Modelling of nonlinearieties

Computational modelling and analysis tools play a key role in high performance machining, and nowadays, they have become essential parts of engineering design especially for complex continuous dynamic systems, which incorporates nonlinearities, such as friction and backlash.

The accuracy of the model depends on the structure and parameters. Backalsh model has been simplified - an initial change in input has no effect on the output. The amount of side-to-side play - deadband is cantered about the output. Friction is categorized according to its pre-sliding and sliding regimes, and the simplest friction models consider only the sliding regime.

FRICTION IN MACHINE TOOLS AND MECHANISMS

Friction, by definition, refers to the resistance to motion during sliding of two opposing objects against one another. Friction is present almost everywhere and it brings benefit in various forms. Braking systems and clutches are two examples of friction effects that benefit us. However, friction is highly undesirable in most industrial and mechanical applications. Uncompensated friction force increases energy consumption and reduces efficiency of these processes.

The model proposed by Al-Bender and Horstein (2005) and used in this research work consists two main control modes. The first mode is dealing with the gross-sliding friction second mode of the controller is activated until the motion leaves the pre-sliding region again. Al-Bender (2005) developed Generalized Maxwell-slip (GMS) friction model and illustrate the superiority of the model with respect to simulation of friction behaviour in both the pre-sliding and sliding regimes. Its main advantage is that it contains a hysteresis function with non-local memory behaviour to describe the pre-sliding regime. The GMS friction model includes the following:

- the Stribeck curve for constant velocity,

- dependence of the friction force Ff on the velocity v = constant - hysteresis function with non-local memory for the pre-sliding regime, - frictional memory for the sliding regime.

- The friction force is constant and only depends on the direction of the velocity. - The Coulomb friction models the dry friction case.

71 Stiction is minor in the model if the external force is known. The friction force simply counteracts the external force and precludes the object from moving. Static friction can usually reach higher magnitudes than dynamic friction and, naturally, it cannot be described as a function of velocity v. At some critical force value 𝐹𝑠𝑡𝑎𝑡𝑖𝑐, which is called breakaway force, the object starts to glide across the surface. If this happens, the friction force drops rapidly to a lower value and from then on has to be described in the context of dynamic friction. (Horstein (2005))

Combining static and dynamic leads to a multi-valued function:

𝐹(f) = {_𝐹 𝐹𝑠𝑡𝑎𝑡𝑖𝑐= 𝐹𝜃if v = 0

𝑑𝑦𝑛𝑎𝑚𝑖𝑐(v) = 𝐹𝑐∗ sign(v) + 𝐹𝑣 𝑒𝑙𝑠𝑒 Equation 3-74

All three functions are multi-valued in the region of static friction (v = 0m/s). Here the friction force is a function of the position rather than of velocity and cannot be properly displayed in the diagrams presended by Canudas et al. (1995). The dynamic behaviour of an elementary slip-block is described by a:

𝑑𝐹_𝑖

𝑑𝑡 = 𝑘𝑖 Equation 3-75

ki spring model with stiffness

On the other hand, slipping occurs if the Fi equals a maximum value Wi = αis(v), of each element during sticking and. During slipping, equation describing the dynamic behaviour of the slip-block is represented on Figure 3-27 as

𝑑𝐹_𝑖

𝑑𝑡 = 𝑠𝑖𝑔𝑛(𝑣)𝐶 (𝛼𝑖− 𝐹_𝑖

𝑠(𝑣)) Equation 3-76

αi is the normalized maximum friction force s(v) is the Stribeck curve

72 The constant parameter C (equals to 1/Vs) in:

𝐹𝑓(𝑣) = {𝐹𝑐+ (𝐹𝑠− 𝐹𝑐)𝑒𝑥𝑝 (− |_𝑉𝑣

𝑠|

𝛿

) + 𝜎|𝑣|} 𝑠𝑖𝑔𝑛(𝑣) Equation 3-77

In pre-sliding regime, friction is dominated by the displacement and behaves as a hysteretic function of displacement with non-local memory behaviour. This behaviour is characterized by the so-called virgin curve. The virgin curve is derived from a sinusoidal excitation of the system. The frequency and amplitude of the sinusoidal input signal are selected to minimize inertia effect and to remain in the pre-sliding regime. (Horstein (2005))

Figure 3-28 GMS Friction SIMULINK model

That indicates the rate at which the friction force follows the Stribeck effect in sliding. The ability to estimate the friction behaviour in pre-sliding regime is the superiority of GMS model, while it does not lose its ability to estimate the friction in the sliding regime. Therefore the GMS model can capture friction behaviour for any working range of displacement and velocity. Al-Bender (2005)

Simple experiment has been conducted by Horstein (2005). His model has been used in CNC feed drive model. It has been found that classical models like Coulomb friction and the Stribeck friction model give satisfactory results for accurate ballscrew system. However, at low displacements, which emphasize the pre-sliding regime, the classical models fail to give a satisfactory therefore it may be useful in future studies. AE features may be useful to investigate frictional parameters. The mayor disadvantage of the GMS model is complexity which complicates its application in real time analysis.

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