Undamped Systems

Setting C2 = 0 in (7.18), the linearized model of the undamped lead screw drive model with compliant threads is obtained as 

The natural frequencies of this system are the roots of the following equation  

The second instability condition, given by (7.22), represents the mode coupling (flutter) instability. The equation for the flutter instability boundary (i.e., coalescence of the two real natural frequencies) is found by replacing the less than sign with the equal sign in (7.22). After some manipulations, one finds 

This equation is quadratic in k and kc and can be solved to find parametric relationships for the onset of the flutter instability (either for as a function of or vice versa). The conditions for the existence of positive real solutions to (7.25) are given by the following lemma. 

Lemma 7.1. The necessary conditions for the mode coupling instability in the model defined by (7.19) are given as follows  

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Figure 9-9. Typical mode shapes of an undamped system.

The natural frequencies of a damped system are essentially the same as the undamped systems for all realistic values of damping. 

Calculation of K In general, the settling time of a system with critical damping is equal to the periodic time of the undamped system, as can be seen in Figure 3.19. This can be demonstrated using equation (3.62) for critical damping... 

The boundary conditions established by the machine design determine the freedom of movement permitted within the machine-train. A basic understanding of this concept is essential for vibration analysis. Free vibration refers to the vibration of a damped (as well as undamped) system of masses with motion entirely influenced by their potential energy. Forced vibration occurs when motion is sustained or driven by an applied periodic force in either damped or undamped systems. The following sections discuss free and forced vibration for both damped and undamped systems. 

For C = 0 (undamped system). The complementary solution is the same as q. (6.93) with the exponential term equal to unity. There is no decay of the sine and cosine terms and therefore the system will oscillate forever. 

In quantum information applications one often treats the system of two qubits being manipulated as part of some information processing. This is modelled by a couple of interacting two-level systems. Following the approach in the present paper, we consider the one system to be strongly damped. In that case it serves as a faked continuum for the other one, and we desire to derive an equation of motion for the originally undamped system. The damping is described by a Markovian term of the Lindblad type. 

The result is re-expressed in terms of the resonant frequency coq = k/m of the undamped system, and the characteristic damping time r = dk of the spring-dashpot combination, giving... 

Because of the linear restoring force of the spring, a deflection of the system will result in a harmonic oscillation. This becomes obvious by considering the free vibration Eq. 1 of the undamped system... 

It will be realized that Equation 15.20 lends itself to matrix notation. The extended notation is used here for sake of more general transparency. Equation 15.20, for the simpler case of an undamped system, becomes ... 

T = period of oscillation of undamped system Typical amplitude- and phase-frequency characteristics are shown in Fig. 4.8. 

This differential equation is descriptive of a second-order undamped system. The U tube resonates at a natural period established by the... 

The use of a damper to suppress stick-slip, although convenient, would seem to be open to the objection that it will produce an average value of the periodic force observed in an undamped system. Our observations reported in our paper on the friction of rubber on ice indicate that the perk value of the frictional force is more appropriate. Without repeating our reasons for believing this, there is the experimental observation that increasing the transducers strong stiffness increases the minimum value of the periodic force without affecting the maximum. It would affect, therefore, that an infinitely stiff mechanical system would produce a non-periodic force equal to the maximum value obtained in a stick-slip measurement. 

Care has been taken to ensure that vibration from the motors used to drive the sample and cleaner do not induce vibrations in the arm and blade assembly. The damper reduces the vibrations which arise from the passage of the blade over the sample. These vibrations can be quite large in an undamped system if the abrasion pattern is coarse. All the measurements have been carried out with a stiff single edge razor blade which is clamped over most of its major surface so that it is not significantly deflected by the forces i nvolved. 

With all the previous data in hand, one can now proceed with solving the initial problem of determining the response of the undamped system to some generic forcing F(t). The complete solution to Eq. 12b, which is a generalized SDOF equation, provides the part of the total response that owes exclusively to the /th mode and can be solved independently of all other modes. As is well known from the solution to an SDOF equation, the modal solution, (0 of Eq. 12c consists... 

Although the mode displacement method was developed typically first, it is not the only option available. One of the most useful alternatives is the so-called mode acceleration method along with its implementation variations. In identical form, though founded on a different basis, this may also be found under the name of static correction method, and the origin of this naming will below become clear. It was first devised for an undamped system, and its main attribute is that instead of modal displacements, one can seek solutions in terms of only modal accelerations for an undamped system or in terms of modal accelerations and velocities for a damped system. A brief presentation is herein put forward following the derivations and critique proposed by Cornwell et al. (1983). 

The natural frequencies (coy M) and the mode shapes (xj e M") of the corresponding undamped system can be obtained (Meirovitch 1997) by solving the matrix eigenvalue problem... 

Since any realistic structures and materials have some dissipati(Hi mechanisms, it seems reasonable to consider a damper with viscous damping. The equations of motion of the damped system will be similar to the one shown in section Undamped System with the additional damping term ... 

Substituting the vaiue of Ut for a half sine shock pulse of acceleration defined by eqn.(1) in eqn. (10), the relative displacement response for an undamped system becomes,... 

Eqn.(15) gives the value of equivalent static acceleration for an undamped system with cOt) <... 

Similar to the above discussions, non-trivial solutions are only possible when the matrix — co M + Ap is singular. Setting the determinant of this matrix to zero gives the characteristic equation of the undamped system ... 

Fig. 3.4 Flutter, undamped system coalescence of two natural frequencies...

Fig. 4.8 Evolution of the real and imaginary parts of the eigenvalues of the undamped system...

Remark 7.3. Lemma 7.1 establishes that for the undamped system (7.19), mode coupling instability can only occur when the lead screw drive is self-locking and the applied force is in the direction of the translation. ... 

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