Spring-mass-damper system

Find the differential equation relating the displaeements X[ t) and Xo t) for the spring-mass-damper system shown in Figure 2.5. What would be the effeet of negleeting the mass ... 

Fig. 2.6 Free-body diagram for spring-mass-damper system.

Henee a spring-mass-damper system is a seeond-order system. If the mass is zero then... 

Find the value of the eritieal damping eoeffieient Q spring-mass-damper system shown in Figure 3.17. 

A spring-mass-damper system has a mass of 20 kg, a spring of stiffness 8000 N/m and a damper with a damping eoeffieient of 80Ns/m. The system is exeited by a eonstant amplitude harmonie foreing funetion of the form... 

Fig. 8.1 Spring-mass-damper system and free-body diagram.

For the spring-mass-damper system given in Example 8.1, Figure 8.1, the state equations are shown in equation (8.13)... 

Flarmonic Excitation of a SDOF System In this section, we will see the behavior of the spring mass damper model when we add a harmonic force in the form below (Fig. 3). 

Based on its operation principle, the resonator can be modeled as a simple spring-mass-dashpot system, as shown in Fig. 2, with the shuttle being the proof mass, folder beams being the spring, and the surrounding air being the dashpot damper. The displacement of the proof mass can then be obtained by solving the second-order differential equation... 

A complex structural system, such as frame structures representing buildings, bridges or mechanical systems, can be assembled from components which are formulated as reciprocal structures. Reciprocal structures are those structures characterized by convex potential and dissipation functions (Stern, 1965). In this section, the concept of reciprocal structures is explained using simple spring-mass-damper-slider models shown in Figure 1. Mixed Lagrangian and Dissipation functions of such systems are derived for various structural components. 

Example 4.1. C Code—Comparison of ideal and approximate solutions of the mass/ spring/damper system. 

Predictably, most systems that produce sound are more complex than the ideal mass/spring/damper system. And of course, most sounds are more complex than a simple damped exponential sinusoid. Mathematical expressions of the physical forces (thus the accelerations) can be written for nearly any system, but solving such equations is often difftcult or impossible. Some systems have simple enough properties and geometries to allow an exact solution to be written out for their vibrational behavior. A string under tension is one such system, and it is evaluated in great detail in Chapter 12 and Appendix A. For... 

Figure 10.1. Nonlinear mass/spring/damper system.

Figure 10.2. Nonlinear mass/spring/damper system response. Initial displacements are 0.1 (left) 1.0 (center) and 10.0 (right).

Track 39] Nonlinear Mass/Spring/Damper System. 

Figure 11.3 A mass/spring/damper system driven by an external force. Here a periodic force is applied, and this will determine the frequency at which the system oscillates. As the driving frequency nears the natural resonance frequency (determined by the mass and spring) the size of the oscillations will increase.

An extensive study into glottal modelling is given in Flanagan [164], which describes various mass/spring/damper systems. These models can be somewhat difficult to model in discrete-time systems, so instead we adopt models that simply generate a time-domain function that has the properties described above. One such model [368], [376] is given by... 

Simple Lumped Models. At frequencies up to several hundred hertz, the biodynamic response of the human body can be represented theoretically by point masses, springs, and dampers, which constitute the elements of lumped biodynamic models. The simplest one-dimensional model consists of a mass supported by a spring and damper, as sketched in Fig. 10.6, where the system is excited at its base. The equation of motion of a mass m when a spring with stiffness k and damper with resistance proportional to velocity, c, are base driven with a displacement x it) is ... 

---