The Lecture Contains: Development of an Active DVA Proof Mass Actutor Application of Active DVA file:///d /chitra/vibration_upload/lecture19/19_1.htm[6/25/2012 12:35:51 PM]
In this section, we will consider the development of an active dynamic vibration absorber. It may be noted that Passive Neutralizer eliminates primary response only at a particular frequency. Use of active element - for example, a hydraulic actuator would increase the advantage of tuned mass damping for a broad frequency range. Figure 19.1: Active dynamic vibration absorber Let us consider the new model as shown in Fig 19.1. Here, denotes the primary mass and the primary stiffness. The damping of the primary system is neglected. The system is subjected to a harmonic excitation. The primary system is attached to a secondary system of fixed mass and stiffness and respectively. However, there is an additional spring element with variable stiffness ' ' representative of a hydraulic actuator. file:///d /chitra/vibration_upload/lecture19/19_2.html[6/25/2012 12:35:51 PM]
The governing EOM of the two DOF system may be written as (19.1) (19.2) Using from Equation 19.2 we get or or (19.3) Similarly, from Equation 19.1, we get or, or Thus, when the hydraulic actuator is switched on the active displacement of the primary mass be written as: may When the hydraulic system is switched off, the passive displacement of the primary mass written as may be file:///d /chitra/vibration_upload/lecture19/19_3.html[6/25/2012 12:35:51 PM]
The ratio of active and passive displacement of the primary mass brings out the efficiency of the new system. Therefore, For a simple case, use As a test case, for and for From these expressions, you can check that the negetive feedback system with better for a wider frequency range. works file:///d /chitra/vibration_upload/lecture19/19_3.html[6/25/2012 12:35:51 PM]
= Active DVA Now, we will consider another active DVA commonly known as proof mass actuator. The basic system is described in Fig. 19.2. Figure 19.2: Proof mass actuator A proof mass m p is connected to a magnetic base with spring k and damper c. The proof mass is placed over a solenoid in which magnetic field could be generated by passing current through coils. The resistance of the coil is R, inductance L, current passing through the coil is i and the proportionality constant corresponding to back EMF is k b. The EOM are provided below. Governing equation for electric system (19.4) Governing equation for mechanical system (19.5) where k a is the current constant Converting Equation 19.4 into frequency domain, (19.6) Similarly, from 19.5 Using Equation 19.6 we get file:///d /chitra/vibration_upload/lecture19/19_4.html[6/25/2012 12:35:51 PM]
Denoting, (19.7) Also, force exerted by the proof mass actuator on the base is, therefore,. Using Equation 19.7 (19.8) This relationship tells us how force F will be generated by the proof mass accelerator upon application of voltage. file:///d /chitra/vibration_upload/lecture19/19_4.html[6/25/2012 12:35:51 PM]
Application of active DVA Consider a SDOF system (undamped, mass m 1 ) subjected to base excitation. The equation of motion may be written as Figure 19.3: SDOF subjected to base excitation (19.9) Transforming into frequency domain this becomes Therefore, (19.10) Plugging the output of the Active DVA into the system and using Equation 19.8, we get the new displacement of m 1 as (19.11) file:///d /chitra/vibration_upload/lecture19/19_5.html[6/25/2012 12:35:51 PM]
A Special case: When, is constant, One can show a wide band amplitude reduction of the primary mass (X 1 ) by suitably choosing k a and k b in equation (19.11) Numerical exercise: Consider,,, Find out the Transfer function and plot. file:///d /chitra/vibration_upload/lecture19/19_6.html[6/25/2012 12:35:52 PM]