Finite element analysis of the platform is performed for static and dynamic load cases. Results for different load cases described in section 4 show the effectiveness of the platform for passive control of vibration. Finite element analysis of the platform is performed using frame elements with six degrees of freedom for each node.
The element (12 x 12) stiffness matrix in the local axes consists of (6 x 6) submatrices and has the form Thus, the actions, stiffness and mass are transformed from to the center of mass of the rigid body as follows: Active sway control of a flexible platform was studied using a finite element model based on Euler-Bernoulli beam theory.
When the input voltage is applied to the piezoelectric actuator in the thickness direction, the electric field and the stress developed by the actuator are given by. where, is the Young's modulus of the piezoelectric layer. In general, the expression for the bending moment at a small cross section of the piezoelectric element is given by. The control force produced by the actuator applied to the beam element is obtained using Eq. 39) which when expressed in terms of the control input to the actuator becomes.
Φ (44) Here Φ is the modal matrix of size ( x 4) and consists of the eigenvectors corresponding to the first four modes. where and are the diagonalized mass and stiffness.
Passive control simulation results
50), is a constant vector that depends on the type of actuator, its location on the beam, its characteristic properties and is given by 53) and is a constant vector that depends on the characteristics of the sensor and its location/position on the beam and is given as An eigenvalue analysis was performed to study the variation of the natural frequency by varying the thickness of the vertical girders. Frequencies were listed as up to 60 Hz, as the platform was expected to take the carrier at a maximum speed of 3,600 rpm.
It is observed that a decrease in the thickness of a vertical beam member decreases the frequency of the platform. In this case, the four horizontal flat bars were placed asymmetrically in relation to the top and bottom folds. In the first type, the horizontal bar is at a distance of 24 mm from the top fold and 36 mm from the bottom fold.
In the second type, the horizontal beam is placed from the upper and lower folds at a distance of 24 mm and 48 mm, respectively. Comparing the results in Table 3, we conclude that increasing the length of the vertical elements leads to a decrease in frequency. We can see that as the number of wrinkles increases, the frequency decreases.
Similarly, when the thickness of vertical elements is reduced from 0.001 m to 0.0006 m, there is a further reduction in frequency. One Fold Two Folds Three Folds Four Folds. i) Increasing the number of folds. ii) Reducing the thickness of vertical members. iii). Results corresponding to the 2nd and 4th degrees of freedom for the platform are shown in Fig.
It is observed that for all degrees of freedom of the platform, the peak responses are small and they damp out in a very short time. The response of the symmetrical flexible platform in the first two degrees of freedom subjected to a harmonic load of 5sin (24.816t) N at the first node is shown in Fig. Hz, it is found that the ratio of the forced frequency to the first natural frequency is 10.797 which amounts to an isolation of 99.0905.
Active control simulation results
For vertical beams with a thickness equal to 0.8 mm and a forcing frequency of 60 Hz, it is found that the ratio of the forcing frequency to the first natural frequency is 10.797, which means isolation where and are the semi-positive definite and positive definite weight of the matrix at the outputs and control inputs, respectively. The dynamic response of the platform was studied for a sinusoidal load of 0.5*sin, where the forcing frequencies are the first four natural frequencies. The control results in a significant reduction in amplitude for all four modes, as shown by the dark line.
13 represents these voltages for the first mode observed to be within the PZT threshold. The stress required for the longer vertical beam is more due to higher bending stress. The eigenvalues for the closed loop system listed in Table 6 show that there is a significant shift in the real part when the.
Eigenvalues of closed loop with external force of N on 2nd dof Eigenvalues of open and closed loop. The system is subjected to an initial displacement of 0.005 m at the second degree of freedom. Although there is a small shift in the real part, the control appears to be significant with the chosen weighting matrices.
Without PZT control With PZT control (actuators and sensors) Eigenvalues Nat.Freq. Hz) Q R Eigenvalues Actuator voltage. The platform is exposed at time 0.1 s, to a sinusoidal input of 0.5 sin( N at 2nd degree of freedom for a duration of 0.4 s where, is the first natural frequency of. This leads to an amplification of m in the response in the uncontrolled state as shown in fig.
The amplitude peak seen as m at the end of 0.5 s does not decay to zero at the end of 2 s for the uncontrolled state. The voltage curves for the first three actuators are given in the figure for this case. The uncontrolled and controlled modal response of the flexible platform to a pulse load of 2 N applied to the 2nd degree of freedom for a duration of 0.4 s is simulated.
The modal response contribution for this input is plotted for the first mode only and is given in Fig. The response peak is seen to be m at the end of 0.5 s and the controlled plot peak is m.
Conclusion
Numerical simulations confirm that vibration isolation of disturbances on board the spacecraft can be achieved passively with this type of platform. Active vibration control of the device using sensors and actuators embedded as a collocated pair on movable arms was studied using an optimal control technique. Vibration reduction has been demonstrated for harmonic loading, impulse loading and initial displacement loading conditions.
Spanos, Vibration isolation and suppression system for precision payloads in space, smart materials and structure. Davis, Line-of-Sight Performance Improvement with Reaction-Wheel Isolation, AAS Paper 86-005, Proceedings of the Annual Rocky Mountain Guidance and Control Conference, Keystone, CO, Feb 1-5, 1986. Neat, micro-precision interferometer: Pointing System Performance in Orbit Disturbance Environment, Proceedings of the 1999, SPIE Smart Materials and Structures Conference, Vol.
Jewell, Hubble Space Telescope Reaction Wheel Assembly Vibration Isolation System, presented at NASA Marshall Space Flight Center, Huntsville, Alabama, 1986. Tupper Hyde, Second Generation Hybrid D-Strut™, presented at the SPIE Smart Structures and Materials Conference, San Diego, CA , 1995. Kyle, An Experimental Investigation of Shape Memory Alloy Springs for Passive Vibration Isolation, AIAA pp.
Preumont, Stewart's Piezoelectric Platform for General Purpose Active Damping Interface and Precision Control, Proc.9th European Space Mechanisms & Tribology Symposium, Liege, ESA SP. Abu Hanieh, A six-axis single-stage active vibration isolator based on the Stewart platform, Journal of Sound and Vibration. 19] Nader Jalili and David W Knowles IV, Structural vibration control using an active resonator absorber: Modeling and control implementation, Intelligent Materials and Structures.
Philippe, Passive and active microvibration control of vibration for space system with very high steering accuracy, Materials and Smart Structures. Hubbard Jr., Active vibration control of dispersed piezoelectric polymer of a cantilever beam, Journal of Guidance, Dynamics and Control. Balamurugan, Finite element modeling of intelligent piezolaminated structures for active vibration control with distributed sensors and actuators, Journal of Sound and Vibration.
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