Multiple Degree Of Freedom Systems Research Articles

On the internal resonance of multi-degree-of-freedom systems with cubic non-linearity

A new method of multiple scales procedure is used to analyze the forced responses of multiple degrees of freedom systems with cubic non-linearity, in which the internal resonance takes place simultaneously together with the forced resonances. Attention is focused on the superharmonic and subharmonic resonance caused by internal resonance as Ω is near Ω 1 or Ω 2 , where Ω is the forcing frequency, and Ω 1 and Ω 2 are the first and the second natural frequencies. It is found that the responses are usually excitation-level dependent and there is a critical excitation amplitude ƒ c . In the case of Ω near Ω 1 , when ƒ>ƒ c the “out-of-phase” superharmonic resonance disappears. In the case of Ω near Ω 2 , when 0<ƒ<ƒ c , there are two branches of response; when ƒ>ƒ c , one branch of responses vanishes. The critical excitation amplitudes in different forced vibration are determined analytically in this paper.

Response of MDOF Systems to Multiple Support Seismic Excitation

The response of general multiple‐degrees‐of‐freedom (MDOF) systems subjected to several components of earthquake at multiple support points is obtained using a random‐vibration formulation. The excitation is modeled as a colored, correlated, vector‐valued, nonstationary random process. A new filter is used, such that the excitation may have multiple predominant frequencies and a wide range of spectral shapes. The time history of the RMS excitation at support points which is the output of the filter is prescribed directly. The corresponding input of the filter, a fictitious piecewise linear strength envelope, is generated accordingly before engaging the filter with the system. This filter also allows the support motion to be prescribed in terms of displacement, velocity, or acceleration, which is not true for other filters that are popular and widely in use. Finally, the correlation between any two components can also be prescribed as a function of time. The concept of “pseudo‐static displacement” used by ...

Analysis of multiply coupled, multimode sonar transducer arrays

Sonar transducer element and array performance predictions using linear analysis methods have been usefully performed for over two decades. These analyses were first performed for single degree of freedom (DOF) piston transducers including only the acoustical coupling through the radiating medium. More recently, single DOF transducers including both acoustical and mechanical coupling through their common mounting surface have been analyzed [D. T. Porter, J. Acoust. Soc. Am. Suppl. 1 78, S73 (1985)]. However, many of the transducer types currently under consideration for use in sonar arrays are multiple DOF systems having two or more resonant modes of significance within their operating frequency band. Flextensional and flexural mode transducers, as well as several types of multimode or multiply resonant piston transducer structures, are examples of multiple DOF elements. A general method for the analysis of multiply coupled, multiple DOF transducer arrays will be presented. The equations describing the general case have the same form as those used previously. As a consequence, many of the lessons learned from experience on arrays of single DOF elements can be extended in a straightforward manner. As an example, the concept of velocity control will be explored in different examples of multimode arrays. It will be shown that velocity control can be achieved in only a special class of multimode transducer elements—those having as many control ports (i.e., electrical) as there are structural modes.

The mechanics of multi-joint posture and movement control.

The dependence of muscle force on muscle length gives rise to a "spring-like" behavior which has been shown to play a role in the execution of single-joint posture and movement. This paper extends this concept and considers the influence of the apparent mechanical behavior of the neural, muscular and skeletal system on the control and coordination of multiple degree of freedom posture and movement. A rigorous definition of "spring-like" behavior is presented. From it a numerically quantifiable, experimental test of spring-like behavior is formulated. It is shown that if the steady-state force-displacement behavior of a limb is not spring-like, this can only be due to the action of inter-muscular feedback, and can not be due to intrinsic muscle properties. The directional character of the spring-like behavior of a multiple degree of freedom system is described. The unique way in which synergistic coactivation of polyarticular muscles may modulate the directional properties of the spring-like behavior of a multiple degree of freedom system is explained. Dynamic aspects of postural behavior are also considered. The concept of mechanical impedance is presented as a rigorous dynamic generalisation of the postural stiffness of the limb. The inertial behavior of the system is characterised by its mobility. As with the stiffness or impedance, in the multiple degree of freedom case it has a directional property. The way in which the apparent kinematic redundancy of the musculo-skeletal system may be used to modify its dynamic behavior is explained. Whereas the inertial behavior of a single limb segment is not modifiable, it is shown that the apparent inertial behavior of a multiple degree of freedom system may be modulated by repositioning the joints. A unified description of the posture and movement of a multi-joint system is presented by defining a "virtual trajectory" of equilibrium positions for the limb which may be specified by the neuro-muscular system. The way in which this approach may lead to a simplification of some the apparent computational difficulties associated with the control of multi-joint motion is discussed.

Deconvolution method between kinematic interaction and dynamic interaction of soil-foundation systems based on observed data

To prove the effect of kinematic interaction experimentally, earthquake records both at the ground and at the foundation are needed. Since the foundation motion is effected not only by the kinematic interaction but also by the dynamic interaction of a soil-foundation system, a general deconvolution method between the kinematic interaction and the dynamic interaction of a soil-foundation system based on the observed data is derived in this paper, where a structure is modelled by a multiple-degree-of-freedom (MDOF) system which represents the dynamic interaction, and the unknown foundation input motion is expressed by a moving average model (MA-model) of the ground motion, which represents the filtering effect of kinematic interaction. Generally, as the parameters of a MDOF system can not be determined exactly, these parameters are treated as random variables on trial using the Monte Carlo simulation method. Next, the coefficients of the MA-model are identified applying the Kalman filter. On the many trials, the result which minimizes the root-mean-square error between the observed response and calculated response is determined as the best estimation. Examples of (1) two single-degree-of-freedom systems arranged in a series and (2) a four-storey reinforced concrete school building are demonstrated to verify the proposed method.

The uncoupling criteria for subsystem seismic analysis

This paper discusses the uncoupling effects of a subsystem from the system based on frequency, mode shape and response variations. The two-mass system is first used to study the problem, and a closed form solution of the frequency variation is derived for the two resonant masses with different mass ratios. Since the coupled and uncoupled analyses have a different number of modes, proper selection of modes for frequency variation check is also discussed. The resonance effect of a coupled analysis is shown in the mode shapes. Thus, the closed form solution of mode shapes for two resonant masses is also derived as a function of mass ratio. Since the response variation is a function of input, the response variation of a two-mass system subjected to white noise input is discussed. Since multiple degree of freedom systems are used in the majority of cases, the results of the two-mass system are extended to the multiple degree of freedom systems by the concept of normal modes. Each normal mode can be represented by a single degree of freedom system with equivalent modal mass and equivalent modal spring. Different equations were used in the past to define the equivalent modal masses depending on the objective of the analysis. The method of defining the equivalent modal mass which takes into account the location of a subsystem, is recommended. Once the equivalent two-mass system of the multiple degree of freedom systems and subsystems is derived, the frequency, mode shape, and response variations of the multiple degree of freedom system and subsystem can be assessed. The above coupling/uncoupling analysis is applied to two different situations. The first one is the building-equipment interaction usually with small mass ratios. The second one is the equipment-equipment interaction where the mass ratio can be large. Here, the equipment includes piping systems, pressure vessels, pumps, etc. The uncoupling analysis of the first case is required because the equipment information is not available during the building analysis. The uncoupling analysis of the second case is required due to practical need to reduce the size of the model. The recommendations of the uncoupling analysis of both cases are presented.