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31 M Galal RABIE, Automatic Control for Mechanical Engineers 2. The reduced mass of the spring is negligible. 3. The structural damping of spring material is negligible. 2 Integrating Element The integrating element has a transfer function of the form G(s)=k/s , where k is constant. The following are some examples of integrating members. 3 shows a scheme of a hydraulic cylinder. Neglecting the inertia of moving parts, the effect of oil compressibility and wall elasticity, the relation between the system input (flow rate q) and output (displacement x) can be deduced as follows.

The structural damping of spring material is negligible. 2 Integrating Element The integrating element has a transfer function of the form G(s)=k/s , where k is constant. The following are some examples of integrating members. 3 shows a scheme of a hydraulic cylinder. Neglecting the inertia of moving parts, the effect of oil compressibility and wall elasticity, the relation between the system input (flow rate q) and output (displacement x) can be deduced as follows. 3 Scheme of a hydraulic cylinder q( t )  A where A is the piston rod-side area.

The following are the pairs of Laplace transform of some key functions: No. F(s) f(t) 1.  δ(t) 2. 3. s k k tn n 1 k sa k 4. 5.  6. k t n e at (s  a)n1 k 7. k sin t 2 s  2 ks 8. k cos t s 2  2 k 9. 2 ( s  a)   k e at sin t 2 k ( s  a) 10. k e at cos t ( s  a) 2   2 k 11. 6 EXERCISE 1. Find Laplace transform of the following functions: a) x( t )  1  2t b) x( t )  4(1  e 2t ) c) x( t )  t 2  2 sin 4t d) x( t )  2e 3 t cos 6t e) x( t )  2e 4 t sin 4t 25 M Galal RABIE, Automatic Control for Mechanical Engineers 2.

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