As can be seen from Eq. (6), the equivalent inertia of the gear is now equal to the sum of the reference shaft inertia I₁₉ and the geared shaft inertia I₁₀ multiplied by the gear ratio, m, squared. It is well known by engineers working with torsional vibration problems of geared systems that one must multiply inertias of the geared shaft by m². The difference is that using bond graphs all this is taken care of systematically. The bond graph modelling approach also offers a straightforward method for solving problems with dynamically dependent elements.

　

5. CLUTCH MODEL

　

Modelling and simulation of the clutch creates some challenging problems. One problem stem in part from the representation of the friction torque. T_f, as a function of the relative angular velocity, (ω_r, between the primary and secondary parts. Neither of the causal input-output T_f=T_f (ω_r) or ω_r=ω_r (T_f) is unique. This means that in the most idealized form, the relations cannot be described in a single input-output form. Part of the time the torque should be zero for any relative velocity and part of the time the relative velocity should be zero for any torque as indicated in Figure 3.

As indicated in Figure 3 the open clutch state is no problem because T_f=0 for all relative velocities, (ω_r and the two inertia elements I₁₉ and I₂₂ can move independently. In the clutch closed state, we must assure that, ω_r is driven to zero and kept near zero. In addition, when the clutch is closed the two above-mentioned inertia elements are rigidly connected which means that the system structures is changed in that we have to remove one equation. That means we have to reformulate the model after each clutching operation. These jump conditions must properly reflect energy losses during clutching as well as conservation of momentum when the inertias are coupled suddenly, and thus not always simple to formulate.

In this paper, we will present a method proposed by Karnopp [1] in which the system structure remains constant for the overall mathematical model of the system, numerical stiffness problems are easily handled and thus the jump conditions after a clutching operation are taken care of automatically. The essence of the method is to maintain constant causality for both states of the clutch. This means that we want a frictional torque, T_f, which is always a function of the relative velocity, ω_r.

When two mating frictional surfaces, one attached to the driving member and the other attached to the driven member, are brought into contact with each other, frictional forces are created because of the normal pressure between the elements. These frictional forces allow torque to be transmitted from one member to the other and enable a load on the driven shaft to be accelerated up to the speed of the driving member.

　

Fig. 3. Ideal constitutive law for clutch.

　

For a friction disk having an outside radius, r₀, and an inside radius, r_i, with a normal force F_n and coefficient of friction, μ, the theoretical equation for the frictional torque may be based on either of two assumptions

　

・ the pressure is uniformly distributed over the surface

・the wear on the surface is uniform

The uniform wear relation is a little more conservative, and will be used in the development, which follows.

If we activate a clutch with, n_p, disks in contact hydraulically by applying a pressure, P_h(t), on an actuating piston with area. A_p, the friction torque is determined by the following expression

As the frictional torque is active but less than the stiction torque, the relative velocity will be driven towards zero. Instead of driving the relative velocity to zero, we use a friction law as indicated in Figure 4.

　

Fig. 4. Model friction law.

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