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The paper presents a mathematical model for analyzing the threshold of stability for rotating machines, where the rotor is linked to the stator by roller bearings, bearing housings and end-shields and where the stator feet are mounted on a soft foundation. The internal (rotating) damping of the rotor is the only source of instability, which is considered in the paper. After the mathematical coherences of the multibody model are described, a procedure is presented for deriving the threshold of stability. Additionally, a numerical example is shown, where the threshold of stability is calculated for different boundary conditions. It could be demonstrated, that the stiffness of the foundation—even if the foundation stiffness is isotropic—can help stabilizing this kind of vibration system in the same way as orthotropic bearing stiffness or orthotropic bearing housing and end-shield stiffness for a rigid foundation.

When designing rotating machines, it is important to calculate the vibration behavior and to consider the influence of the foundation [

The vibration model is a simplified model, which describes the movement in the yz-plane (

The vibrations system consists of two main masses, the rotor mass m w , which is concentrated as a lumped mass in the middle between the two bearings, and the stator mass m s , which is concentrated in the centre of gravity S of the stator with the mass inertia θ s .

Beside these two main masses, two additionally masses are considered, the mass of the shaft journal m v and the mass of the bearing housing m b , mostly to avoid zeros at the main diagonal of the mass matrix. The rotor has the rotor stiffness c and the internal damping d i and rotates with the rotary angular frequency Ω. The rotor is connected to the end-shields by bearing housings and roller bearings, which suppose to be equal for each machine side. Many methods and strategies are described in literature to derive the stiffness of roller bearings, e.g. [_{rz} and horizontal bearing stiffness c r y . Cross coupling coefficients of the roller bearings are neglected as well as damping of the roller bearings. The stiffness and damping of the bearing housing and end- shields is described by the bearing housing and end-shield stiffness and damping matrix C b and D b , which also suppose to be equal for each machine side. The stator structure is here assumed to be very stiff, compared to the foundation stiffness, so the stator structure can be modeled rigid. The stator feet - F_{L} (left side) and F_{R} (right side) - are connected to the ground by the foundation stiffness and damping matrix C f and D f , which are also assumed to be equal for the right side and left side of the machine. When deriving the damping coefficients, it has to be considered, that the natural vibration of the critical mode occurs with the angular natural frequency ω s t a b at the threshold of stability, which is the rotary angular frequency Ω s t a b . Therefore, the whirling angular frequency ω F of the rotor becomes ω s t a b , at the rotary angular frequency of Ω = Ω s t a b :

ω F = ω s t a b (1)

The internal material damping of the rotor d i can be described by the stiffness of the rotor c and mechanical loss factor tan δ i of the rotor, depending on the whirling angular frequency ω F , referring to [

d i ( ω F ) = c ⋅ tan δ i ω F (2)

The same approach is deduced for the damping coefficients of the bearing housing end end-shield and of the foundation:

d b z ( ω F ) = c b z ⋅ tan δ b ω F ; d b y ( ω F ) = c b y ⋅ tan δ b ω F (3)

d f z ( ω F ) = c f z ⋅ tan δ f ω F ; d f y ( ω F ) = c f y ⋅ tan δ f ω F (4)

With the stiffness of the bearing housing end end-shield c b z and c b y and the stiffness of the foundation at each machine side (left and right side) c f z and c f y and the loss factor of the bearing housing and end-shield tan δ b and of the foundation tan δ f .

To get the threshold of stability, it is necessary to derive the homogenous differential equation by separating the vibration system into four single systems: a) rotor mass system, b) journal system, c) bearing house system and d) stator mass system (

The displacements of the stator mass (z_{s}, y_{s}, j_{s}) is small, compared to the dimensions of the machine (h, b, Y), therefore following linearization is possible:

z f L = z s − φ s ⋅ b ; z f R = z s + φ s ⋅ b ; y f L = y f R = y s − φ s ⋅ h (5)

The homogenous differential equation system can be derived by analyzing the equilibrium of at each single system:

M ⋅ q ¨ + D ⋅ q ˙ + C ⋅ q = 0 (6)

with the coordinate vector q :

q = [ z s ; z w ; y s ; y w ; φ s ; z v ; z b ; y v ; y b ] T (7)

with the mass matrix M :

M = [ m s 0 0 0 0 0 0 0 0 0 m w 0 0 0 0 0 0 0 0 0 m s 0 0 0 0 0 0 0 0 0 m w 0 0 0 0 0 0 0 0 0 Θ s x 0 0 0 0 0 0 0 0 0 2 m v 0 0 0 0 0 0 0 0 0 2 m b 0 0 0 0 0 0 0 0 0 2 m v 0 0 0 0 0 0 0 0 0 2 m b ] (8)

with the damping matrix D :

with the stiffness matrix C :

The internal (rotating) damping d i of the rotor in conjunction with the rotary angular frequency Ω leads here to an anti-symmetric stiffness matrix, which causes instability, when the threshold of stability is exceeded ( Ω > Ω s t a b ). The limit of vibration stability Ω s t a b can be calculated, when increasing the rotary angular frequency Ω , and analyzing the eigenvalues. If a real part of one eigenvalue gets zero, the limit of vibration stability is reached. Increasing the rotary angular frequency Ω furthermore will cause a positive real part and the vibration system gets instable. Using the state-space formulation

[ q ˙ h q ¨ h ] ︸ x ˙ = [ 0 I − M − 1 ⋅ C − M − 1 ⋅ D ] ︸ A ⋅ [ q h q ˙ h ] ︸ x (11)

the eigenvalues can be derived. With the formulation x = x ^ ⋅ e λ ⋅ t , the eigenvalues are calculated by:

det [ A − λ ⋅ I ] = 0 (12)

At the threshold of stability the eigenvalue λ of the critical mode gets:

λ = λ s t a b = ± j ⋅ ω s t a b (13)

The real part of the critical eigenvalue λ s t a b is zero and the whirling angular frequency ω F is then identical to ω s t a b , while the rotor is rotating with Ω s t a b . Considering, that the coefficients d i , d b z , d b y , d f z , d f y are depending on the whirling angular frequency ω F , an iterative solution has to be deduced, according to

First, a start value of the whirling angular frequency ω F = ω s t a b , 0 has to be estimated. This can be done e.g. by following estimation, which is based on a ridged mounted machine, without external damping and with the assumption that c r y < c r z and c b y < c b z and that the first natural angular frequency ω y , 0 is here the whirling angular frequency at the threshold of rotor stability:

ω s t a b , 0 = ω y , 0 = c t o t a l m w with : c t o t a l = 1 1 c + 1 2 c r y + 1 2 c b y (14)

With this assumption the damping coefficients d i , d b z , d b y , d f z , d f y can be derived, and therefore also the threshold of stability and the natural angular frequency, leading to Ω s t a b , 1 and ω s t a b , 1 . With this new angular whirling frequency ω F = ω s t a b , 1 the damping coefficients d i , d b z , d b y , d f z , d f y are calculated again, leading to a new threshold of stability Ω s t a b , 2 and a new natural angular frequency ω s t a b , 2 . If the ratio | ω s t a b , 2 − ω s t a b , 1 | / ω s t a b , 1 is less than Δ - an arbitrarily chosen value -the calculation is finished and Ω s t a b = Ω s t a b , 2 and ω s t a b = ω s t a b , 2 . If the ration is larger as the chosen value Δ , a loop has to be run through till the ratio is less than Δ .

Based on the mathematical derivation, a numerical example is shown, where the threshold of stability is analyzed.

The rotating machine consists of a rotor, roller bearings, bearing housings, end-shields and a stator (

In

It can be shown, that at a rotor speed of about 26130 rpm the real part α 3 becomes zero and therefore the threshold of stability is reached. The corresponding natural angular frequency is ω 3 = 394.3 rad / s , which is equal to the whirling angular frequency ω F = ω s t a b at the limit of stability of the critical mode, which is here mode 3. Increasing the rotor speed above 26130 rpm, leads to instability of the vibration system.

Machine data | Description | Value |
---|---|---|

Mass of the stator | ||

Mass inertia of the stator at x-axis | ||

Mass of the rotor | ||

Mass of the rotor shaft journal | ||

Mass of the bearing housing | ||

Height of the centre of gravity | ||

Distance between motor feet | ||

Stiffness of the rotor | ||

Horizontal stiffness of bearing housing and end shield | ||

Vertical stiffness of bearing housing and end shield | ||

Mechanical loss factor of bearing housing and end shield | ||

Mechanical loss factor of the rotor | ||

Bearing data | Description | Value |

Bearing type | Ball bearing; Type 6220 C3 | |

Horizontal stiffness of the roller bearing | ||

Vertical stiffness of the roller bearing | ||

Foundation data | Description | Value |

Type of foundation | Welded steel frame foundation | |

Vertical stiffness of the foundation at each motor side | ||

Horizontal stiffness of the foundation at each motor side | ||

Mechanical loss factor of the foundation |

zero. When increasing the rotor speed furthermore, this real part α 3 gets positive. Therefore mode 3 is the critical mode shape.

Now different cases are investigated, and the threshold of stability n s t a b is calculated as well as the natural angular frequency ω s t a b at the threshold of stability (

If then the bearing stiffness would be changed from isotropic ( c r z = c r y = 2.0 × 10 8 kg / s 2 ) to orthotropic ( c r z ≠ c r y ; c r y = 1.5 × 10 8 kg / s 2 and c r z = 2.5 × 10 8 kg / s 2 ), the threshold of stability can be increased again up to 13020 rpm (case e).

In this section, the influence of the foundation stiffness on the threshold of stability n s t a b and on the whirling angular frequency ω s t a b is analyzed.

Therefore, the foundation stiffness is variated from the rated values in

Case | Description | D of | ||
---|---|---|---|---|

a) | Basic Data (Data | 394.26 | 26130 | 0 |

b) | Data | 393.75 | 25850 | −1.07 |

c) | Data | 391.20 | 24870 | −4.82 |

d) | Data | 344.85 | 3840 | −85.3 |

e) | Data | 340.69 | 13020 | −50.2 |

Now, the influence of bearing stiffness is analyzed for the rated soft foundation (

In this section, the influence of bearing housing and end-shield stiffness is analyzed, for the rated soft foundation (

Here, the influence of the bearing stiffness is analyzed again, but now for a rigid foundation ( c f z = c f y → ∞ ). Therefore, the bearing stiffness is again variated in a range between 1.0 × 10 8 kg / s 2 and 3.0 × 10 8 kg / s 2 (

In this section, the influence of bearing housing and end-shield stiffness is analyzed again, but for a rigid soft foundation ( c f z = c f y → ∞ ). Therefore, the

bearing housing and end-shield stiffness is again variated in a range between 3.5 × 10 8 kg / s 2 and 1.05 × 10 9 kg / s 2 (

In section 4.4 - 4.8 (Figures 7-11) the influence of the foundation stiffness, the bearing stiffness and the bearing housing and end-shield stiffness on the threshold of stability n s t a b and on the whirling angular frequency ω s t a b is analyzed.

described in literature ( [

The innovation of the paper is now, that it can be demonstrated (

In this example, the threshold of stability could be increased even to maximum of about 143000 rpm, at a foundation stiffness of c f z = 5.14 × 10 8 kg / s 2 and c f y = 7.5 × 10 8 kg / s 2 (

The paper presents a mathematical model especially for analyzing the threshold of stability for a special kind of rotating machines, consisting of a rotor, stator, end-shields, bearing housings and roller bearings, mounted on a soft foundation, so that the centre of gravity of the stator is displaced by the height h from the foundation (

Werner, U. (2017) Influence of the Foundation on the Threshold of Stability for Rotating Machines with Roller Bearings―A Theoretical Analysis. Journal of Applied Mathematics and Physics, 5, 1380-1397. https://doi.org/10.4236/jamp.2017.56114