Solution of the inhomogeneous State Equation

= ^{1 2} =  − 

X x x (4.33)

and the fundamental matrix

( )

The general solution of the homogenous equation is therefore with respect to (4.28):

( )

0 ^{10 10 20}

with an arbitrary initial vector x0.

4.3.2 Solution of the inhomogeneous State Equation

If a system is influenced by an outside excitement ^b

( )

^t , we obtain the following state equation:

( )

^t

( )

^t

= +

x Ax b (4.36)

hereafter A will be assumed to be constant. The complete solution of (4.36) consists of the homogenous in (4.37) and the inhomogeneous part xp

( )

t of the outside excitement, i.e. this yields:

( )

^t =

( )

^t ₀+ _p

( )

^{t .}

x Φ x x (4.37)

For the particular solution, we can apply the method of variation of constants for the equation. This makes use of the following basic approach:

( )

^t =

( ) ( )

^{t t}

xp Φ c (4.38)

with a “varied“ constant ^c.

After differentiating (4.38) with respect to time we first obtain:

( )

t =

( ) ( )

t c t +

( ) ( )

t t = p

( )

t +

( )

t

xp Φ Φ c Ax b (4.39)

Because Φ

( )

^{t =}AΦ

( )

^t it yields:

( ) ( )

^{t t =}

( )

^{t ⇒}

( )

^{t = −1}

( ) ( )

^{t t =}

( )

^−t

Φ c b c Φ b Φ b (4.40)

Equation (4.40) can directly be integrated with respect to t and we subsequently obtain

( ) ( ) ( )

0 t

t = ⁰+

∫

−t t td

c c Φ b (4.41)

As one only needs any particular solution, one can equate c₀ =0 and one obtains a general complete solution:

The first summand on the right hand side refers to the free motion of the system with given initial conditions, while the second summand refers to the forced motion expressed by the control vector b. It is very important for this application that the first summand converges to zero with time. This is the case if the free system satisfies certain stability conditions, cp.

Chapter 10. Here, the long-term behavior of the system is only described by the second summand.

Example 4.4: One-dimensional oscillator with excitement

One obtains the following solution for the oscillator-problem described in Example 4.3

( ) ( ) ( )

5 State Space Equations with Normal Coordinates

5.1 Normal Coordinates

Through appropriate transformation one can frequently simplify linear state equations.

Obviously, one practically restricts oneself to linear transformation. Otherwise, one would have to transfer equations from linear to non-linear forms which would rather complicate the calculation.

A linear transformation of a state vector x to a state vector z can generally be described by the following statement

=

x Tz (5.1)

with T as a n n× matrix. Furthermore, there must be a clear inverse transformation:

−1

One applies the transformation (5.1) to the linear state equation:

= +

x Ax Bu (5.4)

Thus, one first obtains

= +

Tz ATZ Bu (5.5)

and then, after left-hand multiplication with T⁻¹, we obtain the following state equation

= +

− .

= ¹

B T B (5.8)

The transformation described in (5.7) is referred to as similarity transformation. The term similarity transformation derives from the fact that the matrices A and A=T AT⁻¹ possess identical eigenvalues.

Theorem 5.1:

The eigenvalues of a similar system are invariant against linear and regular transformations.

The goal of a transformation is such that after transformation, the solution and the interpretation of the system equations are simplified, i.e. the matrix should have a more

“convenient” structure than the initial matrix A. The term “convenient” depends on the respective problem.

A sought after characteristic is that the couplings between the equations is as low as possible. If all eigenvalues of A are different from each other, it would be possible to completely decouple all system equations from each other, i.e. the new system matrix has a diagonal structure.

Theorem 5.2:

If all eigenvalues λ λ1, 2,,λ_n of the n n× matrix A are different from each other, one has a linear transformation with a regular transformation matrix T. Then the system matrix

T AT−1 would have a diagonal characteristic with n eigenvalues arranged diagonally.

This yields

A matrix with these characteristics is referred to as diagonalizable.

One obtains the transformation matrix ^T by inserting the eigenvectors in columns horizontally, i.e. the matrix T is equal to the modal matrix X introduced in Section 4.3.1.

The transformed uncontrolled system

(

^u=0

)

consists of n differential equations which are independent from each other

, 1, ,

i i i

z =λz i=  n (5.10)

and the n-dimensional total motion z(t) is composed of single motions

[

^{0 0} zi ^{0 0}

]

^T, i ^{1, ,}n

= =

zi    (5.11)

The coordinates zi in (5.11) are referred to as normal coordinates, this yields 0, for i j,

= ≠

T i j

z z (5.12)

which means that the vectors zi are standing vertically (normal) on top of each other.

In a system based on normal coordinates each eigenvalue affects exactly one coordinate, i.e.

time-dependent changes of each main coordinate depend on the time-dependent change of the residual main coordinates.

The connection between the original coordinates (state vectors) x and the normal coordinates is given by transformation:

and ⁻ , respectively.

= = ¹

x Xz z X x (5.13)

Annotation 5.1:

Note that if complex eigenvalues (and therefore also complex eigenvectors) appear equations with complex components ensue, which elude themselves from direct precise analysis. This disadvantage can be eliminated if one takes into account that in real matrices complex eigenvalues are generally real or they generally appear as a pair wise conjugated complex:

1/2 i

λ = ±δ ω. (5.14)

The adequate equations would be

( )

After addition and subtraction of (5.15) and (5.16) we obtain

and with the new (real!) coordinates

( )

1 1 2, 2 1 2

y = +z z y =i z −z (5.20)

with the (real) equations

1 1 2

y =δy +ωy (5.21)

2 1 2

y = −ωy +δy (5.22)

These two equations of first order can be summed up to a single equation of second order by differentiating (5.21) with respect to time and by subsequently inserting (5.22)

( )

( )

1 1 1 1 1

y =δy +ω ω− y +_ω^δ y −δy

   .

This represents an equation of a linear homogenous oscillation in the normal form:

(

^{2 2}

)

1 2 1 1 0

y − δy + δ +ω y =

  (5.23)

Summary

Interpretation of the Solution of Linear Systems by means of the Eigenvalues The system matrix A generally possesses ^I real eigenvalues:

, 1, , .

i i i I

λ δ= =  (5.24)

and k pairs of conjugated complex eigenvalues

i , 1, , .

i i i I I k

λ δ= ± ω = +  + (5.25)

Hence, the first ^I equations are equal to the real eigenvalues and can be completely uncoupled. The residual equations are in each case pair wise coupled and can be substituted by k equations of second order by eliminating each second coordinate. Each main

coordinate therefore describes either a non-periodic motion or a (damped or excited) oscillation.

Thus, one can make a complete statement about the possible motions of a linear dynamical system by means of the eigenvalues. The eigenvalues are material to the system behavior.

In graphical representation, the position of the eigenvalues, the root loci (sing. root locus), are drawn into the complex number field, Fig. 5.3. The root loci provide concise information about the change of the eigenvalues with respect to parameter change (root locus curves).

The root locus diagram also directly provides information regarding the stability of the solutions of the linear system. All stable solutions have their eigenvalues on the left hand side of the imagined axis, i.e. the eigenvalues have negative real parts, which guarantees for the decrease of the solution curve against zero, cp. equation (5.17).

Apart from the stability aspect, the root locus diagram also provides information of the long-term behavior of (autonomous) systems. This behavior will be delong-termined by such solutions which experience the slightest damping (if the solutions are stimulated by initial conditions).

The solutions that are affected by the slightest damping are those that are closest to the imaginary axis on the negative side. The eigenvalues of these solutions are referred to as dominant eigenvalues and the appropriate solutions of the state equation which are referred to as dominant solution.

Connection between the Eigenvalues with Eigenfrequency, Damping, Mass and Time Constant of a System

Each real eigenvalue λ has a so-called time constant ^T with T 1

= −λ

Thus, the time response of the state variables of the non-excited system contains a part in the form

Fig. 5.1: Decaying behaviour with real eigenvalues.

In Fig. 5.1 the so-called half-life is drawn in t_0,5.

0,5 ln 2 0, 69215

t = T ≈ T

Conversely, one can also calculate a real eigenvalue λ out of a measured half-life

1 0,5

2e ^t

λ = − = −T ⁻ (5.26)

with a complex eigenvalue pair

, i

λ λ δ ω= ±

the system has a eigenfrequency 1

f 2 ω

= π

with the appropriate damping degree

2 2

D δ

δ ω

= + .

In this case, the time processes of the state variable contains a part in the form

( )

( )t =e^δt cos(2π ft)− sin(2π ft)

s u v

with the appropriate eigenvector belonging to λ

= ± i x u v

the part decays where δ <0, thus D>0. The quotient of two successive amplitudes is

2

2

1 1 1

D

i D

i

s e

s

− π + = − <

Fig. 5.2 represents a corresponding process.

Fig. 5.2: Decaying behaviour with complex eigenvalues.

One can calculate two conjugated complex eigenvalues of the system matrix from an eigenfrequency of the system and the corresponding damping degree:

, 2 2

1 2

f D fi

D λ λ = − π ± π

−

Fig. 5.3: Behaviour of linear systems in dependence of the position of the complex plane.

Example 5.1: Quarter-Car (cp. [9])

Fig. 5.4: Quarter-car.

Subject of the following example is the represented model of a quarter-car, Fig. 5.4.

The unsprang mass of a wheel (rim + hoop + brake + part of the suspension) is summed up to a single substituted mass m_R which is reinforced by a spring on the street. The spring represents the vertical stiffness of the wheel. The damping of the wheel will be neglected.

Between the wheel mass and a further substituted mass mA (pro rata body, assembly, engine, gearbox, etc.) is a spring strut (structure spring and damper). Under the assumption of linear characteristic lines of the spring and damping elements of the wheel

( )

cR and the assembly

(

c dA^, A

)

, the motion equations result from the impulse theorems:

( ) ( )

One chooses the following state vector of the equilibrium position (static position) of the linearized system

therefore, the following equations ensue

[ ]

The structural acceleration xA (comfort degree) and the dynamical wheel load (degree of driving security) are our initial quantities:

( )

0

(

2

)

dyn R S R R S

F =c x −x −F =c x −x with F0 as static wheel load

( )

In matrix form we obtain

  

To analyze the system behavior one chooses the following data

A 380

Fig. 5.5 represents the system in a complex plane. Hereafter, we draw the root locus curves where 1500Ns m/ ≤d_A ≤6500Ns m/ .

Obviously, the four eigenvalues move towards the imagined axis with increasing damping.

Finally they reach real values, which means that the system is asymptotically damped for sufficient large dA.

Fig. 5.5: Eigenvalues of the quarter-car where 1500 Ns/m ≤ dA ≤ 6500 Ns/m.

Worksheet 6: Oscillations of a quarter vehicle model

In document List of Figures (Pldal 78-91)