The Platt-LePage Aircraft Company was a manufacturer of aircraft for the armed forces of the United States of America. Based in Eddystone, Pennsylvania , the company produced the first helicopter to be officially acquired by the United States Army Air Forces .
59-598: Wynn Laurence LePage , a British Aeronautical Engineer living in Pennsylvania, who co-founded the Platt-LePage Aircraft Company in partnership with Haviland Hull Platt, the company's intended purpose being the manufacture of helicopters . LePage, impressed by the performance of the German Focke-Wulf Fa 61 , acquired the manufacturing rights to the aircraft. Two early helicopter prototypes developed by
118-557: 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,
177-410: 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
236-420: 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
295-416: 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
354-477: 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
413-848: 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 )
472-458: 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
531-673: A number of other concepts for helicopters, including the PL-11, an improved civilian version of the XR-1A; the PL-12, a four-passenger variant of the PL-11; and the PL-14, a twin-rotor helicopter based on a Grumman Widgeon fuselage. In addition, a design for a tiltrotor airliner was developed by the company. However, the cancellation of the company's Army contracts had removed Platt-LePage's primary source of income, and without sufficient funds to continue operating,
590-479: A number of other concepts for helicopters, including the PL-11, an improved civilian version of the XR-1A; the PL-12, a four-passenger variant of the PL-11; and the PL-14, a twin-rotor helicopter based on a Grumman Widgeon fuselage. In addition, a design for a tiltrotor airliner was developed by the company. However, the cancellation of the company's Army contracts had removed Platt-LePage's primary source of income, and without sufficient funds to continue operating,
649-475: 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
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#1732872425993708-402: 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
767-418: 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
826-522: 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
885-474: 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
944-459: 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
1003-602: 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
1062-409: 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
1121-468: 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
1180-412: 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
1239-654: The Army with its first helicopter. Having employed designer Grover Loening as a consultant and helicopter enthusiast Frank Piasecki employed as a junior engineer, Platt-Lepage set to work developing the helicopter, using a similar rotor arrangement to that of the Fa 61. The aircraft, designated XR-1, flew three months behind schedule in 1941. The XR-1 suffered from significant teething troubles, including control difficulties, vibration, and resonance issues, and financial difficulties at Platt-LePage caused significant delays in resolving
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#17328724259931298-520: The Fa 61. The aircraft, designated XR-1, flew three months behind schedule in 1941. The XR-1 suffered from significant teething troubles, including control difficulties, vibration, and resonance issues, and financial difficulties at Platt-LePage caused significant delays in resolving the aircraft's problems. Despite this, the USAAF still believed the XR-1 would prove successful, however the improved XR-1A, flying for
1357-918: 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
1416-455: The Platt-LePage aircraft, having successfully passed its flight trials and already entering operational service with the Army. Therefore, in early 1945 the Army elected to cancel its contracts with Platt-LePage. The repaired XR-1A was purchased by Helicopter Air Transport , which intended to operate it for commercial purposes as part of a fleet of 40 helicopters. The company had also developed
1475-523: The aircraft's problems. Despite this, the USAAF still believed the XR-1 would prove successful, however the improved XR-1A, flying for the first time in 1943, proved little better than its predecessor, and was damaged in an accident in early 1944. With Piasecki having left the company, the XR-1's flight testing dragged on through 1944. The Army Air Forces, spurred by Congressional accusations of favoritism towards Vought-Sikorsky , ordered seven YR-1A service-test helicopters. Despite this, Sikorsky's R-4
1534-458: 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,
1593-399: 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
1652-432: 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,
1711-538: The company closed its doors on January 13, 1947. McDonnell Aircraft , which had acquired part of the company in 1942, and a larger share in 1944, purchased the rights to the remainder of the company's intellectual property, including the design for the PL-9, a twin-engine helicopter that McDonnell would develop into the XHJD Whirlaway . Resonance Resonance is a phenomenon that occurs when an object or system
1770-409: The company closed its doors on January 13, 1947. McDonnell Aircraft , which had acquired part of the company in 1942, and a larger share in 1944, purchased the rights to the remainder of the company's intellectual property, including the design for the PL-9, a twin-engine helicopter that McDonnell would develop into the XHJD Whirlaway . Wynn Laurence LePage The Platt-LePage Aircraft Company
1829-492: The company failed to meet with any success; however in 1940, under the terms of the Dorsey-Logan Act, Platt-Lepage was declared the winner of a competition to supply the Army with its first helicopter. Having employed designer Grover Loening as a consultant and helicopter enthusiast Frank Piasecki employed as a junior engineer, Platt-Lepage set to work developing the helicopter, using a similar rotor arrangement to that of
Platt-LePage Aircraft Company - Misplaced Pages Continue
1888-411: The company's intended purpose being the manufacture of helicopters . LePage, impressed by the performance of the German Focke-Wulf Fa 61 , acquired the manufacturing rights to the aircraft. Two early helicopter prototypes developed by the company failed to meet with any success; however in 1940, under the terms of the Dorsey-Logan Act, Platt-Lepage was declared the winner of a competition to supply
1947-469: 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
2006-737: 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
2065-427: 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
2124-474: 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
2183-410: The first time in 1943, proved little better than its predecessor, and was damaged in an accident in early 1944. With Piasecki having left the company, the XR-1's flight testing dragged on through 1944. The Army Air Forces, spurred by Congressional accusations of favoritism towards Vought-Sikorsky , ordered seven YR-1A service-test helicopters. Despite this, Sikorsky's R-4 was proving far superior to
2242-450: 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
2301-445: 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
2360-419: 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
2419-451: 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
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2478-927: 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
2537-548: 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
2596-434: 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
2655-622: 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
2714-413: 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
2773-400: 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
2832-470: 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
2891-455: 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
2950-451: 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
3009-669: 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
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#17328724259933068-671: 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
3127-623: 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
3186-410: 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
3245-413: 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
3304-453: 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
3363-835: 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
3422-578: Was a manufacturer of aircraft for the armed forces of the United States of America. Based in Eddystone, Pennsylvania , the company produced the first helicopter to be officially acquired by the United States Army Air Forces . Wynn Laurence LePage , a British Aeronautical Engineer living in Pennsylvania, who co-founded the Platt-LePage Aircraft Company in partnership with Haviland Hull Platt,
3481-432: Was proving far superior to the Platt-LePage aircraft, having successfully passed its flight trials and already entering operational service with the Army. Therefore, in early 1945 the Army elected to cancel its contracts with Platt-LePage. The repaired XR-1A was purchased by Helicopter Air Transport , which intended to operate it for commercial purposes as part of a fleet of 40 helicopters. The company had also developed
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