Dynamics of a Spatial Multibody Loop

4.6 Computer-Based Derivation of the Equations

4.6.3 Dynamics of a Spatial Multibody Loop

The subsequently exemplarily analyzed spatial multibody loop (Fig.4.2) possesses a spherical joint and four revolute joints with the unit vectorsu1;u2;u3 andu4for each rotational axis. All bodies, with exception of the coupler body (center of mass Sm) and the resting frame, are modeled as massless bodies, in order to simplify the dynamics. The spatial kinematic loop contains seven natural coordinates: the four relative anglesb1;b2;b3;b4of the four revolute joints and three more angles in the spherical joint S1, which are, however, not required in this context. The system possesses one degree of freedom, for which one scalar equation of motion can be formulated. Basically, every coordinate can be selected as a generalized coordinate. Because of the proximity of this example to a technical application, as well as, in order to circumvent singularities, a suitable choice for the input motion is the coordinate b1ẳq. The kinematics of the system, which means the nonlinear

equations for position, velocity, and acceleration are functions of this input variable b1 (which also holds for its derivatives), are formulated in the following, (Hiller1995).

Core system of implicit equations. In order to get an implicit core system of minimal order out of the constraint equations, a characteristic joint pair is chosen having a maximum number of degrees of freedom, (Woernle 1988). In this example, the two revolute joints R2, R3with the two perpendicular axesu2andu3 can be interpreted as a CARDAN joint with two degrees of freedom. The second joint of the characteristic joint pair is the spherical joint S1 with three joint coordinates. The CARDAN joint and the spherical joint compose the characteristic joint pairing of the loop, eliminating in this first step five unknowns. The corresponding (scalar) characteristic constraint parameter is the distanced (Fig.4.2)

ddẳd2: ð4:43ị The vectordcan be represented on one side with respect to the so-called lower segment asdẳr0ỵr1, and on the other side with respect to the upper segment as dẳr3ỵr4. From this, two expressions for the distanced2are gained, which must be identical. The square of the corresponding terms with the projections

r0r1ẳðr1acosaịcosb1; r3r4ẳ ðr3r4ịcosb4; ð4:44ị provide an explicit relation for the angleb4

cosb4ẳ C1cosb1ỵC2; ð4:45ị ( ) = 1( ),…, ( ) ; ( ) = 1( ),…, ( )

input information: mass and moment of inertia tensors , applied forces and torques ,

generalized coordinates

global kinematics (position)

global kinematics (velocity)

global kinematics (acceleration)

equations of motion (in minimal coordinates)

={1 =0 ≠ =

( ) ( ) ( )

+ =

∼

[ ]

] [

∼ ∼ ∼

Fig. 4.1 Equations of motion of complex multibody systems applying kinematic differentials

C1ẳr1acosa r3r4

; C2 ẳr23ỵr42r20r12 2r3r4

: ð4:46ị

The Eq. (4.45) yields two symmetrical solutions for the functionb4ð ị, onlyb1 one of which is compatible with the initial position of the kinematic loop.

Complementary angles. The complementary variables are, in this case, the angles b2 andb3, which can be calculated in simple manner using the projections

cosb2ẳexu3; sinb2ẳezﬄu03ex

; ð4:47ị cosb3ẳ u2e0z; sinb3 ẳu3ﬄe0zu2

ð4:48ị Here, the unit vectorsu3 ande0z can be expressed by using the already known vectors: Foru3 there are two possible solutions:

u3ẳ du2

jdu2j: ð4:49ị Fore0z one gets

e0zẳﬄde0z

du3ﬄde0z

ðdu3ị

d2 ; ð4:50ị

where the projectionsde0zandu3ﬄde0z

can be defined in the systemðx;y;zị0. The explicit evaluation of Eqs. (4.47)–(4.50) provides

4 1

2 1

0 Sm

′

Fig. 4.2 Spatial multibody loop

cosb2ẳ bcosaỵr1cosb1sina

K1ð ịb1 ; ð4:51ị

sinb2ẳ r1sinb1

K1ð ịb1 ; ð4:52ị cosb3ẳK5ðb1;b2;b4ị; ð4:53ị sinb3ẳ K3ðb1;b2;b4ịsinb2K4ðb1;b2;b4ịcosb2; ð4:54ị with the abbreviations

K1ð ị ẳb1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bcosaþr1cosb1sina

ð ị2ỵr21sinb21

h i

; ð4:55ị

K2ð ị ẳb4 r24ỵr232r3r4cosb4; ð4:56ị K3ðb1;b2;b4ị ẳr1sinb1ðr4cosb4r3ị

K2ð ịb4 þr4sinb4

K2ð ịb4 ðasinb2bsinasinb2ỵr1sinb2cosb1cosaị; ð4:57ị

K4ðb1;b2;b4ị ẳðr4cosb4r3ịðbcosaỵr1cosb1sinaị K2ð ịb4

þr4sinb4

K2ð ịb4 ðacosb2bsinacosb2ỵr1cosb2cosaị;

ð4:58ị

K5ðb1;b2;b4ị ẳðr3r4cosb4ịðabsinaỵr1cosb1cosaị K2ð ịb4

þr4sinb4

K2ð ịb4 ðbcosacosb2 ỵr1sinb1sinb2ỵr1cosb1cosb2sinaị:

ð4:59ị All vectors can now be represented with respect to each arbitrarily chosen coordinate system. This completes the position kinematics of this spatial multibody loop.

Time derivatives of the relative angles. For the angular velocity b_4, it follows from the derivation of Eqs. (4.45) and (4.46):

b_4ẳD41b_1; D41ẳ C1

sinb1

sinb4: ð4:60ị

Here, the coefficientD41becomes singular for sinb4ẳ0, which means that the vectorsr3 andr4 are parallel. This can, however, be excluded in the given case.

The angular velocities b_2 andb_3 can be defined through the time derivation of Eqs. (4.47)–(4.48). It is easier, however, to define these values with help of the velocity of the vector d. Two defining equations for this vector were already formulated in the previous step. From the equality of the corresponding derivatives it follows

u1r1

ð ịb_1 ẳðu2dịb_2ỵðu3dịb_3ỵðu4r4ịb_4: ð4:61ị In Eq. (4.61), the two unknown valuesb_2 andb_3 are contained. Note that the unity vectors u3 and u4 are normal to the unity vector u2, and one gets two independent scalar equations forb_2andb_3from Eq. (4.61) through the projection onu3 and onu2 respectively. The corresponding scalar products lead to

b_2ẳD21b_1; b_3ẳD31b_1ỵD34b_4 ð4:62ị with the coefficients

D21ẳu3ðu1r1ị u3ðu2dị; D31ẳu2ðu1r1ị

u2ðu3dị; D34ẳ u2ðu4r4ị

u2ðu3dị

ð4:63ị

or expressed in scalar quantities

D21ẳ r1ðcosb1cosb2ỵsinb1sinb2sinaị

r1ðcosb1cosb2sinaỵsinb1sinb2ị ỵbcosb2sina; D31ẳ r1sinb1cosa

r3sinb3ỵr4sinðb4b3ị; D34ẳ r4sinðb3b4ị

r3sinb3ỵr4sinðb4b3ị:

ð4:64ị

These coefficients become singular for u2ðu3dị ẳ0, which means when the distance vectordis parallel tou2. Again, this special configuration can also be excluded, however, in the given case. If the input variable chosen to beb1, the simulation of the kinematics and dynamics of the multibody loop is then free of singularities. Because of Eqs. (4.62), no further independent conditions for the angular velocitiesb_2 andb_3can be formulated. With the projection of Eq. (4.61) in the direction of vectord, one gets the condition

dðu1r1ịb_1ẳdðu4r4ịb_4; ð4:65ị which is equivalent to Eq. (4.60). This can be shown, after substituting of dẳ r0ỵr1 into the left side of Eq. (4.61), as well as the substitution of dẳr3ỵr4

into the right side of Eq. (4.61). For the angular acceleration, one obtains, using the time derivative of Eq. (4.60):

4ẳD41b

1E4; E4ẳ

C1cosb1 b_2

þcosb4 b_4

sinb4 : ð4:66ị The angular accelerations€b2 andb€3 could be defined in correspondence with the derivation of Eq. (4.62). However, the derivation of Eq. (4.61) leads to a much simpler solution, by repeating the projections which were already used to define the angular velocitiesb_2 andb_3. The differentiation of Eq. (4.61) leads to:

u1r1 ð ịb

1ỵðu1r_1ịb_1

ẳðu2dịb

2ỵﬄu2d_b_2ỵ ðu3dịb

ỵðu_3dịb_3ỵﬄu3d_b_3ỵðu4r4ịb

ỵðu_4r4ịb_4ỵðu4_r4ịb_4:

ð4:67ị

The scalar product of Eq. (4.67) withu3, as well asu2;provides:

2ẳD21b

1þE2; b

3ẳD31b

1þD34b

4þE3:

ð4:68ị

The coefficientsE2 and E3 depend on triple products executed with position vectors and velocities. After the elimination of the vanishing terms u2ðu_3dị andu2ðu_4r4ịyields:

E2ẳu3ðu1r_1ịb_1u3ðu_4r4ịb_4u3ðu_3dịb_3u3ﬄu2d_b_2

u3ðu2dị ;

E3ẳu2ðu1r_1ịb_1u2ðu4r_4ịb_4u2ﬄu3d_b_3

u2ðu3dị :

ð4:69ị The arising velocities can be expressed in dependency of the relative velocities b_1,b_2;b_3 andb_4:

r1ẳðu1r1ịb_1; u_3ẳ u_4ẳðu2u3ịb_2; ð4:70ị _

r4ẳðu2r4ịb_2ỵðu3r4ịhb_3b_4i

: ð4:71ị

The explicit evaluation of the appearing triple products in the corresponding suitable coordinate system provides:

u3ðu1r_1ịb_1ẳr1b_21ðsinb1cosb2cosb1sinb2sinaị;

u3ðu_4r4ịb_4ẳ r4cosðb4b3ịb_2b_4;

u3ðu_3dịb_3ẳb_2b_3ðr4cosðb4b3ịị r3cosb3; u3ﬄu2d_b_2ẳr1b_1b_2ðsinb1cosb2sinasinb2cosb1ị;

u3ðu2dị ẳ r1cosb1cosb2sinabcosb2cosar1sinb1sinb2; u2ðu1r_1ịb_1ẳb_21r1cosb1cosa;

u2ðu4r_4ịb_4ẳr4b_4b_3ỵb_4

cosðb4b3ị;

u2ﬄu3d_b_3ẳr1b_1b_2ðcosb1sinb2cosb2sinb1sinaị; u2ðu3dị ẳr3sinb3ỵr4sinðb4b3ị:

Absolute kinematics. For the derivation of the equations of motion, only the absolute kinematics of the coupler body, the only body with mass in the system, as defined at the beginning of this example, is required. So, only the absolute velocity and acceleration of the center of mass Smas well as the absolute angular velocity and acceleration of the same body will be considered. The corresponding deri- vations yield:

xẳu2b_2ỵu3hb_3b_4i

; ð4:72ị

sẳd_ẵ1j_r4ẳðu1r1ịb_1ẵ1jðxr4ị; ð4:73ị x_ ẳu2€b2ỵu3hb€3€b4i

ỵau; ð4:74ị

€sẳðu1r1ị€b1ẵ1jðx_ r4ị ỵas: ð4:75ị The acceleration termsauandascan be described with the previously defined velocities as

auẳðu2u3ịhb_3b_4i

b_2; ð4:76ị

asẳðu1r_1ịb_1ỵẵ1jðxr_4ị: ð4:77ị

For the evaluation of the corresponding vector expressions, it is convenient to do this in components of the inherent vectors. One gets for example for the angular velocityxin components:

xẳ

cosb2b_3b_4 sinb2b_3b_4

b_2 2

66 4

3 77

5: ð4:78ị

Kinematic differentials. Because the system possesses only one degree of freedom, only a one-column JACOBIAN matrix must be stated. Here, the velocity terms of the former section must again be formulated for the special input~_b1ẳ1.

From the corresponding equations one gets:

~_b4ẳD41;

~_b2ẳD21; b~_3ẳD31ỵD34b~_4;

>;; ð4:79ị

x~_ ẳu2b~_2ỵu3hb~_3~_b4i

;

~_sẳðu1r1ị ẵ1j x~_r4

ﬄ

)

ð4:80ị

The pseudo accelerations (which means the accelerations for the specially given input acceleration~€qẳ0) analogously provide:

~€

b4ẳ E4;

~€ b2ẳE2;

~€

b3ẳD34b~€4ỵE3; 9>

>; ð4:81ị

x~_ ẳu2b~€2ỵu3hb~€3~€b4i þau;

€sẳẵ1jﬄx~_ r4 þas:

)

ð4:82ị

With these expressions, the equations of motion can now be formulated in closed form.

Equations of motion. From d’ALEMBERT’s Principle, the scalar equation of motion follows for this example as:

m€sþmgez

ð ị dsỵðxHSxỵHSx_ị duẳ0; ð4:83ị whereHSindicates the moment of inertia tensor with respect to the center of mass Sm. The virtual displacements in Eq. (4.83) can be represented as linear combi- nations of the virtual displacement, as well as, of the accelerations of the

generalized coordinate qẳb1using the kinematic differentials that have been derived in the previous step:

dsẳ~_sdq;

€sẳ~_s q::ỵ~€s;

duẳxdq;~ x€ ẳx~ q::ỵ~_x:

ð4:84ị

Substitution of Eq. (4.84) into Eq. (4.83) finally leads to the explicit and analytical form of the scalar equation of motion in this example:

M€b1ỵbẳQ: ð4:85ị The generalized massM, the generalized centripetal and CORIOLIS forcesb, and the generalized forcesQfollow as:

Mẳm~_s~_sỵðHSx~ị x;~ bẳm~_s~€sỵﬄHSx~_ỵxHSx

x;~ Qẳ mgez~_s:

;

ð4:86ị

The (numerical) evaluation of Eq. (4.86) can be carried out in an arbitrary coordinate system, because the corresponding expressions only include physical representations of position, velocity, and acceleration vectors. The representation in Eq. (4.86) is also very suitable for physical interpretations.

References

Hiller M (1983) Mechanische Systeme: Eine Einführung in die analytische Mechanik und Systemdynamik. Springer, Berlin u.a. - ISBN 978-3-540-12521-1

Hiller M (1995) Multiloop Kinematic Chains, and Dynamics of Multiloop Systems in Kinematics and Dynamics of Multi-body Systems

Hiller M, Kecskeméthy A and Woernle C (1986–1988) Computergestützte Kinematik und Dynamik für Fahrzeuge, Roboter und Mechanismen. InCarl-Cranz-Kurs, Vol. 1.16, Carl- Cranz-Gesellschaft, Oberpfaffenhofen

Hiller M, Kecskeméthy A and Woernle C (1986) A loop-based kinematical analysis of complex mechanisms. InPresented at the Design Engineering Technical Conference, Columbus, Ohio Kecskemethy A (1993) Objektorientierte Modellierung der Dynamik von Mehrkửrpersystemen mit Hilfe von ĩbertragungselementen [Dr.-Ing.]. Dissertation, Universitọt Duisburg, Dỹssel- dorf: VDI-Verlag

Schiehlen W and Eberhard P (2004) Technische Dynamik- Modelle für Regelung und Simulation; mit 4 Tabellen und 44 Beispielen. 2., neubearb. und erg. Aufl. (ed.), Teubner, Stuttgart u.a. - ISBN 978-3-519-12365-1

Woernle C (1988) Ein systematisches Verfahren zur Aufstellung der geometrischen Schlieòbe- dingungen in kinematischen Schleifen mit Anwendung bei der Rỹckwọrtstransformation fỹr Industrieroboter [Dr.-Ing.]. Dissertation, Universitọt Stuttgart, Dỹsseldorf: VDI-Verlag

Kinematics and Dynamics of the Vehicle Body

Based on Chaps.3 and4, where methods for the derivation of the equations of motion (kinematics and dynamics) of general complex multibody systems have been presented, in this chapter the particular analytical formulation of the kinematics and dynamics of the vehicle body will be developed. From an analytical point of view the vehicle body, consisting of the chassis and the car body, plays the role of a reference body for the subsequent vehicle components, like front wheel suspensions, rear-wheel suspensions or drivetrain.

Kinematical Analysis of the Chassis

Different Types of Wheel Suspension