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Basic vibration analysis
The most basic vibration analysis is a system with a single degree of freedom (SDOF), such as the classical linear oscillator (CLO), as shown in Fig. 1. It consists of a point mass, spring, and damper. This example will be used to calculate the effects of vibration under free and forced vibration, with and without damping. Where needed, refer to this refresher on differential calculus.
Free vibration without damping
The first analysis is free vibration without damping. Using Newton’s second law and D’Alembert’s principle, the equation that describes free vibration without damping (c = 0) is:
The solution to this differential equation is:
The constant multiplying the t is the natural frequency of the system and is:
in radians/unit time. Multiplying by 1/2π gives the natural frequency in cycles/unit time. When Eq. 4 is substituted into Eq. 3, the result is:
The constants C1 and C2 are based on the initial and boundary conditions. If at time 0, x = x0 and ẋ=v0, the initial location and velocity, respectively, the first coefficient is:
which is the initial location. Differentiating once gives:
which, when t = 0, gives the other coefficient:
which is based on the initial velocity. The entire equation then is:
Ex. 1 is an SDOF free and undamped CLO system.
Free vibration with damping
The second analysis of free vibration is with damping (Fig. 2). Using Newton’s second law, the equation that describes free vibration with damping (c ≠ 0) is:
which is rearranged as before to get:
The general solution to this differential equation is:
although the specific solution depends on the value under the square root. When c2 - 4mk = 0, the system is critically damped. Another way to look at this critical damping point is:
Often, the damping coefficient is divided by the critical damping coefficient to get the critical damping ratio:
If ξ > 1, the system is underdamped. When disturbed, the system will experience an oscillating decay. If ξ < 1, the system is overdamped and, when disturbed, will die out without oscillating. If ξ = 1, the system is critically damped and also will not oscillate.
A useful simplifying equation is the "damped" natural frequency, ωd. It is:
If the system is underdamped, that is, if 0 ≤ ξ < 1 , the solution is:
Ex. 2 is an SDOF free and underdamped CLO system.
If the system is overdamped, that is, if ξ>1 , the solution is:
Ex. 3 is an SDOF free and overdamped CLO system.
If the system is critically damped, that is, if ξ=1 , the solution with the initial and boundary conditions is:
Ex. 4 is an SDOF free and critically damped CLO system.
Forced vibration without damping
The next sets of systems have a forcing function driving the vibration. The first of these is a CLO without damping (c = 0), as shown in Fig. 3. The equation of motion for this system with F = F0 sin ωft, a sinusoidally varying force, using Newton’s second law, is:
In this case, there are two terms in the solution, the homogenous or transient term, and the particular or steady-state term. The homogenous term is the same as in a free-vibration case and is solved by setting the forcing function to zero (that is, the free-vibration case, Eq. 9). If the same initial and boundary conditions are applied as before, the solution for the homogenous case is the same as before:
The second term is the effect of the forcing function on the system. This is solved by assuming a particular solution and deriving it back:
Substituting the above equations into Eq. 23 gives:
Collecting the terms gives:
Equating coefficients shows that:
Therefore, the particular solution is:
which can be rewritten as:
where the reciprocal term in the parentheses sometimes is called the magnification factor. The total solution is:
Ex. 5 is an SDOF forced and undamped CLO system.
Note that when the forcing frequency, ωf, matches the natural frequency, ωn, the value of the coefficient is infinity. This is the resonance condition, and it can lead to excessively large displacements (see Fig. 3).
When the forcing frequency is close to but not at the natural frequency, a beating phenomenon occurs. This appears as a low frequency impressed over the frequency of the system. When the engines of a twin-engine aircraft are not quite synchronized, for example, one can hear a beating sound as a low-frequency pulse (the "wow-wow" throb). Fig. 5 illustrates this beating phenomenon.
Forced vibration with damping
The second system with a forcing function driving the vibration is a CLO with damping (c ≠ 0), as shown in Fig. 6. The equation of motion for this system with the same force as before, F = F0 sin ωt, is:
The solution has two parts, as before. It is similar to the last example, except for an additional damping term. The particular solution is solved similarly to the last example in Eqs. 25,26, and 27. Differentiating and substituting into Eq. 35 gives:
Equating coefficients as before yields:
Solving for the constants gives:
which gives the particular solution:
The total solution is homogenous and the particular solutions added together. In this case, if the same initial and boundary conditions are applied as before, the homogenous solution is the same as in the free-vibration case and is Eq. 18. The particular solution is Eq. 42 and is the effect of the forcing function on the system. As noted before, the critical damping coefficient dictates the behavior of the homogenous part of the solution.
Ex. 6 is an SDOF forced and underdamped CLO system.
|c||=||axial damping coefficient, mL/t, lbf-ft/sec|
|ccrit||=||critical damping coefficient, dimensionless|
|C||=||constant of integration, various|
|F||=||axial force, mL/t2, lbf|
|F0||=||initial force, mL/t2, lbf|
|k||=||spring constant, m/t2, lbf/in.|
|keq||=||equivalent spring constant, m/t2, lbf/in.|
|m||=||mass, m, lbm|
|v0||=||initial velocity, L/t, ft/sec|
|x||=||displacement, L, in.|
|xh||=||homogeneous displacement, L, in.|
|xp||=||particular displacement, L, in.|
|x0||=||initial displacement, L, in.|
|ẋ||=||first derivative with respect to time of displacement (velocity), L/t, ft/sec|
|ẋp||=||particular velocity, L/t, ft/sec|
|ẍ||=||second derivative with respect to time of displacement (acceleration), L/t 2 , ft/sec 2|
|ẍp||=||particular acceleration, L/t 2 , ft/sec 2|
|ξ||=||critical damping ratio, dimensionless|
|Φ||=||phase angle, rad|
|ωd||=||damped natural frequency, 1/t, Hz|
|ωf||=||forcing frequency, 1/t, Hz|
|ωn||=||natural frequency, 1/t, Hz|
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