1.3 Differential equation for damped vibration
In actual practice, there is always some damping (e.g., the internal molecular friction, viscous damping, aero-dynamical damping, etc.) present in the system. It causes the gradual dissipation of vibration energy, which results gradual decay of amplitude of the free vibration. Damping is of great importance in limiting the amplitude of oscillation at resonance. Viscous damping force is expressed by the equation
(1.1)
Where c is a constant of proportionality. Symbolically it is designated by a dashpot, as shown in Fig. 1.1
Fig. 1.1: Spring-Mass-Damper system
The differential equation for damped vibration is
(1.2)
If the mass is denoted as m, the viscous damping constant as c, the stiffness as k , and the applied force as F ( t ), for free damped vibration the roots of the characteristic equation are:
(1.3)
The natural un-damped resonant angular frequency is
(1.4)
The critical damping constant is
(1.5)
The critical damping ratio is defined to be
(1.6)
Therefore
(1.7)
So that if we divide the damped vibration equation by mass m, we can write it in terms of ζ and ω nf as
(1.8)
This form of the equation for single DOF systems will be found to be helpful in identifying the natural frequency and the damping of the system.
1.4 Different Cases for damped vibration
1.4.1 Un-damped Case (ζ = 0): When there is no damping then the condition is called as un-damped free vibration. For this case ζ is equal to zero. Using the following equation solution for zero damping can be obtained as,
(1.9)
Fig. 1.2 shows the simulated curve for amplitude vs. time of Eq.(1.9), for ζ = 0, ω nf = 32.91 Hz, = 15.699 m/sec, x (0) = 13.0146 m. In this case also, the motion is considered as periodic, amplitude remains same over time.
Fig. 1.2: Response for free un-damped vibration
1.4.2 Under damped Case ( ζ < 1): The free vibration of an under damped system is oscillatory but not periodic. The vibration would be periodic if the amplitude will not decay with time. Even though the amplitude decreases between cycles, the system takes the same amount of time to execute each cycle.
The general solution is
(1.10)
Frequency of damped oscillation is equal to
(1.11)
Where, x(0), are initial conditions.
The screen-shot of simulated curve of amplitude vs. time for under damping, using Eq.(1.10) is as shown in Fig. 1.3 . The curve was obtained at damping ratio ζ = 0.07142, frequency ω nf = 44.5404 Hz, initial velocity = 24.97 m/sec and initial displacement of x (0) = 5.486 m. From the simulation, it was observed that for under damped case the amplitude decreases over time.
Fig. 1.3: Response for free under damped vibration
1.4.3 Critical damped Case ( ζ = 1): For critical damping case ζ = 1, the roots of the characteristic equation are real and equal to each other.
(1.12)
The correct general solution is:
(1.13)
The simulated response of amplitude vs. time for critical damping is as shown in Fig. 1.4 for the Eq. (1.14). The simulation was observed for ω nf = 5.637 Hz, x(0)=17.8035 m, =4.764 m/sec.
Fig. 1.4: Response for free critically damped vibration
Similarly for , the amplitude vs. time response can be seen from Fig 1.5.
Fig. 1.5: Response for free critically damped vibration
1.4.4 Over damped Case (ζ>1): For overdamped case ζ > 1 and the characteristic equation has two real roots. The general solution is,
(1.15)
Where,
(1.16)
And
(1.17)
For, ζ = 1.6326, ω nf = 13.361 Hz, = 53.942 m/sec, x(0) = 10.8776m,the free motion for over damping is as shown in Fig.1.6. The simulated response is obtained from the Eq. (1.15). The motion is exponentially decreasing function with time which is referred to as aperiodic motion. The response quickly decays after reaching maximum.
Fig. 1.6: Response for free over damped-damped vibration
1.4.5 Negative damped Case (ζ< 0): In this case value of ζ is less than zero. Using the following equation, solution for negative damping can be obtained as,
(1.18)
For, ζ = -0.11224, ω nf = 38.2697 Hz, = 10.53 m/sec, = 15.227 m, the simulated curve of amplitude vs. time for negative damped case is shown in Fig. 1.7. From the simulated curve obtained using Eq. (1.18), it is observed that for negative damped case amplitude increases with time which is a case of opposite in nature of under-damped case.
Fig. 1.7: Response for free negative damped vibration