Free body diagram | EPFL Graph Search

In physics and engineering, a free body diagram (FBD; also called a force diagram) is a graphical illustration used to visualize the applied forces, moments, and resulting reactions on a body in a given condition. It depicts a body or connected bodies with all the applied forces and moments, and reactions, which act on the body(ies). The body may consist of multiple internal members (such as a truss), or be a compact body (such as a beam). A series of free bodies and other diagrams may be necessary to solve complex problems. Purpose Free body diagrams are used to visualize forces and moments applied to a body and to calculate reactions in mechanics problems. These diagrams are frequently used both to determine the loading of individual structural components and to calculate internal forces within a structure. They are used by most engineering disciplines from Biomechanics to Structural Engineering. In the educational environment, a free body diagram is an important step i

New slip synthesis and theoretical approach of CVT slip control

Yves Rothenbühler

Today's vehicle must be efficient in terms of gas (CO2, NOx) emissions and fuel consumption. Due to improvements in material and oil, the continuous variable transmission (CVT) is now making a breakthrough in the automotive market. The CVT decouples the engine from the wheel speed. CVT enables significant fuel gains by shifting the engine operating point for specific power demands. This optimization of the operating point enables a reduction of this fuel consumption. A CVT is constituted of two pulley sheaves, one fixed and the other one movable in its axial direction when subjected to an external axial force, in general hydraulic. The transition from the minimum to the maximum speed ratio is continuous and an infinite numbers of ratio is available between these two limits. An intermediate element (a metallic belt or chain) transmits the power from the input (the primary) to the exit (the secondary) of the CVT or variator. Further improvements of the fuel consumption and gas emission are still required for example by improving the variator efficiency. Increasing hydraulic performance or decreasing mechanical losses by reducing the axial forces are some solutions. The latter method is not without risks. The diminution of the clamping forces increases the slip between pulley sheaves and the intermediate element. If the axial forces decrease too much, high slip values can be reached and cause damage to the pulleys and the intermediate element. Control of the slip is an attractive solution to decrease the clamping forces in order to safely improve the variator efficiency. The objective of this thesis is to understand and model the slip of each pulley and establish analytic tools dedicate to the variator control. The slip study and the theoretical approach of the CVT variator is applied to the slip control of the variator with a chain. The contribution of this work is threefold. Firstly, the slip and the traction coefficient are analyzed for each pulley. The slip analysis of each pulley is then used to define a new slip synthesis as the summation of the slip of each pulley. It is demonstrated that the slip and the traction coefficient are different for each pulley and depend on the speed ratio of the variator. In low ratios, both the secondary pulley and the primary pulley slip, but only the primary reaches macro slip. For middle or higher ratios, only the secondary pulley slips and reaches high values of slip. Experiments show that the pulley with the smallest clamping force limits the system. Secondly, based on kinematics, force equilibrium, elastic deformations of the pulleys and the intermediate element, a detail model of the variator is proposed. The principal results are the estimation of the clamping forces, of the traction curve for each pulley and of the chain efficiency. These results are implemented in a simpler model that describes the variator dynamics. This last model considers the two pulleys and the intermediate element as free bodies. The hydraulic circuit and the actuators, which are important to take into account for control, are also modeled. Thirdly, the new slip synthesis and the results of the dynamic models are applied to the slip control of the variator in order to improve the efficiency. A pole placement law is applied to the actuators to control the flow that enters or exits the pulleys. With this law, the actuators are decoupled and the bandwidth is increased sufficiently for actuators dynamics to be neglected. The primary and the secondary pressures are decoupled and linearized by an input-output feedback linearization. The resulting system is linear and linear control theory can be applied to control the two pressures. The speed ratio is controlled by the primary clamping force. The secondary pressure is chosen as a function of the control mode of the variator: standard mode or slip mode. In standard mode, the intermediate element is overclamped by 30%, whereas in slip mode, the secondary clamping force is set as a function of the desired slip. By controlling the slip at 2%, the mechanical efficiency was increased by more than 2% and the clamping forces reduced by more than 30%. For the slip control, a proportional-integrator law and a model reference adaptive control (MRAC) are presented and the performances compared. The MRAC gives slightly better results.

EPFL2009

Comportement mécanique d'un limon non saturé

Although the majority of soils are unsaturated, their mechanical behaviour is still not well defined. This dissertation presents first an experimental study of an unsaturated silt and then proposes an elasto-plastic constitutive law to model the behaviour of unsaturated soils. The first part of this work deals with the characterisation of the mechanical behaviour of unsaturated soils by means of laboratory experiments on a remoulded silt. Suction is imposed by controlling the excess air pressure in the sample. Several techniques for the measurement of volume changes of samples are developed and compared. The saturated behaviour is studied by means of imposed suction and controlled tests under triaxial and oedometric (uniaxial strain) conditions. Other test performed in Richards cells are used for defining the purely so-called "hydric" behaviour of the soil. The analysis of test results were done in the effective stress plane (σ'=σ-uw,) as well as in the net stress plane (σ*=σ-ua). The hydric tests confirmed the hysteretic nature of the wetting-drying cycles. The hydric and mechanical response changes when the suction raises above the air entry value. Isotropic tests show that the preconsolidation pressure increases with the suction in the log(p')-e plane. However, this increase is rather limited in the net stress diagrams. Deviatoric tests show an increase of stiffness and peak strength with suction at constant mean effective or net pressure. Suction appears to have an effect similar to a mechanical overconsolidation when representing the result in the effective stress plane. Softening appears post peak for the cases of strong suction or weak net mean pressure. The evolution of the critical state is interpreted in terms of effective and net stresses. In the p'-q plane, the critical state line is shown to be independent of the suction (in the studied range of stress and suction); this is not true in the p*-q plane. The yield surfaces F derived from the test results grow with suction and both p'-q and p*-q planes, and the behaviour corresponds to non-associated plasticity. An existing elasto-plastic model originally developed for saturated soils (HISS-δ1) was improved and extended to integrate suction effects. The new proposed model, called δ1-unsat, is described by means of two independent stress state variables: the effective stress σ' and the suction s. Two yield surfaces, coupled through the hardening internal variable, are used in the model; the first one is linked to the mechanical behaviour, while the second one is used for hydric loading. This new model requires five new material parameters: three of them are used for the purely hydric behaviour, one describes the increase of peak strength with the suction and the last one describes the growth of yield surface with suction. The δ1-unsat model is shown to be able to reproduce the principal features of the unsaturated soils: (a) existence on a drying path of a saturated domain with non-zero suction where plastic (irrecoverable) strains appear, (b) elastic behaviour on drying path when the suction is larger then the air-entry value, (c) wetting collapse, (d) influence of mechanical stress state on the hydric behaviour, (e) increase of preconsolidation pressure with suction, (f) increase of peak strength with suction. The quantitative reliability of the δ1-unsat model is good quantitatively. In particular the prediction of the behaviour in the domain of low strains (pre-peak) is well described. The introduction of a damage concept (DSC - disturbed state concept) in the δ1-unsat model allowed the modelling of the post-peak behaviour as well. In conclusion, the mechanical behaviour of an unsaturated silt was characterised. A constitutive elasto-plastic model integrating suction and damage effects was developed and was proved to represent the principal constitutive features of unsaturated soils.

EPFL1999

Vector-Based 3D Graphic Statics (Part II): Construction of Force Diagrams

Corentin Jean Dominique Fivet

This paper presents an approach to the extension of graphic statics force diagrams to the third dimension. The main focus is to build 3D force diagrams out of vectors, so that each edge in the form diagram corresponds to two parallel vectors in the force diagram; this pair of vectors can generally be overlapped to form one single edge. The issue of parallel vectors that cannot overlap in the force diagram is discussed for both planar and spatial geometries and a convenient arrangement of the nonoverlapping vectors is presented. Alternative ways of assembling the force diagrams and their properties are suggested. As a result, the 3D vector-based force diagrams defined in this approach are a direct extension of the 2D force diagrams, taking advantage of their strengths for early design exploration. Moreover the presented approach applies to all typologies of pin-jointed frameworks, (including self-stressed and with external loads).

IASS2016

New slip synthesis and theoretical approach of CVT slip control

Yves Rothenbühler

EPFL2009

Comportement mécanique d'un limon non saturé

EPFL1999