Damping inertia

Component modelling The modelling of the stabiliser trim system includes one-dimensional mechanics with elasticity, damping, inertia and friction hydraulics with bulk modulus, fluid density, turbulent and laminar flow and resistive electrics. One of the developed component models is explained below. 

Inertia foree + Damping foree + Spring foree + Impressed foree = 0 From the previous equation, the displaeement lags the impressed foree by the phase angle 6, and the spring foree aets opposite in direetion to... 

Figure 2.8 shows a reduetion gearbox being driven by a motor that develops a torque T tn(t). It has a gear reduetion ratio of and the moments of inertia on the motor and output shafts are and /q, and the respeetive damping eoeffieients Cm and Cq. Find the differential equation relating the motor torque CmfO and the output angular position 6a t). 

The terms (/q + n /m) and (Co + rP-Cm) are ealled the equivalent moment of inertia /g and equivalent damping eoeffieient Cg referred to the output shaft. 

A field eontrolled d.e. motor develops a torque Tm t) proportional to the field eurrent k t). The rotating parts have a moment of inertia / of 1.5 kg m and a viseous damping eoeffieient C of 0.5 Nm s/rad. 

A torsional spring of stiffness K, a mass of moment of inertia / and a fluid damper with damping coefficient C are connected together as shown in Figure 3.25. The angular displacement of the free end of the spring is 0 ( ) and the angular displacement of the mass and damper is 6a t). 

Fig. 4.41 Angular positional control system. = Error detector gain (V/rad) K2 = Amplifier gain (A/V) Kj = Motor constant (Nm/A) n = Gear ratio Hi = Tachogenerator constant (Vs/rad) H = Load moment of inertia (kg m ) Q = Load damping coefficient (Nms/rad).

If the load eonsists of a rotor of moment of inertia / and a damping deviee of damping eoeffieient C, then the load dynamies are... 

The system is still comprised of the inertia force due to the mass and the spring force, but a new force is introduced. This force is referred to as the damping force and is proportional to the damping constant, or the coefficient of viscous damping, c. The damping force is also proportional to the velocity of the body and, as it is applied, it opposes the motion at each instant. 

The left-hand side of the second equation of (6-186) is the pendulum equation (J being the moment of inertia, D, the coefficient of damping and C, the coefficient of the restoring moment). 

Weiss and Worsham 259 indicated that the most important factor governing mean droplet size in a spray is the relative velocity between air and liquid, and droplet size distribution depends on the range of excitable wavelengths on the surface of a liquid sheet. The shorter wavelength limit is due to viscous damping, whereas the longer wavelengths are limited by inertia effects. 

When damping is ignored, the three forces then acting on the mass are the resistance (K y), (he inertia force (M a), and the external applied force (Ft), The dynamic equilibrium equation for the undamped, clastic system then becomes,... 

Note 3 A damping curve is usually obtained using a torsion pendulum, involving the measurement of decrease in the axial, torsional displacement of a specimen of uniform cross-section of known shape, with the torsional displacement initiated using a torsion bar of known moment of inertia. 

The constants Kp, Kt, and Kd are settings of the instrument. When the controller is hooked up to the process, the settings appropriate to a desired quality of control depend on the inertia (capacitance) and various response times of the system, and they can be determined by field tests. The method of Ziegler and Nichols used in Example 3.1 is based on step response of a damped system and provides at least approximate values of instrument settings which can be further fine-tuned in the field. 

In free vibration methods, the rubber test piece, with or without an added mass, is allowed to oscillate at the natural frequency determined by the dimensions and viscoelastic properties of the rubber and by the total inertia. Due to damping in the rubber, the amplitude of oscillations will decay with time and, from the rate of decay and the frequency of oscillation, the dynamic properties of the test piece can be deduced. 

A qualitative idea of the surface viscosity is given by noting the ease with which talc can be blown about the surface. Most insoluble films have surface viscosities of c. 10 6 kg s l to 10 3 kg s-1 (for films 10-9 m thick this is equivalent to a bulk viscosity range of c. 103 kg m"1 s 1 to 106 kg itT1 s"1). These films can be studied by means of a damped oscillation method (Figure 4.20). For a vane of length / and a disc with a moment of inertia /,... 

There are two distinct modes of flow, laminar and turbulent. Fluid inertia tends to allow fluctuations to grow and give rise to turbulent eddies. Viscosity on the other hand, tends to damp out these fluctuations. A ratio of forces, inertial to viscous, is used to characterise the nature of the flow and is called the Reynolds Number, Re. For pipe flow this takes the form ... 

There is an equivalence between the differential equations describing a mechanical system which oscillates with damped simple harmonic motion and driven by a sinusoidal force, and the series L, C, R arm of the circuit driven by a sinusoidal e.m.f. The inductance Li is equivalent to the mass (inertia) of the mechanical system, the capacitance C to the mechanical stiffness and the resistance Ri accounts for the energy losses Cc is the electrical capacitance of the specimen. Fig. 6.3(b) is the equivalent series circuit representing the impedance of the parallel circuit. 

From an experimental point of view, knowledge of the modes is important because it makes it possible to optimize the location of the detector and supports in the experimental devices in order to minimize effects due to spurious stiffness, inertia, or damping (5, p. 90) (see also Chap. 7). By making = —A. -f m, as corresponds to a free vibration experiment, expressions for the real and imaginary parts of the dynamic modulus and for the loss tangent at the resonance frequency = can be obtained. The resulting equations are... 

The control of dynamic effects at impact rates up to 1 m/s (in some instances somewhat higher) frequently makes use of mechanical damping in the load transmission by placing a soft pad (elastomer or grease) between the striker tup and the specimen [3,5], Above about 1 m/s inertia effects overshadow the true mechanical response of the specimen. Due to such dynamic effects, the applicability of FBA is limited to loading rates up to about 1 to 2 m/s for bending type fracture specimens. 

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