4569846498

114 Mutibody Systems Approach to Vehicle Dynamics
K
6EIyy
GI 4 PEI
PEI
xx z yy 4 y zz
K K K
L ( 1 P ) L L( 1
P ) L( 1
P )
35 2 44 55 66
z
z
where P y 12EI zz A SY /(GAL 2 ) and P z 12EI yy A SZ /(GAL 2 ). Young’s modulus
of elasticity for the beam is given by E and the shear modulus is given
by G. The cross-sectional area of the beam is given by A. The terms I yy
and I zz are the second moments of area about the neutral axes of the beam
cross-section. For a solid circular section with diameter D these would, for
example, be given by I yy I zz D 4 /64 I xx is the polar second moment
of area. Again for a solid circular section this is given by I xx D 4 /32.
The final part of the definition of the terms in the stiffness matrix is the correction
factors for shear deflection in the y and z directions for Timoshenko
beams. These are given in the y direction by A sy A/I 2 yy ∫ (Q A
y/l z ) 2 dA and in
the z direction by A sy A/I 2 zz ∫ A (Q z/l y ) 2 dA. The terms Q y and Q z are the first
moments of area about the beam section y- and z-axes respectively. The
terms l y and l z are the dimensions of the beam cross-section in the y and z
directions of the cross-section axes.
The structural damping terms c 11 through to c 66 in (3.51) may be input
directly or by using a ratio to factor the terms in the stiffness matrix. In a
similar manner to the dampers and bushes discussed earlier it is possible to
define a beam with non-linear properties using more advanced elements
that allow the definition of general force fields.
As with the bush elements the beam will produce an equilibriating force
and moment acting on the J marker using
{F ji } j {F ij } j (3.52)
{M ji } j {M ij } j {d ij } j {F ij } j (3.53)
where {d ij } j is the position vector of the I marker relative to the J marker
referenced to the J marker axis system. An example of the command used
to define a massless beam is
BEAM/0304,I0203,J0504,
,LENGTH250,A315,IXX4021240,IYY2010620,
IZZ2010620
,EMOD200E3,GMOD70E3,ASY1.33,ASZ1.33,
CRATIO0
3.2.10 Summation of forces and moments
Having considered the definition of force elements in terms of model definition
we may conclude by considering the formulation of the equations
for the forces and moments acting on a body. An applied force or moment
can be defined using an equation to specify the magnitude, which may be
functionally dependent on displacements, velocities, other applied forces and
time. Using the example in Figure 3.36 there is an applied force {F A } 1 acting
at point A, the weight of the body m{g} 1 acting at the centre of mass G, a
force {F B } 1 and a torque {T B } 1 due to an element such as a bush or beam
connection to another part. In addition there is an applied torque {T C } 1 acting
at point C. Note that at this stage all the force and torque vectors are
assumed to be resolved parallel to the GRF and that {g} 1 is the vector of
acceleration due to gravity and is again measured in the GRF.
y