Because the flexible multispan shaft in large machines often rotates at supercritical speed, it is desirable to find ways to suppress the resulting bending vibration. In this paper, a novel type of support structure is proposed and investigated, which can suppress the bending vibration using dry friction. This approach is called Smart Spring support (SMSS). A dynamic model for the multispan shaft with SMSS is developed. The relationship between the vibration suppression effect and the control parameters of the SMSS is obtained through a numerical example involving a helicopter tail drive shaft. A structure of the SMSS is designed and examined with a rotor test. The results demonstrate that the SMSS has a significant effect on bending vibration suppression of flexible multispan shafts. The vibration-reduction ratio of the peak amplitude reaches 57.2% in the numerical example and 45.2% in the rotor test.
Multispan shafts are widely used in aviation, vehicles, ships, and large generator units. They are an important part of the transmission chain of many large machines. Together with the trend towards flexible and high-speed rotors, the rotating speed of multispan shafts often exceeds their critical speed, and the problem of supercritical bending vibration becomes increasingly prominent. Therefore, finding an effective bending vibration control method is highly desirable.
The majority of previous studies focused on changing the stiffness and damping characteristics of the multispan shaft supports, using passive vibration dampers to suppress bending vibration. Examples include flexible support and squeeze film dampers (SFD), metal-rubber dampers (MRD), or polymer-based composite structures [
The piezoelectric actuator (PZTA) is low-cost, lightweight, and easy-to-implement [
The Smart Spring is a semiactive control concept to suppress vibration based on the PZTA. It does not use the PZTA to counteract the excitation loads but adaptively varies the stiffness, damping, and mass of the dynamic system [
Nitzsche and other researchers published most of the current literature on Smart Springs. Their research on Smart Springs mainly focuses on the vibration control of the helicopter rotor [
The above research focused on the vibration reduction of helicopter rotor blades. There is little research of the bending vibration suppression of shafts. In 2011, Cavalini et al. established a discrete dynamic model for shafts with a Smart Spring mechanism and performed numerical simulation using on-off control strategies [
In this paper, we apply the Smart Spring to the support structure of a flexible multispan shaft, known as the Smart Spring support (SMSS). The structure of the paper is as follows. First, a novel continuous bending vibration dynamics model of the flexible multispan shaft with a SMSS is introduced. Then, the dynamic differential equation with consideration of the effect of dry friction is derived. In the next section, we perform a numerical simulation of the supercritical bending vibration suppression, using the SMSS, for a helicopter tail shaft. We designed the structure of a type of SMSS, with which we will verify the effect of vibration suppression during rotor testing. Overall, a theoretical model and an experimental reference for the design of an SMSS is provided.
The Smart Spring system is shown in Figure
The Smart Spring system [
As illustrated in Figure
Flexible multispan shaft with the SMSS.
The origin
Coordinate system for the shaft element in motion.
The motion of the shaft in Figure
The shaft section in Figure
The kinetic energy of the entire shaft in Figure
The potential energy of the shaft is mainly composed of two parts: the strain energy induced by the shaft bending and the gravitational potential energy under gravitational acceleration,
In the case, where
If
After substituting (
Equation (
After substituting (
When
When
Equations (
The helicopter tail shaft is a typical multispan shaft, which can be simplified into a combination of continuous hollow shaft sections and supports. A set of typical parameters for the helicopter tail shaft is selected in Table
Typical parameters of a helicopter tail-shaft.
Properties | Value/unit |
---|---|
|
2800/(kg⋅m−3) |
|
7 × 1010/Pa |
|
2.7 × 1010/Pa |
|
90/mm |
|
84/mm |
|
3.1/m |
|
0.85/m |
|
0.75/m |
|
1.15 × 107/(N⋅m−1) |
|
1.5 × 105/(N⋅m−1) |
|
0 |
|
20/(N⋅s⋅m−1) |
|
0.1/mm |
|
4200/(r⋅min−1) |
The approximate critical speed of the tail shaft can be calculated using the transfer matrix method, where the 1st critical speed is 3179 RPM, and the 2nd critical speed is 6562 RPM. The operating speed of the tail shaft
1st mode shape with hinged and elastic boundaries of the helicopter tail shaft.
The SMSS is installed at position
The supercritical bending vibration response of the tail shaft in the
Bending vibration response of the tail shaft in the
Center track of the tail shaft at operating speed.
Relationship between the control force
Relationship between the control force
The peak amplitude of the tail shaft decreases first and increases later with the increase of the control force
In Figure
Considering the results above, we suggest that there is an optimal control parameter for the bending vibration suppression of the flexible multispan shaft with an SMSS. For the helicopter tail shaft in this section, there is a constant optimal control force:
In order to verify the effect of the bending vibration suppression of the SMSS on the flexible multispan shaft, a conceptual structure of the SMSS was designed. Subsequently, a rotor test for vibration suppression was carried out on the multispan shaft test-setup. The arrangement of the test equipment is shown in Figure
Multispan shaft test-setup with a Smart Spring support.
The detailed structure of the SMSS in the test-setup is shown in Figure
Structure of the Smart Spring support.
General structure
Structure of PZTA
We obtained the support parameters for the test-setup in Table
Support parameters of the test-setup.
Properties | Value/unit |
---|---|
|
1.7 × 105/(N⋅m−1) |
|
20/(N⋅s⋅m−1) |
|
1.4 × 105/(N⋅m−1) |
|
22/(N⋅s⋅m−1) |
|
5 × 107/(N⋅m−1) |
|
0 |
|
1.67 × 107/(N⋅m−1) |
|
0 |
|
0.1 |
|
0.15 |
Relationship between the control voltage and the control force
The shaft in the rotor setup starts to accelerate from rest with the angular acceleration
The vibration response of the shaft system for different control voltages.
Voltage = 0 V;
Voltage = 30 V;
Voltage = 60 V;
Voltage = 90 V;
Voltage = 120 V;
Voltage = 150 V;
The peak amplitude of the shaft decreases significantly when the PZTA is activated (see Figures
The peak vibration-reduction ratio
The position of the peak amplitude shifts left first with increasing control voltage (see Figures
The critical speed of rotation
The previous numerical and experimental results suggest that the increase of the radial stiffness of the auxiliary support helps to improve the optimal control parameter (force or voltage) for the SMSS. In addition, we need the PZTA to generate enough control force to accomplish optimal control. The control force
In this paper, a new method for bending vibration suppression of a flexible multispan shaft based on Smart Spring support is investigated. The main conclusions are as follows: (a) The SMSS has a strong effect on the bending vibration suppression of the flexible multispan shaft when it exceeds its critical speed. (b) When the SMSS is in the state of Damping Control, it has better vibration suppression effect than that of Rigidity Control. There exists an optimal control force under the Damping Control state, which causes the peak vibration-reduction ratio
The authors declare that they have no competing interests.
This research work is supported by the National Natural Science Foundation of China (nos. 51375226 and 51505215).