4. Calculation of transients in elastic double-weight electromechanical system of electric drive

4.1 The task

For the generalized rigid and elastic double-weight electromechanical system without and with taking into account the internal viscous friction in the material elastic connection as well as for elastic double-weight mechanical system, shown in the figure 4.1 а, b, c, it’s necessary to:

• write original equation of motion;

• make a structural diagram;

• make a transfer function by control and setting actions;

• write the calculated equation;

•  make a chart of the algorithm solving the problem;

• make the application solution of the problem on a computer;

• modeled on a computer system in the transition process;

• construct graphs transients;

• provide a comparative analysis of the results.

Figure 4.1 Hard electromechanical systems.

Figure 4.2 – Double-weight elastic electromechanical systems.

Figure 4.3 Simplified double-weight elastic electromechanical systems.

In figures 4.1- 4.3 the following notation are used: J₁- moment of inertia of motor armature; J₂- reduced total moment of inertia of elements, which are hard-connected with mechanism; С₁₂ – reduced stiffness coefficient of connections between motor and mechanism; М₁₂ – reduced elastic moment; β₁₂ – coefficient of inner viscous friction; М – electromagnetic moment, which are developed by the motor; М_C1 – moment of motor losses; М_C2 – reduced moment of static resistance of mechanism; ω₁ і ω₂ – angular velocities of motor and mechanism shafts.

4.2 Methodical instructions

4.2.1 Law of voltage change on the armature clips must be accepted in accordance with figure 4.4.

Figure 4.4 – Law of voltage change.

Initial voltage at the armature U₀, velocity of growth К, motor parameters, values of moments of inertia of the motor J₁ and mechanism J₂ and also stiffness coefficient of elastic section С₁₂ of inner viscous friction β₁₂ given in table 4.1.

4.2.2 Value of moment М_с is accepted as equal to the nominal moment of the motor, and value of the moment М_с is equal to the motor moment of losses, which is determined by the value of nominal moment of motor efficiency:

М_C1 = М_H ( 1 / η₁ -1).

4.2.3 Time constant of the armature section:

Тя = Lя / Rя,

where L_Я is the induction of armature section of the motor, which value is calculated by the approximate formula Uman-Linvillya:

,

γ = 0,6 for uncompensated engine;

р = 2 is number of pole parts;

R_я resistance of the armature section of the motor.

Motor constant is found by nominal data of the machine:

С_д = (U_н - І_н R_н) / ω_н.

4.2.4 The calculation must start with tight electromechanical system (Fig.4.1) as the most light, in which there are no elastic ties. Describing the system, consider the parameters of the engine, the law of change of voltage at armature U_я, total moment of inertia J=J₁+J₂ and total moment of static resistance М_С=М_C1+М_C2. When analyzing the transition process should pay attention to the fact that there are no fluctuations in the system, and acceleration is quite fast.

4.2.5 In calculating the elastic two-mass electromechanical system(Fig. 4.2) except for the engine parameters of the law and change the voltage at armature U_я could consider flexible link with the stiffness coefficient C₁₂ and elastic momentМ₁₂. In mathematical description of the system should be guided by the following provisions:

а) elastic connection between the engine and without inertial mechanism and is characterized by a constant stiffness coefficient C₁₂;

b) elastic deformation of the linear chain and corresponds to Hooke law;

c) elastic deformation of the elastic moment is proportional to the level:

М₁₂=С₁₂Δφ = С₁₂(φ₁ - φ₂),

where φ₁ φ₂ – torsion angle of shaft towards the motor and mechanism. It should be noted, that during transient process it increases and the transient becomes oscillatory one. The damping of the oscillations is caused by the damping properties of the motor itself (R_я, L_я).

4.2.6 With inclusion of mechanical damping into the system, caused by internal viscosity friction in material of flexible connection it should be predicted that transients will not become faster. Internal viscosity friction forces are proportional to the difference of the 1^st and the 2^nd mass ω₁ and ω ₂ . Then the braking torque of the viscosity friction М_вт =β₁₂(ω₁ - ω ₂).

The resulting moment of the flexible connection with taking into account internal viscosity friction losses:

М₁₂'== М₁₂ + М_вт = С₁₂ (ω₁ - ω ₂) + β₁₂ (ω₁ - ω₂).

4.2.7 Two-mass flexible electromechanical system is calculated to discover the damping properties of the motor. For this purpose the motor armature circuit is excluded from electromechanical system (fig.4.2) and moment M (fig.4.3) is applied to the output. All the parameters of system remain unchanged, but the moment M value equals to double nominal one (М=2М_Н). It is necessary to make sure that the system is an ideal oscillatory unit. Free oscillation frequency Ω_12. Frequency of free oscillations of the system .

4.2.8 Calculation should be made with the help of computer program. Differential equations of the transients should be substituted to the normal Koshi form, then passing to the finite-difference equations. For example input equation of first mass motion moments М-М_С1-М₁₂=J₁dω/dt, in Koshi form and the design equation will have the form

.

By calculation equation, based on the initial cut-Heeded mustmake diagram of algorithm and calculate the transition tocomputers.

Initial data and conditions

а) initial voltage at armature of the motor U₀, speed increased tension К,  the moment of inertia and other engine parameters and mechanism of the elastic stiffness coefficient level C₁₂, coefficient of internalviscous friction.

б) integration stepΔt, output time  Т, and the end time of calculation of Т_к.

According to the compiled program calculated variable М, М₁₂, ω₁, ω₂, and for the hard-system- М and ω. In print the value of variables and tension at anchor and the time Т, by this data time graphs are built.

4.2.9 At the analysis of obtained results its necessary to:

а) to note on the graphs transients specific points, giving them a score and a physical explanation;

б) to compare between each other the periods (frequencies) of oscillations of all calculated dooouble-weight elastic systems;

c) to explain how the the elastic ties  account can affect the dynamic system of electric drive and when elastic ties can be neglected, considering the system as a rigid mechanical link.

Тable 4.1 – Initial data.	К, V/s	660	645	630	615	600	660	645	630	615	600	660	645	630	615	600
	U0,V	10	15	20	25	30	10	15	20	25	30	10	15	20	25	30
	β12 Nms/ rad	0,0141	0,011	0,0033	0,025	0,036	0,011	0,0144	0,031	0,0042	0,0139	0,011	0,0114	0,011	0,0138	0,025
	Coef. Of stiffnes. С12, Nm/rad	3,53	2,77	0,83	6,3	0,9	2,85	3,6	7,8	1,05	3,45	2,77	2,85	2,77	3,45	6,3
	Red. mom. inertia, J2 kgm^2	0,141	0,11	0,033	0,252	0,036	0,114	0,144	0,312	0,042	0,138	0,111	0,114	0,111	0,138	0,252
	Arm. Mom.of inertia, J1,кkgm^2	0,047	0,037	0,011	0,034	0,012	0,038	0,048	0,104	0,014	0,046	0,037	0,038	0,037	0,046	0,084
	efficiency %	76,5	67	79	76,5	81	77	80	79,5	76	67	74	81	80,5	73	73
	Arm. Resist. Rя,Ohm	2,394	3,544	1,65	2,052	1,326	1,471	1,224	1,362	1,764	2,058	1,918	0,684	0,936	1,338	1,103
	ω, rad/s	110,98	78,53	314,1	265,75	329	167,52	104,7	83,76	230,34	78,55	110,98	247,09	157,05	104,7	78,53
	Armature current, Іа, А	10,5	12,1	11	11,6	11,8	13,2	13,4	13,6	13,8	16,9	17	19,7	21,2	24	23
	U,V	220
	Capacity Р, кWt	1,9	2,0	2	2,1	2,2	2,4	2,5	2,5	2,5	2,8	3	3,7	4	4,2	4,2
	Variant	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

Continue of table 4.1.	К, V/s	660	645	630	615	600	660	645	630	615	600	660	645	630	615	600
	U0,V	10	15	20	25	30	10	15	20	25	30	10	15	20	25	30
	β12 Nms/ rad	0,011	0,0128	0,031	0,011	0,025	0,025	0,011	0,0138	0,025	0,031	0,031	0,06	0,069	0,031	0,0138
	Coef. stiffnes С12, Nm/rad	2,85	3,45	7,8	2,77	6,3	6,3	2,77	3,45	6,3	7,8	7,8	15	17,25	7,73	3,45
	Red. mom. inertia, J2 kgm^2	0,114	0,138	0,21	0,111	0,252	0,252	0,111	0,l38	0,252	0,312	0,312	0,6	0,69	0,309	0,046
	Armature mom. of inertia, J1,kgm^2	0,038	0,046	0,104	0,037	0,084	0,084	0,037	0,046	0,084	0,104	0,104	0,2	0,23	0,103	0,046
	efficiency %	81,5	80,5	77,5	83	79	86,5	85	83,5	83	80	86,5	76,5	79	84,5	86
	Arm. resistance Rя,Ohm	0,399	0,587	0,666	0,47	0,641	0,164	0,281	0,349	0,295	0,469	0,09	0,059	0,042	0,202	0,175
	ω, rad/s	324,57	167,52	83,76	247,09	104,07	314,1	314,1	221,96	157,05	104,7	350,74	78,53	78,53	157,05	314,1
	Armature current, Іа, А	24	29,4	29,6	31,3	32,9	36,5	38,5	39,4	38	42	41,7	50	55	56	57
	U,V	220
	Capacity Р, кWt	4,5	5,5	5,6	6	6	7,1	7,5	7,5	7,5	8	8,1	9	10	11	11
	Variant	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30