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Fem Fundamentals and Chassis Design, spring 2022 course

Lecture Notes

lecture notes, as of 2021-09-07 (big endian).

:!: Errata corrige. :!:

Exam formulary.

Planar beam structure i.p. and o.o.p. loadings

The beam structure centroidal axis lies on a plane, which is also a symmetry plane for the cross-sections.

Symmetric and skew-symmetric loads with respect to such a plane are called in-plane and out-of-plane loads, respectively.

If the superposition of effects holds (e.g., if the structure behaves linearly) each load set only induces an associated subset of the possible components of internal action, see

Symmetric and skew-symm. parts of a general load

(ipe source)

A general load is applied to a symmetric structure in a); in b), c) the loads applied on each half structure is treated separately. In d), e) the action on the loaded portion is halved, and symmetrically propagated to the other portion; those symmetric actions are accumulated in the symmetric part of the load f). In g), h) the action on the loaded portion is halved, and skew-symmetrically propagated on the unloaded portion; those skew-symmetric actions are accumulated in the skew-symmetric part of the load i).

Rollbar-like frame

Quarter ladder frame chassis

OTW profile in torsion

A Phoney monocoque chassis


Suspension link trusses

Solid circular beam sections, ø12mm, aluminum. Essentially rigid with respect to other chassis structures.

Rear framework

Hollow circular section beam, aluminum.

Main structure: outer diameter ø40mm, wall thickness 1.8mm.

Stiffeners: outer diameter ø30mm, wall thickness 1.2mm.

Composite monocoque

Thicker backbone: 1.8mm aluminum sheet, 25.4mm aluminum honeycomb 3003, density 5.2 lb/ft^3 (hex-3003-td.pdf), 1.8mm aluminum sheet.

Thinner panels: 1.8mm aluminum sheet, 6.75mm same aluminum honeycomb, 1.8mm aluminum sheet.

Frontal shock absorber support plate: provisionally as thinner panels, to be defined based on shock.

Sway (anti-roll) bar

outer diameter ø25mm, wall thickness 2mm, extremely stiff (Super-alloy Z, E=E_steel*1e4, nu=0.3); it may be mechanically isolated at need by deactivating one of the connecting elements to the wheel hub carriers.

Such a “deformable but extremely stiff” linkage modeling should be discouraged in favor of an actual kinematic constraining – i.e. an MPC, since excessive stiffness badly impacts the system matrix condition number (or the integration time step, in the case of explicit dynamic simulations); nonetheless, it allowed for a very straightforward implementation.

Inertial elements modeling

The following spreadsheets are used in defining the equivalent rectangular cuboids for each inertially relevant rigid body: engine, wheel assemblies. The driver inertia is modeled through an 80 kg steel bar spanning roughly from the sternum to the pelvis.

Frontal crash absorber collapse loadcase (inertia relief)

At the element faces belonging to the crash_absorber_bearing_area set (an approx. 155×320 mm area at the front bulkhead), a 25 psi = 0,172 MPa distributed pressure is applied which is due to the honeycomb absorber crushing (see datasheet).

How to set a damped response

Click here to expand

Click here to expand

In order to include a small degree of structural damping (eg. 1% of the critical value) into a MSC.Marc/Mentat harmonic response calculation, the following steps may be followed:

  • preemptively define a modulating table 1/ω
    • define NAME as modulate_stiffmatmult
    • set Indipendent variable v1 TYPE as frequency
    • define table through FORMULA and type 1/pi/v1, i.e. $g(f)=\frac{1}{\pi f}$
  • select the various model materials, and for each of them enter the submenu STRUCTURAL → DAMPING and activate DAMPING;
  • leave alone the MASS MATRIX MULTIPLIER value (0 is ok, otherwise some “structural” damping will be associated to rigid body motions),
  • define a STIFFNESS MATRIX MULTIPLIER equal to the desired fraction of the critical value, namely 0.01,
  • set a frequency modulating function, namely TABLE, by hitting the TABLE button on the right of the stiffness matrix multiplier value;
  • select the just defined modulate_stiffmatmult table as the modulating one, hence hitOK and OK again to return back at the material properties menu
  • in this way, I defined damping as a function of the $\alpha$ e $\beta$ coefficients introduced by the Rayleigh proportional damping model, with zero $\alpha$ and hence no contribution of the mass matrix. In particular $\zeta = \frac{1}{2}(\frac{\alpha}{2 \pi f}+2 \pi f \beta)$ with $\alpha=0$ and $\beta= 0.01 \cdot g(f)=\frac{0.01}{\pi f}$, from which $\zeta=0.01$ as desired.
  • enter the MAIN → JOBS menu and create a copy of the undamped harmonic response job by hitting the COPY top left button and by setting a new job name;
  • enter the job PROPERTIES menu, and reach the ANALYSIS OPTIONS submenu; activate the COMPLEX DAMPING options within the dynamic harmonic section, and then exit withOK
  • Enter the JOB RESULTS submenu and deactivate Stress, Equivalent von Mises stress
  • substitute them with the AVAILABLE ELEMENT SCALARS
    • Equivalent Real Harmonic Stress , layers MAX & MIN
    • Equivalent Imag Harmonic Stress , layers MAX & MIN
    • the REAL HARMONIC e IMAG HARMONIC stress resultant equivalents for the beam elements, DEFAULT layer, and the Beam Orientatio Vector
  • insert from the AVAILABLE ELEMENT TENSORS block
    • Real Harmonic Stress , layers ALL
    • Imag Harmonic Stress , layers ALL
  • run the simulation as usual with RUN → SUBMIT
  • open the post file as usual with OPEN POST FILE (RESULTS MENU)
  • The deformed shape may be visualized according to a given phase within the oscillation cycle (see also the DEFORMED SHAPE SETTINGS menu); in the absence of damping the fase was limited to the 0° and 180° values, cases these that may be represented with the bare variation in sign of the stress and displacement components to be monitored.
  • Please note that the real component has a 0° phase ($\cos(\omega t)$ modulation) whereas the imaginary component has a 270° phae ($-\sin(\omega t)$ modulation).
  • Please also note that in resonance conditions the imaginary component becomes dominant and reaches the peak values, whereas the real component vanishes (resonant response is in fact ~90° out of phase with respect to the real, 0° excitation).
  • Lets e.g. collect the displacement in $z$ direction of the node at the center of the excited wheel contact area:
    • enter the POSTPROCESSING RESULTS menu, with opened t16 result file, and proceed within the HISTORY PLOT submenu
    • define the locations for the response sampling with SET LOCATIONS, hence click on the desired node[s], and finalize with END LIST
    • define the range of the sub-increments to be collected with INC RANGE, and then entering at the prompt 0:1 [enter], 0:397 [enter], 1 [enter], as the sampling beginning, end and step.
    • proceed with the definition of collected response diagrams by entering th ADD CURVES menu, and thenALL LOCATIONS (a single location has been selected); select the Frequency global variable as the abscissa, and the Displacement Z Magnitude nodal variable as the ordinate. The FIT scales the axes to contain all the sample points.
    • By hitting RETURN I may return to the HISTORY PLOT menu, where the label density may be reduced SHOW IDS from '1' to '10'; by entering a '0' value labels are hidden.
  • response peaks are now finite (they were theoretically unbounded in the absence of damping), and peaks disappear in correspondence of natural modes that are weakly coupled with the exciting force. In the absence of damping, bounded response at resonance is obtained for strictly uncoupled natural modes only.


Sparse material


Interesting stuff

composite design manual



Reference L-shaped cross section, to verify the coupled bending formulas: maxima worksheet, Oxy and G12 oriented MSC.Marc/Mentat linear models. Large rotation, animated hiE lowE models, gif animation.

free_anticlastic_vs_cylindrical_bending.wxmx maxima worksheet for the four point bending test discussion.

quad4 Mindlin plate elementary modes.


Roller bearing questionable modeling

On the relevance of constraining in dynamic analyses back view side view relevance of (improper) constraints on the dynamic behaviour of a structure. Design is reliable in actual operational conditions (link). Added constraints stiffen up the structure, thus increasing natural frequencies. However, a 0 Hz rigid body mode natural frequency may rise to a finite value due to added positioning constraints; the associate natural mode may be excited in resonance by dynamic loads.

Poor man dynamic response animated view

MSC.Mentat procedure for creating poor man harmonic response animations

MSC.Mentat procedure for creating poor man natural mode animations

Structural damping references





estratto vol. 2, sezione 8 di Soovere, J., and M. L. Drake. Aerospace Structures Technology Damping Design Guide.LOCKHEED-CALIFORNIA CO BURBANK, 1985.


with respect to v002, i) I created the truss elements at the front right suspension, ii) I selected also the rotational dofs at the wheel assebly RBE2, and iii) I created the __crash_absorber_support_area face set
with respect to v004, i) materials and geometric props are associated to the two truss elements that connect the sway bar to the wheel assemblies, ii) one of those trusses is selectively deactivated in the various jobs, iii) the max along layers eq. von Mises stres is asked as a result
wikiffcd2022/start.txt · Ultima modifica: 2022/05/31 10:24 da ebertocchi