Electromagnetic forces in elastic multibody systems
M. Henger1 , R. Schroth2
Robert Bosch GmbH, Electric Vehicle and Hybrid Technology,
Postfach 30 02 40, 70442 Stuttgart, Germany
e-mail: martin.henger@de.bosch.com
1
2
Robert Bosch GmbH, Development Generators,
Postfach 30 02 40, 70442 Stuttgart, Germany
Abstract
This research shows a method to consider electromagnetic forces within the flexible multibody simulation
of a permanent magnet synchronous machine. The force distribution is calculated by FE-Method and transferred into analytical form by description as fourier series. This analytical form is passed towards the multibody system as distributed force. There, the elasticity of substructures is described in terms of the component
mode synthesis. As a result, the electromagnetic force is transformed into modal space and applied to the
flexible structure as modal force.
For significant vibrational frequencies the method is verified by comparison with experimental results.
1 Introduction
Electric machines have gained an important role in modern vehicle technology. Not only due to their high
efficiency but also due to their good performance in lower rotational speed range. Along with new possibilities related to emissions reduction or driving potentials in hybrid or electric vehicle topologies, electric
machines induce novel vibrational behavior into the powertrain, which is mainly caused by electromagnetic
forces.
To simulate the dynamic behavior of complex structures, multibody systems (MBS) are state of the art. The
deformation of elastic bodies is described by means of reduction algorithms, whereas the component mode
synthesis is a widely used approach in commercial MBS tools. However, this reduction algorithm limits the
capability to consider distributed forces.
This paper introduces an approach to implement electromagnetic forces into an elastic multibody simulation
(eMBS) of a permanent magnet synchronous machine by use of modal force formulation. In section 2 the
theory of the electromagnetic force description and the use of modal forces is presented. In 3 the method is
applied onto the synchronous machine. Results are discussed in comparison to experimental results in 4 and
conclusions drawn in 5.
2 Theory
Electric machines produce torque by interaction of stator and rotor fields. They can either be generated by
permanent magnets or electric excitation [1]. In the examined machine the stator field is excited by current,
the rotor field by permanent magnets. These fields induce flux density waves in the airgap of the machine.
There electromagnetic forces are generated, whereas they can be divided in a tangential component ftan that
produces torque and a radial component frad that excites housing vibration.
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According to Jordan [2], the higher amount of those forces does not contribute to torque generation.
frad > ftan
(1)
Regarding dynamic problems, e.g. to solve acoustic thematics, the radial forces are commonly applied to
mechanical FE models and solved in frequency domain [3, 4].
In this paper an approach is investigated, which enables the application of radial forces on elastic multibody
systems and provides the ability of transient simulation. This offers advantages regarding stress and/or
fatigue analysis.
2.1 Electromagnetic force distribution
The distribution of radial force is calculated according to FE Methods, where radial force density is calculated
based on Maxwell’s Stress Tensor according to [5] in the airgap and projected onto the stator teeth’s surface.
Figure 1 shows their distribution along the stator teeth, described by the circumferential angle γ, for a defined
relative angle between stator and rotor α
α = α0 + ωt ,
(2)
where α0 is the initial angle and ωt describes the rotation over time t.
Radial Force F/Fmax
1
Fourier Series
Original
0
0
10
20
30
γ in Deg
40
50
60
Figure 1: Radial force along stator surface. Comparison of original data, generated by FE calculation with
analytical description.
The curve progress in 1 clearly points out the periodicity of the force distribution within every 30◦ . Due to
the machines rotational symmetry, time dependency is periodic as well.
As a result, frad can be modeled as a two dimensional fourier series depending on time kα and space nr γ by
a limited number of coefficients A(r, k). The coefficients are subsequently determined by a discrete fourier
transform (DFT) according to van der Giet et al. [6] for a given number of samples N, M .
frad (γ, α) =
N
−1 M
−1
X
X
A(r, k) cos(nr γ + kα + ϕ(r, k))
(3)
r=1 k=1
nr = r −
N −1
2
(4)
Hereby frequencies are considered to be positive but waves propagate rotating and counter rotating
−
N −1
N −1
< nr <
.
2
2
(5)
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To validate the approach, a second curve is illustrated in figure 1. It shows the force distribution, calculated
by those fourier coefficients of 3 which exceed a specific threshold. As both graphs almost coincide, the
accuracy is considered as sufficient.
2.2 Component mode synthesis and modal force formulation
In flexible multibody systems the component mode synthesis (CMS) is a widely spread reduction algorithm
in order to describe deformational behavior efficiently [7, 8]. It is based on a separation of the state vector x
into master (index m)- and slave (index s) nodes or degrees of freedom (DOF) respectively as described by
Craig and Bampton [9].
x
x= m
xs
(6)
The transformation of the governing equation of structural dynamics for conservative systems with the transformation matrix T
TT MTq̈ + TT KTq = TT F
(7)
leads to reduced mass- and stiffness matrices, with index µ representing modal space.
Mµ = TT MT
(8)
T
Kµ = T KT
(9)
The transformation matrix T is build of static Tstat and dynamic Tdyn deformation modes.
T=
I
Tstat
0
Tdyn
(10)
With (10) the force transformation on the right side of (7) leads to
T
T
Fm
I Fm
=
.
Fs
Fµ,s
(11)
One can see that even in modal space forces on master nodes remain unchanged and can be implemented
directly. Forces on slave nodes instead, are transformed. As the force vector Fµ,s consists of coefficients
describing a linear combination of force shapes, which are analog to deformational mode shapes, this force
is referred to as modal force.
3 Application
The application of electromagnetic forces onto flexible multibody systems by the above described theory
consists of three basic steps, which will be described in this chapter. The first two steps, which are done
in the preface of the simulation are based on FE techniques. The last is implemented in the multibody
simulation tool MSC.ADAMS.
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Electromagnetic force distribution is calculated by FE-Methods with Ansys Multiphysics. In a second
step it is formed as by analytical equations as described in chapter 2.1. The coefficient matrices are
passed to the multibody system. The resulting force upon one node on the stator surface (γ =const.)
is drawn over time and frequency domain in figure 2, whereas Nrot describes the relation between
vibrational fvib and rotational frequency frot
Radial Force
Frad (t)/Frad,max
fvib = Nrot · frot
.
(12)
1
0
0
0.02
0.04
0.06
0.08
0.1
Radial Force
F̂rad (f )/F̂rad,max
Time in s
1
0
0
50
100
Order Nrot of rotational frequency
150
Figure 2: Electromagnetic force acting upon one node on stator teeth over time and rotational order Nrot .
Flexible body modeling of stator and housing is done according to the CMS algorithm. Considering equations (8) - (10) the reduced system matrices have to be calculated by multiplication with the transformation matrix T. This consists of deformational modeshapes, which are generated by a numeric
modal analysis.
Implementation of force distribution is achieved in 4 basic steps, shown in figure 3. At first the current
rotation angle α is calculated and passed towards a C++ based subroutine. There the analytical equations of section 2.1 are implemented, to calculate the force distribution. As those forces are of nodal
shape they cannot be applied to the system directly. Therefore a subsequent calculation of modal forces
is also part of the subroutine. After the force is applied, an integration algorithm, which is provided
by MSC.ADAMS, solves the systems equations.
Determine rotation
angle α
6
Solving MBS equations Compute electromagnetic
force distribution
(Section 2.1)
?
Modal force
implementation
(Section 2.2)
Figure 3: Solution scheme implemented to apply electromagnetic forces on stator surface. Steps in ovals are
calculated inside the MBS Tool, rectangular boxes represent solution steps, based on subroutines.
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4 Results
In this chapter simulated results are verified by comparison with experimental data, which are measured
on a testbench. The machine is decelerated by a load machine (LM). Computational data is generated in
MSC.ADAMS according to section 3.
1
Simulation
Amplitude of Acceleration Â(f )/Âmax
Experiment
0
0
50
100
150
Order Nrot of rotational frequency
Figure 4: Comparison of electromagnetic driven surface vibrations for simulated and measured results at
maximum torque. Results are displayed by order number Nrot according to (12).
Figure 4 shows a comparison of the oscillation at the surface of the housing.
On first sight, the comparison of both curves shows a good consistency for those orders, which emerge clearly
in experimental results. Regarding higher orders, Nrot > 80, simulated results show high amplitudes, that
cannot be seen in experimental data. Moreover a comparison with figure 2 shows that the reason for orders
to emerge strong, especially for Nrot < 80, are not caused by the high excitation but in correlation with
mechanical properties of stator and housing.
A closer look on figure 4 shows:
• Order 48, 60, 84: Good consistency, amplitudes differ below 15 %.
• Order 12, 36, 72: Deviations increase, amplitudes differ up to 50 %. Even though amplitude relation
of the significant orders 36 and 72 are below 10 %.
Âexp (36)/Âexp (72)
Âsim (36)/Âsim (72)
< 110%
(13)
• Order 24 and 108 exist in simulated and experimental results, even though the amplitude shows high
deviations.
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PSfrag
• Order 96, 120, 132, 144 exist in simulated, but not in experimental results.
• Between order 12 and 60 the experiment shows a large number of orders. Except those, discussed
above, these are not seen inside the simulation.
The above described deviations between experimental and computational results are a result of 2 major
causes:
1. The housing contains a water jacket in order to dissipate generated heat. The fluid flow influences
the dynamic by several damping mechanisms. The simulation model does not represent those. As the
effects are not described, only one constant damping factor is applied.
Amplitude of Acceleration
Â(f )/Âmax
2. The load machine induces vibrational orders too. Even though the connection between both is equipped
with damping elements, distorting vibrations are transferred. Figure 5 clearly illustrates the influence
of the load machine at no-load operation for 1000 rpm. Besides additional amplitudes in lower order
range Nrot < 80 the amplitude of existing orders is changed. Even though, one has to consider that
the mean value for Amplitudes in figure 5 approximates 5-10% of the mean value in figure 4.
1
Decoupled LM
Coupled LM
0
0
50
100
Order Nrot of rotational frequency
150
Figure 5: Influence of the load machine on measured vibrations at no-load operation with 1000 rpm for
coupled/decoupled load machine (LM).
5 Conclusions
A method to describe electromagnetic forces analytically and implement those into elastic multibody systems efficiently was proposed. By comparison with experimental results a good verification for significant
orders is achieved. Due to the rough description of damping and the influences of the decelerating machine,
deviations arise for higher orders. From the results seen in this paper several conclusions can be drawn:
• An analytical description of electromagnetic force distribution by fourier series can be considered as
accurate.
• The magnitude for the appearance of different orders is dependent on the correlation between excitation and
• The use of modal force formulation is a proper method to implement electromagnetic forces.
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References
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[3] J. Gieras, J. Lai, C. Wang. Noise of polyphase electric motors. CRC press (2005).
[4] C. Schlensok, B. Schmülling, M. van der Giet, K. Hameyer. Electromagnetically excited audible noise
evaluation and optimization of electrical machines by numerical simulation. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 26, pp.
727–742 (2007).
[5] H. Seinsch. Oberfelderscheinungen in Drehfeldmaschinen. Teubner Stuttgart (1992).
[6] M. van der Giet, R. Rothe, K. Hameyer. Asymptotic Fourier decomposition of tooth forces in terms
of convolved air gap field harmonics for noise diagnosis of electrical machines. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 28
(2009).
[7] U. Sellgren. Component Mode Synthesis–A method for efficient dynamic simulation of complex technical
systems. VISP WP5 Document Data, Stockholm (2003).
[8] P. Koutsovasilis, M. Beitelschmidt. Comparison of model reduction techniques for large mechanical
systems. Multibody System Dynamics, Vol. 20, No. 2, pp. 111–128 (2008).
[9] R. Craig, M. Bampton. Coupling of substructures for dynamic analysis. AIAA journal, Vol. 6, No. 7,
pp. 1313–1319 (1968).
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