyle V_{\text{out}}(s)=(sL+{\frac {1}{sC}})I(s),} V out ( s ) = s 2 + 1 L C s 2 + R L s + 1 L C V in ( s ) . {\displaystyle V_{\text{out}}(s)={\frac {s^{2}+{\frac {1}{LC}}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s).} Using

7569-640: The zeroes of the transfer function, which were shown in Equation ( 7 ) and were on the imaginary axis. Oscillation Oscillation is the repetitive or periodic variation, typically in time , of some measure about a central value (often a point of equilibrium ) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current . Oscillations can be used in physics to approximate complex interactions, such as those between atoms. Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example

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