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I made a list of important learning point from the principles of vibration control, The major points discuss in these lecture notes are:Closed Loop Transfer Function, Basics of Classical Control, Feed-Forward Control, Principles of Active Vibration Control, Single Input Single Output Vibration System, Effect of Vibration, Black-Box, Control Effort
Typology: Study notes
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The figure below shows symbolic representation of a single input single output (SISO) vibration system. Many closed loop control system could be modelled in this form. A typical application of such control system is in the field of engine vibration control. The figure below illustrates such a system. The disturbance force generated at the engine is transmitted to the vehicle through engine mounts. The vibration is sensed by a single accelerometer ( single output), passed to the controller which produces a negative of the output signal (ref, is usually zero in vibration problem). The control signal is amplified through a power amplifier and fed to the actuator. The actuator force is of such a magnitude and direction that it cancels the effect of vibration.
The input-output relationship of a dynamic system is represented as a black-box consisting of frequency domain description of the system transfer function.
Consider a SDOF system subjected to unit step input. Due to flexibility in the system, some part
0.5 is used to control the vibration. Find out the response of the system with and without closed-loop control. The following are the system parameters; m=0.2, c=0.001, k=0.5, H=1 (assume unity feedback).
The open loop transfer function of the system is given by:
The closed loop transfer function of the system is given by:
The open-loop response shows oscillation around 2 with a light damping while the closed-loop response shows damping of vibration within 10 seconds and the system following the step input.
Now you vary the compensator as (a) (b) and compare the closed loop response of the system
You will find with zero K (^) I , the vibration is continuing for a longer time and with zero K (^) D , the system will take more time initially to pick up with the reference value.
To understand this better you are advised to go through quickly through the basics of classical control in the subsequent slides. If you are familiar with this concept then you can directly go to slide 9 of this lecture.
Denoting the right hand side of the above equation as , one can express the ratio of frequency-
domain response X(s) and as
T(s) is also known as transfer function of the system.
In a block diagram form, this can be represented as
The response of a system in time domain could be obtained by carrying out Inverse Laplace Transformation of the transfer function. The inverse Laplace Transform is written as
However, this relationship is seldom used. If F(s) is rational, one commonly uses the method of partial fraction expansion. Consider a rational function F(s) expressed as:
Factoring the numerator and denominator polynomials one can also write
Corresponding to the numerator polynomial, z i s are referred as the^ zeros^ of the transfer function while
the roots of the numerator polynomial p i 's are known as the poles of the transfer function.
Now, the transfer function F(s) may be expressed as
where,
Finally, the response of the system may be expressed as
There are four major specifications that define the response of a second order system.
The swiftness of the response is measured by the rise time (T (^) r ), which is defined as the time required reaching 90% of the reference input.
The closeness of the response signal to the reference is measured in terms of % overshoot ( PO ) such that
PO = ((M (^) p – final value)/final value)*
The time taken to reach the maximum response is denoted by Peak Time (T (^) p ). The % overshoot is
measured at the same location. Finally, the Settling Time (T (^) s ) is defined as the time required for the
system to settle within a certain percentage of the reference input (±d ).
For a second order system, the above four specifications could be defined in terms of natural frequency and damping factor of the system as:
The root locus method is used to study the change in the pole-location of a closed-loop system in the s-plane with respect to the change in the control-gain. A simple closed loop system is described as follows:
Following the block-diagram, the error E(s) is
Again
Hence,
Equation (24.11) is known as the closed-loop transfer function for a negative feed-back system. The characteristic equation corresponding to a unity feedback (H(s) =1) for the above system could be written as
The above equation could be expressed in terms of magnitude and phase as follows:
Using the phase relationship of the above equation, one can plot the locus of the roots of the characteristic equation as K varies from 0 to infinity.
Due to disturbance in the process, the input X(s) to an actuator may generate an output X(s) D(s) which may destabilize the system. In such cases if the input X(s) is passed through a feed-forward compensator C (^) FF (s), and compared with the output X(s)D(s) then with correct estimation of D(s) one may alleviate the effect of the disturbance on the plant G(s). The response of the plant may be sensed by monitoring the signal C(s)H(s). However, such responses are used only to monitor the performance of the filter and not fed back in the loop. It has been shown by Hansen et al ([11-3]) that optimal gain of the filter C (^) FF (s) is C (^) FF (s) = - S (^) FF -1 S (^) DF , Where, S (^) FF denotes the power spectral density of the ref signal X(s) and S (^) DF is the cross spectral density between the disturbance and the filter output.
For an efficient filter, the system should remain stationary. Any unknown delay in the system could destabilize the performance. Since a successful filter has to match the gain and phase contributions by the disturbance signal at all the frequencies present in the signal, the disturbance should be free from near-field noises, which are difficult to estimate. For a system having random disturbance one may attempt to develop a similar feed-forward compensator – however, such a system would be non-causal in nature, hence a purely feed- forward system should be substituted by a combination of feed-forward and feed-back system.
The simplest of the SISO system based on Classical Control is discussed. A typical SDOF system and it's SISO representation is considered.. A quick recapitulation of the classical control system is discussed. Finally, the case of Feed-forward control and its impact is discussed.
