Μέτρηση Δυναμικών Χαρακτηριστικών Μόνωση Ταλαντώσεων Μετρητές Ταλαντώσεων. Απόστολος Σ. Παπαγεωργίου

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1 Μέτρηση Δυναμικών Χαρακτηριστικών Μόνωση Ταλαντώσεων Μετρητές Ταλαντώσεων

2 Μέτρηση Δυναμικών Χαρακτηριστικών MEASUREMENT OF DAMPING Mass and stiffness of a dynamic system can be determined by its physical characteristics, while an estimate of damping resistance can be obtained by experimental measurements of the response of the structure to a given excitation. Experimental Evaluation Techniques of Damping Ratios: Free Vibration Decay Forced-Vibration Response: o Resonant Response Απόκριση Συντονισμού o Width of Response Curve & Half-Power Method o Energy Loss per Cycle Απώλεια Ενέργειας ανα Κύκλο Ταλάντωσης Εύρος Καμπύλης Αποκρίσεως & Μέθοδος Εύρους Ημιισχύος

3 Μέτρηση Δυναμικών Χαρακτηριστικών FREE VIBRATION DECAY LOGARITHMIC DECREMENT: Λογαριθμική Μείωση ln 2 Free vibration of an under-damped system: ln Let: sin Then: 2 sin 2 The ratio to provides a measure of the decrease in displacement over one cycle of motion. The ratio is constant and does not vary with time; its natural logarithm is called logarithmic decrement and is denoted by. [NOTE: The idea is due to HELMHOLTZ, 1862.] Therefore: 2 [For small values of, i.e., 2 ]

4 Μέτρηση Δυναμικών Χαρακτηριστικών If the decay of motion is slow, it is desirable to relate the ratio of two amplitudes several cycles apart, instead of successive amplitudes, to the damping ratio. For lightly damped systems (i.e., small ): 2 To determine the number of cycles elapsed for a 50% reduction in displacement amplitude, we obtain: Therefore: ln It follows that: 2 1 % ln

5 Μέτρηση Δυναμικών Χαρακτηριστικών The damping ratio can be obtained from: or Accelerations are easier to measure than displacement. The damped period of the system can be determined from the free vibration record by measuring the time required to complete one cycle of vibration.

6 RESPONSE TO VIBRATION GENERATOR Απόκριση σε Γεννήτρια Ταλαντώσεων Consider two counter rotating masses,. Harmonic force produced: Ω sin Ω Equation of motion (assuming = mass of structure): Μέτρηση Δυναμικών Χαρακτηριστικών Ω sin Ω Therefore: Ω Ω

7 Μέτρηση Δυναμικών Χαρακτηριστικών FORCED-VIBRATION RESPONSE Εξηναγκασμένη (Αρμονική) Ταλάντωση Resonant Response: Συντονισμός Φάσης At phase resonance, 1& phase angle Therefore, phase resonance can be detected by measuring the phase angle and progressively adjusting the exciting frequency until. Measurement of the phase angle may be somewhat difficult. Therefore, as an alternative, the resonance curve is obtained in the vicinity of resonance and the peak response is measured (amplitude resonance). For viscous damping Συντονισμός Πλάτους Ταλάντωσης The above equation can be used to obtain from the measured value of. For light damping. Usually, the acceleration amplitude is measured and

8 Width of Response Curve Method: Μέθοδος Εύρους Καμπύλης Απόκρισης Μέτρηση Δυναμικών Χαρακτηριστικών The width of response curve near resonance can be used to obtain an estimate of the damping. Measurement of frequencies & at which tan1 Therefore: 1 & Ω Ω The above method relies on the ability to measure the phase angle, which may require sophisticated instrumentation.

9 Μέτρηση Δυναμικών Χαρακτηριστικών If the response curve in the vicinity of resonance has been plotted, the frequencies at which the amplitude is times that at the peak can be measured. As shown in the above FIGURE there are two such frequencies, denoted & and the corresponding ratios &. [The power (~ ) at, is half the power at, hence the name Half-Power Method.] Μέθοδος Εύρους Ημιισχύος (Ζώνημα), NOTE: The ratio is referred to as the quality factor and is a measure of the sharpness of the response curve For small damping Therefore: Ω Ω Ω Ω The above equation is similar to the one obtained by the previous method. In fact, for small damping & are very close to &, respectively.

10 Μόνωση Ταλαντώσεων TRANSMITTED MOTION DUE TO SUPPORT MOVEMENT Μετάδοση Κίνησης λόγω Εδαφικής Διέγερσης A system mounted on a moving support will have some of the support motion transmitted to it. Often the design of such a system requires that the transmitted motion be minimized. Let: Then: sin Ω (The superscript stands for total ; the subscript stands for ground ) Equation of Motion: 0 Ω sin Ω Steady-state Response: sin Ω where: tan

11 Μόνωση Ταλαντώσεων Total displacement : sin Ω sin Ω 2 cos Ω sin Ω 2 cos Ω sin Ω The rotating vector representation can also be used quite effectively to obtain the total displacement (see FIGURE above): 2cos 2cos where: cos & tan Phase angle between & is obtained by applying the cosine identity to triangle : The amplitude ratio vs. is plotted above. This leads to the same value of as the one derived above.

12 Μόνωση Ταλαντώσεων TRANSMISSIBILITY AND VIBRATION ISOLATION Μετάδοση Δυνάμεων & Μόνωση Ταλαντώσεων We know that: & In practical design it is frequently required that the dynamic forces transmitted by a machine to its surroundings be minimized. (Force Isolation) Total transmitted force (assuming steady-state response): sin Ω Ω cos Ω sin Ω It follows that: tan 2 Note that the force balance diagram can be utilized effectively to obtain the transmitted force. where: Ω tan Ω

13 Μόνωση Ταλαντώσεων Transmission Ratio or Transmissibility (TR) Μεταδοτικότης του Συστήματος TR is a measure of the force transmitted to the foundation Note that the ratio is the same as the ratio that we derived for the Transmitted motion due to support movement. If the transmitted force is to be smaller than the applied force, the natural frequency should be selected so that the frequency ratio. Also, for, the transmission ratio decreases with damping, so that theoretically, zero damping will give the smallest transmitted force. In practice, however, some damping should always be provided to ensure that during startup as the machine passes through the resonant frequency, the response is kept within reasonable limits.

14 Μετρητές Ταλαντώσεων MEASUREMENT OF ACCELERATION ACCELEROMETER Επιταχυνσιογράφος Let: sin Ω Equation of Motion: sin Ω Steady-state response: sin Ω where: tan If the instrument is to be designed to measure an input acceleration which may, in fact, have several harmonic components of different frequencies, the measured displacement should be proportional to the input for all values of the input frequency For a satisfactory instrument design, should not vary with. For., stays approximately constant at a value of, provided that.. The time shift for one harmonic is: Ω 1 Ω tan tan It is required that not vary with. 2 1 For., is practically constant, i.e., is a linear function of and hence of (see attached FIGURE).

15 Μετρητές Ταλαντώσεων o

16 Μετρητές Ταλαντώσεων EXAMPLE: Investigate the output of the accelerometer with damping. when used to measure ground motion with the ground acceleration given by: sin Ω sin Ω 2 sin Ω sin Ω 2 Ω Ω sin Ω sin Ω sin Ω sin Ω sin Ω 2 sin Ω 2 For., (where is the natural circular frequency of the accelerometer), so that: Because the time functions in both terms are equal to, the shift of both components along the time axis is equal. Thus, the instrument faithfully reproduces (within a multiplicative factor ) the ground acceleration. & [The natural circular frequency of the instrument has been selected so that,.]

17 Μετρητές Ταλαντώσεων MEASUREMENT OF DISPLACEMENT DISPLACEMENT METER Μετρητής Μετατοπίσεων The transducer must be designed so that the spring is so flexible or the mass so large, or both, that the mass stays still while the support beneath it moves. Such an instrument is unwieldy. Support Displacement: sin Ω Ω sin Ω Equation of Motion: 0 Ω sin Ω sin Ω sin Ω where: For (i.e., sufficiently large), (i.e., independent of ) and (for sufficiently small damping). This implies that although the measured motion is negative of the input motion, there is no shift along the time axis and the output motion, comprised of any number of harmonic components, will be reproduced correctly.

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