A New Mathematical Modeling Method for Four-Stage Helicopter Main Gearbox and Dynamic Response Optimization

Table 1

Four-stage helicopter main gearbox parameters.

Element	Node	Tooth number	Module	Face width (m)	Transmission error (μm)	Initial phase of transmission error	Initial phase of meshing stiffness	Shaft; (m)	Shaft length (m)
Shaft element	1(i)	N/A	N/A	N/A	N/A	N/A	N/A	0.04/0.03	0.8
Meshing element A	2(i)	28	5	0.04	12;	π/3;	0; 0; 0	N/A	N/A
					18;	π/9;
					14	4π/3
	3(i)	86	5	0.04	12;	π/3;	0; 0; 0	N/A	N/A
					18;	π/9;
					14	4π/3
Shaft element	4(i)	N/A	N/A	N/A	N/A	N/A	N/A	0.06/0.05	1.6
Meshing element B	5(i)	38	4.5	0.35	22;	−π/9;	π/3;	N/A	N/A
					17;	−4π/3;	π/9;
					15	0	π/3
	6(i)	88	4.5	0.35	22;	−π/9;	π/3;	N/A	N/A
					17;	−4π/3;	π/9;
					15	0	−π/3
Shaft element	7(i)	N/A	N/A	N/A	N/A	N/A	N/A	0.06/0.04	2.2
Meshing elements C and D	8(i)	65	3	0.42	8;	−π;	0; 0; 0	N/A	N/A
					12;	0;
					10	−π/6
	9	144	3	0.42	8;	−π;	0; 0; 0	N/A	N/A
					12;	0;
					10	−π/6
	10	70	3	0.42	10	−π/6	0	N/A	N/A
Shaft element	11	N/A	N/A	N/A	N/A	N/A	N/A	0.04/0.03	1.4
Meshing element E	12	40	3.5	0.47	12	π/9	0	N/A	N/A
	13	70	3.5	0.47	12	π/9	0	N/A	N/A
Shaft element	14	N/A	N/A	N/A	N/A	N/A	N/A	0.04/0.03	1.1
Shaft element	15	N/A	N/A	N/A	N/A	N/A	N/A	0.14/0.11	0.8
Meshing elements F and G	16	68	5	0.4	13; 8;	−4π/3; π/9;	π; 0;	N/A	N/A
					6; 8;	−π/7; −π/9;	4π/7;
					12; 5	π/3;-π/9	0; −π/3; 0
	17	37	5	0.4	13; 8	−4π/3; −π	π; −π/3	N/A	N/A
	18	37	5	0.4	8; 9	π/9; 0	0; 4π/3	N/A	N/A
	19	37	5	0.4	6; 7	−π/7; 4π/3	4π/7; −π	N/A	N/A
	20	37	5	0.4	8; 6	−π/9; −π/9	0; −4π/3	N/A	N/A
	21	37	5	0.4	12; 6	π/3; π/9	−π/3; 0	N/A	N/A
	22	37	5	0.4	5; 8	−π/9; −5π/3	0; π	N/A	N/A
	23	142	5	0.4		−π; 0;	−π/3;	N/A	N/A
					8; 9; 7	4π/3; −π/9;	4π/3;
					6; 6; 8	π/9; −5π/3	−π; 4π/3;
							0; π
Shaft element	24	N/A	N/A	N/A	N/A	N/A	N/A	0.12/0.09	2.0