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# How To Calculate Resonant Frequency Of An Object

When a vibrating object sets up air vibrations in an enclosed space, the sound vibrations in the air are very weak at some frequencies, and strong at other frequencies. Its units are therefore degrees (or radians) per second. A mechanical analogy to self-resonance is the natural frequency of vibration for an object, given the unavoidable presence of both elasticity and mass in any object. Chirality. When these molecules vibrate naturally they produce a kind of noise that can be annoying some times. Calculate the current in, and the voltage across elements in resonant circuits. Stay safe and healthy. In the physics biz we call the resonant frequency an eignevalue, and the way the object vibrates an eigenmode. By resonance, this will excite all surrounding objects with the same natural frequency. Natural frequency is the frequency at which an object vibrates once it has been set in motion. Adding mass to the system would decrease its resonant frequency. This resonant frequency was found to be extremely close to the blade-passing frequency of the fan system (24 Hz. Using Resonant Frequencies to Create an Unspillable Cup - Duration: 9:16. Ease of Excitation at Resonance. Calculate the velocity ratio of a screwjack of pitch 0. Acoustics: Resonance. The method of measuring the resonance frequencies of an object is comparatively new in the area of nondestructive testing techniques. The sound wave of the resonant frequency moves the molecules of the liquid around the glass making, in essence, small waves. 63x10 13 Hz. correlation between the shift of the resonant frequencies and the hardness of the inspected steel rings. Define resonance. and Antia, H. Vibration Isolation Introduction The purpose of vibration isolation is to control unwanted vibration so that its adverse effects are kept within acceptable limits. 3 The waveguide has a large dimension of a  12. Real circuit elements have losses, and when we analyse the LC network we use a realistic model of the ideal lumped elements in which losses are taken into account by means of "virtual" serial resistances R L and R C. Calculate the natural frequency of a simple harmonic oscillator using the formula: f = √(k / m) ÷ 2π. This frequency is the objects resonant frequency When an object is exposed to from PHY 128 at Thomas Edison State College. Whether it is a guitar sting, a Chladni plate, or the air column enclosed within a trombone, the vibrating medium vibrates in such a way that a standing wave pattern results. For the equivalent RLC circuit, the total impedance is Ztot = Zant + Zs and the resonant frequency is given by the condition LA CS ω 2 = 1. Imaginary FRFs and Mode Shapes. These vibrations can be excited in the presence of a stimulating force – like the ringing of a bell – or even in the presence of a frequency that sets it off – like glass shattering when just the right high-pitched note is sung. Step 3: Measure the voltage over the LC circuit at resonance frequency (f. Definition says something like this: The natural frequency is the frequency at which an object will remain vibrating after hitting it. The rate of the wobbling or precession is the resonance or Larmor frequency. Calculation Preview. RESONANCE: “The property whereby any vibratory system responds with maximum amplitude to an applied force having the frequency equal to its own. This is used to calculate the results of passage through a sequence of resonances for several hypothetical systems. The harmonic ranges will exhibit the most resonance for any given instrument. In a circuit containing both resistance and capacitance, the phase angle will be between 0 and 90 degrees. It is easy to get an object to vibrate at its resonant frequencies, hard to get it to vibrate at other frequencies. Lengthening the radiator will lower the resonant frequency, of course, but sharpening the apex angle will lower both the resonant frequency and the radiation resistance. Infinite array object, specified as a infiniteArray object. 2) You want the resonant frequency of an LC circuit to be 1,000 Hertz. My answer would be to mention that resonance involves a transfer of energy from the driving force to the oscillating object. The data identified a torsional resonant frequency at approximately 23 Hz. An oscilloscope can also be used in place of the voltmeter, but you will have to locate the voltage in relation to the greatest amplitude. frequency, the 7. Use caution to match frequencies of the tuning fork and what you actually hear. Compare your experimental value to the true value. In fact, the number of resonant frequencies is equal to the number of degrees of freedom. There are two big issues involved with destroying stuff using sound, or gentle taps, or whatever you’re using to drive the motion. Natural frequency (fundamental) - Musical instruments and other objects are set into vibration (given energy) when someone or something disturbs the object. Nearby objects will change the resonant frequency of the secondary coil, so you want to be sure you and your equipment are far enough away to not affect the measurements. resonant: filter-type [list] In left inlet: The word resonant sets the filter type of the filtergraph~ object to resonant mode (resonant bandpass filter). The letter "v" stands for the wave velocity, whereas "λ" represents the distance of the wavelength. The important factors associated with this oscillatory motion are the amplitude and frequency of the motion. Electrical wave forms. Trying a lower frequency. Let's take an example, a tuning fork. It is then possible to calculate the resonant frequency given capacitance and inductance with the following equation: f. In designing physical systems it is very important to identify the system's natural frequencies of vibration and provide sufficient damping. As you move above or below that frequency their efficiency rolls off, producing standing waves. The frequency response of the filter is based on three parameters: cf (center frequency) gain, and Q (resonance). ) Such an object will vibrate strongly when it is subjected to vibrations or regular impulses at a frequency equal to or very close to its natural. In Figure 2 you can see how the amplitude of a forced oscillation increases when the frequency of an external force nears the natural frequency of the oscillator. Resonance Condition The interaction energy of a magnetic moment, µ. Thus the resonant motion of the body at the resonant frequency of one of the modes will be dominated by that single mode. Basically, the left top corner and the right side move up at the same time that the remaining area (middle to bottom left corner) moves down, and vice versa, and the salt settles in between. The spectra of benzamide, a conjugated amide (Figure 19), and N-methyl acetamide (Figure 21) clearly identify the Amide I and II bands. Sometimes, the effects of resonance can be powerful. You can alter the length of the string by sliding the stand. one object vibrating or oscillating at the natural frequency of another object forces the other object to vibrate at a frequency higher than its natural frequency. For example, f = 10 MHz. Shorten the antenna. Measure and record mean and standard deviation of the ROI 3. The displacement with time takes the form of a sinusoidial wave. Omnicalculator. @article{osti_22522255, title = {FREQUENCY SHIFTS OF RESONANT MODES OF THE SUN DUE TO NEAR-SURFACE CONVECTIVE SCATTERING}, author = {Bhattacharya, J. A parallel resonant circuit stores the circuit energy in the magnetic field of the inductor and the electric field of the capacitor. It is the best frequency to wavelength and wavelength to frequency calculator. By using the formula of f = velocity of sound/wavelength, you calculated the fundamental frequency is 500Hz, and the next two frequencies are odd multiples of that frequency. • Relate plane waves to spherical waves. To calculate the equivalent centre-mass corresponding to a uniformly distributed mass of m (kg) per metre length we integrate along the span, as before. It is equivalent to the mode 8 message. Each object has a natural frequency at which it vibrates, called the resonant frequency The physical properties of each object determine this resonant frequency The amplitude is quite low, so we don’t usually notice the oscillation Resonant Frequency Dulku –Physics 20 –Unit 4 –Topic J If a forced frequency matches the natural. Adding mass to the system would decrease its resonant frequency. The frequency dependence of the 38 mm PC sensor shows that the Q increases with frequency until the coil becomes self-resonant. The resonant frequency of both enclosures The port length for ported enclosures The reference efficiency in both % and decibels at one watt If any of the fields return 'NAN' or 'infinity', you did not fill one of the fields required to make the calculations. The system has a natural angular frequency of oscillation!0 = 1rad=s. Magnetic resonance, absorption or emission of electromagnetic radiation by electrons or atomic nuclei in response to the application of certain magnetic fields. Resonance is an object’s tendency to vibrate or oscillate at a certain frequency that is basic to its nature. Determine the resonance frequency and the corresponding input current at 24 V. In case of the CL-OCFD, this VNA is able to determine its fundamental resonant frequency at the center, as well as the center load reactance required for resonance on 3. Sound Resonance and Natural Resonant Frequency All objects have a natural frequency, or a group of frequencies, at which they will vibrate. Resonance on Strings A violin sounds a note of F sharp, with a pitch of 370 Hz. Chapter 12 Section 1 Sound Waves Objectives • Explain how sound waves are produced. Acoustics: Resonance. A method of the present invention for stabilizing the resonant frequency of a resonant scanner is that thermal energy of the same amount as the heat generated in an optical scanning mode is applied to the scanner section in a non-scanning mode to insure that the variation of the temperature of the resonant light deflector will lie within a predetermined range, whereby the resonant frequency of. Lengthening the radiator will lower the resonant frequency, of course, but sharpening the apex angle will lower both the resonant frequency and the radiation resistance. (This frequency depends on the size, shape, and composition of the object. This phenomenon is known as resonance. 015 m when a 1. How to Calculate the Resonant Frequencies in a Room Calculate multiples of the first mode until you have reached the tenth mode. Point it at the object. In Figure 2 you can see how the amplitude of a forced oscillation increases when the frequency of an external force nears the natural frequency of the oscillator. Write your answer. Objects which are free to vibrate will have one or more natural frequency at which they vibrate, If an object is being forced to vibrate at its natural frequency, resonance will occur and you will observe large amplitude vibrations. This would explain the reason why some singers…. Vibration Isolation Introduction The purpose of vibration isolation is to control unwanted vibration so that its adverse effects are kept within acceptable limits. You may wonder what does the natural frequency of an object have to do with the resonance in the object?. In the resonance problem posed on the previous page, I asked you to look at a male vocal tract and calculate resonant frequencies. frequency of the sound you're emitting any frequency where the amplitude rises suddenly, would indicate a resonance frequency. Because the FRF response is "normalized" to the input , the peaks in the resulting FRF function are resonant frequencies of the test object. 7, 494, and 983. For example, to create the Bode plot for the measured frequency response of the system and draw a red cursor at. The Action Lab Recommended for you. Using a stop watch determine the length of time a swing needs to complete say 20 cycles. Variable capacitors are more difficult to come by than inductors, so it. (i) With reference to resonance and resonant frequency, comment on the shape of the graph. The notch shows the resonance frequency of the strip. Resonance is when objects with the same natural frequency as the vibrating source also begin to vibrate. This resonant frequency was found to be extremely close to the blade-passing frequency of the fan system (24 Hz. Regardless of the weight of the person swinging on it, the natural frequency will remain the same. Calculate the velocity ratio of a screwjack of pitch 0. The resonant frequency is purely determined by the capacitor having exactly the opposite reactance of the inductor at a particular frequency and the two reactances cancel leaving the series tuned circuit having only resistance at resonance. This formula states that the wave velocity equals the distance of the wavelength multiplied by the resonance frequency. If you ran the tests in the preceding section, you will have seen the system resonate. stant" k are positive. A can of any shape will have resonant frequencies corresponding to various possible modes of oscillations of the electric and magnetic fields. The modes of vibration (associated with resonance) in extended objects like strings and air columns have characteristic patterns called standing waves. Resonance is witnessed in objects that is in equilibrium with acting forces and could keep vibrating for a long time under perfect conditions. Hence, an air column of length L has particular resonance frequencies and will be in resonance with the corresponding driving frequencies. In this chapter, Fano resonances in simple structures with high permittivity such as spheres or core-shell particles are analyzed by Mie theory. Mechanical resonance is the tendency of a mechanical system to respond at a greater amplitude when the frequency of its oscillations matches the system's natural vibration frequency than it does at other frequencies. Objects which are free to vibrate will have one or more natural frequency at which they vibrate, If an object is being forced to vibrate at its natural frequency, resonance will occur and you will observe large amplitude vibrations. Resonance is the name given to a phenomenon when an object shows maximum response when it is subjected to a certain frequency known as 'resonant frequency' or 'natural frequency' of that object. The period of the oscillations is the time it takes an object to complete one oscillation. Consider the particle undergoing simple harmonic motion in Flash 3. As an example, consider a diving board. Two other important characteristics of the oscillation system are period T and linear frequency f. Identify knowns. For each natural frequency, the body takes a certain shape called mode shape. Relate intensity, decibel level, and perceived loudness. Correct trap placement and tuning are important. Divide 20 cycles by the time and you have the swings frequency in cycles per second or Hertz (Hz). This includes nuclear stress tests and nuclear bone scans. Important to note that closed-resonators are able to achieve the same resonant frequency, but at one-half the length. The frequency at which this occurs is called the resonant frequency, and resonance is said to occur. An oscilloscope can also be used in place of the voltmeter, but you will have to locate the voltage in relation to the greatest amplitude. A theory on the effect of temperature on the measurements. In a wave medium, resonance (such as on a string) is standing wave. In order to achieve the goal of usability, you will want to tune the antenna for equal SWR readings at each end of the band. A continuum system has infinite degrees of freedom and so infinite resonant frequencies. It then experiences simple harmonic motion. Use the formula v = λf to find the resonance frequency of a single continuous wave. The natural frequency of that [object] is known as its ‘resonating’ or ‘resonant’ frequency, and the phenomenon is known as ‘Resonance’. At that wavelength, one end of the tube is at a maximum of the wave while the other end is at a minimum. For example, if the longest room dimension was 11. which is that the resonant response of an elastic object depends on several parameters intrinsic to the object, such as the object’s shape, density, elastic constants, and crystallographic orientation. The lowest frequency (n = 1) is called the fundamental frequency and higher frequencies are called overtones. This parameter is known as the speaker\\'s resonant frequency. (This frequency depends on the size, shape, and composition of the object. The measure of natural frequency depends on the composition of the object, its size, structure, weight and shape. Resonance Condition The interaction energy of a magnetic moment, µ. Resonator for speech? pharynx, oral cavity, nasal cavity. An object resonates when a vibration matches the object’s natural frequency. Halogens A: on carbonyl frequencies. Plot the amplitude (of the object's vibration) vs. What do Pa and dB have to do with the natural frequency of an object? Can you add an example of an object whose resonant frequencies you would like to know? $\endgroup$ - Chris Mueller Mar 6 '14 at 13:42 How do you calculate vortex shedding frequency? 2. The fundamental frequency, or f0, is the first harmonic, or H1. Degrees of freedom in the object basically this relates to the supports to the object; Medium in which the object is in like air, water, etc. Some objects tend to vibrate at a single frequency and produce a pure tone. Electrical wave forms. Note as well that while we example mechanical vibrations in this section a simple change of notation (and corresponding change in what the. 4(c) and (d). Use the length reduction procedure for half-wave resonance in Table 6-2 to calculate the resonant frequencies of the two dipoles in Fig. The resonant frequency $$f_0$$ of the RLC circuit is the frequency at which the amplitude of the current is a maximum and the circuit would oscillate if not driven by a voltage source. In our experiment today, we will learn how to calculate the natural frequency of a spring-mass system. And that vibration does not have to be a problem in itself. Even if a natural frequency resides between zero and full speed, it is passed quickly during starting. Then, input light from the waveguide at the resonance frequency undergoes resonant coupling, and is transmitted to the other side with a Lorentzian transmission spectrum. MRs include MRIs and MRAs. Calculate the Q -factor of resonant circuits. The easiest way to find the resonant frequencies is to place the object next to a speaker and also place a microphone attached to an oscilloscope next to the object. The resonant frequency of the resonator depends on the resonator surroundings, relative permittivity and the sample dimensions. For example, if one produces a frequency that is just the right frequency of a wine glass, it shatters. The resonant frequency of an object (or system) is the frequency at which the system will vibrate if it is excited by a single pulse. However, with lower values of resistance the peak shape of the resonance will change but the centre point. I have often to do with experimental data (with a certain length and given sampling frequency) where I obtain trajetories of moving objects in 2d. Frequencies of wood resonance - posted in Physics and Mathematics: Im a recording musician, and as such I have to deal with resonances within my home studio. A Tesla Coil is an air-cored resonant transformer. Open pipes and closed pipes vibrate at a natural resonant frequency. For example, f = 10 MHz. In the resonance problem posed on the previous page, I asked you to look at a male vocal tract and calculate resonant frequencies. Divide 20 cycles by the time and you have the swings frequency in cycles per second or Hertz (Hz). Derived straight from Natural Frequency, is the suspension frequency found in cars. Mass and Tension. Stay safe and healthy. As the base of any geophysical methods is the contrast between the physical properties such as the features, objects, and layers and the surroundings. A resonant sensor is formed by connecting a parallel inductor and capacitor with one of TI's inductance to digital converters. The important factors associated with this oscillatory motion are the amplitude and frequency of the motion. For each natural frequency, the body takes a certain shape called mode shape. Removed references to dubious sources on the Acknowledgements sheet. Also, ﬁnd the resonance frequency and plot the steady state solution when the system is forced at that frequency. The coils can be loosely coupled but demand a high Q factor if energy transfer is to be maintained over several centimeters. Although objects can vibrate at any frequency, each one has a particular frequency at which it vibrates much more powerfully. However, with lower values of resistance the peak shape of the resonance will change but the centre point. A series resonance network consisting of a resistor of 30Ω, a capacitor of 2uF and an inductor of 20mH is connected across a sinusoidal supply voltage which has a constant output of 9 volts at all frequencies. If the object feels vibrational energy at its resonant frequencies, then the motion of the object amplifies the vibrational waves – sometimes to the point of damage. – The others will just move a small amount. Abstract:Feto-infant mortality is increasing worldwide. Every object tends to vibrate at one particular frequency called the natural frequency. “Resonance” is a “driven harmonic oscillation“, where the driving force pushes and pulls at, or near, the “resonant frequency” of whatever it is that doing the resonating. It is common knowledge that the swing moves forward and backwards when pushed. The ailerons, elevators, vertical stabilizers (probably metallic ribs), metallic wing ribs, metallic landing gear, etc. How Science Works Extension: An electronic stroboscope can be used to illuminate the vibrating string. Its chief aim is to understand the hidden subsurface hydrogeological setting correctly and effectively. The period of the oscillations is the time it takes an object to complete one oscillation. The study of natural frequency and resonance and its effect on natural vegetation and trees, agriculture, and man-made constructions is an important and practical application of the fluid dynamics discipline. and possibly a resonant frequency of 333000 Hz as our local space time resonance. The most noteworthy example of resonance occurred in 1940, when resonant vibrations destroyed the Tacoma Narrows Bridge in Washington. To determine the measured resonant frequency, plot the measured magnitude and phase against the plant input frequency. Basically, the left top corner and the right side move up at the same time that the remaining area (middle to bottom left corner) moves down, and vice versa, and the salt settles in between. Solution 332 m/s. Vibrating String Frequencies. The correct lengths for the various models will be displayed in the chart. The rate of the wobbling or precession is the resonance or Larmor frequency. using the calculator please contact us. The frequency that this free vibration naturally wants to occurs at is known as the natural frequency. resonant frequency - the natural frequency at which an object vibrates. The natural frequency, or frequencies, of any physical system or object are the frequencies at which it will vibrate if physically disturbed. The natural frequency in this case is determined by material properties (ie Young's modulus) and the geometry and dimensions of the object (ie a wine glass). Effect of damping on resonance graph. This frequency is known as a structural resonant frequency. Part B shows the transmission response of a strip structure with aspect ratio 1:333 with the magnetic ﬁeld applied along the length as well as the width of the strip, showing different resonance frequencies. For a sound wave, the frequency corresponds to the perception of the pitch of the sound. Choose the velocity of the wave. Resonance and Impedance Antennas can be classified as either resonant or non-resonant, depending on their design. Background When skiing, any type of bump or variation in the surface of. Find the spring constant, the mass. The more damping a system has, the broader response it has to varying driving frequencies. Plot the amplitude (of the object's vibration) vs. The frequency at which this occurs is called the resonant frequency, and resonance is said to occur. Re: relation between "resonant frequency" and "resonant length" alexbugo, for microstrip resonator, it's resonant length is a little smaller than calculated wave-length, because of end-effect. When an object of mass m is attached to the end of the spring , it stretches by a distance y. The less damping a system has, the higher the amplitude of the forced oscillations near. If the forcing frequency is close to the natural frequency of the system, and the system is lightly damped, huge vibration amplitudes may occur. The circuit is said to be resonant when the frequency of the applied voltage is adjusted to produce maximum current while the magnitude of the voltage is held constant. Calculate the wavelengths of sounds at the extremes of the audible range, 20 and 20,000 Hz, in 30. Timothy (Spears) This download is for Registered Users Only. fundamental frequency: the principal component of a sound, which has the greatest wavelength, hence the lowest tone in a sound; sounds are composed of a fundamental tone and overtones or higher tones. In order to generate such magnetic resonance, the team of physicists from UC Riverside and UC Santa Barbara worked with 0. We are most interested in the lowest 3 resonant frequencies. Frequency analysis calculates the natural frequencies and the associated mode shapes. The analogy using static deflection is only intended to help the analyst visualize how to adjust the resonant frequency of a single part or a whole system. Free vibration occurs when an object is given an initial displacement and then allowed to "ring down" with no external force acting to keep it in motion. At these frequencies the vibration amplitude is theoretically. Then enter the numbers, click calculate and your answer is 658. Zooming in, and moving the graph around, I could see a particularly strong one was at 178 Hz. At each value of f, the object eventually reaches a constant amplitude A. Most transducers have more than one natural resonance point. To our knowledge, this theory has never been used to study the full spectrum of modes in piezoelectric disks. 66x10 13 Hz Carbon monoxide 6. Adding mass to the system would decrease its resonant frequency. The period of the oscillations is the time it takes an object to complete one oscillation. Resonance Resonance occurs when the frequency of the vibrating force coincides with the natural frequency of a vibrat-ing system The frequency remains constant but the amplitude – the force felt by the driver – increases greatly. Point it at the object. 25 wavelength EDZ, extended double zepp, with a feedpoint impedance of 189-j1000 ohms. This has a special name, it as known as antenna resonance. Antenna object operating at the specified reference frequency, returned as an antenna object. How do we know when it is at resonance? Does it reflect or absorb more energy of excitation? If we use a spectrum analysis, would we see a spike or a dip at the resonant frequency of the object? It could be a crystal, an iron girder building (ala Tesla) or a bridge with an army marching in step across it. With a speed of sound of 343 m/s, the expression above gives a resonant frequency of 90 Hz. These vibrations can be excited in the presence of a stimulating force – like the ringing of a bell – or even in the presence of a frequency that sets it off – like glass shattering when just the right high-pitched note is sung. Using a Rogowski coil [3] followed by integrator and comparators is an alternative way to control a high-frequency resonant converter SR. What are the frequencies of. I know how to calculate resonant length given the frequency and wavelength, but seeing as i dont have either of those i am stuck. If the object feels vibrational energy at its resonant frequencies, then the motion of the object amplifies the vibrational waves – sometimes to the point of damage. For example, if you got 50 Hertz for the first mode of the resonant calculation, then the second mode would be 100 Hertz. Spectrum analysis is the process of determining the frequency domain representation of a time domain signal and most commonly employs the Fourier transform. In this cool sound resonance experiment, learn about frequency and pitch, and use a tuning fork to calculate sound velocity in air. First click on what you are solving and the units you will need. Point it at the object. Calculate the natural frequency of a simple harmonic oscillator using the formula: f = √(k / m) ÷ 2π. This apparatus allows for exploring both damped oscillations and forced oscillations. (In the diagram at right is the natural frequency of the oscillations, , in the above analysis). It will "pick out" its resonance frequency from a complex excitation, such as an impulse or a wideband noise excitation. one object vibrating or oscillating at the natural frequency of another object forces the other object to vibrate at a frequency higher than its natural frequency. A periodic force driving a harmonic oscillator at its natural frequency produces resonance. and Antia, H. The data identified a torsional resonant frequency at approximately 23 Hz. → Natural frequencies are those that will cause objects to vibrate in standing wave. Resonant frequencies are well-known for man-made objects, but are more difficult to calculate for natural structures, such as Rainbow Bridge. $\begingroup$ I believe the resonant frequency has a lot to do with the speed of sound in the resonant medium (5. f o = resonant frequency in Hz. When a high-frequency magnetic field is required, the resonant technique will reduce the coil impedance and allow for high current to drive the coil with a low-voltage function generator amplifier. Suspension Frequency. Calculate the resonance frequency ω 0 of (a) a 3 kg mass attached to a spring of force constant 240 N/m and (b) a simple pendulum 1. When these molecules vibrate naturally they produce a kind of noise that can be annoying some times. The ailerons, elevators, vertical stabilizers (probably metallic ribs), metallic wing ribs, metallic landing gear, etc. The incident was especially shocking at the time as the structure was designed to withstand winds of up to 120 miles (193 kilometers) per hour and collapsed in a mere 40-mile (64-kilometer) wind. The nonlinear parameter, , is then the slope of the data as dictated by the equation in figure 3 and illustrated in figure 22. So wavelength lambda = 2L. Now, when you release a long string, this long string will vibrate at its natural frequency. Resonance results from an object’s shape, material, tension, and other physical properties. When a vibrating object sets up air vibrations in an enclosed space, the sound vibrations in the air are very weak at some frequencies, and strong at other frequencies. An acoustically resonant object usually has more than one resonance frequency, especially at harmonics of the strongest resonance. Events such as earthquakes and winds can do destruction on objects with the same natural frequency as the event itself. Formula to calculate amplitude of a wave is given by: where, A = Amplitude of the wave [decibels] D = Distance traveled by the wave [meters] F = Wave frequency [hertz] Enter the distance traveled by the wave and wave frequency in our below online amplitude calculator and click calculate button to find the answer. In this video tutorial the instructor demonstrates resonant frequency. When these molecules vibrate naturally they produce a kind of noise that can be annoying some times. This would explain the reason why some singers…. The classic example is of the opera singer breaking a glass with her voice when she hits the right note. Because the FRF response is "normalized" to the input , the peaks in the resulting FRF function are resonant frequencies of the test object. Resonance on Strings A violin sounds a note of F sharp, with a pitch of 370 Hz. Under what kind of condition will resonance happen Each objects tends to have a from ME 2103 at National University of Singapore. For example: microstrip ring resonator lenght is almost the same as calculated from electrical length. The notch shows the resonance frequency of the strip. Angular frequency (ω), also known as radial or circular frequency, measures angular displacement per unit time. Resonant frequencies are well-known for man-made objects, but are more difficult to calculate for natural structures, such as Rainbow Bridge. Natural frequency (fundamental) - Musical instruments and other objects are set into vibration (given energy) when someone or something disturbs the object. This is because the water took up some of the volume that was originally filled with vibrating air. The formula to calculate the resonant frequency is f0 = 1/(2pisqrt(LC))resonance practice problemsProblem: - A RLC tank circuit is composed of components having values as R = 0. I would like to know how I go about calculating the resonance frequency for an object, given its material properties and dimensions. Note Frequency Calculator and Player. 09 mm with flat metallic plates on both ends. First, there’s the frequency generator with a 15-watt linear amplifier which can be used with the contact or electrode method of delivering the frequencies. A theory on the effect of temperature on the measurements. standing wave - a wave produced by the interference of two traveling, identical waves moving in opposite directions. of the circular frequency of a solution, so we will write k/m = � n2 with � n > 0, and call � n the natural circular frequency of the system. A parallel resonant circuit in electronics is used as the basis of frequency-selective networks. These two approximations give us our final equation for the spring resonant frequency, To use this formula we need to know the material's G and all of the spring geometry. It is easy to get an object to vibrate at its resonant frequencies, hard to get it to vibrate at other frequencies. Strength of Materials. If PSG acceptable, use average of 4 ROI σ’s to estimate noise. Thus the resonant motion of the body at the resonant frequency of one of the modes will be dominated by that single mode. It will vibrate easily at those frequencies, and less so at other frequencies. To calculate electrical wavelength, change the velocity factor. A mass of 0:1kg is attached to a spring and the system is then immersed in a liquid. Resonant frequency is the oscillation of a system at its natural or unforced resonance. To calculate the resonant frequency: f=1/(2pi x sqrt(LxC)). harray — Array object operating at specified reference frequency and specified elements array object. A separate article details how to modify the miniVNA PRO to render it truly field proof. A series resonance network consisting of a resistor of 30Ω, a capacitor of 2uF and an inductor of 20mH is connected across a sinusoidal supply voltage which has a constant output of 9 volts at all frequencies. Formula to calculate electric force of charged object. This post is about resonant frequencies of a tube, in the context of speech and the neutral vocal configuration. As the wall is at right angles to the floor, transmission of bass frequencies (and footfalls) are attenuated via this mounting method. (emphasis added). The key risk factors include smoking, increased. 09 mm with flat metallic plates on both ends. 707 of signal at resonant frequency. The notch shows the resonance frequency of the strip. The analogy using static deflection is only intended to help the analyst visualize how to adjust the resonant frequency of a single part or a whole system. If the object is initially displaced 20 cm downward from its equilibrium position and given a velocity of 10 cm/sec upward find the displacement at any time $$t$$. ﬁle 00603 Question 7. The term "acoustic resonance" is sometimes used to narrow mechanical resonance to the frequency range of human hearing, but since acoustics is defined in general terms concerning vibrational waves in matter. At one point in all collisions a maximum amount of energy is abslorbed. Hi everyone, Im learning about the basics of resonance, and I am just wondering why it is not observed more often in everyday objects. Now, all solid objects have a sonic resonance frequency. The frequencies at which the sound vibrations are strong are called resonant frequencies of the system, and these are easily recognized by listening to the sound intensity. Under what kind of condition will resonance happen Each objects tends to have a from ME 2103 at National University of Singapore. How to Calculate Resonant Frequency of an Object? An object exposed to its resonant frequency can vibrate in sympathy with the sound. Mechanical resonance can be observed in any vibrating body such as tuning fork. If you slowly change the frequency you will observe the point of maximum vibration which is the bar resonant frequency. Doing this with strings under tension, we ﬁnd that the string has a variety of modes of vibration with diﬀerent frequencies. Stillbirth is defined as uterofetal death at 20 weeks of gestation or greater. Start with a low frequency and gradually increase the frequency until the tallest wire begins to resonate. The energy differences are small, typically corresponding to radio frequencies (e. A body disturbed from its rest position tends to vibrate at certain frequencies called natural, or resonant frequencies. 2 they turn out to be and. The method of measuring the resonance frequencies of an object is comparatively new in the area of nondestructive testing techniques. Sound Resonance and Natural Resonant Frequency All objects have a natural frequency, or a group of frequencies, at which they will vibrate. When the amount of air in a bottle decreases the resonant frequency goes up. For any two waves to reinforce each other, they must be in phase, which is to say that the peaks and troughs of the Mean Distance vs. Ways to determine resonant frequency: There are few different ways to determine the resonant frequency of your vehicle, all of which requires the use of SPL or RTA meters. A condition of resonance will be experienced in a tank circuit when the reactance of the capacitor and inductor are equal to each other. You should end up with something like this (again, recalling that ω ~ ω 0): This looks useful. Vibration Isolation Introduction The purpose of vibration isolation is to control unwanted vibration so that its adverse effects are kept within acceptable limits. The Override Frequency is one such example, as they serve as a post-campaign. (resonant) frequency. Resonant frequencies are well-known for human-made objects, but are more difficult to calculate for natural structures, such as Rainbow Bridge. The less damping a system has, the higher the amplitude of the forced oscillations near. (Assume that the frequency values are accurate to two significant figures. The resonant frequency is f o. Resonator for speech? pharynx, oral cavity, nasal cavity. Resonance is a forced vibration of energy into molecules of an object that makes those molecules vibrate at their resonant frequency. If the object feels vibrational energy at its resonant frequencies, then the motion of the object amplifies the vibrational waves – sometimes to the point of damage. Every [elastic] object, material, etc has a certain speed of oscillation that will occur naturally when there are zero outside forces or damping applied. frequency of the sound you're emitting any frequency where the amplitude rises suddenly, would indicate a resonance frequency. As a default, our calculator uses a value of 299 792 458 m/s - the speed of light propagating in the air. It is likely that specific waveforms have specific functions. (ii) On the same axes, draw a graph to show the variation with f of A when the frictional force acting on the object is increased. Such an object will vibrate strongly when it is subjected to vibrations or regular impulses at a frequency equal to or very close to its natural frequency. The resonance frequency of an object is a function of its sound velocity - a material characteristic - as well as of its geometry. It is easy to get an object to vibrate at its resonant frequencies, but hard to get it to vibrate at other frequencies. This has a special name, it as known as antenna resonance. spherically symmetric resonant object; it is easiest to evalu-ate the surface integral over a very large sphere centered at the origin. Determine the resonance frequency and the corresponding input current at 24 V. Resonance occurs when driving frequency equals natural frequency f 0 driving frequency amplitude of oscillation Resonance. In this method, waves produced by a short stress pulse and interferring in the specimen are analysed referring to their frequency spectrum. RESONANCE The Silent Killer of Rotating Machinery In every industrial plant there will be failures of rotating machinery due in part to resonance. Parker et al, (2009) indicated that object are only confirmed when. The Coil32 program calculates the losses in the. The equation used to calculate the electrical resonant frequency, f, in an LC circuit is: f=1/(2π√LC) where L is the inductance and C is the capacitance. Or you may also build this Grid dip meter for identifying and setting the resonance frequency. The swaying is especially clear in the following video, likely. In fact, when the two "C" arms are closer, field lines are more dense in such a way that the quality factor is increased. Calculate PSG from ACR accreditation. The fundamental frequency, or f0, is the first harmonic, or H1. That is to say they work best at a single frequency. The Development Quantum Computing Essay The story of computers started with the abacus invented by the Babylonians around 500 B. To our knowledge, this theory has never been used to study the full spectrum of modes in piezoelectric disks. Natural frequencies are different from forced frequencies, which occur by applying force to an object at a specific rate. I have often to do with experimental data (with a certain length and given sampling frequency) where I obtain trajetories of moving objects in 2d. The coils can be loosely coupled but demand a high Q factor if energy transfer is to be maintained over several centimeters. To calculate the equivalent centre-mass corresponding to a uniformly distributed mass of m (kg) per metre length we integrate along the span, as before. The frequency response of the filter is based on three parameters: cf (center frequency) gain, and Q (resonance). Resonance is the phenomenon that occurs when a physical system is periodically disturbed at the same period of one of its natural frequencies. The resonant frequency of both enclosures The port length for ported enclosures The reference efficiency in both % and decibels at one watt If any of the fields return 'NAN' or 'infinity', you did not fill one of the fields required to make the calculations. If you pluck your guitar string, you don't have to tell it what pitch to produce - it knows! That is, its pitch is its resonant frequency, which is determined by the length, mass, and tension of the string. 66x10 13 Hz Carbon monoxide 6. - Then calculate the length of the tube needed. Stay safe and healthy. Ways to determine resonant frequency: There are few different ways to determine the resonant frequency of your vehicle, all of which requires the use of SPL or RTA meters. The harmonic ranges will exhibit the most resonance for any given instrument. Some of these demonstrated that there could be improvements made to ASME PTC 19. With a tiny push on the swing each time it comes back to you, you can continue to build up the amplitude of swing. Acoustics: Resonance. This is referred to as a Single Degree of Freedom (SDOF) system, because it has only one natural frequency/mode of vibration. The other modes, with diﬀerent resonant frequencies will not respond very much. $\begingroup$ The resonance frequency of an object would be the one it makes when you hit it. I have a subwoofer placed under my desk which is constructed of laminated pressboard about 3/4 thick. Then enter the numbers, click calculate and your answer is 658. There is a formula to calculate resonant frequencies of a tube that is closed at one end and open at the other end (like the vocal tract). The lock-in amplifier drives the series resonance circuit with an AC signal at resonance frequency. The ailerons, elevators, vertical stabilizers (probably metallic ribs), metallic wing ribs, metallic landing gear, etc. Frequency analysis calculates the natural frequencies and the associated mode shapes. Another method is to use JavaTC the software design program, which will give a good estimate of the resonant frequency and where you need to tap the primary coil. It will then start decrease when resonant frequency is exceeded Asked. The frequency that this free vibration naturally wants to occurs at is known as the natural frequency. If it is warmer, the speed will be faster. The easiest way to find the resonant frequencies is to place the object next to a speaker and also place a microphone attached to an oscilloscope next to the object. – The others will just move a small amount. For folks with only old tapes and rulers. Core practical 16: Determine the value of an unknown mass using the resonant frequencies of known masses on a spring Objectives. The Override Frequency is one such example, as they serve as a post-campaign. What is the term for the particular frequency at which an object resonates? natural or resonant frequency. I tapped the cookie sheet and watched the spectrum display to see which frequencies the cookie sheet liked to resonate at. object related to the antenna causes a frequency shift that is determined by: F = dφ/dt · 1/2π= 2f/cdr/dt = 2vf/c (2) Where v is the velocity (mutual speed) in meters per second, f is the signal frequency, Hz, and c is again the speed of light. An MR (magnetic resonance) exam uses magnetic and radio waves to take pictures of thin slices of your body from different angles without radiation. This tells you how many oscillations happen per second, which depends on the properties of the spring and the mass of the ball attached to it. If the kid’s excitation load has a frequency which becomes close to the swing’s natural frequency, the swing goes up and up and the oscillations increase. Acoustic resonant frequency equation. However it is possible to calculate the resonant frequency of diatomic (two atom) molecules and some results are shown below. Use the length reduction procedure for half-wave resonance in Table 6-2 to calculate the resonant frequencies of the two dipoles in Fig. To understand resonance, we first have to understand what it is Natural Frequency. Measure and record mean and standard deviation of the ROI 3. Resistor “R1” limits impedance near the resonant frequency. It decreases. Since 2π radians = 360 °, 1 radian ≈ 57. Events such as earthquakes and winds can do destruction on objects with the same natural frequency as the event itself. ﬁle 00603 Question 7. Resonance D. Champlain College Physics 203-NYC 2. An oscilloscope can also be used in place of the voltmeter, but you will have to locate the voltage in relation to the greatest amplitude. The less damping a system has, the higher the amplitude of the forced oscillations near. The natural frequency is the frequency of this oscillation, measured in hertz (Hz). The resonance frequency of an object is a function of its sound velocity - a material characteristic - as well as of its geometry. 10 Because they travel at the same speed in a given medium, low-frequency sounds must have a greater wavelength than high-frequency sounds. (In the diagram at right is the natural frequency of the oscillations, , in the above analysis). It decreases. It disappears. Conceptual Questions. Go back to the graph of power dissipated as a function of driving frequency. The "Inverted Vee" calculator has a slider that starts up with an angle of 45 degrees and can be adjusted down to 20 degrees continuously re-calculating as. (i) With reference to resonance and resonant frequency, comment on the shape of the graph. and possibly a resonant frequency of 333000 Hz as our local space time resonance. Resonance is witnessed in objects that is in equilibrium with acting forces and could keep vibrating for a long time under perfect conditions. [What frequency at which it will change shape]. The relative proximity of the resonant frequencies of these two precious metals allows a low-energy atomic transmutation from 107 Ag to Au to occur near room temperature. Linear frequency is the number of the oscillations per one second. All objects have resonant frequencies at which they naturally oscillate. The natural frequency depends on two things: the stiffness and mass of the system. RESONANCE: "The property whereby any vibratory system responds with maximum amplitude to an applied force having the frequency equal to its own. wavelength = 0. ), indicating the possibility that the torsional resonance excited by the blade-passing per- turbations was a contributing cause of the shaft failure. It is then possible to calculate the resonant frequency given capacitance and inductance with the following equation: f. This is a picture of the left side of the Manchester Mark 1 computer. Resonance occurs when driving frequency equals natural frequency f 0 driving frequency amplitude of oscillation Resonance. In this case, the self-resonance frequency is moved from its actual value for "free" coil. Watch the following presentation before next class and print a copy for your notebook. Example: i = infiniteArray;ia = design(1,1e9,monopole); Designs an infinite array with a monopole antenna element operating at a frequency of 1 GHz. The natural frequency is the frequency of this oscillation, measured in hertz (Hz). Support the mass. English Activity: Setup í Equipment Glass Resonance Tube (Uniform Inside Diameter, With Scale Markings) Rubber Tube Reservoir Stand Low Frequency Generator (or Tuning Fork) Audio Measurement Setup (EA-200, graphic scientific calculator, data. When this body is in contact with an external source of vibration which is vibrating at a frequency which matches the natural frequency of the object, the object starts vibrating with higher amplitude which is called resonance. With a tiny push on the swing each time it comes back to you, you can continue to build up the amplitude of swing. Give an example of resonance in the reception of electromagnetic waves. Compute the peak gain of the higher-frequency resonance in the transfer function s y s = ( 1 s 2 + 0. Not all objects resonate. The important factors associated with this oscillatory motion are the amplitude and frequency of the motion. Halogens A: on carbonyl frequencies. Place the foam base of the resonance apparatus on a flat surface. If you slowly change the frequency you will observe the point of maximum vibration which is the bar resonant frequency. Making use of Equations and , the mean power absorption when the driving frequency is close to the resonant frequency is (120) Thus, the maximum power absorption occurs at the resonance (i. Omnicalculator. In this case, the self-resonance frequency is moved from its actual value for "free" coil. f o = resonant frequency in Hz. Then enter the numbers, click calculate and your answer is 658. In this method, waves produced by a short stress pulse and interferring in the specimen are analysed referring to their frequency spectrum. Since 2π radians = 360 °, 1 radian ≈ 57. resonant frequencies of known masses on a spring Practical activities have been safety checked but not trialled by CLEAPSS. Delivery of the frequencies from the early 1930's had always been done through a large bulb, or tube, filled with helium or argon gas. 5 m in length. Every physical system or object has a frequency that's naturally inherent to it: its resonant frequency. Free vibration occurs when an object is given an initial displacement and then allowed to “ring down” with no external force acting to keep it in motion. The frequency where the inductance peaks is called the self-resonant frequency (SRF), and the sensor must be operated below it to look like an Figure 5. the acceleration is always directed towards the equilibrium position. A theory on the effect of temperature on the measurements. Making use of Equations and , the mean power absorption when the driving frequency is close to the resonant frequency is (120) Thus, the maximum power absorption occurs at the resonance (i. Mass and Tension. Stay safe and healthy. Hint: research the word "resonance" in engineering textbooks, and you are sure to read about some dramatic examples of resonance in action. Note this value for use in the next two steps. The natural frequency in this case is determined by material properties (ie Young's modulus) and the geometry and dimensions of the object (ie a wine glass). If it is warmer, the speed will be faster. In this specific problem, should I be. So if you are designing a system to avoid resonance, you need to consider all of them. Adding mass to the system would decrease its resonant frequency. The resonance of an antenna is directly related to its wavelength. This phenomenon is called « resonance ». Forced Vibraon & Resonance Forced Vibraons : Occurs when a vibrang object forces another object to vibrate. A guitar string resonates at 1260 Hz and 1575 Hz with no resonance frequencies in between. Resonance - The tendency of an object to vibrate most at a particular frequency. At that frequency the resistance of the coil is approximately 4Ω and the Q-factor is calculated as 100. Well, those buildings were full of tens of thousands of people, plenty of whom had cameras. This is how to find the max amount absorbed. The fundamental frequency, or f0, is the first harmonic, or H1. Substitute these values into the wavelength equation λ = v/f. Two formulas are given: the first to calculate the resonant frequencies when the length is known, and second, to calculate the length when the frequency of a formant is known. The absorbed frequencies (at least the low. Therefore to avoid resonance being excited we must change either k or m or both. Resonance and Impedance Antennas can be classified as either resonant or non-resonant, depending on their design. At each value of f, the object eventually reaches a constant amplitude A. A mechanical analogy to self-resonance is the natural frequency of vibration for an object, given the unavoidable presence of both elasticity and mass in any object. So if you are designing a system to avoid resonance, you need to consider all of them. Similarly, all metallic objects have an electromagnetic resonance frequency. The chipless tag presented in Figure 4. Stiffer objects have higher resonant frequencies, whereas more massive ones have lower frequencies. F n = (2n-1)c n = the nth resonant frequency (n=1, n=2, n=3). As the ends were lowered the resonant frequency increased. The fuselage has an electromagnetic resonant frequency. Therefore, the only way to change ultrasound frequency is to change transducers. The lock-in amplifier drives the series resonance circuit with an AC signal at resonance frequency. Then I am interested to find out what the corresponding freqency spectrum is, if the objects are excited by external sources and what the resonance frequencies and amplitudes are. In particular we will model an object connected to a spring and moving up and down. Question: Explain, in terms of resonant sound frequency and resonant light frequency, how an object can produce one dominant frequency. RESONANCE: “The property whereby any vibratory system responds with maximum amplitude to an applied force having the frequency equal to its own. If there is another object nearby that is sensitive to the same frequency, it will absorb the vibrations emanated from. Now, the resonant frequency of an object is simply any frequency at which the waves bouncing inside of the material reinforce each other, instead of cancelling. Another aspect of the motor is the vibration frequency: $$f = \frac{Motor \: Speed \:(RPM)}{60}$$ The displacement is directly affected by the vibration frequency. If you want to design and build a chime set using rods rather than tubes all you have to do is set the inside diameter to zero and enter the outside diameter and type of metal into the DIY calculator. Resonance occurs when the natural frequency of vibration of an object is equal to the driving frequency, giving a maximum amplitude of vibration. 000000322 = 993788819. When calculating the resonant frequency of a room, you will need to double the longest dimension because when sound travels along that route, it will bounce back and cover the same distance again. Hi everyone, Im learning about the basics of resonance, and I am just wondering why it is not observed more often in everyday objects. The LDC or FDC drives the circuit at the resonant frequency and makes a measurement, which is then converted to a high resolution digital value. The frequency at which this occurs is called the resonant frequency, and resonance is said to occur. The less damping a system has, the higher the amplitude of the forced oscillations near. The velocity of the particle can be calculated by differentiating the displacement. Physicist: Nope! "Resonance" is a "driven harmonic oscillation", where the driving force pushes and pulls at, or near, the "resonant frequency" of whatever it is that doing the resonating. 477 mm, and extends a depth c  11. Required activities. “Vibrational frequency” is a number on a logarithmic scale, and it measures how accurately your world view and your emotions, thinking, behavior match reality, to what degree you have departed from human animal and rose to the level of human being. However, after only twodays, it had tobe closed to traffic for safety reasons. The displacement with time takes the form of a sinusoidial wave. It increases. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Resonance is not often a concern when short, rigid shaft pumps are operated at their design speed. By inspection, this corresponds to the angular frequency $$\omega_0 = 2\pi f_0$$ at which the impedance Z in Equation \ref{15. For measurement of the complex dielectric constant in the frequency range of 15 – 22 GHz, a resonant cavity was designed using a section of WR51 waveguide. This happens when the primary coil shoots the current into the secondary coil at just the right time to maximize. Have the speaker play a tone at a given volume, and then without changing the volume slowly change the pitch (or frequency). For structures of arbitrary shapes, temporal coupled-mode theory is applied to explain the Fano. In the resonance problem posed on the previous page, I asked you to look at a male vocal tract and calculate resonant frequencies. But resonating can get out of control. 00 J, an amplitude of 10. Resonance does not happen very often and only affects object close to the vibrating source. Suspension Frequency. 2 ohm; L = 100mH and C="50" μF. 09 mm with flat metallic plates on both ends. Some resonant objects have more than one resonance frequency, particularly at harmonics (multiples) of the strongest resonance. length (L), with several inches added for tuning and for securing to insulators. 25 Compute the frequencies ω for which y00 +y0 +6 y = cos( ωt −φ) produces resonance. The process of low frequency resonance effect can set out four unique frequencies which are beneficial to human biological. Part B shows the transmission response of a strip structure with aspect ratio 1:333 with the magnetic ﬁeld applied along the length as well as the width of the strip, showing different resonance frequencies. When the forced frequency equals the natural frequency, the system is said to experience resonance. formulas in field of Electrical, Mechanical, Chemical, Electronics, Civil, Metallurgy, Oil & Gas, Optical, Plastics, Ceramics, Physics, Maths and many more to come , if you have any questions or difficulty. A child's playground swing is an example of a pendulum, a resonant system with only one resonant frequency. Variable capacitors are more difficult to come by than inductors, so it. Tuning a tesla coil can be done in various ways depending on what equipment you have access to. 2a for the resonance.
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# Probability and Statistics¶

## Outline¶

We will be following Seeing Theory, a fantastic tool created by students at Brown University.

• Introduction to Probability
• Compound Probability
• Probability Distributions
• Frequentist Inference
• Bayesian Inference
• Regression Analysis

## Introduction to Probability¶

### Chance Events¶

🐑 You flip a coin twice. Assume the probability that the coin lands on heads is $p$. What is the probability of have at least one heads?

What is the mathematical definition of independence?

$$P \cap B = P(A)P(B)$$

### Expectation¶

🐑 Using the definition of expectation, calculate the expectation of a single coin flip.

$$E(X) = \sum_{x \in X(\Omega)} xP(X=x)$$

### Variance¶

🐑 Compute the variance of a die roll, i.e. a uniform random variable over the sample space $\Omega = \{1, 2, 3, 4, 5, 6\}$.

## Compound Probability¶

### Set Theory¶

🐑 Prove that $(A\cup B)^c = A^c \cap B^c$

### Counting¶

🐑 In Poker, here are examples of possible hands:

1. Royal Flush: A, K, Q, J, 10 all in the same suit.
2. Straight Flush: Five cards in a sequence, all in the same suit.
3. Four of a Kind: Exactly what it sounds like.
4. Full House: 3 of a kind with a pair.

Calculate the probabilities of the above hands.

### Conditional Probability and Bayes Rule¶

🐑 What is the definition of conditional probability?

🐑 Using the definition of conditional probability, prove Bayes Rule:

$$P(A \mid B) = \dfrac{P(B \mid A) P(A)}{P(B)}$$

🐑 You have two coins in a bag. A biased coin and fair coin. The biased coin lands on heads with probability 0.95. Define the following events:

$A = \{ \text{Picking the biased coin} \}$

$B = \{ \text{Flipping 3 heads out of 3 total flips} \}$

Compute $P(A \mid B)$ using Bayes Rule.

## Probability Distributions¶

### Random Variables¶

Distrete vs continuous (countable vs uncountable)

### Central Limit Theorem¶

🐑 We load on a plane 100 packages whose weights are independent random variables that are uniformly distributed between 5 and 50 kilograms. What is the probability that the total weight will exceed 3000 kilograms?

## Frequentist Inference¶

### Point Estimation¶

How could we estimate $\pi$ ?

### Confidence interval¶

Why is the 95% confidence interval commonly used?

### The Bootstrap¶

Where do you often see bootstrapping used in econometrics?

## Bayesian Inference¶

### Bayes Theorem¶

$P(A \mid B) = \dfrac{P(B \mid A) P(A)}{P(B)}$

Terms: prior, likelihood, posterior

Example exercises

### Likelihood Function¶

How does the likelihood function change as we choose larger samples?

## Regression Analysis¶

### Ordinary Least Squares¶

Graph your data! Check out the Datasaurus Dozen inspired by Anscombe's Quartet.

Derivation of matrix form of OLS.

And we could fit OLS by hand.

You have data on the grades of 10 students in primary school and high school. You would like to estimate the relationship between the grades. Assume that high school grades are linearly related to primary school grades with some idiosyncratic error, $\epsilon$.

Primary school = 5, 2, 3, 4, 8, 9, 10, 8, 5, 6 High School = 6, 4, 3, 4, 6, 7, 8 , 9, 3, 5

The model is Primary = $\alpha$ + $\beta$ High + $\epsilon$

Estimate the value of $\alpha$ and $\beta$ by hand.

### Correlation¶

• Covariance
• Correlation coefficient

🐑 Argue that for given random variables X and Y, the correlation lies between −1 and 1.

Correlation does not imply causation. See this website for examples.

🐑 Independence implies zero correlation but zero correlation does not imply independence.

In [10]:
sns.regplot(x="primary", y="high", data=grades);

In [17]:
results = smf.ols('high ~ primary', data=grades).fit()
results.summary()

/Applications/anaconda/lib/python3.6/site-packages/scipy/stats/stats.py:1390: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=10
"anyway, n=%i" % int(n))

Out[17]:
Dep. Variable: R-squared: high 0.682 OLS 0.643 Least Squares 17.17 Sun, 09 Sep 2018 0.00323 18:12:14 -15.198 10 34.40 8 35.00 1 nonrobust
coef std err t P>|t| [0.025 0.975] 1.6562 1.007 1.645 0.138 -0.665 3.977 0.6406 0.155 4.144 0.003 0.284 0.997
 Omnibus: Durbin-Watson: 0.901 2.214 0.637 0.411 0.465 0.814 2.655 17.1

format: plain


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# Scalar evolution equations for shear waves in incompressible solids: a simple derivation of the Z, ZK, KZK and KP equations

Michel Destrade, Alain Goriely, Giuseppe Saccomandi

## Abstract

We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy–Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then, we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev–Petviashvili (KP), Zabolotskaya–Khokhlov (ZK) and Khokhlov–Zabolotskaya–Kuznetsov (KZK) equations of incompressible solid mechanics.

## 1. Introduction

The Zabolotskaya (Z) equation describes the propagation of finite-amplitude, two-dimensional, linearly polarized, shear waves in nonlinear solids. It can be presented in the following form (owing to Cramer & Andrews 2003): 1.1 where U=U(χ,η,τ) is a measure of the amplitude of the wave, χ is a spatial variable measured in a frame moving with the wave, η is a spatial variable transverse to the wave propagation, τ is a temporal variable and β characterizes the material properties of the solid. Zabolotskaya (1986) derived it via a perturbation scheme from the general equations of elastodynamics. It is a counterpart to the Zabolotskaya–Khokhlov (ZK) equation, itself describing the propagation of weakly nonlinear, weakly diffracting, two-dimensional sound beams (Zabolotskaya & Khokhlov 1969), 1.2 (Note that this form is also due to Cramer & Andrews (2003), and that the nonlinear parameter β is not the same in each equation.) The history of the ZK equation and its variants has recently been retraced by Rudenko (2010).

There are two major differences between the Z and ZK equations. One is obviously the cubic term in equation (1.1) versus the quadratic term in equation (1.2). The other is that the ZK equation governs the motion of the longitudinal displacement and is a scalar equation, while the Z equation governs the propagation of plane-polarized shear waves and is thus a vector equation, a priori. Nonetheless, recent papers (Enflo et al. 2006; Wochner et al. 2008; Pinton et al. 2010) have investigated the propagation of linearly polarized nonlinear shear waves in incompressible solids.

Here, we aim at finding what restrictions must be imposed, if at all, on the form of the strain energy W (say), in order to obtain a scalar Z equation. We also include the effects of pre-strain, similar to Cramer & Andrews (2003) in the compressible case. We point out that another problem of interest related to these model equations concerns the mathematical validation of the various approximations. This problem has been solved in the framework of the Navier–Stokes and Euler system by Rozanova-Pierrat (2009). Moreover, because we are mainly interested in understanding the physical underpinnings at the basement of various modelling assumptions, we adopt here a direct method for the derivation of the model equations, not a general reductive perturbation scheme such as the one proposed by Taniuti (1990), for example.

In §3, we establish that a scalar Z equation can indeed be generated for the whole class of the so-called generalized neo-Hookean solids, for which the strain-energy density does not depend on the second principal invariant of the Cauchy–Green strain tensor. We then try to generalize the study to the case of a generic fourth-order incompressible solid, for which W is expanded up to the fourth order in invariants of the Green strain tensor. We show in §3b that there, the Z equation is necessarily a vector equation that is, only plane-polarized—not linearly polarized—two-dimensional shear waves are possible.

When dissipative and dispersive effects are taken into account for the propagation of longitudinal waves, the ZK equation is generalized to 1.3 where β′ and β′′ are other material constants. This scalar equation covers the purely dissipative (at β′′=0) Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation (Kuznetsov 1971) and the purely dispersive Kadomtsev–Petviashvili (KP) equation (Kadomtsev & Petviashvili 1970). In §4, we consider dissipative and dispersive effects in generalized neo-Hookean solids. We derive the counterpart to equation (1.3) for shear waves as 1.4 a scalar equation that covers the Z equation, the modified Khokhlov–Zabolotskaya–Kuznetsov (mKZK) equation and the modified Kadomtsev–Petviashvili (mKP) equation.

The aim of this paper is to provide a simple derivation of these scalar equations for incompressible materials. This quest is largely motivated by the pressing need to model acoustic-wave propagation in soft tissues and gels, where the combined effects of incompressibility, pre-strain, viscosity and dispersion cannot be ignored. This is a topic of fundamental relevance to ultrasound-based techniques of medical imaging such as elastography.

## 2. Basic equations

Consider a body of incompressible isotropic elastic material, initially at rest in a reference configuration , say. For the study of anti-plane shear motions, we choose a system of Cartesian coordinates (X1,X2,X3) to denote the position of a particle in the body in . We then call (x1,x2,x3) the Cartesian coordinates, aligned with (X1,X2,X3), corresponding to the current position x in the deformed configuration , say. We study the anti-plane motion 2.1 where λ is a positive constant—the axial pre-stretch—and u is a three-times continuously differentiable function—the shear motion.

We call F≡∂x/∂X the associated deformation gradient and B=FFT the left Cauchy–Green strain tensor. The first principal isotropic strain invariant, I1≡trB, is here 2.2 and the second principal isotropic strain invariant, , is equal to 2.3 where u1≡∂u/∂X1 and u2≡∂u/∂X2. In all generality, the strain-energy density W is a function of I1 and I2 only, and the Cauchy stress tensor σ is derived as (e.g. Ogden 1984) 2.4 where W1≡∂W/∂I1, W2≡∂W/∂I2 and p is a Lagrange multiplier introduced by the constraint of incompressibility (which imposes that at all times).

The equations of motion, in the absence of body forces, are 2.5 where ρ is the mass density, which remains constant owing to the incompressibility constraint. Here, the equations of motion reduce to the following system of partial differential equations (see Knowles (1976) for the static case): 2.6 and 2.7 where summation over the repeated index i=1,2 is assumed. The auxiliary quantity q is given by 2.8

## 3. The Z equation

### (a) Results for generalized neo-Hookean materials

The system (2.6) consists of three partial differential equations for the two unknown functions p and u. Therefore, the system is overdetermined and it is not possible, in general, to find a general solution for arbitrary choices of the strain energy W. Progress is possible for some special forms of the strain-energy density, in particular for the Mooney–Rivlin form, which is linear in I1 and I2 (Adkins 1954), but only for static solutions (see Knowles (1976) for details). The dynamic situation for anti-plane motions is much more complex, as discussed by Hayes & Saccomandi (2004) in the Mooney–Rivlin case.

However, if we consider materials for which W=W(I1−3) only—the so-called generalized neo-Hookean materials—then system (2.6) reduces to a well-determined system, 3.1 where W′ is the derivative of W with respect to its argument (I1−3).

#### (i) Derivation of the Z equation

To derive the Z equation, we assume that W is at least of class . We assume that the primary propagation is in the X1-direction and introduce the following new displacement and new variables: 3.2 where ϵ is a small, non-dimensional parameter, giving a measure of the motion amplitude, the plus or minus sign corresponds to left or right running waves and α is a parameter to be adjusted to cancel all terms to order ϵ. (Note that the scalings used here mirror those of Zabolotskaya (1986), and give a consistent asymptotic expansion.) To obtain the Z equation, we expand equation (3.1), that is 3.3 to order ϵ3 (note that the subscript ‘i’ here denotes derivation with respect to Xi).

First, we compute the partial derivatives as follows: 3.4 which leads to 3.5

Second, we expand W′ in ϵ. Since 3.6 we have 3.7 where w(1)=W′(λ2+2λ−1−3) and w(2)=W′′(λ2+2λ−1−3). Third, we use the expansions for W′ and ui to compute each term in equation (3.3) up to order ϵ3. That is 3.8 Equation (3.3) now reads 3.9 Terms to order ϵ disappear by choosing α2=ρ/(2w(1)), which gives to order ϵ3, 3.10

Finally, taking a derivative of this equation with respect to τ and using the substitution U=vτ leads to the scalar Z equation 3.11 where β=3w(2)α5/ρ.

For example, consider the Yeoh strain energy described by 3.12 where μ>0 is the initial shear modulus and γ>0 is a nonlinear elastic constant. In this case, we find 3.13

Note that the apparent difference between equation (3.11) and the Z equation (1.1) is simply owing to the choice of independent variables. Here, we chose the same independent variables as Norris (1998), where τ is a retarded/advanced time, χ is the spatial variable in the propagation direction and η is the transverse spatial variable.

#### (ii) Special solutions

Following Cates & Crighton (1990) and Sionóid & Cates (1994), we perform the transformation 3.14 which reduces the Z equation (3.11) to an equation that, once integrated in ζ, yields 3.15 The solution of this one-dimensional equation does not provide a general solution to the two-dimensional equation (3.11); rather, it gives the solution to the particular Cauchy problems where initial data are given on the parabolic surfaces ζ=const.

A further change of function (from G to ) and of variables (from χ to ), 3.16 reduces equation (3.11) to the autonomous inviscid generalized Burgers equation 3.17 The general integral of this equation is , where F is an arbitrary function. This allows us to recover the following class of exact implicit solution for the Z equation (3.11): 3.18

It is quite remarkable that the transformation (3.14) should work for the Z equation (cubic nonlinearity) because it was initially intended for the KZK equation (quadratic nonlinearity). Through it, a huge class of exact solutions is generated for the Z equation.

General methods to obtain exact solutions for such equations include reduction methods such as the one based on Lie group analysis and its generalizations. Those methods have been systematically applied to the ZK equations with quadratic nonlinearities (e.g. Tajiri 1995; Bruzon et al. 2009), but it seems that there are very few results, regarding the case of a cubic nonlinearity, of interest to incompressible elastic materials.

### (b) Results for general incompressible solids

We return to an important point evoked in §1: that the propagation of finite-amplitude transverse waves cannot be governed by a scalar Z equation (i.e. linearly polarized shear waves exist only in special solids, such as the generalized neo-Hookean materials).

To show this, we take solids with the following Rivlin strain energy: 3.19 (where C10, C01 and C20 are constants), which covers the most general model of fourth-order incompressible elasticity (Ogden 1974), 3.20 with the identifications (Destrade et al. 2010) 3.21 Here, E=(FTFI)/2 is the Green strain tensor and μ, A and D are the second-, third- and fourth-order elasticity constants, respectively.

With this strain energy, the first and second equations of (2.6) read 3.22 where and and are defined as in §2.

Guided by the results of the previous subsection, we know that we must perform the changes of function and variables defined by equation (3.2), if we aim at deriving the Z equation from the last equation in (2.6). Here, those changes give the first terms in the expansion of the two equations (3.22) as 3.23 Now the first of these tells us that must be of order ϵ2: , say. Then, equations (3.22) reduce further to 3.24 at order ϵ2. Clearly, these expressions for the derivatives of Q are not compatible.

We may thus conclude that linearly polarized waves are not possible in general incompressible elastic materials. Indeed, Horgan & Saccomandi (2003) explain that when an anti-plane shear deformation occurs in a general incompressible material, it is necessarily accompanied by secondary motions associated with in-plane deformations. This coupling increases dramatically the complexity of the calculations and is an obstruction to the propagation of linearly polarized two-dimensional shear waves. In general incompressible materials, it is necessary to take into account the in-plane motions coupled to the two-dimensional shear wave.

In fact, Wochner et al. (2008) acknowledge as much when they correct Enflo et al. (2006) and their study of linearly polarized nonlinear shear waves, and conclude that ‘Simply stated, quadratic nonlinearity prevents the existence of a pure linearly polarized shear wave beam’. However, they then state that ‘Cubic nonlinearity can be more pronounced than quadratic nonlinearity due to the quasi-planar nature of wavefronts in directional sound beams’, and go on to ‘consider only the effects of cubic nonlinearity’. But, ignoring quadratic terms in favour of cubic terms runs contrary to the very idea of asymptotic expansions. Upon examination of their equation of motion (eqn (11)), we see that the only way to make quadratic terms vanish is to take 4μ+A=0. According to equation (3.21) here, this is equivalent to saying that C01=0, making W in equation (3.20) belong to the class of generalized neo-Hookean materials.

## 4. Dissipation and dispersion

In our treatment of dissipative and dispersive effects, we use the following constitutive law for the Cauchy stress: 4.1 where 4.2 which separates the purely elastic part σE of the stress from a viscoelastic part σD.

More specifically, σE coincides with the Cauchy stress of an elastic Yeoh material with strain energy as in equation (3.12). We make this choice for convenience, but it is clear that once the asymptotic expansions are performed below, the Yeoh form recovers the entire class of generalized neo-Hookean solids, through the identifications 4.3

The viscoelastic part of the stress, σD, is the stress of a second-grade viscoelastic solid, where ν>0 and α1>0 are constants and A1 and A2 are the first two Rivlin–Ericksen tensors, defined by 4.4 with , the displacement gradient. Here, the superposed dot represents the material time derivative. The first part of σD is νA1=2νD, where D is the stretch tensor; that part describes dissipation by Newtonian viscosity effects, and coincides with the dissipative term in the Navier–Stokes theory. The second part of σD is ; it describes dispersion by accounting for the intrinsic characteristic lengths of the solid (see Destrade & Saccomandi (2006) for details).

Note that the temporal evolution of the canonical energy E(t) for our solid (4.1) and (4.2) is given by , where Ωt is a given region occupied by the material at time t. It is independent of α1 (Fosdick & Yu 1996).

With respect to two-dimensional anti-plane motion, the derivation of the governing equation is not overly complicated by the additional terms. For ease of exposition, we consider that the solid is not pre-strained: λ=1. (It follows that once the asymptotic expansions are performed, the identifications (4.3) reduce to μ=2w(1), γ=2w(2)/w(1).)

We find that the equations qα=0(α=1,2) still apply here, while the last equation in (3.1) is changed to 4.5 In dimensionless form, it becomes 4.6 using the usual scalings.

When the solid is not dissipative (ν=0) and not dispersive (α1=0), the changes of function and variables (3.2) give a governing equation at order ϵ3, when α is chosen as . Here, the expansions of the dissipative and dispersive terms in equation (4.6) start as 4.7

Clearly, in order to be incorporated into the governing equation for the motion, the dissipative and the dispersive effects must be at least of the order and say, where and are constants. With that assumption, we find the following form of the equation of motion for velocity , 4.8 where 4.9 This equation covers not only the Z equation (at β′=β′′=0), but also the dissipative mZK equation (at β′′=0) and the dispersive mKP equation (at β′=0).

To the best of our knowledge, this is the first time that an mKP equation is derived in the continuum mechanics of homogeneous solids.

If we consider the diffusive version of equation (4.8), i.e. 4.10 and we seek solutions that propagate at an angle γ with respect to the τ-direction, it is possible to obtain an interesting reduction. To this end, let us consider solutions that depend only on χ and to reduce equation (4.10) to 4.11 This is the Nariboli–Lin-modified diffusive Burgers equation (Nariboli & Lin 1973). The plane-wave solution of this equation is computed as where V 0 is the speed of the plane wave. This solution is quite different from the plane-wave solution of the classical Burgers equation, which is expressed in terms of the hyperbolic tangent function.

If we consider the purely dispersive case and we rescale some of the variables and and , it is always possible to choose the scaling such that 4.12 the mKP equation. A connection between this equation and the modified Korteweg de Vries equation may be established with a similar transformation to the one used to reduce equation (4.10) to equation (4.11).

It is also possible to seek direct solutions of equation (4.12) in the form This ansatz produces the reduction 4.13 where c0 and c1 are integration constants.

If we consider solitary-wave solutions, the asymptotic boundary conditions require c0=c1=0 and the usual energy integral is given by where again c2 is an integration constant. It is clear that it must be c2=0 and if b2>0, an explicit solitary-wave solution is given by where ζ0 is an integration constant.

If we consider in equation (4.13) c0=0 and c1≠0, it is also possible to find cnoidal wave solutions in terms of elliptic functions.

## 5. Conclusions

We provide a clear and complete derivation of various model equations for shear waves in the framework of nonlinear incompressible elasticity. The rationale for investigating this type of problem is dictated by the use of imaging techniques of improved quality, such as latest generation elastography, to evaluate the nonlinear material properties of soft tissues.

We show that it is possible to derive a pure (scalar) Z equation only in the special case of generalized neo-Hookean materials. For general incompressible materials, the Z equation is coupled with a non-negligible transverse motion, an important issue that is often overlooked in the literature. Clearly, a rigorous evaluation of the validity of the perturbative expansion requires an extension of the existence, uniqueness and stability results that have been provided so far only for model equations with quadratic nonlinearity.

We point out that general structure of the various equations obtained here is as follows: where c is a suitable material parameter, is the second-order derivative (Laplace operator) in an orthogonal direction to the beam axis and is the equation governing plane-wave propagation (inviscid Burgers, Burgers or Korteweg de Vries equation). It would be an interesting problem to study the various transformation and group-analysis properties of such general equations to find exact and explicit solutions for various model equations following a general and complete scheme.

## Acknowledgements

This work is supported by a Senior Marie Curie Fellowship awarded by the Seventh Framework Programme of the European Commission to the first author. This publication is based on work supported in part by award no. KUK-C1-013-04 , made by King Abdullah University of Science and Technology (KAUST) (A.G.), and also based in part upon work supported by the National Science Foundation under grant DMS-0907773 (A.G.).

• Received October 2, 2010.
• Accepted November 12, 2010.

View Abstract


text:
Creating a Scientific Poster with LaTeX and OmniGraffle

05 May 2007

Some might accuse me of doing things the hard way, and I suppose they might be correct. Recently, I had to make a poster for a poster session and I was faced with the problem of choosing the appropriate tool for the job. Most people would probably use PowerPoint, but I've never been pleased with equations in Microsoft products. I've also had issues in the past with PowerPoint converting pdf figures into bitmap images. Adobe InDesign was also an option, but I decided it was too much for this job - I'd spend more time learning to use the program than actually making the poster. In the end, I settled on tools I'm familiar with: LaTeX and OmniGraffle. Perhaps a more complex solution than simply using PowerPoint, but the equations look good and the layout was simple.

At the bottom, you’ll see the finished poster. I’m pleased with the way it came out and it ended up being straightforward to make. The text and equations were typeset in LaTeX directly to pdfs; the headings and title were standard OmniGraffle textboxes. The key component in the process was the a0poster class. This class provides larger than normal font sizes which are necessary for posters. I used normalsize fonts for text (about 25 pt) and Large for equations (about 36 pt). The geometry package was also useful, as I was able to specify the exact size of each textbox as well as the margins; a template is shown below.

\documentclass[a0,landscape]{a0poster}
\usepackage{palatino}
\usepackage{mathpazo}        % For math fonts that match palatino

\usepackage[pdftex]{color}
\usepackage{geometry}
\geometry{paperwidth=12.25in,paperheight=1.75in}
\geometry{left=0.25in,right=0.25in}
\geometry{top=0.25in,bottom=0.25in}

\pagestyle{empty}            % No page numbers

\def\red#1{\textcolor{red}{#1}}
\def\blue#1{\textcolor{blue}{#1}}

\begin{document}
% Insert text here
\end{document}

With all the text as separate pdfs, it was simply a matter of dragging them onto OmniGraffle. Many of the figures were already pdf so I dragged them onto OmniGraffle as well. After moving them around the canvas to get an approximate alignment, I then decided on exact coordinates for the different sets of pdfs and entered those values into the inspector; this was much easier and more accurate than dragging to align.

For the images, I used Photoshop to make them exactly the right size and dpi, e.g., 3 inches wide at 300 dpi or 150 dpi if the original image wasn’t big enough. I also converted each image to sRGB to ensure consistent color.

I would recommend this approach to anyone who wants nicely typeset text and equations (notice the drop cap in the center column) but doesn’t want to wrestle with figure and text placement using LaTeX.


text:
Limited access

Heavy water acts to slow down neutrons in some types of nuclear reactors. The molecular mass of heavy water is 20 AMU (atomic mass units).

Which of the following is the molecular formula of heavy water?

A

${ H }_{ 2 }{ }^{ 18 }O$

B

$D_{ 2 }O$

C

${ H }_{ 2 }{ O }_{ 2 }$

D

$HOD$

Select an assignment template


text:
# A 5.0 kg mass hanging from a spring scale is slowly lowered onto a vertical spring. a) What...

## Question:

A 5.0 kg mass hanging from a spring scale is slowly lowered onto a vertical spring.

a) What does the spring scale read just before the mass touches the lower spring? (N)

b) The scale reads 18 N when the lower spring has been compressed by 2.0 cm. What is the value of the spring constant for the lower spring? (N/m)

c) At what compression length will the scale read zero? (cm)

## Hooke's Law:

The restoring force of a spring is given by Hooke's law

{eq}F=-kx {/eq}

Where,

• {eq}k {/eq} is the spring constant;
• {eq}x {/eq} is the compression distance.

Take note that the restoring force is negative and always points towards the equilibrium.

Given the following quantities:

• mass {eq}m=5.0\text{ kg} {/eq};
• acceleration due to gravity {eq}9.8\text{ m/s}^2 {/eq}

Let the +y-axis be the positive direction. Therefore, downward compression is negative and the restoring force is upward and positive.

a) Just before the mass touches the lower spring, the scale reading is equivalent to the magnitude of the weight of the mass.

{eq}\begin{align} w&=mg\\ &=\left(5.0\text{ kg}\right)\left(9.8\text{ m/s}^2\right)\\ w&=49\text{ N} \end{align} {/eq}

b) When the lower spring is compressed downwarf by {eq}y=-2.0\text{ cm}=-0.020\text{ m} {/eq} the apparent weight of the mass on the scale reads {eq}w'=18\text{ N} {/eq}.

Notice that the apparent weight is equivalent to the difference between the actual weight and the restoring force {eq}F {/eq} of the lower spring.

{eq}w'=w-F {/eq}

and

{eq}\begin{align} F&=w-w'\\ &=49\text{ N}-18\text{ N}\\ F&=31\text{ N} \end{align} {/eq}

From Hooke's law,

{eq}F=-ky {/eq}

Solve for the spring constant {eq}k {/eq},

{eq}\begin{align} k&=-\frac{F}{y}\\ &=-\frac{31\text{ N}}{-0.020\text{ m}}\\ k&=1550\text{ N/m} \end{align} {/eq}

c) The reading in the scale becomes zero when the magnitude of the restoring force is equal to the weight of the mass

{eq}F=49\text{ N} {/eq}

Solving for the compression using Hooke's law,

{eq}F=-ky {/eq}

and

{eq}\begin{align} y&=-\frac{F}{k}\\ &=-\frac{49\text{ N}}{1550\text{ N/m}}\\ y&\approx -0.031\text{ m} \end{align} {/eq}

or

{eq}y\approx -3.1\text{ cm} {/eq}

Hooke's Law & the Spring Constant: Definition & Equation

from

Chapter 4 / Lesson 19
201K

After watching this video, you will be able to explain what Hooke's Law is and use the equation for Hooke's Law to solve problems. A short quiz will follow.


text:
# Solarwinds - Monitoring switch stacks

Today’s post is on using SQL queries in Solarwinds to get more detailed information on the frontpage on stackswitches. Solarwinds tends to average the memory and cpu utilization among all switches in a stack, but this does not always reflect the true statistics on the switches. Sustained CPU above 80% can lead to data packets being dropped. The queries are all build in SWQL studio on a Solarwinds server to verify the information that is returned. In the queries below, also other network devices show up, which is not a problem in my opinion, but you might want to scope a bit more if that is more to your taste.

Please note that the queries below might not seem like valid or efficient queries. They’re not. Solarwinds decided to implement only a subset of SQL, which means that sometimes a lot more code is needed to achieve something simple.

## Top 30 nodes

I want only a top 30 of nodes returned for both performance reasons, but also because I don’t want an endless list stretching the whole front page or even multiple pages. If that would be the case, you’re primary objective is to start fixing those switches!

Three tables are essential to making this query work:

• Orion.Nodes
• Orion.NPM.SwitchStackMember

SQL code to get the top 30 nodes:

SELECT TOP 30 Nodes.DisplayName, Nodes.IPAddress, ssm.MacAddress, ssm.Role, ssm.SwitchNumber, ssm.SwPriority, Nodes.MachineType, ssm.SerialNumber, AvgLoad AS CPU , (DAYDIFF(Nodes.LastBoot, Nodes.NextPoll) ) AS \UptimeDays
INNER JOIN Orion.Nodes AS Nodes ON cpu.NodeID = Nodes.NodeID
LEFT JOIN Orion.NPM.SwitchStackMember AS ssm ON ssm.NodeID = cpu.NodeID  AND  ssm.MemberID = cpu.CPUIndex

The list can be ordered by making it a descending list: ORDER BY AvgLoad DESC

## Excluding devices

This query returns a list of other nodes such as servers as well. We can filter this by utilizing the Orion.NodesCustomProperties table as in this environment all nodes are classified on type. The following code is added to the previous code:

WHERE Nodes.NodeID IN (SELECT ncp.NodeID FROM Orion.NodesCustomProperties AS ncp WHERE ncp.DeviceCategory = 'Network')

NOTE: Be aware that this is a custom property that doesn’t come with the default Solarwinds installation. In your environment, you might have other custom properties. Otherwise, you might need to find another way to distinguish servers from network devices.

Next, in this environment two DWDM devices show up on top of which the cpu util is not relevant for us to monitor. I decided to filter these by ID’s, but if you have more of these devices, you might need to filter on MachineType for example.

AND NOT (Nodes.NodeID = '1867' OR Nodes.NodeID = '1868')

## End result

Code:

SELECT TOP 30 Nodes.DisplayName, Nodes.IPAddress, ssm.MacAddress, ssm.Role, ssm.SwitchNumber, ssm.SwPriority, Nodes.MachineType, ssm.SerialNumber, AvgLoad AS CPU , (DAYDIFF(Nodes.LastBoot, Nodes.NextPoll) ) AS \UptimeDays
INNER JOIN Orion.Nodes AS Nodes ON cpu.NodeID = Nodes.NodeID
LEFT JOIN Orion.NPM.SwitchStackMember AS ssm ON ssm.NodeID = cpu.NodeID  AND  ssm.MemberID = cpu.CPUIndex
WHERE Nodes.NodeID IN (SELECT ncp.NodeID FROM Orion.NodesCustomProperties AS ncp WHERE ncp.DeviceCategory = 'Network')
AND NOT (Nodes.NodeID = '1867' OR Nodes.NodeID = '1868')

Widget:

When you only want nodes above 80% to be displayed, you can add some extra code to the WHERE statement. However, I choose not to do that, because I’d prefer to see nodes doing that ahead of time. In case you do want to cap, use the following code:

AND AvgLoad > 80

Now, in Solarwinds you can add this code to a custom widget. However, perhaps you want e-mail alerting on this as well. In that case, go to Alerts & Activity > Alerts > Manage Alerts (top right) > Add New Alert.

Under Trigger Conditions, the SQL is slightly different, also due to the limited editing capabilities:

INNER JOIN CPUMultiLoad ON Nodes.NodeID = CPUMultiLoad.NodeID
WHERE DATEDIFF(mi, CPUMultiLoad.TimeStampUTC, SYSUTCDATETIME()) < 5
AND Nodes.NodeID IN (SELECT NodesCustomProperties.NodeID FROM NodesCustomProperties WHERE NodesCustomProperties.DeviceCategory = 'Network')

In the reset condition, I’ve configured the load to be lower then 75 (<75) for five minutes. The idea here is that this setting prevents flapping, but the values are up to you.

In the Trigger E-mail, I’ve used some extensive SQL as well. Here is an example, which is not great for reading here, but it does report the IP address, node name, cpu, memory etc. in a readable format when you receive an e-mail:

Automatically generated alert from Solarwinds Orion.

${N=SwisEntity;M=Vendor} device ${N=SwisEntity;M=SysName}  has  >80% CPU utilization  since ${N=Generic;M=DateTime;F=DateTime}. Details Devicename :${N=SwisEntity;M=DisplayName}
IP adress : ${N=SwisEntity;M=IP_Address} Time triggered :${N=SwisEntity;M=LastSystemUpTimePollUtc}
Location: ${N=SwisEntity;M=Location} Device last booted at:${N=SwisEntity;M=LastBoot}
Polling engine: ${N=SwisEntity;M=Engine.ServerName} Severity level:${N=Alerting;M=Severity}
System CPU : ${N=SwisEntity;M=Stats.CPULoad} SystemMem :${N=SwisEntity;M=Stats.PercentMemoryUsed}

Current Statistics
Stack CPU Utilization	:	${N=SWQL;M=SELECT TOP 1 AvgLoad FROM ( SELECT NodeID , CPUIndex , ObservationTimestamp , CPUMultiLoad.AvgLoad , SSM.Role , SSM.SwitchNumber , SSM.Model , SSM.MacAddress , SSM.SerialNumber FROM Orion.CPUMultiLoad AS CPUMultiLoad INNER JOIN Orion.NPM.SwitchStackMember AS SSM ON CPUMultiLoad.NodeID = SSM.NodeID AND CPUMultiLoad.CPUIndex = SSM.MemberID WHERE CPUMultiLoad.NodeID =${N=SwisEntity;M=NodeID}
ORDER BY ObservationTimestamp DESC
)
} %
Role				:	${N=SWQL;M=SELECT TOP 1 Role FROM ( SELECT NodeID , CPUIndex , ObservationTimestamp , CPUMultiLoad.AvgLoad , SSM.Role , SSM.SwitchNumber , SSM.Model , SSM.MacAddress , SSM.SerialNumber FROM Orion.CPUMultiLoad AS CPUMultiLoad INNER JOIN Orion.NPM.SwitchStackMember AS SSM ON CPUMultiLoad.NodeID = SSM.NodeID AND CPUMultiLoad.CPUIndex = SSM.MemberID WHERE CPUMultiLoad.NodeID =${N=SwisEntity;M=NodeID}
ORDER BY ObservationTimestamp DESC
)
}
SwitchNumber	:	${N=SWQL;M=SELECT TOP 1 SwitchNumber FROM ( SELECT NodeID , CPUIndex , ObservationTimestamp , CPUMultiLoad.AvgLoad , SSM.Role , SSM.SwitchNumber , SSM.Model , SSM.MacAddress , SSM.SerialNumber FROM Orion.CPUMultiLoad AS CPUMultiLoad INNER JOIN Orion.NPM.SwitchStackMember AS SSM ON CPUMultiLoad.NodeID = SSM.NodeID AND CPUMultiLoad.CPUIndex = SSM.MemberID WHERE CPUMultiLoad.NodeID =${N=SwisEntity;M=NodeID}
ORDER BY ObservationTimestamp DESC
)
}

Hardware & Software details
IP address		:	${N=SwisEntity;M=IP_Address} MAC Address :${N=SWQL;M=SELECT TOP 1 MacAddress
FROM (
SELECT NodeID , CPUIndex , ObservationTimestamp , CPUMultiLoad.AvgLoad , SSM.Role , SSM.SwitchNumber , SSM.Model , SSM.MacAddress , SSM.SerialNumber
INNER JOIN Orion.NPM.SwitchStackMember AS SSM ON CPUMultiLoad.NodeID = SSM.NodeID AND CPUMultiLoad.CPUIndex = SSM.MemberID
WHERE CPUMultiLoad.NodeID = ${N=SwisEntity;M=NodeID} AND AvgLoad > '80' ORDER BY ObservationTimestamp DESC ) } Model type:${N=SwisEntity;M=HardwareHealthInfos.Model}
IOS Image: ${N=SwisEntity;M=IOSImage} IOS Version:${N=SwisEntity;M=IOSVersion}
Serial Number: ${N=SWQL;M=SELECT TOP 1 SerialNumber FROM ( SELECT NodeID , CPUIndex , ObservationTimestamp , CPUMultiLoad.AvgLoad , SSM.Role , SSM.SwitchNumber , SSM.Model , SSM.MacAddress , SSM.SerialNumber FROM Orion.CPUMultiLoad AS CPUMultiLoad INNER JOIN Orion.NPM.SwitchStackMember AS SSM ON CPUMultiLoad.NodeID = SSM.NodeID AND CPUMultiLoad.CPUIndex = SSM.MemberID WHERE CPUMultiLoad.NodeID =${N=SwisEntity;M=NodeID}
View all node details: ${NodeDetailsURL} View full alert details here:${N=Alerting;M=AlertDetailsUrl}