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[[Berkas:Drum_vibration_mode21.gif|thumb|Salah satu mode getaran [[gendang]]]]
'''Getaran''' adalah suatu gerak bolak-balik di sekitar kesetimbangan. Kesetimbangan di sini maksudnya adalah keadaan dimana suatu benda berada pada posisi diam jika tidak ada [[gaya]] yang bekerja pada benda tersebut. Getaran mempunyai [[amplitudo]] (jarak simpangan terjauh dengan titik tengah) yang sama.
== Jenis getaran ==
'''Getaran bebas''' terjadi bila sistem mekanis dimulai dengan gaya awal, lalu dibiarkan bergetar secara bebas. Contoh getaran seperti ini adalah memukul [[garpu tala]] dan membiarkannya bergetar, atau bandul yang ditarik dari keadaan setimbang lalu dilepaskan.
'''Getaran paksa''' terjadi bila gaya bolak-balik atau gerakan diterapkan pada sistem mekanis. Contohnya adalah getaran gedung pada saat [[gempa bumi]].
== Analisis getaran ==
Dasar analisis getaran dapat dipahami dengan mempelajari model sederhana [[massa]]-[[pegas]]-[[peredam kejut]]. Struktur rumit seperti badan mobil dapat dimodelkan sebagai "jumlahan" model massa-pegas-peredam kejut tersebut. Model ini adalah contoh [[osilator harmonik sederhana]].
=== Getaran bebas tanpa peredam ===
[[Berkas:Mass spring.svg|200px|right|thumb|Model massa-pegas sederhanal]]
Pada model yang paling sederhana redaman dianggap dapat diabaikan, dan tidak ada gaya luar yang mempengaruhi massa (getaran bebas).
Dalam keadaan ini gaya yang berlaku pada pegas ''F<sub>s</sub>'' sebanding dengan panjang peregangan ''x'', sesuai dengan [[hukum Hooke]], atau bila dirumuskan secara matematis:
:<math>
F_s=- k x \!
</math>
dengan ''k'' adalah tetapan pegas.
Sesuai [[Hukum kedua Newton]] gaya yang ditimbulkan sebanding dengan [[percepatan]] massa:
:<math>
\Sigma\ F = ma = m \ddot{x} = m \frac{d^2x}{dt^2} =
</math>
Karena ''F'' = ''F<sub>s</sub>'', kita mendapatkan [[persamaan diferensial biasa]] berikut:
:<math>m \ddot{x} + k x = 0.</math>
[[Berkas:Simple harmonic oscillator.gif|thumb|100px|right|Gerakan harmonik sederhana sistem benda-pegas]]
Bila kita menganggap bahwa kita memulai getaran sistem dengan meregangkan pegas sejauh ''A'' kemudian melepaskannya, solusi persamaan di atas yang memerikan gerakan massa adalah:
:<math>
x(t) = A \cos (2 \pi f_n t) \!
</math>
Solusi ini menyatakan bahwa massa akan berosilasi dalam [[gerak harmonis sederhana]] yang memiliki [[amplitudo]] ''A'' dan frekuensi ''f<sub>n</sub>''. Bilangan ''f<sub>n</sub> adalah salah satu besaran yang terpenting dalam analisis getaran, dan dinamakan '''frekuensi alami takredam'''. Untuk sistem massa-pegas sederhana, ''f<sub>n</sub>'' didefinisikan sebagai:
:<math>
f_n = {1\over {2 \pi}} \sqrt{k \over m} \!
</math>
Catatan: [[frekuensi sudut]] <math>\omega</math> (<math>\omega=2 \pi f</math>) dengan satuan radian per detik kerap kali digunakan dalam persamaan karena menyederhanakan persamaan, namun besaran ini biasanya diubah ke dalam frekuensi "standar" (satuan [[Hertz|Hz]]) ketika menyatakan frekuensi sistem.
Bila massa dan kekakuan (tetapan ''k'') diketahui frekuensi getaran sistem akan dapat ditentukan menggunakan rumus di atas.
=== Getaran bebas dengan redaman ===
[[Berkas:Mass spring damper.svg|200px|right|Mass Spring Damper Model]]
Bila peredaman diperhitungkan, berarti gaya peredam juga berlaku pada massa selain gaya yang disebabkan oleh peregangan pegas. Bila bergerak dalam [[fluida]] benda akan mendapatkan peredaman karena kekentalan fluida. Gaya akibat kekentalan ini sebanding dengan kecepatan benda. Konstanta akibat kekentalan (viskositas) ''c'' ini dinamakan koefisien peredam, dengan satuan N s/m (SI)
:<math>
F_d = - c v = - c \dot{x} = - c \frac{dx}{dt} \!
</math>
Dengan menjumlahkan semua gaya yang berlaku pada benda kita mendapatkan persamaan
:<math>m \ddot{x} + { c } \dot{x} + {k } x = 0.</math>
Solusi persamaan ini tergantung pada besarnya redaman. Bila redaman cukup kecil, sistem masih akan bergetar, namun pada akhirnya akan berhenti. Keadaan ini disebut kurang redam, dan merupakan kasus yang paling mendapatkan perhatian dalam analisis vibrasi. Bila peredaman diperbesar sehingga mencapai titik saat sistem tidak lagi berosilasi, kita mencapai titik '''redaman kritis'''. Bila peredaman ditambahkan melewati titik kritis ini sistem disebut dalam keadaan lewat redam.
Nilai koefisien redaman yang diperlukan untuk mencapai titik redaman kritis pada model massa-pegas-peredam adalah:
:<math>c_c= 2 \sqrt{k m}</math>
Untuk mengkarakterisasi jumlah peredaman dalam sistem digunakan nisbah yang dinamakan [[nisbah redaman]]. Nisbah ini adalah perbandingan antara peredaman sebenarnya terhadap jumlah peredaman yang diperlukan untuk mencapai titik redaman kritis. Rumus untuk nisbah redaman
(<math>\zeta </math>) adalah
:<math>\zeta = { c \over 2 \sqrt{k m} }.</math>
Sebagai contoh struktur logam akan memiliki nisbah redaman lebih kecil dari 0,05, sedangkan suspensi otomotif akan berada pada selang 0,2-0,3.
Solusi sistem kurang redam pada model massa-pegas-peredam adalah
:<math>x(t)=X e^{-\zeta \omega_n t} \cos({\sqrt{1-\zeta^2} \omega_n t - \phi}) , \ \ \omega_n= 2\pi f_n </math>
Nilai ''X'', amplitudo awal, dan <math> \phi </math>, [[Fase (gelombang)|ingsutan fase]], ditentukan oleh panjang regangan pegas.
Dari solusi tersebut perlu diperhatikan dua hal: faktor eksponensial dan fungsi cosinus. Faktor eksponensial menentukan seberapa cepat sistem teredam: semakin besar nisbah redaman, semakin cepat sistem teredam ke titik nol. Fungsi kosinus melambangkan osilasi sistem, namun frekuensi osilasi berbeda daripada kasus tidak teredam.
Frekuensi dalam hal ini disebut "frekuensi alamiah teredam", ''f<sub>d</sub>'', dan terhubung dengan frekuensi alamiah takredam lewat rumus berikut.
:<math>f_d= \sqrt{1-\zeta^2} f_n </math>
Frekuensi alamiah teredam lebih kecil daripada frekuensi alamiah takredam, namun untuk banyak kasus praktis nisbah redaman relatif kecil, dan karenanya perbedaan tersebut dapat diabaikan. Karena itu deskripsi teredam dan takredam kerap kali tidak disebutkan ketika menyatakan frekuensi alamiah.
<!--Grafik di samping menampilkan bagaimana nisbah redaman sebesar 0,1 dan 0,3 akan mempengaruhi bagaimana sistem akan bergetar seiring berjalannya waktu. Yang sering dilakukan dalam praktik adalah mengukur getaran bebas setelah sebuah pukulan (misalnya dengan palu), dan kemudian menentukan frekuensi alamiah sistem dengan mengukur laju osilasi, serta nisbah redaman dengan mengukur laju peluruhan. Frekuensi alamiah dan nisbah peredaman tidak hanya penting dalam getaran bebas, tetapi juga mencirikan bagaimana sistem akan berkelakuan pada getaran paksa. -->
=== Getaran paksa dengan redaman ===
{{sect-stub}}
<!-- In this section we will look at the behavior of the spring mass damper model when we add a harmonic force in the form below. A force of this type could, for example, be generated by a rotating imbalance.
:<math>F= F_0 \cos {(2 \pi f t)} \!</math>
If we again sum the forces on the mass we get the following ordinary differential equation:
:<math>m \ddot{x} + { c } \dot{x} + {k } x = F_0 \cos {(2 \pi f t)} </math>
The [[steady state]] solution of this problem can be written as:
:<math>x(t)= X \cos {(2 \pi f t -\phi)} \!</math>
The result states that the mass will oscillate at the same frequency, f, of the applied force, but with a phase shift <math> \phi </math>.
The amplitude of the vibration “X” is defined by the following formula.
:<math>X= {F_0 \over k} {1 \over \sqrt{(1-r^2)^2 + (2 \zeta r)^2}}</math>
Where “r” is defined as the ratio of the harmonic force frequency over the undamped natural frequency of the mass-spring-damper model.
:<math>r=\frac{f}{f_n}</math>
The phase shift , <math>\phi</math>, is defined by following formula.
:<math>\phi= \arctan {\left (\frac{2 \zeta r}{1-r^2} \right)} </math>
[[Berkas:Forced Vibration Response.png|700px|Forced Vibration Response]]
The plot of these functions, called "the frequency response of the system", presents one of the most important features in forced vibration. In a lightly damped system when the forcing frequency nears the natural frequency (<math>r \approx 1 </math>) the amplitude of the vibration can get extremely high. This phenomenon is called '''[[mechanical resonance|resonance]]''' (subsequently the natural frequency of a system is often referred to as the resonant frequency). In rotor bearing systems any rotational speed that excites a resonant frequency is referred to as a [[critical speed]].
If resonance occurs in a mechanical system it can be very harmful-- leading to eventual failure of the system. 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. As the amplitude plot shows, adding damping can significantly reduce the magnitude of the vibration. 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. If the system cannot be changed, perhaps the forcing frequency can be shifted (for example, changing the speed of the machine generating the force).
The following are some other points in regards to the forced vibration shown in the frequency response plots.
*At a given frequency ratio, the amplitude of the vibration, ''X'', is directly proportional to the amplitude of the force <math>F_0 </math> (e.g. If you double the force, the vibration doubles)
*With little or no damping, the vibration is in phase with the forcing frequency when the frequency ratio ''r'' < 1 and 180 degrees out of phase when the frequency ratio ''r'' >1
*When r<<1 the amplitude is just the deflection of the spring under the static force <math>F_0 </math>. This deflection is called the static deflection <math>\delta_{st}</math>. Hence, when r<<1 the effects of the damper and the mass are minimal.
*When r>>1 the amplitude of the vibration is actually less than the static deflection <math>\delta_{st}</math>. In this region the force generated by the mass (F=ma) is dominating because the acceleration seen by the mass increases with the frequency. Since the deflection seen in the spring, ''X'', is reduced in this region, the force transmitted by the spring (''F''=''kx'') to the base is reduced. Therefore the mass-spring-damper system is isolating the harmonic force from the mounting base—referred to as [[vibration isolation]]. Interestingly, more damping actually reduces the effects of vibration isolation when r>>1 because the damping force (''F''=''cv'') is also transmitted to the base.
==== What causes resonance?====
Resonance is simple to understand if you view the spring and mass as energy storage elements--with the mass storing kinetic energy and the spring storing potential energy. As discussed earlier, when the mass and spring have no force acting on them they transfer energy back and forth at a rate equal to the natural frequency. In other words, if energy is to be efficiently pumped into both the mass and spring the energy source needs to feed the energy in at a rate equal to the natural frequency. Applying a force to the mass and spring is similar to pushing a child on swing, you need to push at the correct moment if you want the swing to get higher and higher. As in the case of the swing, the force applied does not necessarily have to be high to get large motions; the pushes just need to keep adding energy into the system.
The damper, instead of storing energy, dissipates energy. Since the damping force is proportional to the velocity, the more the motion the more the damper dissipates the energy. Therefore a point will come when the energy dissipated by the damper will equal the energy being fed in by the force. 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. If no damping exists, there is nothing to dissipate the energy and therefore theoretically the motion will continue to grow on into infinity.
==== Applying "complex" forces to the mass-spring-damper model====
In a previous section only a simple harmonic force was applied to the model, but this can be extended considerably using two powerful mathematical tools. The first is the [[Fourier transform]] that takes a signal as a function of time ([[time domain]]) and breaks it down into its harmonic components as a function of frequency ([[frequency domain]]). For example, let us apply 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. This type of force has the shape of a 1 Hz [[square wave]].
[[Berkas:Square wave frequency spectrum animation.gif|thumb|300px|How a 1 Hz square wave can be represented as a summation of sine waves(harmonics) and the corresponding frequency spectrum]]
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). The Fourier transform can also be used to analyze non-[[periodic function|periodic]] functions such as transients (e.g. impulses) and random functions. With the advent of the modern computer the Fourier transform is almost always computed using the [[Fast Fourier Transform]] (FFT) computer algorithm in combination with a [[window function]].
In the case of our square wave force, the first component is actually a constant force of 0.5 newton and is represented by a value at "0" Hz in the frequency spectrum. The next component is a 1 Hz sine wave with an amplitude of 0.64. This is shown by the line at 1 Hz. The remaining components are at odd frequencies and it takes an infinite amount of sine waves to generate the perfect square wave. Hence, the Fourier transform allows you to interpret the force as a sum of sinusoidal forces being applied instead of a more "complex" force (e.g. a square wave).
In the previous section, the vibration solution was given for a single harmonic force, but the Fourier transform will in general give multiple harmonic forces. The second mathematical tool, "the principle of [[superposition principle|superposition"]], allows you to sum the solutions from multiple forces if the system is [[linear system|linear]]. In the case of the spring-mass-damper model, the system is linear if the spring force is proportional to the displacement and the damping is proportional to the velocity over the range of motion of interest. 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.
====Frequency response model ====
We can view the solution of a vibration problem as an input/output relation--where the force is the input and the output is the vibration. If we represent the force and vibration in the frequency domain (magnitude and phase) we can write the following relation:
:<math>X(\omega)=H(\omega)\cdot F(\omega) \ \ or \ \ H(\omega)= {X(\omega) \over F(\omega)}.</math>
<math>H(\omega)</math> is called the [[frequency response]] function (also referred to as the [[transfer function]], but not technically as accurate) and has both a magnitude and phase component (if represented as a [[complex number]], a real and imaginary component). The magnitude of the frequency response function (FRF) was presented earlier for the mass-spring-damper system.
:<math>|H(\omega)|=\left |{X(\omega) \over F(\omega)} \right|= {1 \over k} {1 \over \sqrt{(1-r^2)^2 + (2 \zeta r)^2}}, \ \ where\ \ r=\frac{f}{f_n}=\frac{\omega}{\omega_n}</math>
The phase of the FRF was also presented earlier as:
:<math>\angle H(\omega)= \arctan {\left (\frac{2 \zeta r}{1-r^2} \right)}. </math>
For example, let us calculate 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. The values of the spring and mass give a natural frequency of 7 Hz for this specific system. If we apply the 1 Hz square wave from earlier we can calculate the predicted vibration of the mass. The figure illustrates the resulting vibration. It happens in this example that the fourth harmonic of the square wave falls at 7 Hz. The frequency response of the mass-spring-damper therefore outputs a high 7 Hz vibration even though the input force had a relatively low 7 Hz harmonic. This example highlights that the resulting vibration is dependent on both the forcing function and the system that the force is applied to.
[[Berkas:Frequency response example.png|thumb|500px|left|Frequency response model.]]
The figure also shows the time domain representation of the resulting vibration. This is done by performing an inverse Fourier Transform that converts frequency domain data to time domain. In practice, this is rarely done because the frequency spectrum provides all the necessary information.
The frequency response function (FRF) does not necessarily have to be calculated from the knowledge of the mass, damping, and stiffness of the system, but can be measured experimentally. For example, if you apply a known force and sweep the frequency and then measure the resulting vibration you can calculate the frequency response function and then characterize the system. This technique is used in the field of experimental [[modal analysis]] to determine the vibration characteristics of a structure.
-->
== Pranala luar ==
* {{en}}Hyperphysics Educational Website, [http://hyperphysics.phy-astr.gsu.edu/hbase/permot.html#permot''Oscillation/Vibration Concepts'']
* {{en}}Thermotron Industries, [http://www.thermotron.com/resources/vibration_handbook.html''Fundamentals of Electrodynamic Vibration Testing Handbook'']
* {{en}}Nelson Publishing, [http://www.evaulationengineering.com/ ''Evaluation Engineering Magazine'']
* {{en}}[http://structdynviblab.mcgill.ca/index.html Structural Dynamics and Vibration Laboratory of McGill University]
* {{en}}[http://web.mat.bham.ac.uk/C.J.Sangwin/Teaching/CircWaves/waves.html Normal vibration modes of a circular membrane]
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