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Force analysis of a kind of two spring system

introduction

compliant parallel mechanism has potential and wide application in robot assembly and robot force control because of its structural flexibility and sensitivity to load. Its typical application example is robot compliant wrist. The mechanical model of compliant parallel mechanism can be simplified as a spring system, and its force analysis includes two basic problems, namely, the inverse analysis and the forward analysis of force. It is an inverse analysis problem to know the external load and find its equilibrium position; Given the length of each spring, it is a positive analysis to calculate the external load. Because plastic analysis involves the solution of highly coupled nonlinear equations, it is much more difficult than positive analysis. At present, several articles [1 ~ 4, 6] at home and abroad have carried out basic force analysis on several different spring systems. The document [1] makes an inverse analysis of the force of a kind of planar two spring system shown in Figure 1, and makes a preliminary analysis of the behavior of the system with the load. In this paper, a new solution method for the force inverse analysis of this kind of two spring system is proposed by using Dixon's resultant. This method is more general than the solution in literature [1]. Through the analysis of the variation law of the elastic potential energy of this kind of spring system with external force, this paper can choose ppgmt40 or ppgmt30 as the base material, puts forward the discriminant of the catastrophic point of the system, and analyzes the variation law of the value of the discriminant when the mechanism is close to the catastrophic point

Figure 11 inverse analysis which is more attractive than wood 1.1 the establishment of the original equation

is shown in Figure 1. One end of the two springs is connected with the frame through fixed hinges a and B, and the other end is hinged together, which is represented by P. A rectangular coordinate system is established with point a as the coordinate origin, and the X axis passes through point B. The original length of the two springs are L01 and L02 respectively, the length after being loaded is L1 and L2 respectively, the spring stiffness is K1 and K2 respectively, the distance between hinge a and B is D, and the included angle between the two springs and the positive direction of X axis is respectively θ 1 and θ 2。 The external load acts on point P, expressed in FX and FY. The problem to be solved in the force inverse analysis of the system is to know the external loads FX and FY, and solve the length L1 and L2 and position angle of the two springs after being loaded θ 1 and θ 2。 In this paper, L01, L02, K1, K2 and D are called the fixed parameters of the system

at point P, according to the force balance equation, fx=k1 (l1-l01) c1+k2 (l2-l02) C2 (1) fy=k1 (l1-l01) s1+k2 (l2-l02) S2 (2) according to the geometric constraints, l2s2=l1s1 (3) l2c2=l1c1-d (4) in the above formula, c1=cos θ 1,s1=sin θ 1,c2=cos θ 2,s2=sin θ 2。 1.2 elimination and solution of the equation

substitute equations (3) and (4) into equations (1) and (2), eliminate L2, and sort it out to get fx-k1 (l1-l01) c1-k2 (l1c1-d) +k2l02c2=0 (5) fy-k1 (l1-l01) s1-k2l1s1+k2l02s2=0 (6) use C2 multiplication formula (3), use S2 multiplication formula (4), and subtract the two new models, eliminate L2, and get l1s1c2-l1c1s2+ds2=0 (2).7) L2 has been eliminated from equations (5) to (7), and only L1 θ 1 and θ 2 for the three unknowns, the trigonometric function in the above equation is expressed by Euler formula: c1= (t1+t-11)/2, s1= (t1-t-11)/2i when the downward pressure of the economy is greater than the functions of automatic hammer suction, automatic zero setting, automatic lifting and avoiding two shocks,

c2= (t2+t-12)/2, s2= (t2-t-12)/2i, where t1=ei θ 1,t2=ei θ 2. Taking T2 as the compression variable at the same time, the above three equations can be written as follows: F (T1, L1) =e11+e12t1+e13l1+e14t21+e15t21l1 (8) g (T1, L1) =e21+e22t1+e25t21l1 (9) H (T1, L1) =e31t1+e32l1+e33t21l1 (10) the coefficients of equations (8) ～ (10) not only contain the fixed parameters of the system, but also contain unknown variables T2. According to Dixon's resultant principle, the following determinant is constructed: this determinant must contain (t1-u) (l1-v) factors, so there is the following formula δ (t1,L1,u,v)= Δ (T1, L1, u, V)/[(t1-u) (l1-v)] in this formula, the highest power of u is 3, V is 0, T1 is 1, L1 is 1. This formula can be written in the following form, δ (T1, L1, u, V) =t2f1 (T1, L1) u3+t22f2 (T1, L1) u2+

t22f3 (T1, L1) u1+t42f4 (T1, L1) U0 where F1 (T1, L1), F2 (T1, L1), F3 (T1, L1), F4 (T1, L1) contain both fixed parameters of the system and unknown variables T2. No matter what value u takes, for the common root of F (T1, L1) =0, G (T1, L1) =0, H (T1, L1) =0, δ Constant to zero, and because T2 ≠ 0, there must be F1 (T1, L1) =0 (11) F2 (T1, L1) =0 (12) F3 (T1, L1) =0 (13) F4 (T1, L1) =0 (14) equations (11) ～ (14) can be written as (15) where

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