Simulation and Experimental Assessment of the Usability of the Phase Angle Method of Examining the State of Shock Absorbers Installed in a Vehicle

2.1. Structure of the Linear Model Used in the Simulation Tests and Equations of Motion in the Time and Frequency Domains

As in publications [4,11,12,15,16,17,23,26,27,34,37,38,39,40,41], the linear ‘quarter-car’ model presented in Figure 2a was used (cf. with that in [20,28]). In Figure 2b, the inertia of the tester’s vibration plate was taken into account after [17,27,34,35].

The model shown in Figure 2b consists of three mass elements, i.e., ‘sprung mass’ m₁ [kg], ‘unsprung mass’ m₂ [kg], and vibration plate m₃ [kg]. The suspension and tyre stiffness have been denoted by k₁ [N/m] and k₂ [N/m], respectively. Parameters c₁ [N·s/m] and c₂ [N·s/m] represent the suspension and tyre damping coefficients, respectively. The quantity denoted by ζ(t) November is the kinematic excitation resulting from the motion of the vibration plate (exciter), which supports the vehicle wheel. The measuring system measures the dynamic component F_md [N] of the force applied from below to the vibration plate, the static load N_stm resulting from the weight of the mass elements of the model (including the weight of the vibration plate m₃·g) and the force of inertia of the vibration plate F_bp (negative). The static forces are as follows:

$N_{st} = (m_{1} + m_{2}) \cdot g$

(1)

$N_{stm} = (m_{1} + m_{2} + m_{3}) \cdot g$

(2)

The tyre–plate interaction force F_op [N] is equal to the sum of the static load N_st [N] and the dynamic component of the vertical force F_dz [N] in the tyre; the latter is equal to the sum of the dynamic spring force F_dso [N] (measured in relation to the state of static equilibrium, i.e., for the radial deformations of the tyre relative to the static deformation) and the force F_two [N] of viscous damping in the tyre:

The diagnostic tester measures the force F_opm [N], which includes the dynamic component F_md [N] and the static reaction force N_stm [N] recorded at the beginning of the test. The vibration plate’s weight m₃·g [N] and the force F_bp [N] of its inertia (negative) are also taken into account.

$F_{opm} = F_{op} - F_{bp} + m_{3} g = F_{op} + m_{3} \frac{d^{2} ζ (t)}{d t^{2}} + m_{3} g = F_{dz} + N_{stm} + m_{3} \frac{d^{2} ζ (t)}{d t^{2}} = F_{md} + N_{stm}$

(5)

$F_{bp} = - m_{3} \frac{d^{2} ζ (t)}{{dt}^{2}}$

(6)

$F_{md} = F_{dz} + m_{3} \frac{d^{2} ζ (t)}{{dt}^{2}}$

(7)

As mentioned above, F_md [N] is a result of the measurement of the dynamic component (where the static load N_stm [N] is not taken into account) of the force F_dz [N] in the tyre. The result of measurement of the force F_dz (i.e., the value of this force) and, in consequence, the value of force F_op as well, are distorted by the force of inertia F_bp [N] and weight m₃·g [N] of the vibration plate. Moreover, the force F_bp introduces a phase shift of F_md in relation to F_dz. Figure 2b shows the vertical dynamic component F_dz of the tyre–plate interaction force (i.e., the dynamic component of the force in the tyre). This force is directed from the exciter plate to the tyre. The force F_md, i.e., the dynamic component of the vibration-exciting force generated in and measured by the diagnostic test stand, has also been indicated in the drawing.

The equations of motion for the linear model (Figure 2a) have been derived from the principle of dynamic force analysis. Their matrix form is represented by Equation (8), also showing the notation of individual matrices as follows: inertia—M; viscous damping—C; stiffness—K; the action of excitation through the damping in the tyre—C_ζ; the action of excitation through tyre’s radial stiffness—K_ζ. The vectors of generalised coordinates (displacements), velocities, and accelerations have been denoted by

q, \dot{q}, \ddot{q}

, respectively. The said notation has been used to write the equations of motion in a concise form (9).

$\begin{matrix} \underset{\underset{M}{⏟}}{[\begin{matrix} m_{1} & 0 \\ 0 & m_{2} \end{matrix}]} \cdot \underset{\underset{\ddot{q}}{⏟}}{[\begin{matrix} {\ddot{z}}_{1} \\ {\ddot{z}}_{2} \end{matrix}]} + \underset{\underset{C}{⏟}}{[\begin{matrix} c_{1} & - c_{1} \\ {- c}_{1} & c_{1} + c_{2} \end{matrix}]} \cdot \underset{\underset{\dot{q}}{⏟}}{[\begin{matrix} {\dot{z}}_{1} \\ {\dot{z}}_{2} \end{matrix}]} + \underset{\underset{K}{⏟}}{[\begin{matrix} k_{1} & {- k}_{1} \\ {- k}_{1} & k_{1} + k_{2} \end{matrix}]} \cdot \underset{\underset{q}{⏟}}{[\begin{matrix} z_{1} \\ z_{2} \end{matrix}]} = \underset{\underset{C_{ζ}}{⏟}}{[\begin{matrix} 0 \\ c_{2} \end{matrix}]} \cdot \dot{ζ} + \underset{\underset{K_{ζ}}{⏟}}{[\begin{matrix} 0 \\ k_{2} \end{matrix}]} \cdot ζ \end{matrix}$

(8)

$M \cdot \ddot{q} + C \cdot \dot{q} + K \cdot q = C_{ζ} \cdot \dot{ζ} + K_{ζ} \cdot ζ$

(9)

At the next stage, the Laplace transform of Equation (9) was determined for zero initial conditions. The solution obtained after appropriate transformations has the form shown in Equation (10), where the domain s = r + i·ω has a real part r and an imaginary part ω (where i² = −1 and ω is radian frequency [rad/s]).

$q (s) = H_{q} (s) \cdot ζ (s)$

(10)

The operator transmittances (transfer functions) for displacements H_q(s), velocities, and accelerations have been shown as Equations (11), (12), and (13), respectively.

$H_{q} (s) = [\begin{matrix} H_{q_{1}} (s) \\ H_{q_{2}} (s) \end{matrix}] = \frac{q (s)}{ζ (s)} = {(M \cdot s^{2} + C \cdot s + K)}^{- 1} \cdot (C_{ζ} \cdot s + K_{ζ})$

(11)

$H_{\dot{q}} (s) = [\begin{matrix} H_{{\dot{q}}_{1}} (s) \\ H_{{\dot{q}}_{2}} (s) \end{matrix}] = \frac{\dot{q} (s)}{ζ (s)} = s \cdot H_{q} (s)$

(12)

$H_{\ddot{q}} (s) = [\begin{matrix} H_{{\ddot{q}}_{1}} (s) \\ H_{{\ddot{q}}_{2}} (s) \end{matrix}] = \frac{\ddot{q} (s)}{ζ (s)} = s^{2} \cdot H_{q} (s)$

(13)

The linear model describes changes in the dynamic components F_dz and F_md of the forces under analysis (Equations (4) and (7), measured as relative to the values observed in the static equilibrium conditions. The total values of the tyre–plate interaction force F_op and the force F_opm measured in the test stand are described by Equations (3) and (5). They differ from the dynamic component values F_dz and F_md in the constant weight values of the model’s mass elements (N_st and N_stm, respectively, according to (1) and (2)).

The Laplace transform of the dynamic vertical tyre–plate interaction force is expressed by Formula (14).

$F_{dz} (s) = c_{2} \cdot [\dot{ζ} (s) - {\dot{z}}_{2} (s)] + k_{2} \cdot [ζ (s) - z_{2} (s)]$

(14)

On the grounds of Equations (10)–(14), with necessary transformations, the final concise form (15) of the transmittance of the dynamic vertical tyre–plate interaction force was obtained (remembering that q₁ = z₁, q₂ = z₂):

$H_{Fdz} (s) = \frac{F_{dz} (s)}{ζ (s)} = (c_{2} \cdot s + k_{2}) \cdot [1 - H_{q_{2}} (s)]$

(15)

The difference between the dynamic component F_md of the force measured in the test facility and the vertical dynamic component F_dz of the tyre–plate interaction force is caused by the force of inertia of the plate (the last summand in (16)).

$H_{Fmd} (s) = \frac{F_{md} (s)}{ζ (s)} = \frac{F_{dz} (s) + m_{3} \cdot \ddot{ζ} (s)}{ζ (s)} = \frac{F_{dz} (s) + m_{3} \cdot s^{2} \cdot ζ (s)}{ζ (s)} = = (c_{2} \cdot s + k_{2}) \cdot [1 - H_{q_{2}} (s)] + m_{3} \cdot s^{2} = H_{Fdz} (s) + m_{3} \cdot s^{2}$

(16)

The values of spectral transmittances H_Fdz(i·ω) and H_Fmd(i·ω), where ω is the radian frequency of kinematic excitation ζ, are defined for s = i·ω. Their moduli are equal to the ratios of the amplitudes of forces (F_dz and F_md, as appropriate) to the amplitude of excitation ζ, and the arguments are defined by the phase shifts.

By comparing the moduli of the transmittances H_Fdz and H_Fmd with each other (for identical excitation), it will be possible to assess indirectly the impact of mass m₃ of the tester’s vibration plate on the differences between forces F_dz i F_md.

In their publications [27,35], the authors have presented the formulas that define the natural radian and Hertz frequencies of the undamped system (resonance radian frequencies ω₀₁, ω₀₂ [rad/s] and Hertz frequencies f₀₁, f₀₂ [Hz]) as well as suspension’s critical damping c_kr1 [N·s/m] and relative damping ϑ₁ [-].

$ω_{01 / 02}^{2} = \frac{k_{1} \cdot m_{2} + (k_{1} + k_{2}) \cdot m_{1}}{2 \cdot m_{1} \cdot m_{2}} \mp \sqrt{{[\frac{k_{1} \cdot m_{2} + (k_{1} + k_{2}) \cdot m_{1}}{2 \cdot m_{1} \cdot m_{2}}]}^{2} - \frac{k_{1} \cdot k_{2}}{m_{1} \cdot m_{2}}} = {(2 π \cdot f_{01 / 02})}^{2}$

(17)

$c_{kr 1} = 2 \cdot \sqrt{\frac{k_{1} \cdot k_{2} \cdot m_{1}}{k_{2} + k_{1} \cdot (1 + \frac{m_{2}}{m_{1}})}} \approx 2 \cdot \sqrt{\frac{k_{1} \cdot k_{2} \cdot m_{1}}{k_{2} + k_{1}}} for m_{1} ≫ m_{2}$

(18)

where c₁ [N·s/m] is the viscous damping coefficient of the suspension system in the linear model defined previously.

2.2. Structure of the Non-Linear Model Used in the Simulation Tests and Its Equations of Motion in the Time Domain

The structure of the non-linear ‘quarter-car’ model (Figure 2c) is similar to that of the linear one (see also [3,5,35]). It differs from the latter in the description of the spring and damping forces in the suspension system and pneumatic tyre. It reflects the actual characteristic curves measured in laboratory conditions and represents shock absorber and tyre damping forces F_t1 [N] and F_t2 [N], respectively, suspension and tyre spring forces Fs₁ [N] and Fs₂ [N], respectively, sliding friction in the suspension Fts₁ [N], and tyre bouncing (‘wheel hop’) phenomenon. The non-linear model describes changes in the resultant forces F_op and F_opm, taking into account the constant values of the gravity forces of the model masses (N_st and N_stm, respectively, according to Equations (1) and (2)). For the positions of the extremes of the curves under analysis relative to each other to be described (in terms of both the time and phase of the process), the dynamic components of the said resultant forces are important. Therefore, the authors focused mainly on forces F_dz and F_md (see Figure 2b,c and Formulas (4) and (7)).

Relation (20) presents the equations of motion of the non-linear ‘quarter-car’ model. The equations have also been derived from the principle of dynamic force analysis.

$\{\begin{matrix} {\ddot{ζ}}_{1} = \frac{{Fs}_{1} + {Fts}_{1} + {Ft}_{1}}{m_{1}} - g \\ {\ddot{ζ}}_{2} = \frac{{- Fs}_{1} - {Fts}_{1} - {Ft}_{1} + {Fs}_{2} + {Ft}_{2}}{m_{2}} - g \end{matrix}$

(20)

where ${\ddot{ζ}}_{1}$ and ${\ddot{ζ}}_{2}$ are accelerations of the sprung and unsprung mass, respectively; Fs₁, Fts₁, and Ft₁ are forces of elasticity, sliding friction, and viscous damping in the suspension system, respectively; Fs₂ and Ft₂ are tyre spring and damping (substitute viscotic) forces, respectively; and g is gravitational acceleration.

The spring force in the suspension (21) was approximated by a linear function and a third-degree polynomial.

${Fs}_{1} = \{\begin{matrix} \begin{matrix} {A 1 s}_{1} \cdot u_{1} \end{matrix} & for u_{1} \leq u_{1 gr} \\ {A 2 s}_{1} \cdot u_{1}^{3} + {B 2 s}_{1} \cdot u_{1}^{2} + {C 2 s}_{1} \cdot u_{1} + {D 2 s}_{1} & for u_{1} > u_{1 gr} \end{matrix}$

(21)

where A1s₁ and A2s₁ through D2s₁ are coefficients of the functions that describe (as appropriate) the force of suspension elasticity at small and large deflections, respectively; u_1gr is the threshold of applicability of different formulas describing the spring force; u₁ is the suspension deflection defined by Formula (22).

$u_{1} = ζ_{2} - ζ_{1} + {om}_{0}$

(22)

where om₀ is the difference between coordinates ζ₁ and ζ₂, corresponding to ${F s}_{1} = 0$ .

The sliding friction force in the suspension has been expressed by Equation (23).

${Fts}_{1} = \{\begin{matrix} \begin{matrix} {Ats}_{1} \cdot {sgn \dot{u}}_{1} \end{matrix} & for |{\dot{u}}_{1}| > {\dot{u}}_{1 gr} \\ {Ats}_{1} \cdot \frac{{\dot{u}}_{1}}{{\dot{u}}_{1 gr}} & for |{\dot{u}}_{1}| \leq {\dot{u}}_{1 gr} \end{matrix}$

(23)

where Ats₁ is the amplitude of the sliding friction force in the suspension system; ${\dot{u}}_{1 gr}$ is the threshold value of velocity ${\dot{u}}_{1}$ of the suspension deflection, according to (24).

${\dot{u}}_{1} = {\dot{ζ}}_{2} - {\dot{ζ}}_{1}$

(24)

The viscous damping force in the suspension has been expressed by Equation (25).

${Ft}_{1} = \{\begin{matrix} \begin{matrix} {- i}_{z} \cdot c_{o} \cdot {({- \dot{u}}_{1} \cdot i_{z})}^{Wo} \end{matrix} & for {\dot{u}}_{1} < 0 \\ i_{z} \cdot c_{u} \cdot {({\dot{u}}_{1} \cdot i_{z})}^{Wu} & for {\dot{u}}_{1} \geq 0 \end{matrix}$

(25)

where i_z is a coefficient that describes the kinematic relations in the suspension; c_o and c_u are coefficients of rebound and compression damping in the shock absorber, respectively; Wo and Wu are exponents of the functions describing the rebound and compression damping in the shock absorber, respectively.

The spring force in the pneumatic tyre was approximated by a third-degree polynomial (26).

${Fs}_{2} = {As}_{2} \cdot u_{2}^{3} + {Bs}_{2} \cdot u_{2}^{2} + {Cs}_{2} \cdot u_{2}$

(26)

where As₂ through Cs₂ are coefficients of the polynomial describing the spring force in the pneumatic tyre, and u₂ is the radial deflection of the tyre as defined in (27).

$u_{2} = \{\begin{matrix} ζ - ζ_{2} + R & for ζ - ζ_{2} + R \geq 0 \\ 0 & for ζ - ζ_{2} + R < 0 \end{matrix}$

(27)

where R is the free radius of the tyre.

According to (28), the tyre damping force is represented in the tyre model as a linear function of the velocity of radial deflection of the tyre.

${Ft}_{2} = c_{2} \cdot {\dot{u}}_{2}$

(28)

where c₂ is the coefficient of damping and ${\dot{u}}_{2}$ is the velocity of deflection as defined in (29).

${\dot{u}}_{2} = \{\begin{matrix} \dot{ζ} - {\dot{ζ}}_{2} & for ζ - ζ_{2} + R \geq 0 \\ 0 & for ζ - ζ_{2} + R < 0 \end{matrix}$

(29)

The equations of motion (20) were solved in the time domain by means of approximate techniques with numerical integration using an authorial program developed in the Matlab-Simulink environment.

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Jacek Drobiszewski www.mdpi.com

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Simulation and Experimental Assessment of the Usability of the Phase Angle Method of Examining the State of Shock Absorbers Installed in a Vehicle

2.1. Structure of the Linear Model Used in the Simulation Tests and Equations of Motion in the Time and Frequency Domains

2.2. Structure of the Non-Linear Model Used in the Simulation Tests and Its Equations of Motion in the Time Domain

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2.1. Structure of the Linear Model Used in the Simulation Tests and Equations of Motion in the Time and Frequency Domains

2.2. Structure of the Non-Linear Model Used in the Simulation Tests and Its Equations of Motion in the Time Domain

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