7 Simulation and Experimental Results
7.1 Global Dynamic System Behavior
To predict the dynamic behavior of a system, it is necessary to look at the natural properties like eigenfrequencies, mode shapes and damping, as well as at the excitation forces.
Starting the discussion with the excitation forces, one can state that the current rotor system contains three rotating subsystems: the drive shaft, the tumble sleeve and the electric engine.
An obvious excitation mechanism is the natural imbalance of each rotor, which creates speed-dependent excitation forces. However, since the electric engine is precisely balanced and operates sub-critically, the main excitation mechanisms can be expected to arise from the natural imbalance of the drive shaft and from the tumbling movement forced by the sleeve.
Turning now to the natural behavior of the rotor system: In Chapter 5, the natural behavior of the system was evaluated in non-rotating condition. At rest, the system has a vertical (in the xz-plane) and a horizontal (in the yz-plane) bending mode in which the natural frequen-cies are located well within the range of rotational speed of the three spinning subsystems mentioned above. Therefore it can be expected that when the rotational frequency of one of the rotors matches the natural frequency of one of the bending modes, a resonance will occur, leading to large amplitudes. However, in a rotating system, the natural frequencies usually depend, to a smaller or larger extent, on the rotational speed. This means that the resonance frequencies during operation do not necessarily have to match the natural fre-quencies measured or calculated at non-rotating condition. The speed dependency of natural
7 Simulation and Experimental Results
frequencies depends on the ratio between the polar and the diametral moment of inertia of the rotors. In the current case, the rotating components of the diamond coring systems are relatively slender; so no large speed dependency is to be expected.
According to the results in Chapter 6.2, the mode shape of the first bending mode(s) is mainly a bending of the drive shaft against the rest of the tool, which performs an almost rigid body motion. Therefore, during operation, the largest displacement amplitudes in the case of a resonance can be expected at the tip of the core bit or at the rear handle. Figure 7.1 shows the time signal of the displacement amplitude of the core bit tip in vertical direction (x-direction) relative to the housing during a slow run-up and run-down of the system. The same plot also contains the time signals of the rotational speed of the drive shaft and the tumble sleeve. The signals are recorded according to the experimental setup described in Chapter 4.
Figure 7.1: [Meas] Vertical displacement of core bit tip relative to housing during slow run-up and run-down
The plot shows two major resonance phenomena during run-up and run-down. The first, and in this case smaller peak, is caused when the rotational frequency of the drive shaft matches the natural frequency of the first bending mode. The second resonance peak occurs, when the rotational frequency of the tumble sleeve matches the natural frequency of the bending mode. Looking closely at the resonance peaks, one can see that each of them shows a smaller side peak caused by the excitation of the horizontal bending mode. The unwanted, yet in the current case unavoidable, cross-sensitivity of the triangulation lasers causes a “cross-talking”
between the horizontal and vertical channels, as discussed in Chapter 4.1.1 and visualized
7.1 Global Dynamic System Behavior
in Figure 4.2a. Since the horizontal bending mode has a slightly higher natural frequency, the corresponding resonance during operation appears a little delayed in time during run-up, or at a slightly higher rotational speed, respectively. During run-down it is the other way round.
Assigning the resonance peaks to a certain excitation mechanism becomes easier by plot-ting the data (i.e. displacement or acceleration) over rotational speed instead of time. In Figure 7.2, a vibration order analysis according to Chapter 4.3.2 has been performed to calculate the overall displacement level, as well as the share that can be assigned to the first order of the drive-shaft, the tumble sleeve and the electric engine. Since the ratio of the rotational speeds between the three different rotating subsystems is fixed, one of them can arbitrarily be chosen to act as a reference and the other two can be expressed as multiples of the reference. For example, the first order of the drive-shaft can be considered as then-th order of the sleeve, according to Equation 3.1. This allows for the plotting of all three orders on the same scaling of the x-axis. In Figure 7.2, the tumble sleeve acts as a reference. Since no torsional oscillations between shaft and tumble sleeve could be observed in any experi-ment performed within the current work, the signal of the rotational speed of the core bit is simply a multiple of the signal obtained from the tumble sleeve, and is therefore redundant information. As a consequence, all further plots will present information about rotational speed using the signal of the tumble sleeve instead of using both tacho signals.
10 15 20 25 30 35 40 45
0 10 20
rotational speed [r]
displacement[d]
overall level
1st order of drive shaft 1st order of tumble sleeve 1st order of electric motor
Figure 7.2: [Meas] Overall-level and order-plot of vertical displacement of core bit tip relative to housing during slow run-up
7 Simulation and Experimental Results
Comparing the share of the orders to the overall displacement level indicates the following with regard to the three different main excitation mechanisms :
• The first resonance occurring during run up is excited almost solely by the natural imbalance of the drive shaft.
• The second resonance is caused almost completely by the excitation through the tumble movement.
• The excitation by the first order of the electric engine (imbalance) can be neglected with regard to the dynamic displacement of the core bit.
The two resonances appear to be well separated, meaning that each resonance is dominated by only one of the excitation mechanisms, the imbalance of the drive shaft and the forced tumbling movement by the sleeve. This is confirmed when regarding the corresponding Campbell diagram in Figure 7.3: The relevant frequency of the first bending mode is clearly dominated by the two excitation mechanisms described above. The two resonances appear at the same frequency, and – interestingly – at the same frequencies as the natural bending frequencies (first vertical and horizontal bending) when the system is not rotating. This is a double indication of the fact that the natural frequencies of the first bending mode are not influenced by the rotational speeds of the rotors. The reason for this is the fact that the rotors are very slender, meaning that the polar moment of inertia is significantly smaller than the diametral one.
1st order of drive shaft 1st order of sleeve
0 20 40 60 80 100
10 20 30 40
frequency [f]
rotationalspeedofsleeve[r]
0 10 20
displacement[d]
Figure 7.3: [Meas] Campbell diagram of vertical displacement of core bit tip relative to hous-ing durhous-ing slow run-up
Chapter 6.2 deals with the modal behavior of the whole system, pointing out that the first global bending mode is mainly a counter-phase movement of the core bit against the