Getaran: Perbedaan antara revisi

[[Berkas:Drum_vibration_mode21Drum vibration mode21.gif|thumbjmpl|Salah satu mode getaran [[gendang]]]]

[[Berkas:Mass spring.svg|200px|rightka|thumbjmpl|Model massa-pegas sederhanal]]

[[Berkas:Simple harmonic oscillator.gif|thumbjmpl|100px|rightka|Gerakan harmonik sederhana sistem benda-pegas]]

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, namuntetapi besaran ini biasanya diubah ke dalam frekuensi "standar" (satuan [[Hertz|Hz]]) ketika menyatakan frekuensi sistem.

[[Berkas:Mass spring damper.svg|200px|rightka|Mass Spring Damper Model]]

Dengan menjumlahkan semua gaya yang berlaku pada benda kita mendapatkan persamaan

Solusi persamaan ini tergantung pada besarnya redaman. Bila redaman cukup kecil, sistem masih akan bergetar, namuntetapi 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.

(<math>\zeta </math>) 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, namuntetapi frekuensi osilasi berbeda daripada kasus tidak teredam.

Frekuensi alamiah teredam lebih kecil daripada frekuensi alamiah takredam, namuntetapi 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.

:<math>r=\frac{f}{f_n}</math>

The phase shift , <math>\phi</math>, is defined by following formula.

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).

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.

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]].

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]].

<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>

:<math>\angle H(\omega)= \arctan {\left (\frac{2 \zeta r}{1-r^2} \right)}. </math>

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.

* {{en}}Hyperphysics Educational Website, [http://hyperphysics.phy-astr.gsu.edu/hbase/permot.html#permot''Oscillation/Vibration Concepts''] {{Webarchive|url=https://web.archive.org/web/20230605171506/http://hyperphysics.phy-astr.gsu.edu/hbase/permot.html#permot |date=2023-06-05 }}

* {{en}}Thermotron Industries, [http://www.thermotron.com/resources/vibration_handbook.html''Fundamentals of Electrodynamic Vibration Testing Handbook''] {{Webarchive|url=https://web.archive.org/web/20070824053405/http://www.thermotron.com/resources/vibration_handbook.html |date=2007-08-24 }}

* {{en}}Nelson Publishing, [http://www.evaulationengineering.com/ ''Evaluation Engineering Magazine''] {{Webarchive|url=https://web.archive.org/web/20071031003535/http://www.evaulationengineering.com/ |date=2007-10-31 }}

* {{en}}[http://structdynviblab.mcgill.ca/index.html Structural Dynamics and Vibration Laboratory of McGill University] {{Webarchive|url=https://web.archive.org/web/20220204022357/http://structdynviblab.mcgill.ca/index.html |date=2022-02-04 }}

* {{en}}[http://web.mat.bham.ac.uk/C.J.Sangwin/Teaching/CircWaves/waves.html Normal vibration modes of a circular membrane] {{Webarchive|url=https://web.archive.org/web/20050518033722/http://web.mat.bham.ac.uk/C.J.Sangwin/Teaching/CircWaves/waves.html |date=2005-05-18 }}

[[Kategori:TeknikRekayasa struktur]]