Force Transmissibility 6k4m4j

Introduction Consider a machine with rotating unbalance. During its operation, the vibration produced by it will be transmitted to the foundation upon which it is installed. eg: Industrial motor installed on a cantilever section Now consider a second case where the foundation is disturbed with a particular frequency. This disturbance will be transmitted to the mass resting on the foundation. eg: Wall clock in a moving bus In both the cases, the transmission of the disturbance is generally not desirable. FORCE TRANSMISSIBILITY (for the first case) is defined as the ratio of the force transmitted to the foundation to the force impressed upon the system. Experimental Apparatus: i. ii. iii. iv.

Body of mass ‘M’ Spring of stiffness value ‘k’ Dashpot with damping coefficient ‘C’ Foundation which is excited by a force F = Fo sinwt, where Fo = amplitude of impressed force w = frequency of the impressed excitation to the foundation

The is expected to enter ‘non-negative, real number values’ for M, k, C, Fo and w in the spaces provided in the GUI.

Various Parameters Required: Natural frequency, Wn: The number of oscillations of the mass per second in case of undamped condition. It is given by: Wn = √ Critical damping coefficient: The value of C which gives equal roots to the characteristic equation of the differential equation for damped vibration: Mx’’ + Cx’ + kx = 0 It is given by:

Cc = 2MWn = 2√

Damping ratio, C/Cc: It is ratio of the damping coefficient of the system to the critical damping coefficient.

Damped natural frequency, Wd: The number of oscillations executed by the mass per second in damped condition. It is given by: Wd = √

x Wn

Peak natural frequency, Wpk: The maximum number of oscillations executed by the mass in one second. Wpk = √

x Wn

W/Wn: It is the ratio of the angular frequency of impressed disturbance (W) to that of the natural frequency (Wn) of the system.

Derivation of the transmitted force Ftr: The four forces acting on the system: Fo, CwX, kX and Mw2X where X is the displacement of the foundation. From D’Alembert Principle, we get,

or,

Fo2

= (CwX)2 + (kX – Mw2X)2

X

=

…………….1

√

The impressed force, F is being transmitted to the mass through the spring and the dashpot. Hence the forces common to the mass and the foundation are: kX and CwX. or,

Ftr

=√ =X√ =

√ √

………From 1

INFERENCES FROM THE PLOT

   

The plot starts at unity value of transmissibility, i.e. for W=0, whole of impressed force F is transmitted to the system. The curves through unity transmissibility again for W/Wn = √ The transmitted for tends to zero as W tends to 0. The plot can be divided into three regions: o Region I  This region is stiffness controlled. Higher value of ‘k’ gives a higher value of natural frequency and consequently a lower value of W/Wn. o Region II  The transmitted force is always higher that the impressed force.  This region is damping controlled. Increasing damping value brings down the amplitude of Ftr/F0. o Region III  The transmitted force is always lower than the impressed force.  This region is mass controlled. Larger value of mass gives reduced value of Wn, therefore, higher value of W/Wn. (Hence, for effective vibration isolation, Wn<<W.)  Increasing the damping in this region aids in better transmissibility.

Hence, if Ftr > F0, the is advised to increase the value of (i) mass or (ii) stiffness to minimize force transmissibility,