The Mechanical Structure Contact Mechanism Analysis Considering Modified Tangential Stiffness with Friction’s Effect

Abstract

This paper proposes a modified tangential contact stiffness model considering friction’s effect, which is the first key step to establish the dynamic model of the fixture-workpiece system, and this is the foundation of vibration suppression for the manufacturing process of aerospace blades. According to Love’s elastic deformation, the model’s derivation process starts with the potential function in each coordinate axis’s direction respectively. The generalized Hertz contact theory is employed to calculate the contact forces in this model. The symmetrical characteristic of the contact area has simplified the derivation process to obtain the eventual tangential contact stiffness model. A validation experiment focusing on a tangential stiffness measuring is achieved by putting two spherical objects in contact together to get the tangential contact stiffness. Based on the data collected in this experiment, a comparison with a most similar existed model is carried out, and the result shows that the relative error of this modified model are all less than 10%, while the original model’s (the most similar model) relative error exceeding 50% captures more than 3/4 of the 30 data sets randomly selected in each experiment group, and that means the modification of this paper brings great improvement to the contact stiffness model.

Effect of asymmetric shroud gap and shroud friction coefficient on vibration response of shrouded blades

“…Where M , C , and K are mass matrix, damping matrix, and stiffness coefficient matrix respectively, and F r is the nonlinear collision force between shrouds, F S is the tangential friction between the shrouds. So far, many mathematicians have used the nonlinear Hertz contact force to describe the contact and collision force between shrouds 18–20 :…”

Self-shrouded torsional blade model equivalence and blade vibration response analysis

“…In the equation, M, C, K are the mass matrix, damping matrix and stiffness coefficient matrix, respectively, F r is the nonlinear collision force between shrouds, θ is the shroud inclination angle, l is the vertical distance from the point of excitation to the shroud, L is the length of the blade body, β L is the torsion angle of the blade, F 0 is the excitation force amplitude, and F s is the tangential friction force between shrouds. At present, many scholars use the generalized Hertz contact force to describe the contact collision force between shrouds: [19][20][21]…”

“…The total equation of motion of the oblique collision between the shrouds of the self-shrouded blades is:In the equation, M , C , K are the mass matrix, damping matrix and stiffness coefficient matrix, respectively, Fr is the nonlinear collision force between shrouds, θ is the shroud inclination angle, l is the vertical distance from the point of excitation to the shroud, L is the length of the blade body, βL is the torsion angle of the blade, F0 is the excitation force amplitude, and Fs is the tangential friction force between shrouds. At present, many scholars use the generalized Hertz contact force to describe the contact collision force between shrouds: 19–21 where kh is the equivalent Hertz contact stiffness, δ is the normal relative displacement of the inter-shroud contact surface, Ch is the damping coefficient function, n is the exponent, and n≥1, δ˙ is the normal relative inter-shroud velocity.

Figure 1.Spring-mass block model.

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