A universal nonlinear model for the dynamic behaviour of shock absorbers

Abstract

Modern hydraulic shock absorbers display a wealth of nonlinear effects such as hysteresis and instabilities at high flow rates. Despite their wide application in practically all vehicles, both on- and off-road, a universal analytical model that captures the essential shock absorber dynamics and compressibility effects for various common damper architectures is lacking. This paper presents such a model and derives its system of equations from first principles for dampers of monotube- and piggyback-type. By applying the model to a typical suspension configuration, all relevant system variables, such as pressure drop, shim stack deflection, and damping force, are computed. Nonlinear oscillations and hysteresis loops, which might prove dangerous during operation, can be predicted effortlessly. The results achieved with the mathematical model are validated and agree well with test bench measurements. Furthermore, the presented approach is shown to be easily modifiable to describe the physical processes within other hydraulic shock absorber geometries in use today.

Appendices

Appendix A: Baseline system properties

The properties listed in Tables 1, 2 and 3 for the shock absorber geometry and the damper fluid are used as baseline for the results presented in Sect. 4. Where other values are used, this is explicitly stated in the text. For the shim stacks the following setup of circular annular shims is used:

• The compression stack is built up of shims with radii $\left[ a_{k}\right] =\left[ 12,13,14,15,16,18,19,20,21,22,\right.$ $\left. 22,22,22,16,22,22,22,22,22.5\right] \times 10^{-3}\,\hbox {m}$ and thicknesses $\left[ t_{k}\right] =\left[ 0.20,0.25,0.25,0.20,0.20,\right.$ $0.20,0.20,0.20,0.25,0.20,0.20,0.20,0.20,0.10,0.20,$ $\left. 0.20,0.20,0.20,0.20\right] \times 10^{-3}\,\hbox {m}$ with a clamp radius of $a_{c}=11\times 10^{-3}\,\hbox {m}$.

• The rebound stack consists of shims with radii $\left[ a_{k}\right] =\left[ 13,14,15,16,17,18,12,14,20,20,20,\right.$ $\left. 20,20\right] \times 10^{-3}\,\hbox {m}$ and thicknesses $\left[ t_{k}\right] =\left[ 0.30,0.30,\right.$ $\left. 0.30,0.30, 0.25,0.25,0.10,0.10,0.15,0.20,0.20,0.20,0.20\right] \times 10^{-3}\,\hbox {m}$ with a clamp radius of $a_{c}=11.5\times 10^{-3}\,\hbox {m}$.

• The compression stack on the check valve piston is built up of shims with radii $\left[ a_{k}\right] =\left[ 5,7,8,9,11,11,6,11,11,11\right] \times 10^{-3}\,\hbox {m}$ and thicknesses $\left[ t_{k}\right] =\left[ 0.20,0.20,0.20,0.20,0.20,0.20,0.10,0.20,0.20,0.20\right. ]\times 10^{-3}\,\hbox {m}$ with a clamp radius of $a_{c}=4\times 10^{-3}\,\hbox {m}$. Moreover, an additional spring with a pretension of $F_{0}={43.2}\,\hbox {N}$ and a stiffness of $k={16000}{\,\hbox {N m}^{-1}}$ is applied.

• The check valve is one annular shim of an inner radius of $4\times 10^{-3}\,\hbox {m}$, an outer radius of $6.3\times 10^{-3}\,\hbox {m}$, and a thickness of $0.30\times 10^{-3}\,\hbox {m}$. The valve is spring-loaded with a pretension of $F_{0}={0.35}\,\hbox {N}$ and a stiffness of $k={200}{\,\hbox {N m}^{-1}}$.

Table 1 Properties of the shock absorber pistons and compartment

Appendix B: Integration coefficients for shim stack stiffness

The following coefficients after Roark and Young [22] are used for the computation of linearized deflections at a radius r of a flat shim plate of outer radius a as a result of pressure or contact force loads at radius $r_{0}$:

$$\begin{aligned} C_{8}&= \frac{1}{2} \left[ 1+\nu +\left( 1-\nu \right) \left( \frac{b}{a}\right) ^{2}\right] , \end{aligned}$$

(B1)

$$\begin{aligned} C_{9}&= \frac{b}{a} \left\{ \frac{1+\nu }{2}\ln {\frac{a}{b}}+\frac{1-\nu }{4}\left[ 1-\left( \frac{b}{a}\right) ^{2}\right] \right\} , \end{aligned}$$

(B2)

$$\begin{aligned} F_{2}&= \frac{1}{4} \left[ 1-\left( \frac{b}{r}\right) ^{2}\left( 1+2\ln \frac{r}{b}\right) \right] , \end{aligned}$$

(B3)

$$\begin{aligned} F_{3}&= \frac{b}{4r} \left\{ \left[ \left( \frac{b}{r}\right) ^{2}+1\right] \ln \frac{r}{b}+\left( \frac{b}{r}\right) ^{2}-1\right\} , \end{aligned}$$

(B4)

$$\begin{aligned} G_{3}&= \frac{r_{0}}{4r} \left\{ \left[ \left( \frac{r_{0}}{r}\right) ^{2}+1\right] \ln \frac{r}{r_{0}}+\left( \frac{r_{0}}{r}\right) ^2-1\right\} \left<r-r_{0}\right>^{0}, \end{aligned}$$

(B5)

$$\begin{aligned} G_{11}&= \frac{1}{64} \left\{ 1+4\left( \frac{r_{0}}{r}\right) ^2-5\left( \frac{r_{0}}{r}\right) ^4-4\left( \frac{r_{0}}{r}\right) ^2\right. \nonumber \\&\left. \left[ 2+\left( \frac{r_{0}}{r}\right) ^2\right] \ln \frac{r}{r_{0}}\right\} \left<r-r_{0}\right>^{0}, \end{aligned}$$

(B6)

$$\begin{aligned} L_{9}&= \frac{r_{0}}{a} \left\{ \frac{1+\nu }{2}\ln {\frac{a}{r_{0}}}+\frac{1-\nu }{4}\left[ 1-\left( \frac{r_{0}}{a}\right) ^{2}\right] \right\} , \end{aligned}$$

(B7)

$$\begin{aligned} L_{17}&= \frac{1}{4} \left\{ 1-\frac{1-\nu }{4}\left[ 1-\left( \frac{r_{0}}{a}\right) ^4\right] -\left( \frac{r_{0}}{a}\right) ^2\right. \nonumber \\&\left. \left[ 1+\left( 1+\nu \right) \ln {\frac{a}{r_{0}}}\right] \right\} . \end{aligned}$$

(B8)

Here, the expression $\left<r-r_{0}\right>^{0}$ is equivalent to $\left( r-r_{0}\right) \forall \, r>r_{0}$ and $0 \,\forall \, r \le r_{0}$.

Table 2 Properties of the shim stacks and valves

Table 3 Properties of the damper mineral oil

Appendix C: Modelling of crossover shim contact

For the case of an introduced crossover shim at position m within a stack, the shim deflections and stiffness are initially computed in the same way as presented in Sect. 3.2.3 until contact is reached between the adjacent layers of the crossover. At this point the difference between the displacements of neighbouring shims equals the total thickness of the crossover, $\delta _{m+1}-\delta _{m-1}=t_{m}$, and the new contact force $w_{a}$ needs to be taken into account:

$$\begin{aligned}&\delta _{1}\left( w_{1}\right) = \epsilon _{2}\left( w_{1}\right) +\epsilon _{2}\left( w_{2}\right) , \end{aligned}$$

(C1)

$$\begin{aligned}&\delta _{2}\left( w_{1}\right) + \delta _{2}\left( w_{2}\right) = \epsilon _{3}\left( w_{2}\right) +\epsilon _{3}\left( w_{3}\right) , \end{aligned}$$

(C2)

$$\begin{aligned}&\vdots \nonumber \\&\delta _{m-2}\left( w_{m-3}\right) + \delta _{m-2}\left( w_{m-2}\right) =\epsilon _{m-1}\left( w_{m-2}\right) \nonumber \\&+\epsilon _{m-1}\left( w_{m-1}\right) \nonumber \\&+\epsilon _{m-1}\left( w_{a}\right) , \end{aligned}$$

(C3)

$$\begin{aligned}&\epsilon _{m-1}^{m}\left( w_{m-2}\right) +\epsilon _{m-1}^{m}\left( w_{m-1}\right) +\epsilon _{m-1}^{m}\left( w_{a}\right) \nonumber \\&= \delta _{m}\left( w_{m-1}\right) + \delta _{m}\left( w_{m}\right) , \end{aligned}$$

(C4)

$$\begin{aligned}&\delta _{m}\left( w_{m-1}\right) + \delta _{m}\left( w_{m}\right) = \epsilon _{m+1}\left( w_{m}\right) +\epsilon _{m+1}\left( w_{m+1}\right) \nonumber \\&\quad +\epsilon _{m+1}\left( w_{a}\right) , \end{aligned}$$

(C5)

$$\begin{aligned}&\delta _{m+1}\left( w_{m}\right) + \delta _{m+1}\left( w_{m+1}\right) +\delta _{m+1}\left( w_{a}\right) \nonumber \\&=\epsilon _{m+2}\left( w_{m+1}\right) +\epsilon _{m+2}\left( w_{m+2}\right) , \end{aligned}$$

(C6)

$$\begin{aligned}&\delta _{m+2}\left( w_{m+1}\right) + \delta _{m+2}\left( w_{m+2}\right) \nonumber \\&=\epsilon _{m+3}\left( w_{m+2}\right) +\epsilon _{m+3}\left( w_{m+3}\right) , \end{aligned}$$

(C7)

$$\begin{aligned}&\vdots \nonumber \\&\delta _{n-2}\left( w_{n-3}\right) + \delta _{n-2}\left( w_{n-2}\right) = \epsilon _{n-1}\left( w_{n-2}\right) \nonumber \\&+\epsilon _{n-1}\left( w_{n-1}\right) , \end{aligned}$$

(C8)

$$\begin{aligned}&\delta _{n-1}\left( w_{n-2}\right) + \delta _{n-1}\left( w_{n-1}\right) = \epsilon _{n}\left( w_{n-1}\right) \nonumber \\&+\epsilon _{n}\left( q\right) , \end{aligned}$$

(C9)

$$\begin{aligned}&\delta _{m+1}\left( w_{m}\right) + \delta _{m+1}\left( w_{m+1}\right) +\delta _{m+1}\left( w_{a}\right) \nonumber \\&=\epsilon _{m-1}^{m+1}\left( w_{m-2}\right) +\epsilon _{m-1}^{m+1}\left( w_{m-1}\right) \nonumber \\&\quad + \epsilon _{m-1}^{m+1}\left( w_{a}\right) + t_{m}. \end{aligned}$$

(C10)

Here, the notation $\epsilon _{m-1}^{m}$ stands for the intermediate deflection of the shim $\left( m-1 \right)$ before the crossover at the same radius of the crossover shim m. Eq. (C10) is added to the original equation system to account for the additional contact force $w_{a}$. Using Eq. (C1)–(C10), the matrix ${\hat{A}}$ of linear coefficients is built up as in Eq. (36) for the non-crossover case and the algebraic system ${\hat{A}}\cdot \vec {w}=\vec {b}$ with

$$\begin{aligned} \vec {w}&= \left[ {\begin{array}{c} w_{1} \\ w_{2} \\ \vdots \\ w_{n-1} \\ w_{a} \\ \end{array} } \right] , \end{aligned}$$

(C11)

$$\begin{aligned} \vec {b}&= \left[ {\begin{array}{c} 0 \\ \vdots \\ 0 \\ \epsilon _{n}\left( q\right) \\ -t_{m} \\ \end{array} } \right] . \end{aligned}$$

(C12)

is solved for all unknown forces $\vec {w}$. The above approach can be readily generalized for multiple additional contact forces or intermediate crossovers as shown in Fig. 5.