# A free fall of a material point

A free fall of a material point is being examined. In order to find the solution of the considered example formulas for an accelerated motion and kinetic and potential energy in the gravitational field are used.

$$v = v_0 + a \cdot t$$
$$s = v_0 \cdot t + \frac{a \cdot t^2}{2}$$
$$E_k = \frac{m \cdot v^2}{2}$$
$$E_p = m \cdot g \cdot h$$

A free fall of a material point

# Transmittance and state space representation equations designation

The considered electrical circuit is analyzed as an object which is to be controlled automatically, therefore, its transmittance and state space representation equations will be written down.

Transmittance is given by formula

$$H(s) = \frac{Y(s)}{U(s)}$$

State space representation equations

$$\dot{ \textbf{x} } = A \cdot \textbf{x} + B \cdot \textbf{u}$$ $$\textbf{y} = C \cdot \textbf{x} + D \cdot \textbf{u}$$

An example which is solved step by step for the considered electric circuit can be found here

Transmittance and state space representation

# An electrical circuit solved with the branch current method

In the examined electric circuit the currents and voltages will be computed. The Kirchhoff’s current law (KCL) and the Kirchhoff’s voltage law (KVL) will be employed.

branch current method – electric circuit

# A dynamic model of a series RLC circuit

An example where a transfer function H(s) and the state space representation equations are defined for a series RLC circuit. As a result the dynamic model of the considered electrical system will be obtained in two forms. The considered electrical circuit is composed of three components: resitor, capacitor and inductor.

http://www.mbstudent.com/control-theory-state-space-representation-RLC-circuit-example-1.html

# Total capacitance

Capacitance C is a one of basic parameters of electric circuits next to resistance R and inductivity L. Capacitance C is defined as relation of charge Q to voltage V → C=Q/V. The measurement unit of capacitance is Farad → [C]=1F, Farad is a derived unit of SI system. Sometimes it is essential to calculate capacitance of electrical circuit which contains a few capacitors in its topology, therefore, it is often said that total capacitance of electrical circuit is computed. Sometimes during circuits analysis a subject is to calculate the total capacitance which is seen from specific circuit’s terminals.

Total capacitance

# Nodal analysis

Nodal analysis is one of methods used for electrical networks analysis. Nodal analysis is based on Kirchhoff’s current law. Main idea of this method is to calculate electrical potentials of every node. This will allow to calculate voltages in branches since voltage is a difference of potentials. This approach has one rule which requires to assume that potential of one chosen node has be equal zero volts. Symbolically this chosen node is connected to the ground on electrical diagram.

Node voltage method

# Dynamics of robot

During robot’s motion various kinds of torques and forces works on robot. Some components of load is worn by robot’s construction. Rest of loads have to be equivalent by robot’s drives. If we construct a robot we have to know values of forces and torques because that knowledge is a base information for servodrives control. In this project dynamic forces and torques will be calculated for considered robot.

# Equilibrium equations

Beam is hanged to the ceiling via two ropes. Masses of ropes are so small that it is possible to omit them in computation. Beam has mass m[kg]. Force F[N] is placed in specified beam’s point. Static equilibrium equations for system have to be calculated. Reaction forces in ropes also have to be calculated. Beam is in gravity field. Beam has length 8·a. Gravity force which works on beam has to be placed in beam’s gravity center C.

# Oblique throw – motion equation

Material point is thrown with start velocity v0. Between ground plane and vector of velocity v0 is an angle α. Material point is inside gravity field described by gravity acceleration g. Vector of gravity acceleration g is perpendicular to ground plane. It is assumed that move takes place in vacuum. It means that there is no air resist. Subject of example is to find equation of trajectory and value of angle which will provide maximal range of throw.