Question 1.5: An automotive damper similar to that shown in Fig. 1.9 is st......
An automotive damper similar to that shown in Fig. 1.9 is stated by the supplier to produce a linear damping force, in both directions, defined by the equation F = c \dot{x}, where F is the applied force in newtons, \dot{x} is the stroking velocity, in m/s, and the constant c is 1500 N/m/s. A test on the unit involves applying a single-peak sinusoidal force of ±1000 N at each of the frequencies, f = 1.0, 2.0 and 5.0 Hz. Calculate the expected single-peak displacement, and total movement, at each of these frequencies.
The applied force is F = c \dot{x}, where F = 1000 \sin (2\pi ft). The expected velocity, \dot{x}, is therefore:
\dot{x} = \frac{F}{c} =\frac{1000 \sin (2\pi ft)}{1500} (A)
The displacement x is given by integrating the velocity with respect to time:
x=-\frac{1000\cos (2\pi ft)}{1500(2\pi f)} + x_{0} (B)
where the initial displacement x_{0} is adjusted to mid-stroke. The single-peak displacement, \left|x\right| , measured from the mid-stroke position, is therefore:
\left|x\right|=\frac{1000}{1500\times 2\pi f}=\frac{0.106}{f} m (C)
The expected single-peak displacement and total movement at each test frequency are given in Table 1.2.
| Table 1.2 | ||
| Frequency f (Hz) | Single-peak displacement \left|x\right| (m) | Total movement of damper piston (m) |
| 1 | 0.106 | 0.212 |
| 2 | 0.053 | 0.106 |
| 5 | 0.021 | 0.042 |