The intensity of vibration is measured by the amount of vibration, which can be the displacement, speed or acceleration of the vibrating body. If the vibration exceeds the allowable range, mechanical equipment will produce large dynamic load and noise, which will affect its working performance and service life, and even lead to early failure of parts. For example, the fracture of turbine blades caused by vibration can lead to serious accidents. As the modern mechanical structure becomes more and more complex, the movement speed becomes faster and faster, and the harm of vibration becomes more and more prominent. On the contrary, mechanical equipment working on the principle of vibration should be able to produce the expected vibration. In the field of mechanical engineering, besides solid vibration, there are also fluid vibration and coupled vibration of solid and fluid. Surge of air compressor is a kind of fluid vibration.
The simplest mechanical vibration is the simple harmonic vibration of a particle. Simple harmonic vibration is a kind of motion that changes with time as a sine function. This vibration can be regarded as the result of the projection of a point moving in a uniform circle on a vertical plane on the vertical axis in this plane. Its vibration displacement is x (t) = asin ω t.
Where a is the amplitude, that is, the maximum deviation from the equilibrium position, that is, the maximum vibration displacement; T is time; ω is the circular frequency (2π times the sine frequency). Its vibration speed is
dx/dt=ωAsin(ωt+π/2)
Its vibration acceleration is
d2x/dt2=ω2Asin(ωt+π)
Vibration can also be represented by vectors. The vector rotates counterclockwise with equal angular velocity ω. The modulus of displacement vector is amplitude a, the modulus of velocity vector is amplitude ωA of velocity, and the modulus of acceleration vector is amplitude ω2A of acceleration. The velocity vector is 90 before the displacement vector and the acceleration vector is180 before the displacement vector. If the particle is not in an equilibrium position at the beginning of vibration, its displacement can be expressed by the following formula.
x(t)=Asin(ωt+ψ)
Where psi is the initial phase. The time required to complete a vibration is called a period. The reciprocal of the period is the number of vibrations per unit time, which is called frequency. The vibration with a fixed period returns to the state of the beginning of the period after one period, which is called periodic vibration. As long as certain conditions are met, any periodic function can be expanded into Fourier series. Therefore, anharmonic vibration can be decomposed into a series of anharmonic vibration. Vibration without a fixed period is called aperiodic vibration. For example, the rotating machinery first appears aperiodic vibration during the starting process, and then produces periodic vibration when the rotating machinery rotates at a uniform speed.
I hope it can help you solve the problem.