Mechanical Vibration
Web Book
Forced Vibration
Learning objectives
Determine the equivalent mass, equivalent stiffness and equivalent damping in a vibrating system
Sections
4.1 Forcing frequency
4.2 Undamped forced vibration
4.3 Damped forced vibration
4.4 Vibration measurement
4.2 Undamped forced vibration
Having known how to derive the equation of motion of a SDOF system as in Chapter 3, now we have an undamped SDOF system, with an external force, f(t) acting on the system as shown in Figure 4.2.1.
The equation of motion is given by
m\ddot{x}(t)+kx(t)=f(t)\phantom{xxxxxxxxxxxx}{\color{red}(4.2.1)}
where m is the mass, k is the stiffness and x is the displacement with t is time, and where \ddot{x}=\text{d}^2/\text{d}t^2 .
If the force acting on the system is a simple harmonic force (single frequency of \omega=2\pi f ) given by
f(t)=F\sin(\omega t)\phantom{xxxxxxxxxxxxxx}{\color{red}(4.2.2)}
where F is the magnitude of force, then the response of the system will also be harmonic expressed as
x(t)=X\sin(\omega t)\phantom{xxxxxxxxxxxxxx}{\color{red}(4.2.3)}
where X is the magnitude of displacement.
Figure 4.1.1 Forced vibration in a car from engine and road input.
Figure 4.1.1 Forced vibration in a car from engine and road input.
4.1.1 Forcing frequency vs. natural frequency (Example 1)
To illustrate the difference between the forcing frequency (or excitation frequency), and the natural frequency, let us see Figure 4.1.2.
A rotating machine with mass M is attached at the end of a beam-like structure. The stiffness of the beam at the location of the motor is k.
Considering the effective mass of the beam, m_{b_e} at the location of the attachment, the natural frequency of the system is
f_n=\displaystyle\frac{1}{2\pi}\sqrt{\frac{k}{M+m_{b_e}}}\phantom{xxxxxxx}{\color{red}(4.1.1)}
Thus the natural frequency relates to the physical parameters of the system. Note that at this stage, the machine is not operating yet.
Figure 4.1.2 A rotating machine with variable speed on a beam-like foundation.
When we turn on the machine at a certain operating speed (assume to have mass unbalance to create a greater dynamic force), the beam is then forced to vibrate at that particular forcing frequency, relating to the machine speed (see again Animation 1.2.1). This forcing frequency can be lower, the same or greater than the natural frequency of the system.
The natural frequency relates to the physical paramaters of a vibrating system.
The vibration amplitude will depend on the frequency of vibration (i.e. the forcing frequency) relative to the natural frequency, and the amplitude of the dynamic force providing by the vibration source (in this example, the rotating machine). At resonance, the damping plays its role to significantly reduce the vibration amplitude.
You will understand this better through the graph of the Frequency Response Function (FRF) in Section 4.2.
4.1.2 Forcing frequency vs. natural frequency (Example 2)
Figure 4.1.3 shows a pipe section carrying a process fluid, which is common in oil and gas industries. The vibration frequency of the pipe section will be dependent on the characteristic of the fluid flow inside the pipe (e.g. the velocity, the degree of turbulence, the presence of bubbles, etc). Usually the greater the velocity of the fluid, the higher the vibration frequency.
Meanwhile, the pipe structure itself has its natural frequency f_n=(1/2\pi)\sqrt{k/m} , which is determined by the mass of the pipe, m (including the mass of the fluid) and the stiffness of the pipe, k .
The stiffness depends on how the pipe is supported, which determines the flexibility of the pipe structure to move due to the internal force from the fluid.
Figure 4.1.3 Illustration of the forcing frequency on a process pipe.
Figure 4.1.3 Illustration of the forcing frequency on a process pipe section.
Figure 4.1.3 illustrates that it is possible for the pipe to vibrate at any forcing frequency.
The magnitude of vibration at the corresponding forcing frequency will depend on the magnitude of force from the fluid and the forcing frequency relative to the natural frequency of the pipe (determined by the mass and stiffness of the pipe).
For example, the vibration will be high if the fluid injects large force magnitude at frequency below the natural frequency (f<f_n) and at the same time, the pipe structure has very low stiffness.
However if the fluid excites the pipe with the frequency very close to the natural frequency of the pipe, then the pipe will vibrate with high magnitude, regardless the force magnitude of the fluid (case of resonance, and little damping).
This is what we will prove with math in this chapter.
Figure 4.1.3 In oil and gas industries, piping vibration is one of the engineering problems which needs to be controlled and monitored.
Watch my video that summarizes the introduction of forced vibration.
