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**Tuned Mass Dampers a mass that is connected to a structure**

by a spring and a damping element without any other support,in order to reduce vibration of the structure.

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**Tuned mass dampers are mainly used in the following applications:**

tall and slender free-standing structures (bridges, pylons of bridges, chimneys, TV towers) which tend to be excited dangerously in one of their mode shapes by wind, Taipeh 101

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**stairs, spectator stands, pedestrian bridges excited by marching or**

jumping people. These vibrations are usually not dangerous for the structure itself, but may become very unpleasant for the people,

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**steel structures like factory floors excited in one of their natural**

frequencies by machines , such as screens, centrifuges, fans etc.,

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**ships exited in one of their natural frequencies by the main engines**

or even by ship motion.

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SDOF System eigenfrequency: damping ratio of Lehr:

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Thin structures with low damping have a high peak in their amplification if the frequency of excitation is similar to eigenfrequency → High dynamic forces and deformations Solutions: Strengthen the structure to get a higher eigenfrequency Application of dampers Application of tuned mass dampers

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**Strengthen the structure to get a higher**

eigenfrequency Eigenfrequency of a beam: Doubling the stiffness only leads to multiplication of the eigenfrequency by about 1.4. Most dangerous eigenfrequencies for human excitation: Hz

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**Application of dampers**

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**Application of tuned mass dampers**

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2 DOF System

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**differential equations:**

solution:

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**linear equation system by derivation of the solution**

and application to the differential equations:

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static deformation: eigenfrequencies: ratio of frequencies: Damping ratio of Lehr: mass ratio:

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**All lines meet in the points S and T**

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→ → →

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**for the smallest deformation:**

Optimisation of TMD for the smallest deformation: → Optimal ratio of eigenfrequencies: → Optimal spring constant Minimize → Optimal damping constant:

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Ratio of masses: The higher the mass of the TMD is, the better is the damping. Useful: from (low effect) up to 0.1 (often constructive limit) Ratio of frequencies: Damping Ratio of Lehr:

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**Adjustment: Different Assumptions of Youngs Modulus and Weights**

Increased Main Mass caused by the load

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Problem: Large displacement of the damper mass Plastic deformation of the spring Exceeding the limit of deformation

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Realization

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**damping of torsional oscillation**

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400 kg - 14 Hz

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Millennium Bridge

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**Mass damper on an electricity cable**

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Pendular dampers

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Mechanical Vibrations In many mechanical systems: The motion is an oscillation with the position of static equilibrium as the center.

Mechanical Vibrations In many mechanical systems: The motion is an oscillation with the position of static equilibrium as the center.

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