Dynamics Part Two

8/12/2019 Dynamics Part Two

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CHAPTER 4. DYNAMICS 232

4.4 Application of Computational Procedures in Dy-

namic Analysis

4.4.1 Introduction

An important numerical task in dynamic problems is the time integration of the

equation of motion. The aim is to get the time history of the systems motion due to

a given excitation or load profile. Different numerical codes are available in various

programme packages. The routines offered in MATLAB are one-step and multistep

methods with implicit or explicit time schemes. Routines for differential-algebraic

equations (DAE), called as solvers for singular mass matrix, are available as well.

Only a general advice can be given how to choose a code appropriate to the givenparticular problem, see also chapter 5. Nevertheless, a critical interpretation of the

obtained solution is necessary and highly recommended since the reliability of the

results depends on various conditions like truncation and round off errors of the

computer, stability characteristics of the integration code, chosen error tolerances,

configuration of the mechanical system and initial conditions.

Example 1.1: Planar pendulum, ODE representation

The simple example of a planar pendulum, shown in Fig. 4.17, shall illustrate the

influence of numerical errors to the behaviour of the computed solution.

x

y

Figure 4.17: Planar pendulum

The pendulum consists of a mass attached on a rod of length where a torque


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is applied. Considering the planar motion in the x-y plane, the equation of

motion can be derived using Newtons and Eulers equations

(4.117)

(4.118)

(4.119)

In connection with two imposed geometrical constraints

(4.120)

(4.121)

the system possesses

degree of freedom. Choosing the angle

as

the generalised coordinate, the constraint forces can be eliminated and the equation

of motion reads as

(4.122)

An other way to derive this equation is the application of Eulers equation with

respect to the fixed pivot point A

(4.123)

For the comparison of the numerical solution obtained by time integration with theanalytical solution, we choose the trivial case of static equilibrium position as shown

in Fig. 4.18.

Figure 4.18: Different positions of planar pendulum


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In a static equilibrium position

, it is , which leads to the applied

torque in the equilibrium state

(4.124)

For numerical solution, the integration code ode45 of MATLAB is used choosing

a tolerance of RelTol = AbsTol = 10e-5 for the relative and the absolute error, re-

spectively. The specific parameters of the pendulum considered are ,

and (mathematical pendulum).

Case I: Equilibrium position

Fig. 4.18 shows the pendulum in the considered position. Starting the integra-

tion with the initial conditions

and

and looking at the solution

within a time period of 30 s it remains zero as it is expected. This means the

solution is exact within the precision of the machine.

An artificial error is now introduced into the initial conditions

and and the result is shown in Fig. 4.19.

An exponential-like deviation can be seen indicating an unstable behaviour. It

can be shown by applying stability theory that the chosen position is a mechanically

monotone unstable equilibrium.

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

9

(t)[rad]

t [s]

Figure 4.19: Mechanically unstable motion of pendulum initiated by numerical er-

rors, position I


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Case II : Equilibrium position

As indicated in Fig. 4.18, the mass of the pendulum is below the bearing, which

is a mechanically stable position (more precisely, it is marginally stable). The inte-

gration with the exact initial conditions again leads to a solution exact in the range

of machine precision. An artificial error as above causes the solution as plotted in

Fig. 4.20 showing an oscillatory instability behaviour.

0 5 10 15 20 25 306

4

2

0

2

4

6x 10

5

(t)[rad]

t [s]

Figure 4.20: Unstable oscillation in position II

Case III: Equilibrium position

The hanging pendulum is mechanically marginally stable, of course.

a) The integration with the exact and with the disturbed initial conditions leads

to the same behaviour as we could see in case II.b) Changing the coordinate of description to leads to a sligthly

changed equation of motion.

Now the corresponding initial condition is

. With this condition the

computed solution shows the same oscillatory instability as in the case of the dis-

turbed condition in a). Obviously the truncation error in the representation of the

number is sufficient to cause this unacceptable behaviour.

In addition to the stability properties of the mechanical system, the stability proper-


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ties of the numerical procedure itself has to be taken into account when a numerical

solution is computed. The details can not be provided here and reader ist referredto [8],[9]. Other illustrative examples can also be found in chapter 5.

As it can be seen the stability behaviour depends on the configuration of the

system. The sensitivity of the system on errors in the initial conditions (round off

error due to numeric representation of ) is different. The numerical errors which

cause the instability in case III do not cause unstable behaviour in case II.

However, although the numerical representation of the initial conditions may

influence the quality of solution, the characteristics and parameters of the chosen

integration procedure (stability of the algorithm, relative and absolute tolerances)

are cruical for obtaining reliable results.

The programs Integpendode.mandOdefuncpend.mfor time integration reads:%

% ODE-INTEGRATION PROGRAM: PLANAR PENDULUM

% ---------------------------------------

%

%----------------------------------------------------------------------------

%

%

load Date_pend % load the saved parameters and

% % initial conditions

%

%

%% set options for the integration subroutine

%

% RelTol - relativ tolerance of integration (default {1e-3});

% AbsTol - absolut tolerance of integration (default {1e-6});

%

%

options = odeset(RelTol,1e-5,AbsTol,[1e-5 1e-5]);

%

%-----------------------------------------------------------------------------

%

% Integration with solver ode45:

% -----------------------------

%

% ode45 - MATLAB ODE nonstiff solver, based on an

% explicit Runge-Kutta (4,5) formula,

% the Dormand-Prince pair

%

% Vector [t,x] is to create, whereas:

%

% t - time, [0 3] - time interval;

% x - statevector [alpha, omega];

% odefuncpend - extern function, that provides right

% hand side of ODE system


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% incondvect_ode - initial condition vector

% options - see above%

[t,x]=ode45(odefuncpend,[0 30], incondvect_ode,options);

%

%----------------------------------------------------------------------------

%

% Function f of ODE system rhs

%

%

% odefuncpend Name of function (responds to name in

% integrpendode)%

% t, x time, statevector (integration variables)

%

%

%

%-------------------------------------------------------------------------------

%

function rhs = odefuncpend(t,x)

%

%-------------------------------------------------------------------------------

%

% read parameters% ---------------

%

%

%

load Date_pend % load the saved parameters

%

%

% calculate the torque for equilibrium position

%

Mom = m*g*l*sin(alphades);

%

%

rhs = [x(2); (Mom - m*g*l*sin(x(1)))/(m*l^2)]; % right hand side

%---------------------------------------------------------------

Example 1.2: Planar pendulum, DAE representation

In this example the integration in time domain is shown using a solver for differential-

algebraic systems DAE.


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The planar pendulum is considered as shown in Fig. 4.17 where the configura-

tion coordinates are given by the coordinates

(4.125)

In contrary to the minimal form of equation of motion (4.114), the governing equa-

tions consist of the Newton-Euler-equations (4.109-4.111) plus the kinematical con-

straints (4.112, 4.113) as listed below

(4.126)

(4.127)

These equations form a system of coupled differential and algebraic equations

( DAE system) of index 3 because the geometrical constraints in Eq. (4.119) are

formulated at position level. Although it can be integrated in this form, many nu-merical procedures demand a formulation of the DAE in index 1 - form.

The first differentiation of Eq.(4.119) provides constraints at velocity level

(4.128)

By combining Eq. (4.118) and Eq. (4.120), DAE of index 2 is established.

Further differentiation of constraint equation results in

(4.129)

representing the geometrical constraints at acceleration level.

To formulate DAE of index 1, dynamical equations (4.118) has to be combined

with Eq. (4.121).

Since this DAE system contains time derivatives of second order, it has to be

reformulated to obtain the form

(4.130)

as it is demanded for utilisation of MATLABroutinesODE15s or ODE23t.


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With this aim in view, a state vector is defined containing the unknown positions

and the constraint forces . To formulate a first order differential equation, thevelocities

have to be also included in the state vector.

However, to formulate the right hand side , where only coordinates of are

allowed to appear, the accelerations is introduced in the state vector as well.

Finally it reads as

(4.131)

Then the governing equations are

(4.132)


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or

. . .

...

(4.133)

The internal structure of is depending on how the structure of is

chosen by the user.

The representation used above show particular features, some of them are im-

portant for the effective application ofODE15sand ODE23t.

The mass matrix is singular. This is a general fact for DAE-system.

has a diagonal form. Therefore the DAE-system is a semi-explicit DAE.

By putting all variables on the right hand side, is a constant matrix. Mo-

rover, is not sparse. This has positive consequences for the solver when

building the Jacobian

.

The programs Intependdae.m and daefuncpend are listed. The corresponding datafile is already given in example 4.1.


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%

% daefuncpend - Name of function (responds to name in% integrpendode)

%

% t, y - time, statevector (integration variables)

% flag - string indicating the type information

% that should be returned

%

%-------------------------------------------------------------------------------

%

function rhs = daefuncpend(t,y,flag)

%

%-------------------------------------------------------------------------------

%

% read parameters

% ---------------

%

%

%

load Date_pend % load the saved parameters

%

%

%-------------------------------------------------------------------------------

%

% DAE solver demands:

% -------------------

%% event1: right hand side f of DAE is required

%

switch(flag)

case

%

% calculate torque for equilibrium position

%

Mom = m*g*l*sin(alphades);

%

% calculate the right hand side f

%

rhs = [y(4); y(5); y(6); y(7); y(8); y(9); -m*y(7) + y(10); -m*y(8) - m*g+ y(11); -I*y(9) + Mom + y(2)*y(10) - y(1)*y(11); y(7) + y(5)*y(6)

+ y(2)*y(9); y(8) - y(4)*y(6) - y(1)*y(9)];

%

%

% event2: mass matrix of DAE is required

%

%

case mass

%

% calculate the mass matrix


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%

rhs = diag([1,1,1,1,1,1,0,0,0,0,0])%

% end o f switch

%

end

%

%----------------------------------------------------

The results of a time integration using ODE15s with the default tolerances are

discussed.

a) Equilibrium positionsSimilar to example 1.1, the three cases shown in Fig. 4.18 are considered. Us-

ing the exact initial conditions, in all three cases the numerical solutions show no

difference to the analytical ones within the considered time interval of

s.

Using the disturbed initial condition with an error of in the

positions, in all three cases the numerical solution converged to the exact one. Even

in case I, representing the mechanical unstable position, the artificial error

was damped out by the integrator.

b) Oscillatory motion

Considered is the case II with the statical equilibrium position

and aninitial condition

. A time integration delivers the solution

is shown in

Fig. 4.21.

It is a periodical motion but not a harmonical one because the problem is kine-

matically nonlinear (the lower reversal point is not at ).

The full descriptor formulation contains also the constraint forces

an

which are plotted in Fig.4.22 with respect to time. A frequency double to that of the

pendulum can be observed in the vertical component

of constraint force.

With these examples the utilisation of a DAE solver has been demonstrated.

The particular properties of this procedure like damping out of numerical errors and

delivering constraint force could be seen. Attention has to be drawn on the long

term stability of computed solutions since drift effects may occur where energy is

transfered between the motions in particular coordinates.


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0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

[rad]

t [s]

Figure 4.21: Time evolution of the angle

0 0.5 1 1.5 2 2.5 3 3.5 42

1

0

1

2

3

4

5

6

7

t [s]

Fx

[N],

Fy

[N]

Fx(t)

Fy(t)

Figure 4.22: Pendulum constraint forces


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Remark:

In practice, sometimes small vibrations with respect to a reference motion haveto be investigated.

In that case the equations of motion are linear with respect to the unknown

variables. To obtain these equations a particular procedure must be followed or

the nonlinear equation of motion have to be linearised. For that in the equation of

motion (4.114) of the pendulum the coordinate

is introduced.

With the assumption of small vibration , it is and , follows.

By use of these relations all nonlinear functions of

are represented in a power

series expansion. Neglecting all terms of quadratic and higher order, then

hold.

Introducing these relations into Eq. (4.114) it follows

In particular cases the reference motion may be a constant

= const, as it was in

the example above.)

4.4.2 Dynamics of robot

In this chapter dynamic analysis of planar robot with two rigid links, that follows

specified trajectory, is presented. Several case-studies (inverse and forward dynam-

ics of 2 DOF robot, forward dynamics of 2 DOF robot that contains PD - control

loop, inverse and forward dynanmics of robot whose end-effector is mechanically

constrained) are performed and analysed within following examples.Mathematical modelling of 2 DOF robot, is schematically depicted in Fig. 4.23.

The governing equations are derived in the full desriptor and minimal form as fol-

lows. The mathematical models so obtained are basis for the particular examples

presented in this chapter.


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x

y

Robot: 2 DOF MBS

Full descriptor form: Minimal form:

Mathematical modellingleading to full descriptorform

Mathematical modellingleading to minimal formvia LE of 2nd kind

Figure 4.23: Modelling of 2 DOF robot

Full descriptor form

generalised Cartesian coordinates: 3 coordinates per body ( , , )

dynamical equations of the free-body diagram


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x

y

A

B

x

y

x

y nth body

generalised Cartesian coordinates

Figure 4.24: Derivation of governing equations

(4.138)

(4.139)

(4.140)

(4.141)

(4.142)

(4.143)


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kinematical constraint equations

(4.144)

(4.145)

(4.146)

(4.147)

dynamical equations of the free body diagram in matrix form

(4.148)

(4.149)

constraint forces can be expressed using "geometrical approach" via

constraint matrix

and Lagrange multipliers

:

(4.150)

obviously:

,

,

,

.

(4.151)

derivation of the constraint matrix


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kinematical constraints in matrix form:

(4.152)

(4.153)

(4.154)

(4.155)

dynamical equations of the free body diagram in the matrix form

expressed via transposed constraint matrix and Langrange multipli-

ers

kinematical constraints in the matrix form

(4.156)

Minimal form

Governing equations in the minimal form (equations of motion of the system)

have been derived via transformation from the full descriptor form.

the chosen generalised coordinates (2 DOF): absolute orientation angle

of the first body, angle of the relative orientation of the second body with

respect to the first one

Jacobian matrix that specifies relation between two sets of coordinates

has been established using NEWEUL multibody package


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Example 2.1: Inverse dynamics of 2 DOF robot

Inverse dynamics of the robot that follows a specified trajectory

Determination of the applied forces (torques of the actuators of the force

type) that have to be imposed to the robot to follow specified trajectory as

well as the constraint forces due to the joints.

Kinematical data which is the basis of inverse dynamic analysis has been cal-

culated on the basis of the specified trajectory of the robots end-effector and

the kinematical structure of robots mechanism (length of linkages). Inertia

parameters and lengths of linkages are specified as follows:

kg,

,

kg,

,

m,

m.

Inverse dynamics problem of the robots motion is solved using mathemati-

cal model in full descriptor form, formulated above. Trajectory of the end-

effector is prescribed by the equation

,

.

The time history of the generalized coordinates

and

, calculated on the

basis of the specified trajectory of robots end-effector are shown in Fig. 4.25.

The driving torques and constraint forces in two cases (with and without gravity)

are shown in the Fig. 4.26 - Fig. 4.31. It can be observed that the forces due to the

statical origin are dominant.


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Case I: gravity is included

0 0.5 1 1.5 2 2.5 31

0.5

0

0.5

1

1.5

M

1[Nm];M2

[Nm]

t [s]

M1(t)

M2(t)

Figure 4.26: Robot driving torques needed to follow a specified end-effector trajec-

tory. Calculated on the basis of robots prescribed motion


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0 0.5 1 1.5 2 2.5 30.2

0.15

0.1

0.05

0

0.05

0.1

Fbx

[N],Ffx

[N]

t [s]

Fbx

(t)

Ffx(t)

Figure 4.27: Constraint force in -direction: joint A (basis joint) and joint B

(flying) joint

0 0.5 1 1.5 2 2.5 34

5

6

7

8

9

10

Fby

[N],Ffy

[N]

t [s]

Fby

(t)

Ffy

(t)


(flying joint)


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Case II: gravity is not included

0 0.5 1 1.5 2 2.5 30.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

0.05

M1,

M2

[Nm]

t [s]

M1 (t)

M2(t)

Figure 4.29: Robot driving torques needed to follow a specified end-effector trajec-

tory. Calculated on the basis of robots prescribed motion

Example 2.2: Forward dynamics of 2 DOF robot,

In this example forward dynamics of 2 DOF robot, based on the marhematical

model in minimal form (two dimensional ODE system) is performed. The explicit

Runge-Kutta method (within M ATLABcode: ode45integration algorithm) has been

used. The tolerances have been set as follows: RelTol=1e-3, AbsTol=1e-6 (default

values).

The simulation results are shown in Fig. 4.32. In Fig. 4.33, the difference be-

tween simulation results and reference values is depicted. It is visible that absolute

error within the whole integration domain does not exceed the magnitude of 4e-6.Further analysis has shown that time integration with imposed more strict tolerances

provides more accurate integration results.

The proof-check if integration results converge when more strict tolerances are

imposed on integration routine, is in the most case-studies a necessary step in ob-

taining reliable simulation results.


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0 0.5 1 1.5 2 2.5 30.2

0.15

0.1

0.05

0

0.05

0.1

Fbx

[N],Ffx

[N]

t [s]

Fbx

(t)

Ffx(t)


(flying joint)

0 0.5 1 1.5 2 2.5 30.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

Fby

[N],Ffy

[N]

t [s]

Fby

(t)

Ffy(t)

Figure 4.31: Constraint force in

-direction: joint A (basis joint) and joint B

(flying joint)


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0 0.5 1 1.5 2 2.5 31

0.5

0

0.5

1

1.5

1

[rad],2

[rad],

t [s]

1(t)

2(t)

Figure 4.32: Simulation results: generalised coordinates

,

Example 2.3: Forward dynamics of 2 DOF robot,

Here, the same robot is considered as in the Example 2.2, with the only difference

that gravity forces act on the robot links. Integration routine ode45is used again.

In contrary to the results obtained in the case (Fig. 4.32, Fig. 4.33), an

integration with default tolerances leads to a very strongly divergent solution.

Even when more strict tolerances (i. e. 1e-8) are imposed to the integratin

routine the errors are reduced but the solution is still divergent, Fig. 4.34.

A further restriction of tolerances to 1e-10 leads to further reduction of errors,

but the divergent characteristics are still the same, as it can be seen in Fig. 4.35.

The reason for such a behaviour lies in the fact that analysed system is me-

chanically unstable (the proof is left to the reader) and numerical errors that are

unvoidable during integration routine (discretisation errors due to the characteris-tics of the chosen routine and round-off errors) initiate an unstable behaviour of the

system (see also Example 1.1 and Example 1.2).

The consequences of the errors in the joint angles on the position of end-effector

is illustrated in Fig. 4.36 (tolerances of magnitude 1e-8 have been used) and Fig.

4.37 (tolerances of magnitude 1e-10).


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0 0.5 1 1.5 2 2.5 34

3

2

1

0

1

2

3x 10

6

1

(t)[rad],

2

(t)[rad]

t [s]

1(t)

2(t)

Figure 4.33: Difference between simulated results (RelTol=1e-3, AbsTol=1e-6) and

reference data: generalised coordinates

,

Example 2.4: Control via PD-feedback

From the previous examples it became clear, that mechanically unstable or marginally

stable systems are difficult to simulate by numerical integration. In reality such

feedforward calculation of the desired driving torques by inverse dynamics and ap-

plying to the robot also does not result in a satisfactory motion, i.e. it is not possible

to drive the robot along a prescribed path, since disturbances act on the robot.

Therefore a feedback control is necessary, see Fig. 4.38. From control theory

a broad variety of control laws are known and may be applied. The choice of a

specific or an optimal control law is not considered here.

In the following example it is assumed that the positions

and

as well as the

corresponding velocities can be measured, e.g. by resolvers and tachogenerators,

respectively. Therefore, a complete state variable feedback can be performed. In the

case of rigid robot arms and ideal joints, this allows the full reduction of deviations

of the end effector position

(4.157)

from its desired motion.


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0 0.5 1 1.5 2 2.5 30.2

0.15

0.1

0.05

0

0.05

0.1

1

(t)[rad],

2

(t)[rad]

t [s]

1(t)

2(t)

Figure 4.34: Difference between simulated results (RelTol=AbsTol=1e-8) and ref-

erence data: generalised coordinates

,

The control torques from the feedback due to a PD-controller reads as

(4.158)

where the gains of state variable feedback control have to fulfil the stability condi-

tion that the poles must be in the left half-plane.

With a desired modal damping of

the gains are

(4.159)

Here

has been chosen arbitrarily as

leading to a stable system. For closer

details of design of controllers see [14].

The standard routineode45

with the default tolerances ( RelTol = 10e-3, AbsTol= 10e-6 ) is used to calculate the trajectory of the robot, and the deviations of the

angles

,

,

and

from their desired values were observed. The integration

delivered satisfactory results, the errors are bounded and smaller than the chosen

tolerance of the numerical integration process.

To show the stability of the entire system, the initial condition is now disturbed

by a

e

rad.

First, the case

is shown. It turns out that the initially given deviation can

not be compensated by the PD-controller. Its feedback gain are too weak to cancel

the destabilising forces, as illustrated in Fig. 4.39.


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1.5 2 2.5 35

4

3

2

1

0

1

2

3x 10

3

1

(t)[rad],

2

(t)[rad]

t [s]

1(t)

2(t)

Tol: 1010

Tol: 108

Figure 4.35: Comparison between simulation results obtained using integration rou-

tines with different tolerances, RelTol=AbsTol=1e-8, RelTol=AbsTol=1e-10. Dif-

ference between simulated results

,

and reference data are depicted

In the cases

and

, the simulations clearly show a decay of the

initially given deviations. The remaining error is bounded within the precision of

the integration routine. Depending on the parameters of the feedback control, the

decay behaviour is different as it is shown in Fig. 4.40, Fig. 4.41 and Fig. 4.42. The

position error (position of end-effector is expressed in worldcoordinates) is within

acceptable range.

On the basis of these numerical experiments, it could be seen that the feedback

controller law stabilises the solution. This happens due to the fact that additional

terms appear in the "stiffness matrix", leading to an asymptotic stable system.

From the mechanical point of view, such terms can be caused by- a fictive mounting of springs and dampers between the real links of the robot

and guiding device (not existing in reality) or

- installation of a controller system (consisting of sensors, control law, amplifiers

and actuators) to get a stable behaviour of the robot. It has to be mentioned that

for very high feedback gains the system becomes stiff and beyond a certain limit

instability may occur.

Concerning the numerical integration of differential equations, this controller

system can be considered as a kind of stabilisation procedure as mentioned in chap-


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0 0.5 1 1.5 2 2.5 30.005

0

0.005

0.01

0.015

0.02

0.025

xe

[m],ye

[m]

t [s]

xe(t)

ye(t)

Figure 4.36: Difference betweeen simulated results and referent values: end-

effector position coordinates

0 0.5 1 1.5 2 2.5 30.5

0

0.5

1

1.5

2

2.5x 10

4

xe

[m],ye

[m]

t [s]

xe(t)

ye

(t)

Figure 4.37: Difference betweeen simulated results and referent values: end-

effector position coordinates


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Feedforward

Controler

Robot

Figure 4.38: Closed loop structure of the 2-DOF robot

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5 x 10

4

1

[rad],

2

[rad]

t [s]

1(t)

2(t)

Figure 4.39: Unstable behaviour for feedback gain

ter 4.3.3. (This type of stabilisation was proposed e.g. by Baumgarte.)Note: Real technical systems have elastic links and joints and their deformation

may influence the position of the end effector considerably. If this position can

not be measured directly, an observer based on the equations of motion has to be

applied to estimate the elastic deformation and include it in the control.


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0 0.5 1 1.5 2 2.5 33

2

1

0

1

2

3

4

5

6

7

x 106

xe

[m],ye

[m]

t [s]

xe(t)

ye(t)

Figure 4.42: Stable behaviour for feedback gain

, deviations

and

of

end-effector


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Example 2.5: Dynamics of robot with mechanically constrained end-effector

x

y

Full descriptor form:

Minimal form:

Integration of ODE

Inversedynamics

constraintand appliedtorques and

forces Forwarddynamics

Figure 4.43: Modelling of 1 DOF robot

MBS with 1 DOF

Chosen generalised coordinate: absolute orientation angle of the first body or

angle of the relative orientation of the second body with respect to the first

one


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System constraint forces: constraint forces at the joints, constraint force at

the mechanically constrained end-effector

INVERSE DYNAMICS

Since robot (see Fig. 4.43) possesses 1 DOF, its motion can be controlled by

just one force-actuator. There is a choice between two cases:

a) Case I - torque acts at joint A (

),

b) Case II - torque acts at joint B (

).

However, actual magnitude of constraint force that acts on end-effector (as well

as other applied and constraint forces) depenf on that choice.

In the Case I the constraint force that acts on the end-effector is shown in Fig.

4.44. On the contrary, if the actuation is provided via

actuator (Case II), thetime-history of the end-effector constraint force is different as it is depicted in Fig.

4.45.

In copmpariison to the Case I, the constraint force is of much higher magnitude

and with changing orientation. By comparing Fig. 4.44 and Fig. 4.45, it is visible

that the actuation via is preferable, if the criteria of having small magnitude of

constraint force acting on end-effector force is of importance.

0 0.5 1 1.5 2 2.5 32.42

2.43

2.44

2.45

2.46

2.47

2.48

2.49

F

end

[N]

t [s]

Figure 4.44: Constraint force that acts on the end effector, Case I


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0 0.5 1 1.5 2 2.5 315

10

5

0

5

Fend

[N]

t [s]

Figure 4.45: Constraint force that acts on the end effector, Case II


36/69


FORWARD DYNAMICS

In Fig. 4.46 the result of time integration (difference between integrated resultsof end-effector

coordinate and reference values, based on the mathematical model

in minimal form (1 ODE), are shown for Case I. Integration routine ode45 with

imposed tolerances RelTol=1e-8 and AbsTol=1e-8 has been applied.

By dealing with Case II, the errors are considerably greater (Fig. 4.47). The

errors of such magnitude exceed the precision demanded in robotics.

Because of the inherent mechanical instability of the analysed system (see Ex-

ample 1.1 and Example 2.4), the integration results deviate of the accurate values.

This deviation is strongly dependent on the error tolerances that have been imposed

on the integration routine.

0 0.5 1 1.5 2 2.5 30.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

5

xe

(t)[m]

t [s]

Figure 4.46: Difference betweeen simulated results and referent values, Case I


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0 0.5 1 1.5 2 2.5 30.5

0

0.5

1

1.5

2

2.5 x 103

xe

(t)[m]

t [s]

Figure 4.47: Difference betweeen simulated results and referent values, Case II


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4.4.3 Dynamics of slider-crank mechanism

Kinematics of slider-crank mechanism has been analysed in Chapter 2 (Excersise

2.4.9). In Exampe 3.1 and Example 3.2 the results of dynamic analysis of slider-

crank mechanism are presented.

Example 3.1: Inverse dynamics of slider-crank mechanism

Inverse dynamics of slider-crank has been solved using mathematical model in full

descriptor form. In Fig. 4.48, time history of the torque that is needed to drive

mechanism with the crank constant angular velocity

is shown. Two

different cases are calculated and compared: a) mass of the piston,

kg b)

mass of the piston kg.The components of constraint forces in crank bearing and piston-rod bearing as

well as the transversal force of piston guidance are shown in Fig. 4.49 and Fig.

4.50.

0 0.5 1 1.5 2 2.5 320

15

10

5

0

5

10

15

20

t [s]

Mm[

Nm]

mp= 0.8 kg

mp= 4.0 kg

Figure 4.48: Driving torque of slider-crank mechanism: two cases with different

masses of the piston

Example 3.2: Forward dynamics of slider-crank mechanism

Forward dynamics of slider-crank mechanism has been performed by:


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0 0.2 0.4 0.6 0.8 1250

200

150

100

50

0

50

100

150

200

Fbx

[N],Fby

[N]

t [s]

Fbx

(t)

Fby

(t)

Figure 4.49: Components of constraint force in crank bearing

0 0.2 0.4 0.6 0.8 130

20

10

0

10

20

30

40

50

Ffx

[N],Ffy

[N],F

px[N

]

t [s]

Ffx

(t)

Ffy

(t)

Fpx

(t)

Figure 4.50: Components of constraint force in piston-rod bearing and piston guid-

ance

integration of governing equations formulated in minimal form: 1ODE


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using 1 generalised coordinate: crank angle

deriving equation of motion via transformation from the full descriptor form

Systems free-run has been studied (no driving torque is applied).

In Fig. 4.51 time history of the angular velocity of crank shaft is presented for

two cases: a) mass of the piston,

kg b) mass of the piston

kg.

A comparison of the piston displacement travel with an ideal cos time-function

is presented in Fig. 4.52. It becomes obvious that the piston does not follow an

hatmonic function due to the nonlinearity in crank-slider mechanism

0 0.5 1 1.5 2 2.5 35.8

5.85

5.9

5.95

6

6.05

6.1

6.15

6.2

6.25

6.3

t [s]

[rad/s]

mp= 0.8 kg

mp= 4.0 kg

Figure 4.51: Angular velocity of crank: casses with different masses of the piston

MATLAB program for forward dynamics of slider-crank mechanism in minimalform / main program reads as

% Integration of the equations of motion of the slider% crank model in minimal form

% integration interval

t0=0; tf=1;

% inital conditions, x(1)=positions, x(2)=velocities

x0=[0, 0]

% call of ode45 integrator

[t,x]=ode45(Eqslide,t0,tf,x0)

% description of the system: Eqslide.m

% End


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0 0.5 1 1.5 2 2.5 30.2

0.4

0.6

0.8

t [s]

yp

[m]

yp

ycos

Figure 4.52: Piston displacement travel: comparison with the cos time-function


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Example 8: Cable way

Typical questions concerning the dynamics of a mechanical system are discussed

on a cabin of a cable way.

The cabin of a cable way as shown in Fig. 4.53 is considered with its carriage

and suspension. The roller of the carriage by a spring (stiffness constant

). A

bearing (mass

, centre of mass

) is visco-elastically hinged (stiffness constant

, damping constant ) in the trolley. It carries the cabin itself (mass

, centre of

mass

, moment of inertia

).

Forward dynamics is carried out to get an insight into the dynamical behaviour

using time integration. Since large oscillations are expected the nonlinear equations

of motion were used for the integration in time domain. In this example the excita-

tion is assumed as a step-like switch on of the winding drum at

:

for

,

for .

g

Figure 4.53: Cable way

EQUATION OF MOTION

Obviously, the system has

degrees of freedom. An appropriate vector of


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generalised coordinates is

By writing Newtons equations for trolley, bearing and cabin and adding Eulers

equation for cabin the Newton-Euler equation reads as

By premultiplying with the transposed global Jacobian


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the equation of motion is received as

sym.

.

This premultiplication reduces the dimensions of the matrices and vectors to the

dimension , it garantees a symmetrical mass matrix and eliminates the generalised

constraint forces . This is due the orthogonality of the matrices and .

PARAMETERS OFCABLEWAY

All physical constants are kept in the seperate m-file values.m. Note that

is

denoted as w0 in all m-files.

% values.m

% Physical constants for cabin-problem:

% Make constants accessible in all modulesglobal m1 m2 m3 g alpha w0 R r cf cs d Dbeta I3 l y0

m1 = 50; % Mass of trolley [kg]

m2 = 10; % Mass of cabin bearing [kg]

m3 = 300; % Mass of cabin [kg]

% possibly + 200 kg for persons

I3 = m3*1; % Cabins moment of inertia [kg*m^2]

g = 9.81; % Gravity [m/(s^2)]

alpha = rad(35); % Inclination towards horizontal [rad]

w0 = rad(180); % Rotary speed of drive roll [rad/s]

R = 1; % Radius of drive roll [m]

cf = 8e4; % Spring stiffness for cabin bearing [N/m]

cs = 5e4; % Spring stiffness for towing rope [N/m]

d = 1e3; % Damper constant for bearing [N*s/m]

Dbeta = 800; % Rotational damper constant [N*m*s]

l = 2; % Distance from bearing to [m]

% cabins centre of gravity

r = 0.6; % Distance rope - damper [m]

% Initial condition

% y0=[s;ds;u;du;beta;dbeta]

y0=[0;0;0;0;0;0];


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% Define constants for field-names to increase readability

global fn_s fn_ds fn_u fn_du fn_beta fn_dbetafn_s = 1;

fn_ds = 2;

fn_u = 3;

fn_du = 4;

fn_beta = 5;

fn_dbeta = 6;

The incluence of parameter variation on the dynamical behaviour is shown on

the time history of characteristic coordinates, a linearisation of the equations of

motion and an eigenvalue analysis is not performed.

The two short auxiliary functions deg and rad are used to convert radian to

degree and vice versa. Each function is defined in a separate m-file.

% deg.m

% Convert rad into degree

function angle_deg=deg(angle_rad)

angle_deg=angle_rad.*180./pi;

% rad.m

% Convert degree into rad

function angle_rad=rad(angle_deg)

angle_rad=angle_deg.*pi./180;

The right hand side of the ODE system is generated in the Fcabin.m:

% Fcabin.m

% ODE-File for cabin-problem

% Defines the ODE-system

% y(1)=s

% y(2)=ds=dy(1)

% dds=dy(2)

% y(3)=u

% y(4)=du=dy(3)

% ddu=dy(4)

% y(5)=beta

% y(6)=dbeta=dy(5)

% ddbeta=dy(6)

function dy = Fcabin(t,y,command)

% get access to physical constants and field-names

global m1 m2 m3 g alpha w0 R cf cs d Dbeta I3 l


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y0(fn_s)=-(m1+m2+m3)*g*sin(alpha)/cs;

y0(fn_u)=m3*g*cos(alpha+y0(fn_beta))/cf;

% Solve the system

[T,Y]=ode45(Fcabin,time_range,y0,options);

% Calculate desired path of trolley

s_des = w0.*R.*T-(m1+m2+m3)*g*sin(alpha)/cs;

% Store data for plot in another matrix

solution(:,1)=Y(:,fn_s)-s_des;

solution(:,2)=Y(:,fn_u);

solution(:,3)=deg(Y(:,fn_beta)); % give beta in degree

% Prepare graphic window

figure(1)

clf

% Plot all solutions in a single window

subplot(3,1,1)

plot(T,solution(:,1));

title(Excursions versus time using default error tolerance)

% Define plot layout

axis tight % Scale axes

grid on % Switch grid on

xlabel(Time t [s]) % Label axesylabel(s(t) - s_d (t) [m])

subplot(3,1,2)

plot(T,solution(:,2));




xlabel(Time t [s]) % Label axes

ylabel(u(t) [m])

subplot(3,1,3)plot(T,solution(:,3));





ylabel(\beta(t) [\circ])

This program calculates a solution by using default tolerances for the integrator

and shows the results in a single window. The first graph shows the difference


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0 1 2 3 4 5 6 7 8

0.15

0.1

0.05

0

0.05

0.1

Excursions versus time using default error tolerance

Time t [s]

s(t)sd

(t)[m]

0 1 2 3 4 5 6 7 8

0.02

0

0.02

0.04

0.06

0.08

0.1

Time t [s]

u(t)[m]

0 1 2 3 4 5 6 7 8

15

10

5

0

5

10

15

20

25

Time t [s]

(t)[]

Figure 4.54: Solution of ODE by use ofode45

between the actual location of the cable ways carriage and the desired location

. The desired location can be expressed as

(4.160)

The second graph shows the excursion of the suspension relative to the

carriage. The third graph depicts the swing motion of the cabin relative to its

suspension. As can be seen, the solution is already stable when using default error

tolerance. To check the reliability of the solution, stricter error tolerances of 10e-6

for relative and 10e-8 for absolute error have been chosen.For doing this, only very few lines have to be modified in cabin.m:

% cabin_H.m


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...

% Define options used for integration

% relative error tolerance

options = odeset(reltol,1e-6);

% absolute error tolerance

options = odeset(options,abstol,1e-8);

...

The result obtained from the modified m-file looks very similar to the pre-

vious one. To show the actual difference between the two solutions, the m-file

cabin_lh.m can be used:

% cabin_lh.m

% Initialisation:

clear;

values; % read constants

% Define time interval of integration,

% use steps of 0.01 seconds for output.

time_range = 0:0.01:8;

% Define options used for integration

% relative error tolerance (default)


% absolute error tolerance (default)options = odeset(options,abstol,1e-6);

% Calculate the initial static equilibrium



% Solve the system


% Switch to high accurancy

% relative error tolerance


% absolute error tolerance

options = odeset(options,abstol,1e-8);

% Solve the system using new error tolerances

[T,Y_H]=ode45(Fcabin,time_range,y0,options);

% Calculate difference between the two solutions

solution(:,1)=Y(:,fn_s)-Y_H(:,fn_s);

solution(:,2)=Y(:,fn_u)-Y_H(:,fn_u);

% Convert beta to degree

solution(:,3)=deg(Y(:,fn_beta))-deg(Y_H(:,fn_beta));


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...

% Print maximum differences as numbers

max(solution)

0 1 2 3 4 5 6 7 8

1

0.5

0

0.5

1

x 104 Differences between default and small error tolerance

Time t [s]

s(t)sd

(t)[m]

0 1 2 3 4 5 6 7 8

8

6

4

2

0

2

4

6

8

x 105

Time t [s]

u(t)[m]

0 1 2 3 4 5 6 7 8

5

4

3

2

1

0

1

2

x 103

Time t [s]

(t)[]

Figure 4.55: Comparison of solution with default and high accuracy

Note: Here it is mandatory to use a fixed width of the time steps. Otherwise

the output matrices of the different solvers could have different dimensions and

therefore can not be subtracted from each other. The program prints out the maximal

differences of the two solutions. These are less than 2e-4 meters for

and

,

and less than 3e-3 for

.

Now the integrator ode45 is compared to ode15s. Therefore the m-file has to

be changed slightly:


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% cabin_45_15s.m

...

% Solve the system using ode45


% Solve the system using ode15s

[T,Y_H]=ode15s(Fcabin,time_range,y0,options);

% Calculate difference between the two solutions

solution(:,1)=Y(:,fn_s)-Y_H(:,fn_s);

solution(:,2)=Y(:,fn_u)-Y_H(:,fn_u);

% Convert beta to degree

solution(:,3)=deg(Y(:,fn_beta))-deg(Y_H(:,fn_beta));

...

Again, the comparison shows that the absolute difference is acceptable from the

engineering point of view. Therefore, ode45with default error tolerance is used for

further calulations.

PARAMETER VARIATION

Now it is investigated how variations of several parameters influence the be-

haviour of the cable way. To present the data in graphs, the function plotcabin

is used. This function controls different line types, 2D and 3D-plots, automatic la-

beling and some other things. Control arguments are Y for the solution including

the time vectors, and Param for the varied parameter. The other arguments are for

describing the plots and are explained in the file.

% plotcabin.m

% Plot three graphs containing 3d-lines

function plotcabin(Y,Param,ParamUnit,ParamName,TitleSuff, AZ,EL)

% input arguments:

% Y 3-dim. matrix containing solutions of several

% diff. equations:

% Each 2D-matrix contains solutions in column-

% vectors, each column contains values for a single

% parameter value.

% y{i}(:,1) Deviation from desired position for i-th

% parameter value

% y{i}(:,2) Excursion of bearing for i-th parameter value

% y{i}(:,3) Angle beta (in DEGREE!) for i-th parameter value

% Y{i}(:,4) Time vector for i-th parameter value

%

% Param Matrix containing values of varied parameter

% ParamUnit Name of measure unit for Parameter

% ParamName Name of varied parameter


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0 1 2 3 4 5 6 7 8

5

0

5

10

x 104 Differences between ode45 and ode15s

Time t [s]

s(t)sd

(t)[m]

0 1 2 3 4 5 6 7 8

6

4

2

0

2

4

6

8

x 104

Time t [s]

u(t)[m]

0 1 2 3 4 5 6 7 8

0.02

0.01

0

0.01

0.02

Time t [s]

(t

)[]

Figure 4.56: Comparison of solution with ode45and ode15s

%

% TitleSuff Suffix for title

%

% AZ Azimuth angle for viewing, default: 40 degree

% EL Elevation angle for viewing, default: 50 degree

%Apply default values if necessary if nargin < 7, EL=50; end;

if nargin < 6, AZ=40; end;

if nargin < 5, TitleSuff = ; end;

if nargin < 4, ParamName =Parameter; end;

if nargin < 3, ParamUnit =; end;

% Deside wether to use different line-styles or not


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if (length(Param) < 5) % less than 5 lines ?

UseDiffStyles = 1;else

UseDiffStyles = 0;

end;

% Prepare graphic window

figure(1)

clf

% Plot all results in a single window

subplot(3,1,1)

title([Excursions versus time TitleSuff])

DoPlot(Y,1,Param,s(t) - s_d (t) [m],ParamName,ParamUnit,...

AZ,EL,UseDiffStyles)

subplot(3,1,2)

DoPlot(Y, 2,Param, u(t) [m], ParamName,ParamUnit, AZ,EL,...

UseDiffStyles)

subplot(3,1,3)

DoPlot(Y,3,Param,\beta(t)[\circ],ParamName,ParamUnit,...


if UseDiffStyles == 1 % Show legend if needed

subplot(3,1,1);

for n = 1:length(Param)

legendtext{n}= sprintf(%s = %.2e %s,ParamName,Param(n),...

ParamUnit);

end;legend(legendtext,1); % Place legend in upper right-hand corner

end;

function DoPlot(Y,Sel,Param,V_Name,ParamName,ParamUnit,...


% Sub-function used for drawing actual plot

% Y Solution array

% Sel Selector for data to plot

% Param Values of parameter

% V_Name Name of plotted value

% ParamName Name of parameter

% ParamUnit Name of measure-unit% AZ, EZ Azimuth and elevation angles

% UseDiffStyle Set to 1 to use different line-styles

% Define array containing different line-styles:

% [Parameter for plot3d-function, line-width]

linestyles = {r-,g-,b:,m:};

linewidths = { 0.5, 1, 0.7, 1, 1};

% Rotate coordinate system

view(AZ,EL)


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% Keep existing graphicshold on

% Plot one curve for each parameter value

for n = 1:length(Param)

if UseDiffStyles == 1

linestyle = linestyles{n}; % use different line-styles

linewidth = linewidths{n};

else

linestyle = b-; % use same line-style if more than 4 lines

linewidth = 0.5;

end;

% Plot line in space

% Y{n}(:,4) contains always the time vector

L=plot3(Y{n}(:,4),ones(1,length(Y{n}(:,4))).*Param(n),...

Y{n}(:,Sel),linestyle); set(L,LineWidth,linewidth);

Style=get(L);

if AZ ~= 0

% If azimuth angle is different from zero, draw pointer at the

% end of integration interval

% Find global minimum of plot

for i = 1:length(Y)

mins(i) = min(Y{i}(:,Sel));end;

last=length(Y{n}(:,4));

L=line([Y{n}(last,4), Y{n}(last,4)], [Param(n), Param(n)],...

[Y{n}(last,Sel) , min(mins)]);

set(L,Color,Style.Color);

set(L,LineStyle,--);

set(L,LineWidth,Style.LineWidth);

end

end

hold off





if isempty(ParamUnit)

ylabel(ParamName);

else

% Concatenate strings to form description

ylabel(strcat(ParamName, [, ParamUnit, ]));


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end;

zlabel(V_Name)

With this powerful function the main programs become much shorter. To get

smooth plots, the step size of the output is automatically controlled by the ODE-

solver. Then the time vectors differ between different solutions. In order to store

the results in a single variable despite of this, a cell-array is used, and the time vec-

tor is stored together with each solution. Then the variable containing the results is

then passed to the function plotcabin.

Mass of cabin

Obviously, in practice

may change widely. This refers to the load of the

cabin. The output showing the influence of the mass of the cabin is generated bythe m-file cabin_m3.m:

% cabin_m3.m

% Vary mass of cabin m3:

% Initialisation:

clear;

values; % Read constants

% Define time interval of integration

time_range = [0, 6];

% Define range and step for m3

range = 300:150:600;

for n = 1:length(range)

% Set parameter for current loop

m3 = range(n);

% Calculate initial spring excursions



% Clear previous dimensions of T and Y

clear T Y

% Solve the system

[T,Y(:,:)]=ode45(Fcabin,time_range,y0);

% Calculate desired position for trolley


% Store data that should be plotted in a cell-array

solution{n}(:,1)=Y(:,fn_s)-s_des;


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solution{n}(:,2)=Y(:,fn_u);

solution{n}(:,3)=deg(Y(:,fn_beta)); % give beta in degreesolution{n}(:,4)=T; % Store time-points for each solution

end

% Draw plots, set azimuth and elevation angle to zero

% to draw 2d-plot

plotcabin(solution,range,kg,m3,, variation of m3,0,0);

0 1 2 3 4 5 6

0.15

0.1

0.05

0

0.05

0.1

Time t [s]

Excursions versus time, variation of m3

s(t)sd

(t)[m]

m3 = 3.00e+02 kgm3 = 4.50e+02 kgm3 = 6.00e+02 kg

0 1 2 3 4 5 6

0

0.05

0.1

0.15

Time t [s]

u(t)[m]

0 1 2 3 4 5 6

20

10

0

10

20

Time t [s]

(t)[]

Figure 4.57: Variation of mass of cabin

An increasing mass

leads to increased amplitudes and larger periods in trol-

ley oscillation. Suspension excursions are significatly larger bringing the damp.

The swing motion of the cabin shows increased amplitudes whereas the time period


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is not affected.

Stiffness of towing rope

% cabin_cs.m

% Vary stiffness cs:

...

% Define range and step for cs

range = 1e4:4.5e4:10e4;


% Set parameter for current loopcs = range(n);

...

plotcabin(solution,range,N/m,c_s,, variation of c_s,0,0);

The trolley carries out large and slow motion for low stiffness of the towing

rope. The suspension shows only a small amplitude resulting in weak damping.

This is seen also in the large, weakly damped swing of the cabin.

Stiffness of suspension

% cabin_cf.m

% Vary stiffness cf:

...

% Define range for cf

range = 1e4:2e4:5e4; % starting value



cf = range(n);

...plotcabin(solution,range,N/m,c_f,, variation of c_f,0,0);

A stiffer suspension leads to a larger exxcursion of the trolley. The amplitudes

of the bearing are significantly reduced and the time periods become shorter. Nearly

no influence on the swing motion is found.

Damping constant of suspension

To find an optimal damping is an important task in optimisation. The variation

of the damping constant shall give a rough information about the sensivity.


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0 1 2 3 4 5 6

0.15

0.1

0.05

0

0.05

Time t [s]

Excursions versus time, variation of cf

s(t)sd

(t)[m]

cf= 1.00e+04 N/mc

f= 3.00e+04 N/m

cf= 5.00e+04 N/m

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

Time t [s]

u(t)[m]

0 1 2 3 4 5 6

15

10

5

0

5

10

15

20

25

Time t [s]

(t)

[]

Figure 4.59: Variation of stiffness of suspension

plotcabin(solution,range,N*s/m,d,, variation of d, 0,0)

A direct influence can be seen on the motion of the bearing. Also the trolley is

considerably damped. The high frequency vibrations in the swing motion are de-

caying.

Rotational damper constant of cabin

All the previous alterations of design parameters influence the slow swing mo-

tion of the cabin only very weakly. Therefore, the constant of the damper between

the suspension and the cabin is increased now.

The two cases and are shown:


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0 1 2 3 4 5 6

0.15

0.1

0.05

0

0.05

Time t [s]

Excursions versus time, variation of cf

s(t)sd

(t)[m]

cf= 1.00e+04 N/mc

f= 3.00e+04 N/m

cf= 5.00e+04 N/m

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

Time t [s]

u(t)[m]

0 1 2 3 4 5 6

15

10

5

0

5

10

15

20

25

Time t [s]

(t)

[]

Figure 4.60: Variation of stiffness of suspension

% cabin_dbeta1.m

% Vary damper constant dbeta for alpha = 20 degree:

% Initialisation:

clear;


% Set alpha to 20 degree and convert to rad

alpha = rad(20);

% Define time interval of integration


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0 1 2 3 4 5 6

0.15

0.1

0.05

0

0.05

0.1

Time t [s]

Excursions versus time, variation of d

s(t)sd

(t)[m]

d = 5.00e+02 N*s/md = 1.50e+03 N*s/md = 2.50e+03 N*s/m

0 1 2 3 4 5 6

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Time t [s]

u(t)[m]

0 1 2 3 4 5 6

15

10

5

05

10

15

20

25

Time t [s]

(t)[]

Figure 4.61: Variation of damp constant of suspension

time_range = [0, 6];

% Define range and step for dbeta

range = 8e2:15e2:38e2;



Dbeta = range(n);

...

plotcabin(solution,range,N*m*s,d_{\beta},...

, variation of d_{\beta} for \alpha = 20 \circ,0,0)


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0 1 2 3 4 5 6

0.15

0.1

0.05

0

0.05

Time t [s]

Excursions versus time, variation of dfor = 20

s(t)sd

(t)[m]

d= 8.00e+02 N*m*sd= 2.30e+03 N*m*s

d= 3.80e+03 N*m*s

0 1 2 3 4 5 6

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

Time t [s]

u(t)[m]

0 1 2 3 4 5 6

15

10

5

0

5

10

15

20

25

Time t [s]

(t)

[]

Figure 4.62: Variation of rotational damper constant,

An effective decay of the swing motion is obtained. It can be seen that increasing

the damping coefficient of the suspension is strongly influencing the motion of thetrolley as well as of the suspension.

Previously the elevation angle of the guiding rope was . Now the eleva-

tion angle is considered to be degree. For this, replace with :

% cabin_dbeta2.m

% Vary damper constant dbeta for alpha = 70 degree:

% Initialisation:

clear;


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% Set alpha to 70 degree and convert to rad

alpha = rad(70);

...

0 1 2 3 4 5 6

0.2

0.1

0

0.1

0.2

Time t [s]

Excursions versus time, variation of dfor = 70

s(t)sd

(t)[m]

d= 8.00e+02 N*m*s

d= 2.30e+03 N*m*s

d= 3.80e+03 N*m*s

0 1 2 3 4 5 6

0.02

0

0.02

0.04

0.06

Time t [s]

u(t)[m]

0 1 2 3 4 5 66

4

2

0

2

4

6

8

10

Time t [s]

(t)[]

Figure 4.63: Variation of rotational damper constant,

Here again the effective reduction of the swing amplitudes of the cabin can be

seen. Because of the high inclination angle, the coupling to the trolley and the sus-

pension oscillations is weak, there is nearly no influence on the swing damper.

Elevation of the guiding rope


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The influence of the slope of the guiding rope is considered.

% cabin_alpha.m

% Vary alpha:

...

% Define range and step for alpha and convert into rad

range = rad(20:25:70);



alpha = range(n);

...

% Convert parameter alpha back to degree

plotcabin(solution,deg(range),\circ,alpha,...

, variation of alpha,0,0)

For a steep slope nearly the whole mass is hanging at the towing rope, both

dampers are only weakly articulated which results in weakly damped motions.

DIFFERENT SOLVERS FOR DIFFERENT SYSTEM CONFIGURATIONS

Here a variation of the angle of the guiding rope is carried out to show thedependence of the sensivity on numerical errors. The two integration routines ode45

and ode15swith default tolerances are used.

% cabin_alpha_dif.m

% Vary alpha and compare different integration methods:

% Initialisation:

clear;


% Define time interval of integration. Here it is mandatory to

% use a fixed step-width, otherwise later the two solutions

% cannot be compared

time_range = 0:0.01:6;

% Define range and step-width for alpha and convert into rad

range = rad(20:25:70);

% First, find a solution with ode45



alpha = range(n);


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0 1 2 3 4 5 6

0.2

0.1

0

0.1

0.2

Time t [s]

Excursions versus time, variation of alpha

s(t)sd

(t)[m]

alpha = 2.00e+01 alpha = 4.50e+01

alpha = 7.00e+01

0 1 2 3 4 5 6

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

Time t [s]

u(t)[m]

0 1 2 3 4 5 6

15

10

5

05

10

15

20

25

Time t [s]

(t)[]

Figure 4.64: Variation of the elevation of the guiding rope





clear T Y

% Solve the system using default error tolerance

[T,Y(:,:)]=ode45(Fcabin,time_range,y0);




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% Store data that should be plotted in a cell-arraysolution1{n}(:,1)=Y(:,fn_s)-s_des;

solution1{n}(:,2)=Y(:,fn_u);

solution1{n}(:,3)=deg(Y(:,fn_beta)); % give beta in degree

solution1{n}(:,4)=T; % Store time-points for each solution

end

% Then integrate by using ode15s



alpha = range(n);





clear T Y

% Solve the system using default error tolerance

[T,Y(:,:)]=ode15s(Fcabin,time_range,y0);



% Store data that should be plotted in a cell-array

solution2{n}(:,1)=Y(:,fn_s)-s_des;

solution2{n}(:,2)=Y(:,fn_u);

solution2{n}(:,3)=deg(Y(:,fn_beta)); % give beta in degree

solution2{n}(:,4)=T; % Store time-points for each solution

end

% Calculate difference between the two solutions.

% Time-vector must stay untouched!

for i = 1:length(range)

diff{i}(:,1:3) = solution1{i}(:,1:3)-solution2{i}(:,1:3);% Time-vector must stay untouched!

diff{i}(:,4) = solution1{i}(:,4);

end;

% Convert parameter alpha back to degree

plotcabin(diff,deg(range),\circ,alpha,...

using different integration methods (plotted: difference!),...

35,50)

In Fig. 4.65 the difference between the obtained solutions is plotted. Obviously,

the error is depending on the system configuration. The errors are maximal in the


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BIBLIOGRAPHY 298

01

23

45

6 20

30

40

50

60

702

0

2

x 103

alpha []

Excursions versus time using different integration methods (plotted: difference!)

Time t [s]

s(t)sd

(t)[m]

alpha = 2.00e+01

alpha = 4.50e+01

alpha = 7.00e+01

01

23

45

6 20

30

40

50

60

705

0

5

x 104

alpha []

Time t [s]

u(t)[m]

01

23

45

6 20

30

40

50

60700.04

0.02

0

0.02

alpha []

Time t [s]

(t)[]

Figure 4.65: Comparison ofode45and ode15sfor different configurations

case of a steep slope. Of course, for a steep angle large trolley amplitudes occur

whereas only a small swing amplitude is induced. On the other hand, the damping

of the motion of the trolley is very low.

Bibliography

[1] Klaus-Jrgen Bathe. Finite-Elemente-Methoden. 1990. (in German).

[2] D. Bestle. Analyse und Optimierung von Mehrkrper-systemen. 1994. (in

German).


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BIBLIOGRAPHY 299

[3] A. F. DSouza and V. K. Garg. Advanced Dynamics, Modeling and Analy-

sis. Prentice-Hall International Editions, Englewood Cliffs, New Jersey, USA,1984.

[4] W. Schiehlen (ed.). Multibody Systems Handbook. Springer-Verlag, Berlin,

Heidelberg, Germany, 1990.

[5] W. Schiehlen (ed.).Advanced Multibody System Dynamics. Kluwer Academic

Publishers, Dordrecht, The Netherlands, 1993.

[6] R. Fletcher. Practical Methods of Optimization. John Wiley & Sons, Chich-

ester, UK, 1987.

[7] N. Gershenfeld. The Nature of Mathematical Modelling. University Press,

Cambridge, 1999.

[8] E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equa-

tions I. Springer-Verlag, Berlin, Germany, 1987.

[9] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II.

Springer-Verlag, Berlin, Germany, 1991.

[10] J. L. Junkins and Youdan Kim.Introduction to Dynamics and Control of Flex-

ible Structures. AIAA, American Institute of Aeronautics and Astronautics,

Inc., Reston, VA, USA, 1993.

[11] K. Kreuzer and W. Schiehlen. NEWEUL. 1988. (in German).

[12] K. Magnus and H. H. Mller. Grundlagen der Technischen Mechanik. 1974.

(in German).

[13] P. C. Mller and W. O. Schiehlen. Linear Vibrations. Martinus Nijhoff Pub-

lishers, Dordrecht, The Netherlands, 1985.

[14] Nattick. Control Toolbox Users Guide. The MathWorks, 1996.

[15] D. E. Newland. Mechanical Vibration Analysis and Computation. Longman

Scientific and Technical, Essex, England, 1989.

[16] P. E. Nikravesh. Computer-Aided Analysis of Mechanical Systems. Prentice-

Hall International Editions, Englewood Cliffs, New Jersey, USA, 1988.

[17] J. C. Polking.MATLAB Manual for Ordinary Differential Equations. Prentice-

Hall, Englewood Cliffs, NJ, USA, 1995.

[18] W. Schiehlen. Technische Dynamik. 1986. (in German).


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BIBLIOGRAPHY 300

[19] A. A. Shabana. Dynamics of Multibody Systems. John Wiley & Sons, New

York, USA, 1989.

[20] Irving H. Shames. Engineering Mechanics. Statics and Dynamics. Prentice

Hall, Upper Saddle River, New Jersey, USA, 1996.

[21] V. Stejskal and M. Valek. Kinematics and Dynamics of Machinery. Marcel

Dekker, Inc., New York, USA, 1996.

[22] Z. Terze, D. Lefeber, and O. Muftic. Null Space Integration Method for Con-

strained Multibody Systems with No Constraint Viola tion, in: Multibody Sys-

tem Dynamics. 2001. (in print).

Dynamics Part Two

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Transcript of Dynamics Part Two