The nose landing gear is modeled as a simplified mechanical system incorporating the strut’s lateral bending and torsional characteristics.

In Figure 1 , L_g represents the height of the landing gear strut, and H₁ denotes the distance from the installation point of the diagonal strut and shimmy damper to the landing gear mounting point. The shimmy damper is simplified as a massless spring-damper system with lateral stiffness K₁ and damping C₁. The nose landing gear is simplified as a mass-spring-damper system with mass m, lateral bending stiffness K_s, and damping C_s.

Figure 2 shows the top view of the dynamic model for the landing gear system shimmy analysis. Below, the calculation process for the parameters in the shimmy dynamic equations is introduced.

Assuming that the angular displacement at each point in the lateral bending of the strut is equal, the lateral bending stiffness and damping coefficients of the diagonal strut are derived based on the moment equilibrium equation and energy conservation theorem:

$K_{eff}=K_1{\left(\frac{H_1}{L_g}\right)}^2$, $C_{eff}=C_1{\left(\frac{H_1}{L_g}\right)}^2$ (1)

The equivalent lateral bending stiffness and damping coefficient of the strut in the nose landing gear system are K_s1 and C_s1, respectively:

$K_{s1}=K_s+K_{eff}$, $C_{s1}=C_s+C_{eff}$ (2)

Considering the presence of a rake angle $\delta$, the equivalent stability margin L_eff from the wheel center to the strut axis is calculated as follows:

$L_{eff}=L+R\tan \left(\delta \right)$ (3)

Assuming the upper end of the nose landing gear strut is fixed, the system is analyzed with torsional freedom θ around the strut axis, lateral displacement y at the lower end of the strut, and lateral tire deformation $\Delta$. The mass of the wheel and strut assembly is concentrated at their center of gravity, and the tire is assumed to experience no slip relative to the ground. The shimmy dynamic equations for the landing gear system are derived as follows:

$m\ddot{y}+m{L}_{cg}\ddot{\theta }+C_{s1}\stackrel{\dot{}}{y}+K_{s1}y+F_n=0$ (4)

$I\ddot{\theta }+m{L}_{cg}\ddot{y}+\left(C_{\theta }+C_{sh}\right)\stackrel{\dot{}}{\theta }+K_{\theta }\theta +F_n{L}_{eff}+M_n=0$ (5)

$V\theta +\stackrel{\dot{}}{y}+L_{eff}\stackrel{\dot{}}{\theta }-\stackrel{\dot{}}{\Delta }=0$ (6)

Assuming the torsional angle θ is small, it can be approximated as $\sin \theta \cong 0$, $\cos \theta \cong 1$. The lateral force $F_n$ acting on the nose landing gear tire can be expressed in terms of the lateral tire deformation $\Delta$, denoted as $F_n=K_{\Delta }\Delta +C_{\Delta }\stackrel{\dot{}}{\Delta }$ The aligning moment $M_n$ can be expressed in terms of the tire torsional angle $\phi$ denoted as $M_n=K_{\phi }\phi +C_{\phi }\stackrel{\dot{}}{\phi }$ where $\phi \cong \Delta /\sigma$ are the lateral stiffness and torsional stiffness of the tire, respectively.

The nose landing gear strut is simplified as a cantilever beam fixed at one end and free at the other. Below, the lateral bending stiffness, torsional stiffness, lateral bending damping, and torsional damping coefficients of the strut are derived.

Lateral Bending Stiffness Coefficient:

The lateral bending stiffness coefficient of the landing gear strut describes its ability to resist bending deformation under lateral forces. For a circular cross-section beam, the lateral bending stiffness coefficient can be calculated using the following formula. By simplifying the nose landing gear strut as a cantilever beam fixed at one end, the lateral bending stiffness coefficient K_s is given by:

$K_s=\frac{3E\pi d^4}{64L_g^3}$ (7)

where E is the elastic modulus of the material, I is the moment of inertia of the cross-section, and L_g is the height of the strut. For a circular cross-section, the moment of inertia $I=\pi d^4/64$, where d is the diameter of the circular cross-section.

Lateral Bending Damping Coefficient:

The lateral bending damping coefficient of the landing gear strut describes its ability to dissipate energy during lateral vibrations. The damping coefficient is typically related to the material’s damping properties. The lateral bending damping coefficient C_s is given by:

$C_s=\zeta d^2\sqrt{\frac{3E\pi m}{4L_g^3}}$ (8)

where ζ is the damping ratio and mm is the mass of the system. Substituting Equation (5) into this expression yields the lateral bending damping coefficient $C_s=2\zeta \sqrt{K_{s}m}$.

Torsional Stiffness Coefficient:

The torsional stiffness coefficient of the landing gear strut describes its ability to resist torsional deformation under torque. For a cylindrical strut, the torsional stiffness coefficient $K_{\theta }$ is given by:

$K_{\theta }=\frac{G\pi d^4}{32L_g}$ (9)

where G is the shear modulus of the material, J is the polar moment of inertia of the cross-section, and L_g is the height of the strut. For a circular cross-section, the polar moment of inertia $J=\pi d^4/32$ Substituting this into the formula yields the torsional stiffness coefficient.

Torsional Damping Coefficient:

The torsional damping coefficient of the landing gear strut describes its ability to dissipate energy during torsional vibrations. The damping coefficient is typically related to the material’s damping properties. The torsional damping coefficient $C_{\theta }$ is given by:

$C_{\theta }=\frac{\zeta \pi d^4}{16}\sqrt{\frac{G}{L_g}}$ (10)

where ζ is the damping ratio, J is the polar moment of inertia, and $K_{\theta }$ is the torsional stiffness coefficient. Substituting the polar moment of inertia and torsional stiffness coefficient into this expression yields the torsional damping coefficient $C_{\theta }=2\zeta \sqrt{K_{\theta }J}$.

The test utilized the PSVQTec3D ultra-low noise 3D scanning laser vibrometer. The QTec scanning laser vibrometer is one of the most advanced non-contact vibration measurement systems available today. This equipment is widely used in scientific research and development for determining vibration modes and characteristic frequencies in fields such as NVH (Noise, Vibration, and Harshness), acoustics, structural dynamics, ultrasonics, FEM (Finite Element Method) modal validation, and NDT (Non-Destructive Testing). The parameters of the single-wheel nose landing gear model used in the experiment are shown in Figure 5 .

For each test condition, several different forward speeds of the nose landing gear were tested: 5 m/s, 10 m/s, 15 m/s, 20 m/s, 25 m/s, and 30 m/s. These speeds were selected because they cover the range at which shimmy commonly occurs in both heavy and light aircraft. The reported peak amplitude refers to the first positive peak, i.e., the maximum amplitude after the impact [27] - [30] .

The first condition involves the forward speed test on a smooth runway surface. All single-wheel nose landing gear parameters are consistent with those in Table 2 .

Below are the experimentally measured relationship graphs for frequency vs. amplitude, frequency vs. velocity, and frequency vs. acceleration at the corresponding speeds.

The relationship graphs for a speed of V = 5 m/s are shown in Figure 6 :

The relationship graphs for a speed of V = 10 m/s are shown in Figure 7 :

The relationship graphs for a speed of V = 15 m/s are shown in Figure 8 :

The relationship graphs for a speed of V = 20 m/s are shown in Figure 9 :

The relationship graphs for a speed of V = 25 m/s are shown in Figure 10 :

The relationship graphs for a speed of V = 30 m/s are shown in Figure 11 :

As shown in the figures above, the vibration response of the landing gear tire changes with increasing speed. When the tire’s taxiing speed is 20 m/s, the vibration amplitude reaches its maximum value of 42 micrometers, corresponding to a frequency of 2 Hz. When the speed increases to 30 m/s, the vibration frequency rises to 91 Hz, while the amplitude slightly decreases to 34 micrometers. In terms of velocity response, at a taxiing speed of 30 m/s, the peak velocity response reaches approximately 20 mm/s, occurring at a frequency of 92 Hz. In contrast, at a speed of 25 m/s and a frequency of 78 Hz, the peak velocity response is only 8 mm/s. The acceleration response shows more pronounced changes, reaching a maximum value of 12 m/s² at a taxiing speed of 30 m/s, with the peak corresponding to a frequency of 91 Hz. At 25 m/s and a frequency of 78 Hz, the acceleration response is 3.4 m/s².

Further comparison of Figure 6 , Figure 9 , and Figure 11 reveals that as the taxiing speed increases, the growth trends of the vibration parameters become evident:

1) Amplitude Variation: Compared to the low speed of 5 m/s, when the taxiing speed increases to 20 m/s, the amplitude expands from 7 micrometers to 42 micrometers, a 6-fold increase. When the speed further increases to 30 m/s, the amplitude slightly decreases but remains at a relatively high level. This indicates that the amplitude initially increases with speed within a certain range, reaches a peak, and then begins to decrease.

2) Velocity Response: At a taxiing speed of 5 m/s, the velocity response is 2.6 mm/s, while at 30 m/s, the peak velocity response reaches 20 mm/s, an increase of approximately 9 times. The experimental data show that the velocity response exhibits a more significant increase trend at high taxiing speeds.

3) Acceleration Variation: At the low speed of 5 m/s, the acceleration response is 1.8 m/s², while at 30 m/s, the maximum acceleration increases to 12 m/s², a 6.7-fold increase. The acceleration shows a growing trend with increasing taxiing speed, particularly in the high-frequency range.

The experimental results reveal the following characteristics of the landing gear tire’s vibration response:

1) The amplitude significantly increases with taxiing speed but begins to decrease after reaching a peak at 20 m/s. The experimental results show that the system’s amplitude significantly increases when approaching the critical shimmy speed, which aligns with the theoretically calculated critical shimmy speed range (10 m/s - 20 m/s).

2) Both velocity and acceleration responses increase exponentially with taxiing speed, showing more pronounced peak responses at high speeds. The higher the speed, the higher the peak, indicating that high-frequency vibrations have a greater impact on the system’s dynamic performance.

3) Amplitude, velocity response, and acceleration response all exhibit peaks corresponding to specific speeds and frequencies, indicating that the system tends to resonate under certain conditions.

In terms of acceleration alone, it increases with the speed of the landing gear tire. As the speed increases, the vibration frequency corresponding to the acceleration changes also shifts. The graphs show that each experiment exhibits three distinct regions of significant acceleration changes. The higher the speed, the higher the frequency at which these changes occur, and the acceleration peaks show a clear backward shift.

From this experiment, it is evident that as the landing gear system is driven by the flywheel, the tire gains speed. When the speed increases from 10 m/s to 20 m/s, the landing gear tends to become unstable, with the amplitude increasing by at least 6 times. However, when the speed increases to 30 m/s, compared to 20 m/s, the frequency at which instability occurs shifts backward to 16 Hz, and the amplitude decreases from 42 micrometers to 32 micrometers, though it remains significantly higher than 7 micrometers. This shows that while an increase in tire speed does not necessarily lead to a larger amplitude and instability in the strut’s midsection, it does increase the acceleration.