###### frequency ratio vibration

, Let both masses have a mass of 1 kg and the stiffness of all three springs equal 1000 N/m. ), where N corresponds to the number of degrees of freedom. {\displaystyle {\tfrac {1}{2}}kx^{2}} The eigenvalues provide the natural frequencies of the system. Animated deformation shapes derive from frequency domain measurements such as auto-spectra or frequency response functions. In a lightly damped system when the forcing frequency nears the natural frequency ( { For lightweight structures with span above 8 m (24 ft) vibrations may occur. Also, the magnitude can be reduced if the natural frequency can be shifted away from the forcing frequency by changing the stiffness or mass of the system. The granite megaliths of the kings chamber – in the valley of the kings weigh up to 60 tons each and were transported over 500 miles from the Aswan quarry. The natural frequency is proportionate to the total spring stiffness of the system and the total mass of the system. [ 2 ) the amplitude of the vibration can get extremely high. The magnification factor is defined as the point at intersection between an underdamped system to a critical damped system, which is always constant having a value of that correspond to each eigenvalue are called the eigenvectors. ] Failure of slow speed machinery can result in catastrophic machine damage, lost production, and worker safety hazards. If challenged to move this mammoth block of stone today – modern engineers would not be able to transport it. impulses) and random functions. and The number fn is called the undamped natural frequency. Vibration Analysis (VA), applied in an industrial or maintenance environment aims to reduce maintenance costs and equipment downtime by detecting equipment faults. It would only be necessary to step up the vibrations of the machine to fit the natural vibration of the building and the building would come crashing down. Create the discrete-time transfer function. “If you want to find the secrets of the universe, think in terms of energy, frequency and vibration.”. 1 The mass–spring–damper model described above is called a single degree of freedom (SDOF) model since the mass is assumed to only move up and down. Particularly, whenever Effect of vibration frequency on packing density for various large particle fractions at size ratio of 0.15. Using the natural frequency of a harmonic oscillator = ... For underdamped vibrations, the damping ratio is also related to the logarithmic decrement via the relation = + ≜ ⁡ where and are the vibration amplitudes at two successive peaks of the decaying vibration. Every vibrating system has one or more natural frequencies that it vibrates at once disturbed. Later, more sophisticated analog and then digital controllers were able to provide random control (all frequencies at once). Suddenly all the heavy machinery in the place was flying around. The existence of a rigid-body mode results in a zero natural frequency. Free vibration occurs when a mechanical system is set in motion with an initial input and allowed to vibrate freely. For this example, consider the following discrete-time transfer function with a sample time of 0.01 seconds: s y s (z) = 5 z 2 + 3 z + 1 z 3 + 6 z 2 + 4 z + 4. {\displaystyle {\begin{bmatrix}M\end{bmatrix}},} Answer Explanation ANSWER: When frequency ratio is less than unity . The matrices are NxN square matrices where N is the number of degrees of freedom of the system. vibration test was c arried out again , to derive the dynamic characteristics of the repaired pier model. f The plots to the side present how 0.1 and 0.3 damping ratios effect how the system “rings” down over time. Archaeologists and scientists claims these ‘primitive’ civilizations used brute strength to move these colossal stones … but how do they explain megaliths of Peru weighing 400 tonnes each being transported from a quarry more than 200 miles away… through a steep mountain incline, up to 15,000 feet? The increase of the vibration frequency improves the mobility of individual particles by means of kinetic energy, allowing the particles to achieve a more stable position. The vibration characteristics of materials are typically represented by a graph of transmissibility versus frequency. free vibration). z Fig. f − [ 1 e [1] Other "response" points may experience higher vibration levels (resonance) or lower vibration level (anti-resonance or damping) than the control point(s). The second mathematical tool, "the principle of superposition", allows the summation of the solutions from multiple forces if the system is linear. A natural frequency will always occur in a free and forced vibration.   In more complex systems, the system must be discretized into more masses that move in more than one direction, adding degrees of freedom. Measure the damping ratio and damping coefficient “c” using rate of decaying approach. e ] • Each mode is defined by its natural frequency, During the early history of vibration testing, vibration machine controllers were limited only to controlling sine motion so only sine testing was performed. = 3) A slight shift of the driving frequency can lead to significant reduction in vibration level. At forcing frequencies between about 0.5 and 1.4 times the natural frequency, large amplifications of the vibratory input occur producing dynamic farces, which added to the static laadweight, may cause compressive overloading and failure of the container. The system is subjected to a sinusoidal base input with a variable frequency. MIL-STD-810G, released in late 2008, Test Method 527, calls for multiple exciter testing. Figure 6: Normalized amplitude vs. frequency ratio Many notes can be deduced from this plot, mainly: 1) As the damping ratio is increased, the maximum vibration decreased. The natural frequency range is provided from the specification list. The curves show that as the damping ratio increases, the maximum value of magnification factor decreases. All rights reserved. FREQUENCY RATIO ( f / fn ) T R A N S M I S S I B I L I T Y (G o u t / G i n) TRANSMISSIBILTY MAGNITUDE SDOF SYSTEM SUBJECTED TO BASE EXCITATION Figure 3. These properties can be used to greatly simplify the solution of multi-degree of freedom models by making the following coordinate transformation. Transmissibility is the ratio of vibration of the isolated surface to that of the source. The eigenvectors have very important properties called orthogonality properties. ω When Eq. {\displaystyle {\begin{bmatrix}M\end{bmatrix}}^{-1}{\begin{bmatrix}K\end{bmatrix}}={\begin{bmatrix}A\end{bmatrix}}} The word comes from Latin vibrationem ("shaking, brandishing"). x [ The Power of Frequency and Vibration Nikola Tesla was a genius. In our case the first mode shape vector is saying that the masses are moving together in phase since they have the same value and sign. The ratio of the frequency for Mode 2 to Mode 1 is approximately 4.34 to 1.0. 5.033 The vibration spectrum provides important frequency information that can pinpoint the faulty component. ╲ Example . ϕ The remaining components are at odd frequencies and it takes an infinite amount of sine waves to generate the perfect square wave. x The exponential solution is only used because it is easier to manipulate mathematically. The negative sign indicates that the force is always opposing the motion of the mass attached to it: The force generated by the mass is proportional to the acceleration of the mass as given by Newton's second law of motion: The sum of the forces on the mass then generates this ordinary differential equation: {\displaystyle {\begin{bmatrix}K\end{bmatrix}}} • Vibration of a real structure can be represented as a series of modal contributions. The studies of sound and vibration are closely related. Humans have an optimal frequency–as does everything else in the universe–that occurs when each of the cells in our body vibrates at the frequency it was designed to. ) of the mass-spring-damper model is: For example, metal structures (e.g., airplane fuselages, engine crankshafts) have damping factors less than 0.05, while automotive suspensions are in the range of 0.2–0.3. These 4 elements are generally denoted as F, D,V,A. The mode shape vectors are not the absolute motion, but just describe relative motion of the degrees of freedom. This key statement supports theories that the ancient civilizations such as the ancient Egyptians used sound frequencies to lift incredibly heavy, giant blocks of stone – megalith’s. such as Unbalance, Misalignment, rolling element bearing faults and resonance conditions. m Consequently, one of the major reasons for vibration analysis is to predict when this type of resonance may occur and then to determine what steps to take to prevent it from occurring. 6) Which of the following statements is/are true?  Hz The following are some other points in regards to the forced vibration shown in the frequency response plots. FIGURE 2.9 Effect of damping upon the ratio of displacement maxima of a damped free vibration. The mass and stiffness matrix for this problem are then: Then After learning about these interesting facts, Nikola Tesla’s famous quote about sound, frequency and vibration holding the secrets to the Universe begin to make a little more sense and evokes a lot more curiosity. There are three key features of this system. ⋯ As discussed earlier, when the mass and spring have no external force acting on them they transfer energy back and forth at a rate equal to the natural frequency. Ψ T The excitation frequency of the vibration actuator, specifically for the device manufactured by Company A, had a range of 0–600 Hz. ω Damping: The dissipation of energy in an oscillating system. With so much wisdom lost in the book-burning pyres of the middle ages, it is apparent that the some of the secrets of frequency and vibration have been lost… for now. This example highlights that the resulting vibration is dependent on both the forcing function and the system that the force is applied to. In rotor bearing systems any rotational speed that excites a resonant frequency is referred to as a critical speed. ] New generations of sensors, instruments, and analysis techniques are available for low frequency measurements. An unrestrained multi-degree of freedom system experiences both rigid-body translation and/or rotation and vibration. Vibration and Fourier Analogy • Any periodic function can be represented as a series of sinusoidal functions. ] Magnification factor is minimum at resonance 2. M It is useful for cases where a force is applied and dwells at a specific frequency. This damping ratio is just a ratio of the actual damping over the amount of damping required to reach critical damping. , For example, the vibrational motions of engines, electric motors, or any mechanical device in operation are typically unwanted. ] C The common symbol is f or v, and the unit is second -1. For linear systems, the frequency of the steady-state vibration response resulting from the application of a periodic, harmonic input is equal to the frequency of the applied force or motion, with the response magnitude being dependent on the actual mechanical system. (Note: Since the eigenvectors (mode shapes) can be arbitrarily scaled, the orthogonality properties are often used to scale the eigenvectors so the modal mass value for each mode is equal to 1. Frequency ratio: The number of vibrations completed per unit of time is the amount that describes the frequency of reciprocating motion of a vibrating object. Why have been limits imposed on the frequency ratio for thermowells? The building would have been about our ears in another few minutes.Outside in the street there was pandemonium. Click here to discover how frequencies are replacing acupuncture needles – allowing pain sufferers to treat themselves at home. ω 0. Figure 2.0 is a typical plot of a material’s transmissibility curve. ] X Nikola Tesla, “the father of modern electricity” told what had happened as follows: through a steep mountain incline, up to 15,000 feet? 2000 (ζ=0), it reaches to infinity at the resonance i.e. For example, the above formula explains why, when a car or truck is fully loaded, the suspension feels ″softer″ than unloaded—the mass has increased, reducing the natural frequency of the system. [ The Fourier transform of the square wave generates a frequency spectrum that presents the magnitude of the harmonics that make up the square wave (the phase is also generated, but is typically of less concern and therefore is often not plotted). Hence, the solution to the problem with a square wave is summing the predicted vibration from each one of the harmonic forces found in the frequency spectrum of the square wave. To start the investigation of the mass–spring–damper assume the damping is negligible and that there is no external force applied to the mass (i.e. Examples of this type of vibration are pulling a child back on a swing and letting it go, or hitting a tuning fork and letting it ring. [ = Although boats are an obvious means for moving the stones for most of the distance, fitting them perfectly in place within the pyramid is another matter. I asked my assistants where did the sound come from. With recent breakthroughs using ultrasound to break up kidney stones and even destroying tumors. The major concepts of multiple degrees of freedom (MDOF) can be understood by looking at just a 2 degree of freedom model as shown in the figure. The measured response may be ability to function in the vibration environment, fatigue life, resonant frequencies or squeak and rattle sound output (NVH). cannot equal zero the equation reduces to the following. Sine (one-frequency-at-a-time) tests are performed to survey the structural response of the device under test (DUT). r Vibrations are never completely eliminated, but they can be greatly reduced. An example of animating mode shapes is shown in the figure below for a cantilevered I-beam as demonstrated using modal analysis on ANSYS. 1 • Each mode is defined by its natural frequency, damping, and mode shape. – Nikola Tesla. It is unlike the displacement transmissibility. When these eigenvalues are substituted back into the original set of equations, the values of Frequency is plotted on the X-axis in Hertz (Hz) and transmissibility is plotted on the Y-axis in decibels. In practice, this is rarely done because the frequency spectrum provides all the necessary information. m Bruce Tainio, a famous researcher and … Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. and Therefore, the damped and undamped description are often dropped when stating the natural frequency (e.g. – A megalith is a massive stone that has been used to construct a structure or monument, either stand-alone or together with other giant stones. Frequencies are used to determine and differentiate vibrational patterns. When frequency ratio is less than unity b. {\displaystyle \phi .}. Multiply the ratio by the first-order tire frequency. i When considering the property of damping, the equation is rewritten as Equation (2). The Fourier transform is almost always computed using the fast Fourier transform (FFT) computer algorithm in combination with a window function. To simplify the solutions coming up, we define the critical damping c c, the damping ratio z, and the damped vibration frequency w d as, where the natural frequency of the system w n is given by, Note that w d will equal w n when the damping of the system is zero (i.e. Hence, attempts to reduce noise are often related to issues of vibration. ELFRAD (Extremely Low Frequency Research And Development), a self-funded research organization which tracks radiation on Earth through an array of antennas, has found an extremely low frequency signal at phi, or 1.618033 Hertz, or cycles … 8.717 First, obtain the rear axle ratio. Free damped-logarithmic decrement vibrations simulation aims to: 4. Excerpt from the New York World Telegram, July 11, 1935, Nikola Tesla revealed that an earthquake which drew police and ambulances to the region of his laboratory at 48 E. Houston St., New York, in 1898, was the result of a little machine he was experimenting with at the time which ”you could put in your overcoat pocket.”. When there are many degrees of freedom, one method of visualizing the mode shapes is by animating them using structural analysis software such as Femap, ANSYS or VA One by ESI Group. Summing the forces on the mass results in the following ordinary differential equation: The solution to this equation depends on the amount of damping. ) • Each individual sinusoid is define by its amplitude, frequency and phase. Vibration measurements can tell us about the condition of a gearbox, if properly done. [ Therefore, this relatively simple model that has over 100 degrees of freedom and hence as many natural frequencies and mode shapes, provides a good approximation for the first natural frequencies and modes†. Vibration means the state of an object moving repetitively back/forward, right/ left or up/down and is generally expressed by Frequency, Displacement, Velocity, and Acceleration. How Vibration and Frequency Affect Your Wellbeing. VA is a key component of a condition monitoring (CM) program, and is often referred to as predictive maintenance (PdM). Trending of data should be helpful, but only if the data being trended is collected consistently. = This becomes more difficult as the DUT gets larger and as the test frequency increases. The rated speed of JW6314 is 1440 r/min, converted into frequency is 24 Hz, which is far less than the first-mode frequency of the CFRP motor frame. For this example, consider the following discrete-time transfer function with a sample time of 0.01 seconds: s y s (z) = 5 z 2 + 3 z + 1 z 3 + 6 z 2 + 4 z + 4. { In general - as a rule of thumb - the natural frequency of a structure should be greater than 4.5 … {\displaystyle \phi ,} Theses feats of engineering aren’t limited to Egypt. x In this case, the finite element method was used to generate an approximation of the mass and stiffness matrices by meshing the object of interest in order to solve a discrete eigenvalue problem. Generally, one or more "input" or "control" points located on the DUT-side of a vibration fixture is kept at a specified acceleration. 2 The fundamentals of vibration analysis can be understood by studying the simple Mass-spring-damper model. The equations of motion of the 2DOF system are found to be: A more compact form of this matrix equation can be written as: where Although modern scientists are just now beginning to understand this quote from Tesla, ancient civilizations around the world had known of the power of sound, frequency, and vibration for thousands of years. { 2 Scientists are stumped when it comes to explaining how these megaliths were moved, but they’re equally confused on how they were cut so perfectly. Now that we have the tire frequency, we can calculate the driveshaft vibration for a rear wheel drive vehicle. The ratio of the vibration transmitted after isolation to the disturbing vibration is described as transmissibility and is ex-pressed in its basic form in Equation (1). ( For example, by applying a force to the mass–spring–damper model that repeats the following cycle – a force equal to 1 newton for 0.5 second and then no force for 0.5 second. Frequency Ratio ... amplitude, frequency and phase. 2 I told my assistants to say nothing. 1. 2 The figure illustrates the resulting vibration. The eigenvalues and eigenvectors are often written in the following matrix format and describe the modal model of the system: A simple example using the 2 DOF model can help illustrate the concepts. as shown in Figure 3. = The steady state solution of this problem can be written as: The result states that the mass will oscillate at the same frequency, f, of the applied force, but with a phase shift Sound, or pressure waves, are generated by vibrating structures (e.g. At this point, the system has reached its maximum amplitude and will continue to vibrate at this level as long as the force applied stays the same. Resonance is simple to understand if the spring and mass are viewed as energy storage elements – with the mass storing kinetic energy and the spring storing potential energy. Frequency-based ODS provides information on the rela­tive levels of structural deformation. When the frequency ratio β > 1 then ω n < ω and the total motion is characterized as the forced response oscillating about the free vibration portion of the response as indicated in Fig. Open Live Script. 1000 The phase of the FRF was also presented earlier as: For example, calculating the FRF for a mass–spring–damper system with a mass of 1 kg, spring stiffness of 1.93 N/mm and a damping ratio of 0.1. Applying a force to the mass and spring is similar to pushing a child on swing, a push is needed at the correct moment to make the swing get higher and higher. Applying the 1 Hz square wave from earlier allows the calculation of the predicted vibration of the mass. Upon the ratio vibration test was c arried out again, to derive the dynamic of!. }. }. }. }. }. }. }. } }! To an output exceeding its final, steady-state value speed that excites a resonant frequency.... 9 ] of angular frequency using the fast Fourier transform can also be written in terms of its of. Able to provide random control ( all frequencies at once disturbed the curve below shows time... Any periodic function can be modeled as a series of sinusoidal functions span/dept ratio less than 25 limitations on Y-axis. Sine motion so only sine testing was performed to issues of vibration _____ - -. ] = [ 2000 − 1000 − 1000 2000 ] damage, lost production, the... Damping required to reach critical damping of 7 Hz that overall natural frequency objects, but also how! This example that the resulting vibration is undesirable, wasting energy and,,! Break step crossing a bridge. ” frequency of the resonances that may be in. Different from the undamped ) \phi, } is defined by its natural frequency oscillations about. A gearbox, if properly done defined vibration environment study indicate s that overall natural frequency the... Damping and no applied forces ( i.e testing, '' SVIC Notes, 17. Frequency fn = 100 Hz and amplification factor Q=10 how much Each mode  participates '' the... Calculation results to understand in general no problems with vibrations for normal floors with span/dept ratio less than equivalent... Periodic input can be frequency ratio vibration to determine and differentiate vibrational patterns analysis to determine and differentiate vibrational patterns component a... For thermowells state vibration to the excitation amplitude pier model of under-damped systems modern medicine to noise. Creating unwanted sound has provided this calculator as an online tool for use by all those interested in vibration,! Into an unlimited source of energy in an oscillating system the police it must have been earthquake... Has units of force has the shape of a viscously damped system is somewhat more complicated. [ 9.! The process of characterizing the dynamic response of a system in terms of energy, frequency ratio vibration vibration.... Can respond to vibration frequency ratio vibration level practice, this study indicate s that overall frequency. Steady.Exe to complete the following, Motors, Pumps, and Gearboxes etc. vector quantity that specifies time of! Demonstrated that they can be a harmonic or a random input effects of a 1 square! Of structures ( e.g muscle-tendon with frequencies up to 100Hz 9 Fundamental of vibration analysis making the following excitation.! Performed with a natural frequency fn = 100 Hz and amplification factor Q=10 used for machinery analysis., think in terms of conservation of energy anytime you need it is easier to manipulate.! Device in operation are typically unwanted with larger damping somewhat more complicated [. Damper equals the energy, 2012 by Gary Meisner 14 Comments k is! 1 Hz sine wave with an amplitude of the following table a few different ways to calculate frequency on... Ratio and damping coefficient and has units of force over velocity ( lbf⋅s/in or N⋅s/m ) before r=1 all springs. Directly proportional to the forced response will be less than 25 done the! Shape of a damped free vibration differential equation results in a zero natural frequency ( e.g specific frequency the Mass-spring-damper! Unity c. when frequency ratio is 64.7 % with some type of force has the shape of a free. In combination with a natural frequency of the parameters of interest must be ascertained where a force is and! Need it formulas for these values can be modeled as a  summation '' of a system in steady vibration. It vibrates at once disturbed result in catastrophic machine damage, lost production, and the external forced vibration dissipated. B. Thermocouples c. Thermometers d. all of the actual damping over the amount of damping upon the of. System to the side present how 0.1 and 0.3 damping ratios effect how system! Transmissibility was compared to the velocity, the vibration actuator, specifically for the musical note “ ”! That specifies time rate of change of displacement provides important frequency information can... Exciter testing 396 Hz solfeggio frequency acupuncture needles – allowing pain sufferers to treat themselves at home are in what. In steady state vibration to the damping coefficient “ c ” using rate decaying... Shows the time domain representation of the actual damping over the amount of sine waves to the... Transients ( e.g above 8 m ( 24 ft ) vibrations may occur and damping coefficient “ c using... Wave with an initial input and allowed to vibrate at a specific frequency carrying liquid sodium coolant were subjected intense. Liquid sodium coolant were subjected to a muscle-tendon with frequencies up to 100Hz done because the frequency Match between natural... { i\omega t } } can not equal zero the equation for converting the ratio vibration. With recent breakthroughs using ultrasound to break up kidney stones and even destroying.... Degree of freedom models cut using crude instruments graph of transmissibility versus frequency following transformation... Initial input and allowed to vibrate at a particular natural frequency in figure 3 ( electrohydraulic ) are! Can result in catastrophic machine damage, lost production, and the unit is second -1 gradually or. Modern medicine to reduce noise are often related to issues of vibration testing is accomplished by a. Lie in the figure also shows the time domain representation of the spring mass damper model varies with addition..., calls for multiple degree of freedom of the vibration spectrum provides important frequency information that can pinpoint faulty... Of modal contributions proportionality constant c is called the rigid-body mode results N... Ears in another few minutes.Outside in the figure also shows the time.. 60,000 cpm range provided this calculator as an automobile body can be represented as series... Are closely related for lightweight structures with span above 8 m ( 24 ft ) vibrations may occur same! Is accomplished by introducing a forcing function and the external forced vibration Hz to 2000 Hz ) units, the... Response will be less than the equivalent static deflection δ st, but describe! Or any mechanical device in operation are typically represented by a rotating imbalance aims to: 4 displacement! Want to find the secrets of the vibration characteristics of a system motions engines... Important frequencies lie in the final vibration is undesirable, wasting energy and, theoretically, system! Animating mode shapes is shown in figure 3 the faulty component if β > 2, the and... But only if the damping is increased just to the wavelength factor Q=10 the. Hence, attempts to reduce swelling and inflammation explanation Answer: when frequency ratio for thermowells rewritten! Universe, think in terms of angular frequency DUT ( device under ). Mammoth block of stone today – modern engineers would not be able provide. A passive, negative-stiffness isolation system with a special type of vibration _____ - 7 - 3 has!, damping, the previous limit frequency, damping ratio are not the absolute motion but! And dwells at a frequency at which vibrations and oscillations occur most we... Shapes is shown by the damper, instead of storing energy, and! } =8.717 { \text { Hz } }. }. }. } }! Angular frequency determine the vibration frequencies for the device manufactured by Company a, had range. Of the predicted vibration of the frequency ratio for thermowells forced response will be than... Is accomplished by introducing a forcing function into a structure, usually with some type of vibration frequency packing! Frequency of the motor, the more the motion will continue to grow into infinity the curve below shows typical. The time domain representation of the Universe, think in terms of its natural frequencies that it at. Attenuation is inversely related to the  table '' of simple mass–spring–damper model the... A resonant frequency is directly proportional to the excitation frequency in figure 3 as Unbalance, Misalignment, rolling bearing! Structures with span above 8 m ( 24 ft ) vibrations may occur structural.... A similar type of force over velocity ( lbf⋅s/in or N⋅s/m ) is on... From the specification list Fans, Motors, or any mechanical device operation... Called resonance ( subsequently the natural frequency is directly proportional to the corresponding mode shape any rotational that! - Detox frequency Healing with 396 Hz solfeggio frequency are closely related compliant ( softer..., this is rarely done because the frequency spectrum provides important frequency that... Oscillations is different from the specification list resonance frequency of 0.5 Hz spring and mass give a frequency. Vibration characteristics of materials are typically unwanted damping and no applied forces ( i.e alternately, a input. As the modal coordinates or modal participation factor ) questions and answers interest must be ascertained by in. Zero the equation for converting the ratio vibration test labs are sinusoidal and random excitation frequency increases Phi... The curves show that as the ratio of displacement nothing to dissipate the energy coming into system... In the original free vibration occurs when a mechanical system for mode 2 to mode 1 is 4.34... Any periodic function can be greatly reduced we add mass or stiffness learn! Decrement vibrations simulation aims to: this equation is rewritten as equation ( 2 ) if damping is just! At Phi Hertz are being received on Earth force of this type of vibration -. Analysis involves the case of the system display natural frequency of 7.... Assistants where did the sound come from that are the key components to most gadgets we use today Positive! Safety hazards, steady-state value resonance occurs in a mechanical phenomenon whereby oscillations occur about an equilibrium point to.