Vibration analysis is the study of mechanical oscillations of a system, often performed to evaluate its performance, reliability, and stability. It identifies potential issues like resonance, fatigue, or instability, and helps design systems that minimize unwanted vibrations.
This study is for educational purposes only. No commercial purposes intended.
An industrial pump impeller is being designed to operate at high rotational speeds. To ensure safe and efficient operation, it is critical to determine the impeller's critical speed, defined as the rotational speed at which the system experiences resonance due to its natural frequency. Resonance can lead to excessive vibrations, mechanical failures, and reduced operational life.
The impeller has the following properties:
Mass (m) of the impeller: X kg (STAINLESS STEEL 304)
Radius of gyration (k): Y m
Stiffness of the supporting shaft (ks ): Z N/m
Determine the critical speed (Nc ) in revolutions per minute (RPM), considering the natural frequency of the system and assuming the impeller behaves as a simple spring-mass system.
Initial values to be examined.
Bearing Stiffness and damping coefficients acting on a bearing.
Black Outline is the original position of the impeller.
Black Outline is the original position of the impeller.
Black Outline is the original position of the impeller.
Characteristic of the vibration for damped frequency (red), stability (green) and modal damping ratio (blue) in Hertz (Hz).
The initial campbell diagram result with the four points at 100, 250, 500 and 5000 radians per second from mode 1 to 6 with corresponding whirl direction and instability state.
Forward Swirl (FW)
Direction: Same as rotor direction
Frequency Range: Vibration frequency is more than or equal to rotor rotational speed.
Backward Swirl (BW)
Direction: Opposite to rotor direction
Frequency Range: Vibration frequency is less than rotor rotational speed.
Note:
Damping Ratio - Higher damping ratio decreases damped frequency.
Stiffness increases damped frequency while Mass decreases it.
Damping ratio is a key determinant of stability:
Underdamped (0 < DR < 1): Oscillatory response with exponential decay.
Critically damped (DR =1): Fastest return to equilibrium without oscillation.
Overdamped (DR > 1): Non-oscillatory return to equilibrium, slower than critical damping.
Unstable (DR < 0): Oscillations grow over time, leading to system failure.
Since the deformation is too large, we will start to investigate the characteristic of K11 stiffness while holding C11 constant focusing on the first mode.
The first bending mode typically coincides with the lowest critical speed, where the effects of unbalance are most pronounced.
Rotors tend to have the largest deflection in this mode, making it a primary concern for vibration and stability.
If the rotor operates below the first critical speed, the unbalance response is dominated by the first mode.
If the rotor operates above the first critical speed, higher modes might become significant, but the first mode still often contributes the most energy.
Different Design Points of stiffness at constant damping coefficient.
Adding more design points helps to analyze the trend of minimizing the coefficients in this geometry. It is interesting to note that the intersection of P1 and P2 is the minimum range to minimize the deflection.
Lowering stiffness shifts the natural frequency (ωn) of the system. If this moves the operating frequency away from resonance, deformation can reduce despite lower stiffness.
Reducing damping sometimes prevents over-damping, where excessive energy dissipation slows the return to equilibrium, thereby minimizing transient deformation.
REVISED CAMPBELL DIAGRAM (BELOW) WITH TARGETED DAMPING AND STIFFNESS COEFFICIENTS:
Undetermined Whirl Direction:
Whirl direction is either not relevant, or the mode does not involve significant gyroscopic effects.
This typically occurs in modes that are axial (along the rotor axis) or purely structural (not influenced by rotor spin).
Mode Stability:
The mode is classified as stable, meaning there is no divergence or instability (no energy growth over time).
Indicates positive damping (DR>0).
No Critical Speed:
The mode does not cross the excitation frequency line, indicating no resonance.
Often occurs when the mode frequency is constant or negligible relative to operational speeds.
0 Hz Frequency at Various Rotor Speeds (10, 500, 1000, 5000 rad/s):
A frequency of 0 Hz suggests no oscillatory behavior.
This is typical mode where the damping causes complete suppression of oscillations.
REVISED CAMPBELL DIAGRAM
Each mode represents a specific pattern of vibration (mode shape) with a corresponding natural frequency.
Mode 1 has the lowest natural frequency, followed by higher modes with increasing frequencies. In this case, we will use 1764.1 Hz as our based frequency for Harmonic Response.
Lower modes (e.g., Mode 1 and Mode 2) often dominate the response, especially in systems with low excitation frequencies, as they are easier to excite.
Avoiding Resonance:
Compare the operating frequency range (e.g., rotational speed in Hz) and excitation frequencies (e.g., Blade Passing Frequency, BPF or 2× rotational speed) with the natural frequencies of all modes.
The natural frequency closest to the operating range or excitation frequency is most critical to avoid resonance.
In modal analysis results, the natural frequencies correspond to the frequencies associated with each mode of vibration.
Below are the results of frequency response and phase response due to 3 Mpa pressure which is the operating pressure of the pump.
The frequency response describes the amplitude of the system's steady-state output (e.g., displacement, velocity, acceleration, or pressure) as a function of the excitation frequency.
Amplitude vs. Frequency:
Shows how the system amplifies or attenuates the input excitation at various frequencies.
Peaks in the frequency response indicate resonance, where the excitation frequency matches a natural frequency of the system, resulting in amplified output.
The phase response describes the relative phase shift (ϕ) between the input excitation and the system's output as a function of the excitation frequency.
Phase vs. Frequency:
At low frequencies, the output is nearly in phase with the input (ϕ≈0∘).
Near resonance, the phase shift rapidly changes, typically lagging by −90∘ at resonance which is not observed.
At high frequencies, the phase shift approaches −180∘ in systems dominated by inertia or damping which is not applied during analysis.
Rotor dynamics is the study of the dynamic behavior of rotating machinery components, such as shafts, rotors, and discs, under operating conditions. It focuses on understanding and predicting the response of these components to various forces, vibrations, and instabilities during rotation. Rotor dynamics is a critical field in mechanical engineering, particularly for designing and maintaining high-speed rotating machinery like turbines, compressors, pumps, generators, and engines.
Unbalanced mass at end of rotor: .45 kg Radius: 72.2mm from the center of impeller showing the amplitude of displacement to check any interference with pump casing.
An unbalanced mass of 0.45 kg is located at the end of the rotor, with a radius of 72.2 mm (measured at the edge of the impeller's diameter) from the center of the impeller. The amplitude of displacement is analyzed to ensure there is no interference with the pump casing, to account for manufacturing tolerances, and to verify compatibility with bearing clearances.
It is important to note that the origin of the rotating force may need adjustment based on post-manufacturing measurements of the impeller dimensions or high probable area of inconsistencies of cast or mold, especially in cases where there are discrepancies between the CAD geometry and the actual manufactured impeller.
Deformation phase response lags behind the applied load of unbalanced mass.
Pump frame vibration analysis is crucial for ensuring the reliability, efficiency, and safety of rotating machinery. Vibration in pumps can indicate misalignment, imbalance, bearing issues, or other mechanical faults. In this study, we will just consider the modal analysis based on the weight of the pump and the motor based on the pump's fabricated structural frame below.
The modal masses and kinetic energies below are calculated with unit normalized modes including the hight participation factor along Y axis for translation and X axis for rotation.
With remote poins for pump and pump motor center of gravity weighing 350 kg for pump and 450 kg for the motor exluding the moment of inertia.
Applying acceleration profile (half-sine pulse) on Y-axis on the frame with 2% damping
In this case, the frame on the pump side needs to be strengthened due to high stresses including one of the foundation bolt and 2 of the motor bolt locations. However, The low factor of safety indicates that the frame needs to be redesigned.
Viscoelastic materials are commonly used in vibration control because of their ability to dissipate energy through internal friction when subjected to dynamic loads. These materials exhibit a combination of elastic (spring-like) and viscous (damper-like) behavior, making them ideal for applications where both vibration isolation and energy dissipation are required. in this case, we will just model the viscoelastic material Using Prony Shear Relaxation:
Density 2.2e-06 kg/mm3
Young's Modulus 25 Mpa
Poisson's Ratio 0.45
Bulk Modulus 83.333 Mpa
Shear Modulus 8.6.207 Mpa
Prony Shear Relaxation with relative moduli at 0.15 at 0.5s, 0.15 @ 0.05s and 0.2 at 0.005s.
We will consider a simple geometry used to damp vibrations of a small pump (household application) using M24 Type 8675 and 8674 Bolt and Nut and washer UNI 5910 M24x2x60:1, with an axial force of 50 N acting on the tip of the bolt while having a frictionless support on a foundation. Investigate the displacement at 100 to 200 Hz.
Although, the phase angle reverses abruptly, the amplitude of the vibration did not react and there is no observed resonance in the defined frequency range showing high damping dominance which suppresses resonance peaks by dissipating energy.
Maximum amplitude is 0.77242mm, Maximum deformation (along X) is 1.5113mm at 116 Hz with a phase angle of 100.57 deg.
The geometry with viscoelastic material absorbing the load of 50 axial force on the bolt which shows sufficient strength to damp the vibration.
With the same frequency and load applied, the phase response shown above how the material dissipates energy and reacts to the applied load. This phase lag between the applied stress and resulting strain is a key characteristic of viscoelastic behavior. Q.E.D.
