Single-degree-of-freedom Systems

A quick review of system torsional response may help explain why a resilient coupling works. Figure 9-14 is a torsional single degree of freedom system with a disk having a torsional moment of inertia J connected to a massless torsional spring K. 

The shear wall is effectively a single degree of freedom system,... 

This section considers the response measurement of a single-degree-of-freedom system or a single-channel measurement of a multi-degree-of-freedom system. Discrete data is sampled with a time step Ar and y denotes the measured response at time t = nAt. The measurement is different from the model response ... 

In the literature and the previously mentioned identification methods, the input is either measured or modeled as a prescribed parametric stochastic model (even though the parameters may be unknown). This seems to be a necessary condition for model identification purpose. For example, consider a linear single-degree-of-freedom system. In the frequency domain, the response X is equal to the input 7, magnified by the transfer function of the oscillator Jf ... 

From our previous work, we know that this system (a single degree of freedom system) can be solved for r as a function of t and the initial conditions. Once r = r t) is known, we may obtain 6 by integration ... 

The modified energy for the Verlet method for a single degree of freedom system with energy H = pV2 -h U q) is... 

The static deflection of the tie rod due to the forces shown in Figure 12-2 was calculated to be approximately 0.007 in. Thus, the amplification, or resonance, factor was found to be 2.18. In comparing this factor with that which would be expected from a single degree of freedom system with no damping (amplification factor equals 6.5, Reference 4), it was found that the effect of the tension in the rod served to reduce the amplification factor by 2.98. 

For a single degree of freedom system at resonance, the amplification factor is approximately 45 for damping equal to 1.2 per cent of critical... 

Because not all structures can be modeled as a single degree of freedom system this approach has to be expand formulti degree of freedom systems (Schueller 1981). For this purpose the equation of motion has to be expressed in matrix form, size of matrix is the number of degree of freedom of the structure. To solve the differential equation the modal analysis is advantageous which is based on the separation of the single equations. With the help of the impulse force in the frequency domain the square of the absolute structural reaction function for every n-th natural vibration mode can be expressed, see (Clough and Penzien 1975)... 

In a fundamental theoretical model, the dynamic absorber (ma, fca, Ca) is attached to a single degree of freedom system (mi, ki) as shown in Fig. 8.63. The equations of motion are... 

In Fig. 1, the displaced dam from its initial position is defined by the displacement u(y,t), which is the Siam of the horizontal ground excitation Ug(t) and the relative displacement Y< (t) (y). The dam is assumed to be elastic and as a first approximation, it is assumed to deflect in its first mode with mode shape (y). For this dynamic system the equation of motion can be derived by the principle of virtual displacement in the form of a generalized single degree of freedom system. In terms of the maximum displacement Y Ct) at the top of the dam and the generalized force Pc(t), the equation is... 

For a lightly damped single-degree-of-freedom system, subjected to a narrow-band excitation at frequency (uf, the mean square system response is given by... 

A typical segment and evolutionary spectral densities of the ice force record are shown in Figs. 7 and 8. Typical physical spectra for the record are plotted in Fig. 9. A mean square response curve for the single-degree of freedom system, considered in Example 1, for an assumed duration of the window function, T 10 sec, is shown in Fig. 10. 

Hasselman, T.K., Transient Response of a Linear Single-Degree-of-Freedom System to a Nonstationary Narrow-Band Stochastic Process, M.S. Thesis, Unlv. Calif., Los Angeles, Calif., 1967. 

Consider a example of a linear single degree of freedom system under dynamic excitation, the excitation is modeled as Gaussian white noise, the intensity S, the duration D of the excitation and the natural frequency/, damping ratio f are uncertain and modeled as random variables as shown in Table 1. For a simple limit state of a given threshold level of the displacement being exceeded, the failure probability with deterministic system parameters is... 

It has been shown that existence and uniqueness problems arised even for a single-degree-of-freedom system and the conditions under which such problems occur have been determined. A graphical method has been... 

The variation presented above is called Griffith s variation in [2]. As a matter of fact, this variation was used by all authors who applied energy considerations to cracked bodies. However, most authors applied this not to the principle of virtual work but to that of energy balance. Both approaches are equivalent for single-degree-of-freedom systems with potential energy. The principle of virtual work is applicable for multi-degree-of-freedom and nonpotential systems. 

For illustration purposes, we present the methodology for calculating fragility curves for the single-degree-of-freedom system in Eq. 10. In this context, we define the seismic fragility as... 

Lin W-H, Chopra AK (2003) Earthquake response of elastic single-degree-of-freedom systems with nonlinear viscoelastic dampers. J Eng Mech ASCE 129(6) 597-606... 

This equation treats the superstmcture as a rigid mass single degree-of-freedom system oscillator whose stiffness and damping are entirely defined by the isolation system. This base shear represents the total lateral force delivered to the superstmcture given the frequency and damping of the isolation system. This base shear may be... 

Earthquake Response Spectra and Design Spectra, Fig. 2 Single-degree-of-freedom system under earthquake excitation... 

For the safety evaluation, the N2 method defines a bilinear representation of the capacity curve corresponding to an equivalent single-degree-of-freedom system. According to the Italian code (IBC 2008), this representation includes a first straight line that passes by the origin and intercepts the capacity curve of the actual system for a base shear force of 70 % of the maximum and a second line, horizontal and defined in mode that the area below the envelope of the actual system equalizes the area under the bilinear idealized response. The bilinear representation allows to obtain the reference period of vibration for the computation of the target displacement. 

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