4.2. Theoretical Derivation
Equation (4) represents the temperature rise for each cycle under adiabatic conditions [
19], where
is the damping coefficient,
is the maximum displacement of the damper, and
is the specific heat capacity under constant pressure. Traditionally,
is commonly regarded as a constant.
The internal pressure of the damper, as expressed in Equation (6), can be separated into two components: the pressure increase from the previous cycle
and the pressure difference between the two sides of the piston in the current cycle
. The overall pressure can be determined by solving for each component individually.
Based on the dynamic viscosity–temperature and dynamic viscosity–pressure characteristics, the combined influence of oil dynamic viscosity and temperature [
29] and the pressure variation [
11,
13,
30] can be represented by Equation (7), where
is the initial temperature,
T is the temperature following the temperature rise,
stands for the dynamic viscosity at
,
referred to as the dynamic viscosity–temperature coefficient,
represents the dynamic viscosity–pressure coefficient, and p signifies the fluid pressure.
In the above equations, both the pressure variable p and temperature variable t exist, which makes the treatment of practical problems inconvenient. The flow of liquid through clearances involves not only a pressure drop but also heat generation. The pressure drop and temperature rise occur simultaneously within a unified system, and their relationship can be analyzed from an energy perspective.
The energy relationship when oil flows through the clearance per unit time is as follows: (1) The pressure drop (
dp) across the flow results in a loss of energy
q·dp, where
q represents the liquid flow rate. (2) A part of the energy is lost because of the liquid friction caused by the viscosity during the flow. (3) The losses of these two types of energy are converted into heat energy, leading to an increase in the oil temperature. The heat gained by the oil is given by
q·ρ·c·dT, where
ρ is the mass density of the liquid and
c is the specific heat capacity of the liquid. (4) Another portion of the heat energy is dissipated into the surrounding environment through the cylinder wall.
Overall, considering only the pressure-differential flow, the energy dissipated into the surrounding environment was much smaller than the heat generated by the temperature increase. Therefore, it can be assumed that the heat dissipation is balanced by E2, and all energy from the pressure drop is used to increase the oil temperature. Let q represent the liquid flow rate and P1 and T1 represent the oil pressure and temperature at the gap inlet, respectively. Assuming that the oil pressure and temperature at distance x from the inlet are denoted as P and T, respectively, we can observe a pressure drop of (P1 − P) and a temperature rise of T − T1 within this length x.
As a result
where
represents the dynamic viscosity of the fluid when
and
;
represent the dynamic viscosity of the fluid when
and
; and
denotes the thermodynamic pressure, which remains constant for a fixed fluid
.
Substituting Equation (12), and simplifying, we get
Equation (13) enables the computation of the dynamic viscosity of the damping fluid after each temperature increase cycle under long-duration loads.
The changes in temperature and pressure not only significantly influence the dynamic viscosity of the damping fluid, but also induce deformation in the damper cylinder, resulting in an enlargement of the damper gap. The radial deformation caused by pressure during damper operation is also a significant factor to consider.
In Equation (14), represents the operating pressure of the damper, is the inner diameter, is the outer diameter, denotes the elastic modulus of the damper material, and is Poisson’s ratio of the damper material.
The relationship between the temperature and pressure can be approximated as linear. By substituting Equation (1) into Equation (14), the radial deformation of the damper can be determined as
By Equation (15), the radial deformation quantity of the damper sleeve after the next cycle of temperature rise can be calculated.
Equation (4) reflects the temperature increase in the damper after a cycle. In Equation (4), parameter
C is considered constant. However, owing to the influence of the temperature and pressure rise on the dynamic viscosity of the damper fluid, as well as the deformation of the cylinder caused by the temperature and pressure rise, the damping coefficient
C experiences slight variations for each cycle. As a result, the temperature increase for each cycle decreased slightly, and the outcome was dependent only on the damping coefficient
C. By substituting Equations (13), (15) and (16) into the damping coefficient calculation formula in Equation (4), the ratio of the temperature increase between two consecutive cycles is given by Equation (17). Through iterative summation, the temperature increase after 30 cycles could be determined, allowing for the calculation of the increase in pressure, as follows:
Dynamic viscosity can be obtained through Equation (13), and subsequently, the consistency coefficient k can be determined. By employing Equation (16), the gap h is computed. And then, based on Equation (17), the temperature rise for each cycle can be determined. Finally, Equation (6) yields pressure results, and through an iterative approach, the corrected theoretical pressure values are obtained. This method is of paramount importance in evaluating the mechanical performance of dampers.