The simulation time is maintained by the SIMULINKTM block ‘Clock’. The triangular blocks contain numbers and parameters progressively multiplied. Operating parameters like the spin speed (ω), mass (m) and viscous damping (c) are placed in such blocks. Trigonometric function blocks
‘cos’ and ‘sin’ perform the assigned trigonometric function on the incoming signal. Unbalance phase ‘β’ is placed in constants block. Summation blocks perform algebraic operations over the input signals in the sequence in which the + and – sign appears inside the block. Integrator and differentiator blocks perform the calculus operation over the input signal. Dynamics of main blocks of the above model are
Unbalance Force: For x direction the trigonometric block ‘cos’ has input ωt+β. Multiplied with
me ω
2, the output is meω2cos( tω +β). For y direction the output is meω2sin( tω +β).Crack Force: Input to this sub-block is ωt. Sub-block outputs 12s t( )∆k22(1+cos 2ωt)δx and
1
2s t( )∆k22(sin 2ωt)δx, in the x and y directions respectively, as provided for in Eqn. (2.22).
Summation Block: This block performs algebraic summation of all force signals in accordance to Eqn. (2.26). The block accumulation amounts to
mx ɺɺ and my ɺɺ
, respectively, for the x and y directions.Integration Blocks: These blocks perform numerical integration over the input signal. The sum total of the summation block is the force term. Before the integration block, multiplication by 1/m is performed to obtain the acceleration term. The progressive integration blocks output velocity and displacement, respectively, for both the orthogonal directions.
PID Blocks: The displacement output from the integration block is the input to the PID block. This block executes the simulation for the PID controller in accordance to Eqn.(2.24) and outputs the control current.
Reference Signals Block: This block implements a complex reference signal of the form
cos( i t ω ) jsin( + i t ω )
. The input to the block is ωt and the individual harmonic sub-block has multiplier i (the harmonic) before passing on the signal to trigonometric blocks. A representative SIMULINKTM block implementing ith complex reference signal is presented in Figure 2-6.Figure 2-6 One harmonic of multi-harmonic complex reference signal
Bus Creator: Bus creator stacks the multiple incoming signals as a single signal line, with individual signals stacked as individual vectors in the Bus output.
The model offers wide choice for simulation conditions particularly, with or without any of three conditions, viz. the crack, unbalance and AMB, simply by disconnecting the signal flow across the individual component. The effect of relative positioning of crack and unbalance can also be
simulated by varying the angle
β
in the input. The simulation data used for the response generation is summarized in Table 2-1.
Table 2-1 The rotor and AMB system data for the numerical simulation
Parameters Values Parameters Values
Disc mass, m 2 kg Actuator Factors
Intact shaft stiffness, k0
7.6×105 Nm-1 Force – current factor, ki
42.1 N/A Additive crack
stiffness, ∆k22
3×105 Nm-1 Force – displacement factor, ks
105210 N/m
Viscous damping, c 76 Nsm-1 Controller Gains
Phase of unbalance, β 30° deg. Proportional, KP 12200 A/m Shaft deflection, δx 2.6×10-5 m Derivative, KD 3 A-s-m-1
Disc eccentricity, e 24 µm Integral, KI 2000 A/(m-s)
Numerical values of the intact shaft stiffness – k0 and the shaft static deflection – δx, are based on mild steel shaft material of length 500 mm and 16 mm diameter carrying a disc mass of 2 kg and reported in Shravankumar and Tiwari (2014). Gains of the PID controller used in the simulation are based on the performance optimization study on gain parameters by Bordoloi and Tiwari (2013). The response is generated with a fourth-order Runga-Kutta integration solver. A typical response of the vibration displacement and the AMB control current obtained at the shaft spin speed of 1500 rpm (25 Hz) is presented in Figure 2-7.
Figure 2-7 Generated response (a) x-displacement (b) y-displacement (c) x-current (d) y-current (e) current orbit (f) shaft centreline orbit
The effect of presence of AMB on the rotor behaviour can be studied by comparison of the response of the rotor with and without AMB support. The response of vibration displacements at 1500 rpm in two directions and the shaft centreline orbit are generated with rotor properties as detailed in Table 2-1, in absence of AMB in the support system. The response of the rotor with and without AMB in the support system is presented in Figure 2-8 and it may be seen that the presence
4 4.5 5
-5 0 5 10x 10
-6
Time (second) (a)
Displacement (m)
4 4.5 5
-10 -5 0 5x 10
-6
Time (second) (b)
Displacement (m)
-5 0 5
x 10-6 -2
0 2 4 6 8
x 10-6
x - Disp (m) (f)
y - Disp. (m)
4 4.5 5
-0.05 0 0.05 0.1 0.15
Time (second) (c)
Current (A)
4 4.5 5
-0.1 -0.05 0 0.05 0.1
Time (second) (d)
Current (A)
-0.1 -0.05 0 0.05 0
0.05 0.1
x - Current (A) (e)
y - Current (A)
of AMB reduces the magnitude of the vibration displacement. The centreline orbit also gets diminished.
Figure 2-8 Rotor response without AMB (- - -) and with AMB ( ): (a) x-displacement (b) y-displacement (c) shaft centreline orbit
Since the unbalance force increases with the square of angular velocity of the shaft, at higher spin speeds, the response due to the crack is overshadowed by response due to the unbalance, in time domain signal. In such cases, the shaft centerline orbit will not exhibit the characteristic double loop discussed in literature, for instance – Sinou and Lees (2005) and Shravankumar and Tiwari (2013). At lower speeds the crack effect is visible in time domain also, as seen in displacement response at a spin speed of 900 rpm (15 Hz) in Figure 2-9.
4 4.1
0 4 8 12
x 10
-6
Time (s)
Displacement (m)
4 4.1
-4 0 4 8
x 10
-6
Time (s)
Displacement (m)
-5 0 5
x 10
-6
-5 0 5 10 15 20
x 10
-6
x - Disp. (m)
y - Disp. (m)
(a) (b) (c)
Figure 2-9 Response generated at 900 rpm (a) x-displacement (b) y-displacement
In general, the shape of the orbit plot is indicative of the type of rotor defects and the size of the orbit plot indicates the severity of defects, for example – a circular orbit plot indicates a dominant unbalance in the rotor, the magnitude of which is directly proportional to the radius of the orbit.
Since, for the rotor health monitoring, the motion of the rotor is characterized by the forward and backward whirls; more information about the rotor condition could be obtained by analyzing the participation of individual harmonics that constitutes the response signal. The analysis of participating harmonics could be performed by the regression analysis as detailed in Section 2.3.1 or by full spectrum of the frequency transformed response as detailed in Section 2.3.2. A comparison of the speed, the accuracy and adequacy of both methods is presented in the next section.