Especially mechatronic systems or systems which incorporate components from different disciplines are based on system equations in implicit form, Section 3.1.
( , , )t =
F x x 0 (8.1)
If equation (8.1) can clearly be solved forx, we can reproduce a statement ( , )t
= x f x
and the methods represented in the previous sections can be applied again. But this is impossible in most cases and (8.1) must be solved in its given form. This lecture now concentrates on the case that (8.1) can be solved as such
( , , ) ( , , )
d d a
d a
t t
=
=
x f x x 0 g x x
(8.2)
This special form of DAE is referred to as semi-explicit.
In this statement ( , , )
d = d a t
x f x x (8.3)
is a system of differential equations while ( d, a, )t
=
0 g x x (8.4)
represents a system of purely algebraic equations.
If (8.4) can explicitly be solved for xa, then xa can be eliminated from (8.3) and can be inserted back into (8.4), which would then lead to a simple system of differential equations.
But this is generally not the case. It is sometimes possible that xa does not even exist in (8.4) In this case, the system of DAE must be solved in another way. For that purpose, we first of all define the index of the DAE.
Definition 8.1:
The (differentiation) index i of a DAE is the minimal number of differentiations of the equation 0=g x x( d, a, )t in order to transform (8.3) and (8.4) into a regular differential equation system.
Index 1:
In this case one differentiation is by definition sufficient in order to transform (8.3) and (8.4) into an ODE:
Differentiation of (8.6) with respect to time yields:
a d 0
Note that
a
∂
∂ g
x must not be singular to guarantee, that it actually deals with a system of index 1.
Consider the following system ( , , )
Differentiating with respect to time; then
0 d ( d, a, )
Index 3:
Analogous to index 2 systems, we obtain the same statement (8.13) after double differentiation.
In this case, we obtain an equation system with the form ( , , )
where the function h depends on the index of the system.
The systems which occur in mechatronics are normally index 3-problems.
In many cases it is more efficient to solve the DAE directly instead of reducing it to an ODE:
This is especially true for DAEs of index 1. Therefore, in many simulation programs the index is reduced to 1 and afterwards the resulting DAE is solved. In most cases either implicit Runge-Kutta- or BDF-methods are applied to solve DAEs. At first, the solution of DAEs with the help of BDFs is presented.
In case of ordinary differential equations their solution is obtained by solving Eq. (7.86) for xn+1. For this purpose, the last j points xn j− as well as xn+1 are necessary, whereby x can be easily determined by evaluating f.
In case of DAEs it is not possible anymore to determine x by evaluating f. Use Eq. (8.1) instead. Since Eq. (8.1) cannot be solved for x in general, every calculation of x requires the solution of a nonlinear system of equations. Additionally, a nonlinear system of equations has to be solved due to the implicit calculation rule. However, this enormous calculation cost can be avoided by solving Eq. (7.86) for xn+1 and inserting it into Eq. (8.1). In this case, merely the nonlinear system of equations
0
has to be solved in order to obtain xn+1. Therefore, only the solution of a nonlinear system of equations of dimension n of the state vector x is necessary by inserting the integration rule into the model equations.
The frequently used method for the solution of DAEs is DASSL. This method uses the BDF formulas of order 1 to 5, as well as a step size control. DASSL is very well suited for the solution of stiff DAEs and is therefore surely capable of solving non-stiff problems. In this case however, DASSL is relatively inefficient due to the bad error coefficients of the BDFs. If a problem is not stiff, it is more efficient to use an implicit Runge-Kutta method. The most popular implicit Runge-Kutta method for solving ODEs and DAEs is the Radau method of fifth order. This method has the Butcher table
4 6 88 7 6 296 169 6 2 3 6
The solution of DAEs with the help of implicit Runge-Kutta methods resembles the solution of ODEs. Merely the steps are calculated with the help of the DAE formulation
3
coefficient of Radau’s method is much smaller than the coefficients of the BDF methods, for which reason Radau’s method allows the choice of greater step sizes. This, especially for non-stiff systems Radau’s method is better suited. Additionally, Runge-Kutta methods can be more easily started, as described above.
Notice that beside the initial values for x also initial values for x have to be provided in order to apply DASSL and Radau’s method. These initial values must comply with Eq. (8.1) and are in this case called consistent. The determination of consistent initial values is closely related to index reduction, but is not discussed in this lecture.
9 Numerical Solution of Non-linear Sysem Equations
9.1 Nonlinear Equations
The need to solve non-linear system of equations frequently occurs in mechatronic systems, e.g.
• In the determination of equilibrium positions,
• In the solution of kinematic ties,
• In the estimation of system parameters.
We consider a non-linear equation system to discuss the appropriate solution methods
1 1
The solution of a non-linear equation system is a vector x which exactly satisfies (9.1); x is also the zero point of f.
Example 9.1:
( )
21 sin .
n= f x = x
Solving non-linear equation systems frequently causes problems in practice.
The reasons are
• generally, there are no analytical solution, i.e. a numerical approximation (iteration) is necessary,
• the convergence of an approximation procedure can generally not be guaranteed,
• there are generally no applicable methods to determine all zero points.
Example 9.2: Planar Four-bar Mechanism, Fig. 9.1
Fig. 9.1: Planar fourbar mechanism.
With a given set of measurements, the angle β2 relative to (dependent on) β1, should be
In this case, there exists an analytical (but not unique) solution
( )
with D=A2+B2−C2
The solution is not unique, see Fig. 9.2:
Fig. 9.2: Solutions of the planar four-bar mechanism.
Example 9.3: Five-link Wheel Suspension, Fig. 9.3
Fig. 9.3: Five-link wheel suspension.
The figure above Fig. 9.3 illustrates a five-link wheel suspension with one degree of freedom which is described by the spring deflection cz.
We have to calculate the remaining coordinated cx and cy of the wheel carriage in the vehicle fixed coordinate system as well as the Bryant angle ψ θ ϕ, , of the wheel carriage.
To determine the geometry, the vector c will be introduced at the origin of the vehicle fixed coordinate system
{
OF,xF,yF,zF}
which has the following coordinatesFurthermore, the vectors
, 1, ,5
of the wheel carriage and the vehicle fixed coupling point of the guide are given by the construction data.
Assuming that the guides have identical lengths, we obtain the following 5 constraints
( ) ( )
22 2
, , , , , 0
i c c cx y z ψ θ ϕ = + i− Fi − =li
l c p r (9.5)
These are the 5 equations of the variables , , , ,
x y z
c c c ψ θ and ϕ In written form we obtain
The equation system (9.5) together with (9.6) cannot analytically be solved. Instead, we must insert a numerical method.
Fig. 9.4: Vectors of the five-point wheel suspension.
Fig. 9.5: Wheel centre trajectory of the five-point real axle wheel suspension.
Worksheet 7: Kinematics of a rear five-link wheel suspension