The tangent stiffness matrix in rigid multibody vehicle dynamics

Figures & data

Figure 1. Revolute joint.

Figure 2. Point-on-plane joint.

Figure 3. Rolling disk contact.

Figure 4. Wheel on tipping table.

Table 1. Wheel on table eigenvalue analysis – no preload.

Number	Frequency (Hz)	Damping ratio	Time constant (s)	Wavelength (s)
1	1.2328 × 10^0	3.8730 × 10^−1	3.3333 × 10^−1	8.7983 × 10^−1
2	1.2328 × 10^0	3.8730 × 10^−1	3.3333 × 10^−1	8.7983 × 10^−1
3	–	–	–	–
4	–	–	–	–

Note: Oscillatory roots are listed twice and negative time constants denote unstable roots.

Table 2. Wheel on table eigenvalue analysis – preload added.

Number	Frequency (Hz)	Damping ratio	Time constant (s)	Wavelength (s)
1	1.5507 × 10^0	2.8368 × 10^−1	3.6181 × 10^−1	6.7251 × 10^−1
2	1.5507 × 10^0	2.8368 × 10^−1	3.6181 × 10^−1	6.7251 × 10^−1
3	–	–	3.2324 × 10^−1	–
4	–	–	−3.8146 × 10^−1	–

Figure 5. A-arm suspension.

Table 3. Body location and properties.

Body name	Location	Mass	Inertia $\left(I_{\mathrm{x}\mathrm{x}},I_{\mathrm{y}\mathrm{y}},I_{\mathrm{z}\mathrm{z}}\right)$	Inertia $\left(I_{\mathrm{x}\mathrm{y}},I_{\mathrm{y}\mathrm{z}},I_{\mathrm{z}\mathrm{x}}\right)$
Chassis	0.500, 0.000, 0.300	250.000	1.000, 1.000, 1.000	0.000, 0.000, 0.000
Upright	0.500, 0.800, 0.300	5.000	0.100, 0.100, 0.100	0.000, 0.000, 0.000
Wheel + hub	0.500, 0.900, 0.300	10.000	2.000, 4.000, 2.000	0.000, 0.000, 0.000
Lower A-arm	0.500, 0.600, 0.150	5.000	1.000, 1.000, 2.000	0.000, 0.000, 0.000
Upper A-arm	0.500, 0.600, 0.400	5.000	1.000, 1.000, 2.000	0.000, 0.000, 0.000
Bell-crank	0.450, 0.350, 0.450	1.000	0.050, 0.050, 0.050	0.000, 0.000, 0.000
Push-rod	0.500, 0.550, 0.300	1.000	0.011, 0.003, 0.008	0.000, −0.005, 0.000
Tie-rod	0.340, 0.550, 0.280	1.000	0.021, 0.003, 0.024	0.008, 0.002, 0.001

Note: All values are given in standard metric units, i.e., m, kg, kg·m², and inertias are defined as the positive integral over the body, e.g., $I_{\mathrm{x}\mathrm{y}}=+\int r_{\mathrm{x}}{r}_{\mathrm{y}}dm$.

Table 4. Connection location and properties.

Connection Name	Location	Unit axis	Stiffness	Damping
Wheel bearing	0.500, 0.900, 0.300	0.000, 1.000, 0.000	–	–
Lower ball joint	0.500, 0.800, 0.150	–, –, – -	–	–
Lower A-arm pivot, rear	0.300, 0.100, 0.150	–, –, –	–	–
Lower A-arm pivot, front	0.700, 0.100, 0.100	1.000, 0.000, 0.000	–	–
Bell-crank pivot	0.400, 0.400, 0.400	0.000, 0.707, 0.707	–	–
Upper A-arm pivot, front	0.700, 0.300, 0.350	1.000, 0.000, 0.000	–	–
Upper A-arm pivot, rear	0.300, 0.300, 0.350	–, –, –	–	–
Upper ball joint	0.500, 0.750, 0.450	–, –, –	–	–
Lower push-rod end	0.500, 0.700, 0.200	0.000, −0.832, 0.555	–	–
Upper push-rod end	0.500, 0.400, 0.400	–, –, –	–	–
Inner tie-rod end	0.250, 0.300, 0.260	–, –, –	–	–
Outer tie-rod end	0.430, 0.800, 0.300	0.338, 0.938, 0.075	–	–
Chassis slider	0.500, 0.000, 0.500	0.000, 0.000, 1.000	–	–
Tire, vertical	0.500, 0.900, 0.000	0.000, 0.000, 1.000	150,000	0
Tire, horizontal	0.500, 0.900, 0.000	0.000, 0.000, 1.000	0	100
Suspension spring	0.000, 0.300, 0.500	–, –, –	20,000	1,000
Suspension spring	0.450, 0.300, 0.500	–, –, –	–	–

Note: All values are given in standard metric units, i.e., m, N/m, Ns/m.

Table 5. Quarter car eigenvalue analysis – no preload.

Number	Frequency (Hz)	Damping ratio	Time constant (s)	Wavelength (s)
1	1.2003 × 10^1	1.3490 × 10^−1	9.8295 × 10^−2	8.4084 × 10^−2
2	1.2003 × 10^1	1.3490 × 10^−1	9.8295 × 10^−2	8.4084 × 10^−2
3	1.0344 × 10^0	1.5120 × 10^−1	1.0176 × 10^0	9.7796 × 10^−1
4	1.0344 × 10^0	1.5120 × 10^−1	1.0176 × 10^0	9.7796 × 10^−1
5	–	–	4.4437 × 10^−1	–
6	–	–	–	–

Table 6. Quarter car eigenvalue analysis – preload added.

Number	Frequency (Hz)	Damping ratio	Time constant (s)	Wavelength (s)
1	1.2154 × 10^1	1.3385 × 10^−1	9.7832 × 10^−2	8.3024 × 10^−2
2	1.2154 × 10^1	1.3385 × 10^−1	9.7832 × 10^−2	8.3024 × 10^−2
3	1.1871 × 10^0	1.2531 × 10^−1	1.0700 × 10^0	8.4911 × 10^−1
4	1.1871 × 10^0	1.2531 × 10^−1	1.0700 × 10^0	8.4911 × 10^−1
5	–	–	–	–
6	–	–	4.4439 × 10^−1	–

Figure 6. Chassis vertical displacement vs. time, non-linear solution.