Appendix B: Major Exercises
This appendix provides three major exercises, not specific to any particular chapter
of the book. To solve them you need to integrate subjects dealt with in several
chapters. The exercises can be solved step-by-step while proceeding through the
relevant chapters, or as a final check-up or as course work. Most sections of
questions form entities, related to specific chapters, which can be solved more or
less independently of other questions.
B.1 Tension Control of Rotating Shafts
Background. Lightweight transmission shafts may operate at supercritical operating speeds. Hence, such shafts may have to pass one or more resonance frequencies during startup and shutdown. Accelerations through resonance can cause
excessive vibrations, which in turn may cause considerable damage or produce
unacceptable noise. Some suggestions for preventing vibrations building up during
passage of resonance are based on controlling the stiffness of the shaft system. By
this approach the natural frequencies are displaced in a controllable manner during
startup and shutdown. The stiffness of the system can be controlled by varying the
stiffness of the supports (e.g., by using shape memory alloys). Alternatively the
bending stiffness of the shaft can be controlled.
Turkstra and Semercigil (1993) suggested varying the bending stiffness of shafts
by applying controlled tension forces. They proposed changing the stiffness
instantaneously at pre-determined moments of the startup/shutdown cycle. This is
an open-loop control, since no feedback of measured performance is involved in the
decision on control action.
Fig. B.1 shows the system as suggested by Turkstra and Semercigil, though,
with two extensions: 1) The pneumatic cylinder delivering axial force can provide
compression as well as tension; 2) The control is supplemented by a closed-loop
© The Editor(s) (if applicable) and The Author(s), under exclusive license to
Springer Nature Switzerland AG 2021
J. J. Thomsen Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9
457
This appendix provides three major exercises, not specific to any particular chapter
of the book. To solve them you need to integrate subjects dealt with in several
chapters. The exercises can be solved step-by-step while proceeding through the
relevant chapters, or as a final check-up or as course work. Most sections of
questions form entities, related to specific chapters, which can be solved more or
less independently of other questions.
B.1 Tension Control of Rotating Shafts
Background. Lightweight transmission shafts may operate at supercritical operating speeds. Hence, such shafts may have to pass one or more resonance frequencies during startup and shutdown. Accelerations through resonance can cause
excessive vibrations, which in turn may cause considerable damage or produce
unacceptable noise. Some suggestions for preventing vibrations building up during
passage of resonance are based on controlling the stiffness of the shaft system. By
this approach the natural frequencies are displaced in a controllable manner during
startup and shutdown. The stiffness of the system can be controlled by varying the
stiffness of the supports (e.g., by using shape memory alloys). Alternatively the
bending stiffness of the shaft can be controlled.
Turkstra and Semercigil (1993) suggested varying the bending stiffness of shafts
by applying controlled tension forces. They proposed changing the stiffness
instantaneously at pre-determined moments of the startup/shutdown cycle. This is
an open-loop control, since no feedback of measured performance is involved in the
decision on control action.
Fig. B.1 shows the system as suggested by Turkstra and Semercigil, though,
with two extensions: 1) The pneumatic cylinder delivering axial force can provide
compression as well as tension; 2) The control is supplemented by a closed-loop
© The Editor(s) (if applicable) and The Author(s), under exclusive license to
Springer Nature Switzerland AG 2021
J. J. Thomsen Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9
457