18 Modal Modal Analysis of PLate in Ansys 19. Feb 27, 2023 · Understanding Modal Frequency Analysis Setup. k is the spring constant for the spring. 60. The 14. Critically Damped System. If the amplitudes of the vibrations are large enough and if natural frequency is within the human frequency range, then the vibrating object will produce sound waves that A modal analysis calculates the undamped natural modes of a system, characterised by their modal frequency and mode shape. 1) Modal Calculation the more exact eigenvalue extraction method and 2) Raleigh Frequency Method, the approximation method Here we will discuss about Modal Calculation. 1. Figure 1: Left – Modal test setup for an aircraft, Right fundamental natural mode. The natural frequency of a system is dependent only on the stiffness of the structure and the mass which participates with the structure (including self-weight). The lighter the mass of the moving elements in the exciter, the less this problem will be. , the j-modal damping ratio increases as the natural frequency increases. 1. This yields the approximate value of ω1 2. is the stiffness matrix (which includes initial stiffness effects if the base The modal analysis module of Caesar II dynamic analysis is also used to calculate the natural frequency of pipe systems connected to compressors and reciprocating pumps. For a sinusoidal wave, the angular frequency refers to the angular displacement of any element of the wave per unit of time or the rate of change of the phase of the waveform. ” [1] Why do we perform modal analysis? Modal analysis is the most fundamental type of structural dynamics analyses. The frequency of a mode is often called a “modal frequency”, “resonant frequency”, or “natural frequency”. So hitting (impulsive forcing function) a 256 Hz tuning fork (system) on a table will cause the Eigenvalue extraction. You can find the spring constant for real systems through experimentation, but for most problems, you are given a value for it. Now, we can write down the solution for x: Overdamped System. It plays a very important role, as we shall see below. The value I used in the first model was 80 MPa so the natural frequency will drop significantly. 5*E*Strain*Strain. The default value is 1. The characteristic frequency is known as the natural frequency of the system. The formula for the natural frequency fn of a single-degree-of-freedom system is. Get the natural frequency from STAAD. 6. Vibrating objects can have one or more normal modes, depending on how they’re Jul 19, 2023 · Natural frequency is the performance of structural natural characteristics, while resonance frequency is the performance of structural response under external forces. 03 Differential Equations, Supplementary Notes Ch. 18 Hz and mode 2 is 0. (A-28) The mass term m is simply the mass at the end of the beam. Oct 30, 2007 · resonance (natural) frequency of a cantilever beam is given by f=[kn/2pi][sqrt(EI/wL^4)] where, kn=3. These modes are numbered, from 1, in order of increasing frequency. Calculate corresponding time period. One cycle per second is equivalent to 1 Hertz. For example, for a particular input (like an applied load of certain amplitude and frequency), what are the limits of the system’s response (for example, when and what is the maximum displacement). Mar 4, 2020 · The natural frequency of a structure is the frequency at its free or natural vibration. 2. Each “mode” of the structure has a specific frequency, damping, and deflection shape associated with it. Every structure has the tendency to vibrate at certain frequencies, called natural or resonant frequencies. Also, natural frequency and resonance are explained. Young). 2 . , resonant frequency = natural frequency. This presentation covers: What is a natural frequency and why do they exist; How to conduct a basic bump test with a single channel analyzer; What is a modal analysis and what additional information does it provide Dec 6, 2020 · To calculate the natural frequency using the equation above, first find out the spring constant for your specific system. For the oscillating system to maintain a constant amplitude of oscillation, it is necessary to apply Aug 17, 2021 · Once the data is all in the frequency domain, some additional mathematical operations are performed on it to obtain a vibration spectrum plot (displacement versus frequency). There is a standard, and useful, normalization of the second order homogeneous linear constant coefficient ODE. If it is made in the design phase, numerical simulations on virtual models or experiments on physical models or prototypes can predict the resonant frequencies [ 2 ]. Estimate the slope at a small strain, like at 0. Jun 5, 2019 · Without damping, the load and displacement flip from being perfectly in phase below the natural frequency to being 180° out-of-phase above the natural frequency. (this is from Formulas for Stress and Strain, 5th edition by Raymond J. 13. The "natural" frequency of a system, is that system's response to an impulsive forcing function. Acoustic range factor. Single-degree-of-freedom System Figure 1. Modal Calculation to be used to obtain full scale Oct 1, 2008 · The exact values at which the first natural frequency vanishes are called the critical speeds and afterwards the system is unstable about the zero equilibrium. Complex systems, with several natural frequencies, load the vibration exciter each time the excitation frequency coincides with a natural frequency. 4. function [freqs,modes] = compute_frequencies (k1,k2,k3,m1,m2) Mar 17, 2022 · Note that these curves are normalized with respect to the natural frequency. The attached file is an R18. Dependence: Frequency depends on the external force or input that drives the system, whereas natural frequency is solely determined by the system's physical properties. 5 Natural Frequencies and Mode Shapes. Roark and Warren C. Feb 7, 2012 · Therefore Strain energy = 0. Feb 23, 2023 · The signal level of the force may, in fact, drop to the noise floor in the instrumentation. M MN M M N. Because Here, the modal class is the data interval with the highest frequency. It is not Aug 11, 2023 · Natural frequency is the rate at which a body vibrates when disturbed without being subject to a driving or damping force. These fixed frequencies of the normal modes of a system are known as its natural frequencies or Resonance. Frequency Analysis. m is mass; k is stiffness; Reciprocal of the natural period of a building is nothing but the natural frequency and its unit is Hertz(Hz) Thank you. May 22, 2022 · Figure 10. 03 as the constant for the second mode, The frequency or frequencies at which an object tends to vibrate with when hit, struck, plucked, strummed or somehow disturbed is known as the natural frequency of the object. fn = modalfit(frf,f,fs,mnum) estimates the natural frequencies of mnum modes of a system with measured frequency-response functions frf defined at frequencies f and for a sample rate fs. Angular frequency formula and SI unit are given as: Where, ω = angular frequency of the wave. May 1, 2013 · Where Ω 1 is the natural frequency of mode one i n (rad. Natural frequency and damping ratio. We saw that the spring mass system described in the preceding section likes to vibrate at a characteristic frequency, known as its natural frequency. k. July 19th, 2023. 3. To calculate natural frequency: a) Determine both ‘k’ (stiffness of spring) and ‘m’ (mass attached). 3 Natural Frequencies and Mode Shapes. On the other hand, natural frequency is calculated Identification of natural frequencies, modal damping, and mode shapes of a structure based on FRF measurements is called Modal Analysis. expresses the frequency domain relationship between an input (x) and output (y) of a linear, time 18. the upper frequency fu and the lower frequency fl: fl = [fc]/[ 2 1/12], fu = [fc ][2 1/12], where fc is the center frequency. Thus, the mode can be calculated by the formula: Mode = L+h (fm−f1) (fm−f1)+(fm−f2) L + h ( f m − f 1) ( f m − f 1) + ( f m − f 2) where, L is the lower limit of the modal class. Here, l = Lower limit of the modal class. Using the Results of Modal Analysis Figure 12 shows that natural frequency on the Y-axis rotation modal sensitivity with the increase of the leg 4 stiffness k d increased significantly, the inherent frequency movement modal of X-axis and Y-axis modal, Z-axis movement mode, the X-axis rotation mode, the Z-axis rotation modal sensitivity with the increase of the stiffness k d leg 4 Oct 10, 2022 · In Eq. is the mass matrix (which is symmetric and positive definite); KMN K M N. A modal analysis calculates the frequency modes or natural frequencies of a given system, but not necessarily its full-time history response to a given input. 7: Natural frequency vs mode number Natural Frequency. 2 Object Excitation by Means of an Impulse Hammer The reliability and performance of the mechanical systems can be optimized, and potential design issues can be identified by analyzing its frequency response. Hence, C-type composite leaf springs possess the highest natural frequency although the G-type composite leaf springs have the least natural frequency. Then you can input the frequency and the damping ratio and the program will calculate β for you. [M] = global mass matrix. 9 and 1. , by finite element software, you For the better explanation I will mention the steps. In an experimental modal, a physical structure is tested, and its modes of vibration are identified ( Figure 1 ). /sec); C 1 = 3. Calculate design spectral acceleration for the time period using the formula from code or the acceleration plot. The eigenvalue problem for the natural frequencies of an undamped finite element model is. The peak acceleration for a particular frequency is the exact value for that exact frequency, . The natural frequencies of a structure are affected by tensile or compressive stresses resulting from applied loads. For the calculation, the elastic modulus E of the beam should be This is a very important observation, and we will expand upon it below. m is the mass of the ball. 7 we can see that there is no particular trend of increasing or decreasing order. 1 Hz band thus has lower and upper limits of 13. The acoustic range factor must be greater than 0. Use modal analysis to calculate the natural frequencies and mode shapes of your model. The Maximum Displacement Magnitude vs Frequency plot is showing extremely high displacement values. c) Solve for f. Increase of amplitude as damping decreases and frequency approaches resonant frequency of a driven damped simple harmonic oscillator. used to identify the resonant frequencies, damping and mode shapes of a physical structure. Introduction The purpose of this report is to discuss frequency response functions. A Frequency Response Function (or FRF) is: is a frequency based function. axially moving speeds for different k 4: (a) the first mode and (b) the second mode. I would like to derive this formula. 18 . where is known as the damped natural frequency of the system. Feb 24, 2005 · to specify the modal frequency and significant wave height, this spectrum can be used for sea states of varying severity from developing to decaying. 400 Fig. There are as many natural frequencies as natural modes. To extract the ith frequency and mode shape, use. In all the preceding equations, are the values of x and its time derivative at time t=0. 1 Ł The following static fundamental frequency estimation method can be used as a “ballpark” check for grounded finite element model modal analysis frequencies. Mechanical resonance is the tendency of a mechanical system to respond at greater amplitude when the frequency of its oscillations matches the system's natural frequency of vibration (its resonance frequency or resonant frequency) closer than it does other frequencies. Apr 30, 2024 · The previous study suggests that vibratory energy stabilizes when the modal frequency and modal damping are equalized across both modes of vibration, while keeping the remaining parameters constant [25, 27, 28]. frf is assumed to be in dynamic flexibility Hydrodynamic Mass, Natural Frequencies and Mode Shapes Flow-Induced Vibration Handbook for Nuclear and Process Equipment Transverse Free Vibration Analysis of Hybrid SPR Steel Joints A good understanding of natural frequencies and the tests necessary to identify them will help solve these vexing situations. 2. 1 3 EI fn. Low-rise buildings have high natural frequencies Figure 13. The Frequency Domain, Modal () study and study step are used, for example, to compute the response of a linear or linearized structural mechanics model subjected to harmonic excitation for one or several frequencies. (−ω2M MN+KMN)ϕN =0, ( - ω 2 M M N + K M N) ϕ N = 0, where. The pattern or shape of this vibrating motion is the corresponding mode of the body’s or system’s vibration, known as the normal mode. K - structure stiffness; m0 - reduced mass of the structure. The purpose of modal testing is to identify the natural frequencies, damping ratios, and May 25, 2022 · A modal basis is the series of structural modes (mode shape + natural frequency) associated to a linear structure within a given frequency range. The graphs above for a mechanical oscillator show how the peak in Aug 30, 2016 · Once the lowest (or fundamental) frequency has been fixed by choosing the weight, tension and length of the string, then all the other frequencies are whole-number multiples: if the first is f, then the second is 2f, the third 3f and the nth is nf . 3. But these names are only naming convention after the case that the damping factor equals 0 (the only case we can state that w is the natural frequency, as it is the only factor Global Technical Support. These functions are used in vibration analysis and modal testing. A spring with a higher constant is stiffer and requires additional force to extend. For damping proportional to stiffness only, 0, (structural damping) and 2 2 j nj j jj K KM (13b) i. Download : Download full-size image; Fig. Aug 31, 2023 · The Modal Assurance Criterion Analysis (MAC) analysis is used to determine the similarity of two-mode shapes. fn. [1] [2] In physics, resonance refers to a wide class of phenomena that arise as a result of matching temporal or spatial periods of oscillatory objects. The Frequency, Domain Modal study consists of two study steps: an Eigenfrequency study step for computing the eigenfrequencies ratio ζ, and natural frequency ωn, where ! k (1. Feb 2, 2017. Oct 9, 2020 · This article explains some of the key steps involved in performing a modal test, from start to end. Use modalfrf to generate a matrix of frequency-response functions from measured data. Natural frequency of the system is the frequency at which it will vibrate freely. , the j-modal damping ratio decreases as the natural frequency increases. A similar result is obtained for the modes of vibration of a continuous system such as a beam. m. In practice, any value between 0. Calculation: Frequency can be calculated using the formula f = 1 / T, where 'T' represents the period of the oscillation. Calculate design horizontal seismic coefficient Ah. Then, the first four-grades natural frequency of the testing beam model at −40 °C, −20 °C, 0 °C, 20 °C, 40 °C, and 60 °C was successively obtained. Results also indicate that damping ratios depend on both the natural frequency and the mode shapes. Description. Every physical system has natural frequencies associated with it, which depend on the system’s mass, damping and stiffness properties. This factor applies only to structural-acoustic problems and is used to set the maximum frequency for the acoustic stage of the uncoupled eigenproblem as a multiple of the nominal maximum frequency of interest. It is common to use the finite element method (FEM) to perform this analysis because, like other calculations using the FEM, the object being analyzed can have arbitrary shape and the results of the calculations are acceptable. 7) Second natural frequency (4. Response at the natural frequency The frequency response at ω = ωn, β = 1, consists of phase angle ϕ(ωn) = − 90 ∘ regardless of the value of viscous damping ratio ζ, and magnitude ratio X(ωn) / U = 1 / (2ζ). These are the normal modes of the system, and the ω’s are the natural frequencies. Modal is the simplest analysis and the only thing it does is telling you what are the “resonance frequencies” of your geometry. Jul 30, 2023 · For this, it must be achieved that the natural frequency of the suspension is as far away as possible from the frequency ranges to be investigated, e. Truss Bridge | Modal Analysis | How to find Natural Frequency | ANSYS WorkbenchThis video shows how to find the natural frequency of a truss bridge using Mod Oct 23, 2022 · The natural characteristics are natural frequency, damping, and mode shapes. 5. Mechanical resonance. frf is assumed to be in dynamic flexibility Comparison of Results:Natural frequency vs mode number have been compared for different materials. It is represented by ω. I then ran a Modal Frequency Analysis on the bracket to evaluate the displacement at the different Natural Frequencies. Modal analysis. The first natural frequency is now 0. With damping, the transition in phase shift is smooth, as shown in the graph below. Here, the lower-frequency sounds are emitted by the large speaker, called a woofer, while the higher-frequency sounds are emitted by the small speaker, called a tweeter. Above the resonant frequency, the base and mass move out of phase. Analytical calculations of the natural frequency of the plate are c This is the mode formula for grouped data in statistics. The free motion described by the normal modes takes place at fixed frequencies. At the natural frequency, the base and mass move 90 degrees apart, which creates a kind of “bucking” motion causing the high levels of vibration. Underdamped System. Sep 17, 2019 · The slope of the blue line is the linear elastic property of Young's Modulus. From Table IV and Fig. For an oscillatory dynamical systems driven by a Normal mode. • Solving the free-free modal analysis numerically, e. A closed form of the circular natural frequency ω nf, from above equation of motion and boundary conditions can be written as, (4. 8) Third natural frequency (4. = the eigenvalue for each mode that yields the natural frequency =. The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration. f 0 = Frequency of the class preceding the modal class. example. In other words, higher modes are increasingly more damped than lower modes. Each natural frequency is associated with a certain shape, called mode shape, that the model tends to assume when vibrating at that frequency. You should see that Then the natural frequency wn in radians per second is: wn=sqrt(k/m) and the cyclic natural frequency fn (in Hz) is calculated with this equation fn=wn /(2*pi). 16 Modal damping generated by the under-platform damper and its distribution versus kinetic energy considering the uncertainties of contact parameters [87] Krack 等 [88] Fig. Dec 12, 2017 · If you want to use Damping Ratio instead, click on the pull down on Stiffness Coefficient Define By and select Damping vs Frequency. The frequencies are called the natural frequencies or overtones, and this simple numerical Abstract. Again, certain RLC circuits will have similar curves, while others (e. 0 is considered a good correlation. Resonance frequencies change due to the shape of your model and the way it Increasing the mass reduces the natural frequency of the system. Irrespective of the damping level, the phase shift at the undamped natural frequency is always 90°. Modal Analysis Method Inventor Nastran determines the natural frequency by solving the eigenvalue problem: where: [K] = global linear stiffness matrix. I ran a Normal Mode analysis on a bracket to find out its Natural Frequencies. This is expected as there is no boundary condition. It is observed that structural steel gives the maximum frequency. Formula to find out natural period is, Tn =2π√(m/k) where, Tn is a natural period of building. One can show that these structural modes are Apr 21, 2022 · ω = √ (k ÷ m) This, in turn, adjusts our formula to the following: f = √ (k ÷ m) ÷ 2π. • Zero natural frequency means that the structure can have rigid movement without any excitation. Flextural rigidity = E*I (for any beam or rod element) So when E increases, Flextural rigidity increases. Fig. 2 √ km The natural frequency ωn is the frequency at which the system would oscillate if the damping b were zero. For a simple mass-spring system, the natural frequency is given by Equation (1); f = 1/2π √ ( k / m ) —– (1) Where f is the natural frequency (Hz), and k and m are the stiffness and mass respectively. e. 1 and it would come out to about 1. We measure the spring constant in Newtons per meter. 48 Hz. Mode Formula Class 10 For a free-free modal analysis, theoretically, the first 6 natural frequencies must be 0. The MAC number is defined as a scalar constant, between 0 and 1, expressing the degree of consistency between two mode shapes. The Ochi Spectrum (equation 16) is a three parameter spectrum that allows the user to specify the significant wave height, the modal frequency, and the steepness of the spectrum peak. 15. b) Use the formula provided above. The natural frequencies vs. 40) ωn = m (1. Spikes with wide bases appearing in the plot point to the natural frequencies of the structure. AN INTRODUCTION TO FREQUENCY RESPONSE FUNCTIONS By Tom Irvine Email: tomirvine@aol. Rayleigh’s method The above equation can be used to find an approximate value of the first natural frequency of the system. The modeling of a n-DOF mechanical system leads to a set of n-coupled 2nd order ODEs, Hence the motion in the direction of one DOF, say k, depends on or it is coupled to the motion in the other degrees of freedom, j=1,2…n. The natural frequency fn is m k 2 1 fn S Frequency is defined as so many cycles in a given time period. When a vibration load applied to the structure matches the natural frequency, it can be dangerous leading to the destruction of The following can be observed: Below the natural frequency, the base and mass move together in phase. Most of us have played with toys where an object bobs up and down on an elastic band, something like the paddle ball suspended from a finger in Figure 14. In this case, the frequency of natural vibrations will be equal to: f = [K / m0] 1/2. f is the natural frequency. Since stiffness is directly proportional to Natural frequency, and Strain energy and Stiffness is directly Apr 12, 2023 · Calculate the natural frequencies of your structure in Structural 3D Dynamic Frequency Analysis, or modal analysis, is needed to determine the natural frequencies (or resonant frequencies) of a structure in order to predict its maximum response. This turns out to be a property of all stable mechanical systems. Mar 11, 2024 · In contrast to quasi-static and dynamic, modal analysis provides an overview of the limits of the response of a system. 8 MPa. Increasing the stiffness of the spring increases the natural frequency of the system; Increasing the mass reduces the natural frequency of the system. = the eigenvector for each mode that represents the natural mode shape. 526 is the constant for the first mode; hence eq uation one above with C 2 = 22. sometimes referred to a “transfer function” between the input and output. Mechanical. The natural frequency of the cantilever beam with the end-mass is found by substituting equation (A-27) into (A-28). f 2 = Frequency of the class succeeding the modal class. , the series RLC circuit) will have curves that always peak at the natural frequency, i. When a structure is properly excited by a dynamic load with a frequency The frequency extraction procedure: performs eigenvalue extraction to calculate the natural frequencies and the corresponding mode shapes of a system; will include initial stress and load stiffness effects due to preloads and initial conditions if geometric nonlinearity is accounted for in the base state, so that small vibrations of a preloaded frequency knows as a natural frequency. The information extracted from modal analysis often acts as inputs to other types of analyses such as: Response Spectrum Analysis; Random Vibrations The natural frequency is determined by the roots of the differential equation, which in turn are characterized by the "damping factor" alpha and the "frequency" omega. In this calculation, a cantilever beam of length L with a moment of inertia of the cross-section Ix and own mass m is considered. A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. 8 Because they travel at the same speed in a given medium, low-frequency sounds must have a greater wavelength than high-frequency sounds. Frequency Calculation by STAAD. m mass k stiffness Consider the SDOF system shown in Figure 1. Harmful vibrations will result when the pipe’s natural frequency is close to that of connected rotary equipment. The analysis may be done either experimentally or mathematically. The eigenvalue is related to the system’s Description. 52 for mode 1, E is Young's modulus, I is moment of Inertia, w is beam width, L is beam length. 9 Hz, respectively. 39) and b ζ =. Pro There are two method one can calculate Frequency of Structures. Modal analysis is the study of the dynamic characteristics of a system independent from the loads applied to that system. This indicates that the peak energy levels in both modes reach a state of minimal energy, resulting in an optimal response. For this reason, Autodesk Fusion includes an option to Compute Preloaded Modes. T = time period of the wave. Jun 21, 2021 · The first four-grades natural frequency of the testing beam model obtained at a fundamental frequency temperature of 20 °C were taken. It may cause violent swaying motions and potentially i. Insert this value into the spot for k (in this example, k = 100 N/m), and divide it by the mass Formula: Natural Frequency (f) = (1 / 2π) * √(k / m) Where: – k is the spring constant or stiffness – m is the mass attached to the spring. natural frequency of the suspension at most 1/5 of the first mode of the structure to be investigated. 2 archive (Domenico, you have to upgrade to Restore Archive). Nov 18, 2016 · Modal analysis calculates the natural frequencies of the system alone. under the assumption that both the “mass” m and the “spring con stant” k are positive. Since every real oscillating systems experiences some degree of damping, if no external energy is supplied, the system eventually comes to rest. 9) The natural frequency is related with the circular natural frequency as The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance, and a system being driven at its natural frequency is said to resonate. Modal analysis is widely used to describe the dynamic properties of a structure in terms of the modal parameters: natural frequency, damping factor, modal mass and mode shape. g. This first mode is also called V mode as illustrated in Figure 1. The modes are identified by natural frequency, damping, and mode shape. , \({w}_{n}\) is the natural frequency, \(k\) is the spring rate, and \(m\) is the weight. omega = sqrt (D (i,i)) X = V (:,i) For example, here is a MATLAB function that uses this function to automatically compute the natural frequencies of the spring-mass system shown in the figure. The damping ratio ζ is the ratio of the actual damping b to the critical damping bc = 2 √ km. These fixed frequencies of the normal modes of a system are known as its natural Description. mx ̈ + bx ̇ + kx = 0. 3 Hz and 14. Damped frequency is lower than natural frequency and is calculated using the following relationship: wd=wn*sqrt (1-z) where z is the damping ratio and is defined as the ratio of the system damping to the critical damping coefficient, z=C/Cc where Cc, the critical damping coefficient, is defined as: Cc=2*sqrt (km). Free vibration of single degree of freedom spring system […] Shane. And when E increases, as seen in the previous equation, Strain energy also increses. 1: Frequency response functions for standard 2 nd order systems with viscous damping ratios ζ varying from 0 to 1. Apr 11, 2024 · Modal analysis is a procedure to estimate the natural frequencies of structures and its mode shapes, to get information about the dynamic behavior of the structure . Any numerical matrix method–such as MATLAB– will yield both the λi’s (called the eigenvalues) and the Xi’s, called the eigenvectors for a particular matrix [A]. For this, we select a trial vector X to represent the first natural mode X(1) and substitute it on the right hand side of the above equation. where. You can also see the response to the natural frequencies of your model when it is subjected to time-dependent and/or oscillatory/vibration loads by running any dynamic analysis: dynamic time, dynamic frequency, dynamic random, or dynamic shock Jun 18, 2020 · The natural period is one of the properties or building which is controlled by mass and stiffness. Modal analysis is a standard technique, well-documented in the literature: we give here a brief description of the underlying theory. It isn’t related to a loading at this stage, only to the geometry. com August 11, 2000. h is the size of the class interval. h = Size of the class interval (assuming all class sizes to be equal) f 1 = Frequency of the modal class. 6) Where So, First natural frequency (4. For example if a simply supported beam is excited at its first natural frequency, it will deform by following its first mode shape. pb gk wv cx dh vl ww zv wk sf