Exercise 4 INVESTIGATION OF THE ONEDEGREEOFFREEDOM SYSTEM


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1 Eercise 4 IVESTIGATIO OF THE OEDEGREEOFFREEDOM SYSTEM 1. Ai of the eercise Identification of paraeters of the euation describing a onedegreeof freedo (1 DOF) atheatical odel of the real vibrating syste.. Test stand The test stand consists of a plate undergoing oscillations forced by springs connected to an eccentric device ounted on an electric otor shaft (Fig. 4.1). Viscous daping coes fro an oil daper and the syste of supporting springs akes the syste stiffness. In Fig. 4.1 there are vibration ass, which can rotate about the ais of support bearings 3, set of springs 4, oil reservoir 14, and oil daper 9 foring the support of ass. On base 1 there is a forcing subsyste, which consists of electric otor 6 with constant speed control, eccentricar device 7 and spring 8. Fig Schee of the test stand The approiate freuency of forcing is easured with tooth disc 5 placed on the shaft of otor 6, photoelectric transducer 1 and analogueeasuring device 11 of the ZPA19 type. The easureent of the ass vibration is ade with transforer 1 of the OT3 type, easuring device 11 and PC icrocoputer 13 euipped with the AMBEX LC data acuisition syste collecting data fro displaceent transducer 1. These data are sent then directly to the PC eory for further processing. 3. Theoretical introduction For further discussion, the real syste is odelled with a siplified one, having 1 DOF under haronic ecitation. Instead of the rotational otion, which takes place on the stand, we approiate its otion as the linear one. This is done by reducing rotations in the particular point (displaceent sensor position) to the linear (vertical) ones, on assuption that the rotation angles are sall. Then, siple trigonoetric relations can be applied. A odel of the euivalent syste is shown in Fig. 4..
2 Fig. 4.. Model of the eperiental euivalent syste otations in Fig. 4.:  reduced ass, k 1  reduced stiffness of spring set 4, k  reduced stiffness of spring 8, c  reduced daping coefficient, e  vibration aplitude of the point E (reduced value of the eccentricity),  ecitation force radial freuency eual to the otor rotational speed,  reduced displaceent of the ass. All the values are reduced with respect to the point of the easureent (the contact point of easureent device 1 with the vibrating ass  Fig. 4.1). The euation of otion of the ass is as follows: & = ( k1+ k ) + c & + k e sin t (4.1) Euation (4.1) can be arranged as: & + h & + = sin t, (4.) where: c k1+ k ek h=, =, =, (4.3) The particular solution to the euation of otion presents the state of the syste when natural vibration disappears: = Asin( + ϕ), (4.4) where A is the aplitude of forced vibration, is the forcing freuency of the syste, and ϕ represents a phase shift between the forcing and the resulting displaceent of the ass : h A =, ϕ = arctan. (4.5) + 4h ( )
3 A A Fig Resonance graph The aplitude A of the syste vibration described with the function shown in the first euation of Es. (4.5) can be presented in a typical for as shown in Fig. 4.3, and is usually called a resonance graph. During the eperient, one can obtain a siilar graph recording the appropriate pairs of data (, A). The paraeters,, h are to be found by coparison of both graphs, the theoretical and the eperiental one, assuing the curves are to be as close each other as possible. Matheatically, one can set the conditions possible to forulate the euations describing their values. Due to the fact that we need to deterine the values of three paraeters,, h in Es. (4.5), three different conditions for the adjacency of both graphs should be posed. It is useful to choose the following conditions: 1) both curves have to pass the point (, ) (see Fig..4), ) both curves have to coe through the point (, A ), the peak of the resonance graph, 3) the theoretical graph is to be tangent to the eperiental one in the point (, A ). If we replace and A with and, respectively, in the first euation of Es. (4.5), the above conditions can be shown atheatically as: =, (4.6) then, the first of the aboveentioned conditions will be satisfied. Siilarly, after replacing and A with and A respectively, A = ( ) + 4 h (4.7) the second condition will be fulfilled. The third condition eans that the derivative A/ disappears in the point (, A ). Deriving the first euation of Es. (4.5), we get: A = [ ( )( ) + ] 8 h 3 [( ) + 4 h ] =, (4.8)
4 which results in: + + h =. (4.9) Solving the syste of euations (4.6), (4.7) and (4.9) with respect to the paraeters,, h, we obtain: = h= 1 A (, = = )= 1 A 1 1 A, 1. (4.1) A A T HEORY A i THEORY A i EXP i E XPERIMET i Fig Theoretical and eperiental resonance graphs Fro the above euations one can see that the necessary condition is to deterine the coordinates of the graph peak (, A ) and the displaceent under static conditions. We find here the situation shown in Fig. 4.4 both the theoretical and eperiental curves pass through the points (, ) and (, A ), and, additionally, they have a coon tangent value in the point (, A ). If the investigated syste is linear, then both curves in Fig. 4.4 should be identical, but it is rather ipossible that the eperiental data are ideal. To evaluate how the discussed linear odel approiates the real syste, we eploy the following forula: = ( i) ( A i) (4.11)
5 Soe rearks on the processing of the data. 1) The data fro the easuring device have the for of an electric signal. Voltage values are easured, but before collecting the data, there is a need to scale the device. Applying the known displaceent to the probe, its output value is to be easured and the characteristic curve as a function U = f() is obtained. The resulting curve should for a broken line, which should be approiated with a line and its slope gives the recalculating coefficient of voltage to displaceent values. Using different distance plates, we set up different known displaceents and ake 5 easureents of appropriate voltage at least. ) There is a need to identify the ecitation freuency in each point of the eperiental resonance curve. Since the reading of the freuency on the ZPA19 device is not accurate, this has been done applying the Fast Fourier Transfor to the easured displaceent signal. Its behaviour follows the ecitation for and its freuency is eual to the ecitation. The highest peak in the calculated spectru is assued to be such freuency. The initial values set with the device 1 are to be used as approiate. 3) The progra used in the eperient deterines the theoretical resonance graph using the above described assuptions in two characteristic points (, ) and (, A ). For the first one in, a parabola is constructed on the basis of the first two data points of the recorded set with the condition for the tangent in to be horizontal (slope = ). For the deterination of the peak of the resonance curve, another parabola is used. Using coordinates of 3 eperiental points found to be closest to the highest value, a parabola is atheatically constructed and its peak is assued to be the eperiental resonance curve peak, hopefully approiating better the value (see Fig. 4.5). Due to such a procedure, the peak coordinates are closer to the real ones the control setup not always allows us to set the ecitation freuency, which in fact gives the aiu displaceent. 4) The aplitude of vibration of the ass is taken as a half of the detected peak values fro probe 1 and ultiplied by the coefficient deterined in point 1. A peak value A a easureent points parabolic curve a Fig Iproved ethod for finding the peak value of the resonance graph 5) In soe range, an oil reservoir position allows us to change the aount of daping in the syste. The lowest position of the oil reservoir drains alost the whole volue of oil off the piston area, lowering thus the coefficient of viscous daping etreely. Such a position can result in dangerous behaviour of the syste under forcing. The highest position (vertical) causes that
6 viscous daping is aial, liiting also oveents of the ain ass. Other positions should allow ean values of daping. 4. Course of the eercise Initial part 1) Start the PC coputer and then the control device 11 (its knobs should be set to the iniu, i.e., to the left). ) Run the progra called CW13.BAT, choose either the basic or epanded version. Scaling the displaceent probe 1) Turning the shaft of the otor by hand (otor turned off!), set the eccentric device in echanis 7, its arker should be in the highest position. It is assued to be the zero position of the displaceent probe. Take care not to shift this position during the whole process. ) Run the progra option called Scaling. 3) Run the easureent for the zero value of the displaceent probe. 4) Put the thinnest plate under probe 1, enter its value, run the easureent. 5) Repeat point 8 for other plates. 6) Reove the last used plate and choose the option: E (End of scaling). The coefficient used in the calculation should be displayed on the screen. Eperiental resonance graph 1) Oil reservoir of daper 14 is to be set in the vertical position (Fig..1). ) Turn the otor drive on, using the knobs on the ZPA19 control device. 3) Set an approiate value of the ecitation freuency 4) Suggested values of the ecitation freuencies: 15,, 5, 3, 34, 36, 38, 4, 4, 45, 5, 7, 9 rad/s. Detailed easureents should be done around the resonance value. The easureents for the lowest values should be ade especially carefully (deterining the value of ). These values should be set fro the range: 515 rad/s. 5) Setting the values close to the resonance, watch carefully indicators of the ZPA19 device. The values should be set a little lower and a little higher, but giving a visible difference on the scale. 6) Choose the option Resonance graph to start easuring the resonance graph. 7) Each easureent is started by pressing the key P (Measureent) on the coputer keyboard. After collecting the data in the upper left corner, two traces of easured signals are visible. The upper part is used for the easureent of the aplitude of the signal. The lower one is used by the FFT procedure to calculate the ecitation freuency. 8) If everything is satisfactory, the key Y (Yes) should be used. Otherwise, one should press the key (o) and repeat the easureent by pressing P again. 9) Repeat points 15 and 16 for other ecitation freuencies. 1) On copleting this part of easureents, turn off the otor using the knobs. 11) Finish the easureents choosing the option K (End) with the confiration by T (Yes). 1) Then, to analyse the real resonance graph, Analysis is to be perfored. 13) To deterine coordinates of the peak value (copare point 3), select 3 points closest to the aiu value with the cursor and ark the with the Z (Mark) key. Press the K key to finish this part. Pressing any other key eans the beginning of the analysis. 14) The values of the paraeters, α, h and the theoretical graph are obtained autoatically. Epanded case: Characteristics at different daping levels 1) To obtain the dependence of resonance peak values on daping, change daping to a slightly lower value by decreasing the angle of the daper cylinder reservoir with respect to the base level.
7 ) After such a change, the collection of data points for the net resonance curve is to be perfored, repeating procedure described in points 1 . There should be at least 3 to 5 easureents at different daping values, and then the final analysis is needed to obtain the dependence. 3) Using the option A (Analysis), the graphical for of results is presented. It shows the dependence of the natural daped freuency of the syste on the daping paraeter level. 4) Print out the Report, and then draw your conclusions. References 1. Rao S.S.: Mechanical Vibrations, AddisonWesley, Y, Tse F.S., Morse I.E., Hinkle R.T.: Mechanical Vibrations  Theory and Applications, Allyn and Bacon Inc., Kapitaniak T.: Wstęp do teorii drgań. Wydawnictwo Politechniki Łódzkiej, Łódź 199.
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