Trifilar Suspension
HERIOT-WATT UNIVERSITY SCHOOL OF ENGINEERING AND PHYSICAL SCIENCES (MECHANICAL ENGINEERING) MECHANICAL ENGINEERING SCIENCE 3 (B58EC1) DYNAMICS LABORATORY: TRIFILAR SUSPENSION
Objective: To calculate the polar moment of inertia of an assembly and using the result to predict the periodic time of a trifilar suspension of the assembly. Theory: The moment of inertia of a solid object is obtained by integrating the second moment of mass about a particular axis. The general formula for inertia is:
where
Ig m k
I g = mk 2 = inertia in kg.m2 about the mass centre = mass in kg = radius of gyration about mass centre in m.
In order to calculate the inertia of an assembly, the local inertia Ig needs to be increased by an amount mh2. where m h = local mass in kg = the distance between parallel axis passing through the local mass centre and the mass centre for the overall assembly.
The Parallel Axis Theory has to be applied to every component of the assembly. Thus
I = ∑ I g + mh 2
(
) mr 2 2
The polar moments of inertia for some standard solids are: Cylindrical solid Circular tube
Square hollow section
I Cylinder =
I tube =
I sq =
( r : radius of cylinder)
m 2 ro + ri 2 2
(
)
( ri and ro : inside and outside radius)
m 2 ( ao + ai2 ) 6
( ai and
ao
: inside and outside length)
An assembly of three solid masses on a circular platform is suspended from three chains to form a trifilar suspension. For small oscillations about a vertical axis, the periodic time is related to the Moment of Inertia.
Page 1/4
1
φ
L
φ
1 2
φ
θ
O 3 2
θ
3
θ
Ø600
Figure 1. Trifilar suspension From Figure 1, the equation of motion is: I d 2θ mgR 2 + θ =0 dt 2 L (1)
Comparing this to the standard equation (2nd order differential equation) for Simple Harmonic Motion (SHM), d2y + ω2 y = 0 dx 2
(2)
the frequency ω in radians/sec and the period T in seconds can be calculated by:
ω= and mgR 2 LI
Objective: To calculate the polar moment of inertia of an assembly and using the result to predict the periodic time of a trifilar suspension of the assembly. Theory: The moment of inertia of a solid object is obtained by integrating the second moment of mass about a particular axis. The general formula for inertia is:
where
Ig m k
I g = mk 2 = inertia in kg.m2 about the mass centre = mass in kg = radius of gyration about mass centre in m.
In order to calculate the inertia of an assembly, the local inertia Ig needs to be increased by an amount mh2. where m h = local mass in kg = the distance between parallel axis passing through the local mass centre and the mass centre for the overall assembly.
The Parallel Axis Theory has to be applied to every component of the assembly. Thus
I = ∑ I g + mh 2
(
) mr 2 2
The polar moments of inertia for some standard solids are: Cylindrical solid Circular tube
Square hollow section
I Cylinder =
I tube =
I sq =
( r : radius of cylinder)
m 2 ro + ri 2 2
(
)
( ri and ro : inside and outside radius)
m 2 ( ao + ai2 ) 6
( ai and
ao
: inside and outside length)
An assembly of three solid masses on a circular platform is suspended from three chains to form a trifilar suspension. For small oscillations about a vertical axis, the periodic time is related to the Moment of Inertia.
Page 1/4
1
φ
L
φ
1 2
φ
θ
O 3 2
θ
3
θ
Ø600
Figure 1. Trifilar suspension From Figure 1, the equation of motion is: I d 2θ mgR 2 + θ =0 dt 2 L (1)
Comparing this to the standard equation (2nd order differential equation) for Simple Harmonic Motion (SHM), d2y + ω2 y = 0 dx 2
(2)
the frequency ω in radians/sec and the period T in seconds can be calculated by:
ω= and mgR 2 LI
References: – Books or publications to support theory and discussion. Lecture notes cannot be used as reference. Identify where the materials are used in your report. Appendix (Optional) – any other information to support the report. Attendance to lab session is compulsory. Any report does not have all the above sections will be rejected and returned, and it will be considered as non-submission. Marks will be deducted for late submission. Page 4/4