Control Engineering (2151908) G-14 Equivalent Mechanical circuits Prepared by:- 130110119005

The equivalent mechanical network is drawn as follows:- Total number of nodes=Total number of Displacements= Total number of masses Take one reference node in addition to represent the ground line(or reference) (a) Mass has only one displacement x(t). Connect it between the node x and the reference node. (b)Spring and Damper have two possible displacements x1 (t) And x2(t). Connect these elements between x1(t) and x2(t) or between x1(t) and the reference node.

 

4. Draw the nodes and connect the different elements accordingly 5 4. Draw the nodes and connect the different elements accordingly 5. The excitation to the system f(t) is represented alongwith the direction at the appropriated node where it is acting. 6. Apply the usual method of nodal analysis (as in KCL) and develop the system equations at each node using Newton’s third law of motion

Example 1 Mechanical translation system. Consider the spring -mass-dashpot system shown in Fig. . Let us obtain the transfer function of the system by assuming that the force F(t) is the input and the displacement y(t) of the mass is the output. (7.7)

 

Example 2 Seismograph. figure shows a schema of seismograph. A seismograph indicates the displacement of its case relative to the inertial reference. It consists of a mass m fixed to the seismograph case by a spring of stiffness k , b is a linear damping between the mass and the case. Let us define: xi = displacement of the seismograph case relative to the inertial reference (the absolute frame) x0 = displacement of the mass m within the seismograph relative to its case y = x0 – xi = displacement of the mass m relative to the case. Note that y is the variable we can actually measure.Since gravity produces a steady spring deflection, we measure the displacement x0 of mass m from its static equilibrium position. Considering xi as input an y as output, the transfer function is

 

Multivariable mechanical system. Example 3 Multivariable mechanical system.  Consider the system shown in Fig. We assume that the system, which is initially at rest, has two inputs F1(t) and F2(t) and two outputs x1(t) and x2(t).In this problem we would like to demonstrate formulation of a multivariable system transfer-function description.

where the transfer-function matrix The equations describing the multipole diagram model of the system given in  are      F1 F2 1 b1 + sm1 + k1/s –b1 –1 2 b1 + sm2 + k2/s

Dynamic Vibration Absorber Example 4 Dynamic Vibration Absorber A dynamic vibration absorber consists of a mass and an elastic element that is attached to another mass in order to reduce its vibration. The figure is a representation of a vibration absorber attached to a cantilever support

Example 5 Rolling Machine: A model of a commercial rolling machine (for metals processing) has been created and is shown above. The model comprises a cylinder with a mass, m, with a radius, r that spins about an axle. The cylinder rolls without slip on the lower surface. Attached to the axle housing are a damper (b) , a spring (k) and a force source( f).

Cont.  

 

From MATLAB, plotting TF for step input, we get,

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