EXPERIMENT 2 FREE VIBRATIONS OF A CANTILEVER BEAM WITH A LUMPED MASS AT FREE END 2.1 Objective of the experiment: To experimentally obtain the fundamental natural frequency and the damping ratio of a cantilever beam having lumped mass at free end and to analyze the free vibration response of a cantilever beam subjected to an initial disturbance. This virtual experiment is based on a theme that the actual experimental measured vibration data are used. 2.2 Basic Definitions Free vibration takes place when a system oscillates under the action of forces inherent in the system itself due to initial disturbance, and when the externally applied forces are absent. The system under free vibration will vibrate at one or more of its natural frequencies, which are properties of the dynamical system, established by its mass and stiffness distribution. In actual practice there is always some damping (e.g., the internal molecular friction, viscous damping, aero-dynamical damping, etc.) present in the system which cause the gradual dissipation of vibration energy and it result gradual decay of amplitude of the free vibration. Damping has very little effect on natural frequency of the system, and hence the calculations for natural frequencies are generally made on the basis of no damping. Damping is of great importance in limiting the amplitude of oscillation at resonance. The relative displacement configuration of the vibrating system for a particular natural frequency is known as the mode shape (or eigen function in continuous system). The mode shape corresponding to lowest natural frequency (i.e. the fundamental natural frequency) is called as the fundamental (or the first) mode. The displacements at some points may be zero. These points are known as nodes. Generally nth mode has (n-1) nodes (excluding end points).The mode shape changes for different boundary conditions of the beam.
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Free Vibrations of Cantilever Beam With a Lumped Mass at Free End
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EXPERIMENT 2
FREE VIBRATIONS OF A CANTILEVER BEAM
WITH A LUMPED MASS AT FREE END
2.1 Objective of the experiment:
To experimentally obtain the fundamental natural frequency and the damping ratio of a cantilever beam
having lumped mass at free end and to analyze the free vibration response of a cantilever beam subjected
to an initial disturbance. This virtual experiment is based on a theme that the actual experimental
measured vibration data are used.
2.2 Basic Definitions
Free vibration takes place when a system oscillates under the action of forces inherent in the system itself
due to initial disturbance, and when the externally applied forces are absent. The system under free
vibration will vibrate at one or more of its natural frequencies, which are properties of the dynamical
system, established by its mass and stiffness distribution.
In actual practice there is always some damping (e.g., the internal molecular friction, viscous damping,
aero-dynamical damping, etc.) present in the system which cause the gradual dissipation of vibration
energy and it result gradual decay of amplitude of the free vibration. Damping has very little effect on
natural frequency of the system, and hence the calculations for natural frequencies are generally made on
the basis of no damping. Damping is of great importance in limiting the amplitude of oscillation at
resonance.
The relative displacement configuration of the vibrating system for a particular natural frequency is
known as the mode shape (or eigen function in continuous system). The mode shape corresponding to
lowest natural frequency (i.e. the fundamental natural frequency) is called as the fundamental (or the first)
mode. The displacements at some points may be zero. These points are known as nodes. Generally nth
mode has (n-1) nodes (excluding end points).The mode shape changes for different boundary conditions
of the beam.
2.3 Theoretical natural frequency for cantilever beam
Figure 2.1 (a) A cantilever beam
Figure 2.1 (b) The beam under free vibration (without mass at free end)
A cantilever beam with rectangular cross-section is shown in Figure 2.1(a), the bending vibration can be
generated by giving an initial displacement at the free end of the beam. Figure 2.1(b) shows a cantilever
beam under the free vibration.
When a system is subjected to free vibration and the system is considered as a discrete system in which
the beam is considered as mass-less and the whole mass is concentrated at the free end of the beam. The
governing equation of motion for such system will be
..
0m y t ky t (1.1)
where m is a concentrated mass at the free end of the beam and k is the stiffness of the system. The
transverse stiffness of a cantilever beam is given as (using strength of materials deflection formula,
Timoshenko and Young, 1961)
3
3EIk
l (1.2)
Where E is the Young’s modulus of the beam material (it can be obtained by the tensile test of the
standard specimen). The fundamental undamped circular natural frequency of the system is given as
nf
k
m (1.3)
Where, m is an equivalent mass placed at the free end of the cantilever beam (of the beam and sensor
masses), on substituting equation (1.2) into equation (1.3) we get
3
3nf
EI
ml (1.4)
The undamped natural frequency is related with the circular natural frequency as
2
nf
nff (1.5)
I the moment of inertia of the beam cross-section and for a circular cross-section it is given as
4
64I d (1.6)
Where, d is the diameter of cross section, and for a rectangular cross section
3
12
bdI (1.7)
Where b and d are the breadth and width of the beam cross-section as shown in Figure 2.2. Dimensions of
the beam material are given in Table 2.2.
b
d
Fig 2.2 A rectangular cross-cross of the beam
In case of the test specimen, the beam mass is distributed over the length. By taking a one-third of the
total mass of beam at the free end (Thompson. 1961), the system can be assumed as discrete system.
Hence,
33
140bm m (1.8)
Where bm is the mass of beam and is given as
bm V bdl
Where,
is the mass density of the beam material and V is the volume of the beam from the fixed end to
the free end.
The equivalent tip mass of a cantilever beam can be obtained as follows. Consider a cantilever beam as
shown in Fig.2.3 (a). Let 1m be the mass of the beam per unit length, l is the length of the beam,
1bm m l is total mass of the beam, and
maxv is the transverse velocity of the free end of beam and f is
the force applied, E is the young’s modulus of the beam and I is the moment of inertia of the beam.
Fig. 2.3(a) A cantilever beam with distributed mass Fig. 2.3(b) The cantilever beam with a tip mass
Consider a small element of length dx at a distance x from the free end (Fig.2.3 (a)). The beam
displacement at this point is given by (Timoshenko and Young, 1961)
3
2 3
3
1( ) 3
2 3
fly x lx x
l EI (1.9)
Here
3
3
fl
EI is the deflection at free end of the cantilever beam. Now the velocity of the small element at
distance x is given by
2 3
max3
3( )
2
lx xv x v
l
Hence, the kinetic energy of the element is given by
22 3
max3
1 3
2 2
lx xdT Adx v
l
and the total kinetic energy of the beam is
22 22 3 2 5 6 7
2 4 5 6max maxmax3 6 6
0 0 0
2 72 2maxmax max6
1 3 9 69 6
2 2 8 8 5 6 7
33 1 33 1 33
8 35 2 140 2 140
ll lAv Avlx x l x lx x
T A v dx l x lx x dxl l l
Av lAl v Al v
l
(1.10)
where 1 bm l m . If we place a mass of
33
140bm at the free end of the beam and the beam is assumed to be
of negligible mass, then
Total kinetic energy possessed by the beam = 2
max
1 33
2 140bm v (1.11)
Hence two systems are dynamically same. Therefore the continuous system of cantilever beam can be
changed to single degree freedom system as shown in Fig.2.3(b) by adding the 33
140bm of mass to its free
end.
Values of the mass density for various beam materials are given in Table 2.1. If any contacting type of
transducer is used for the vibration measurement, it should be placed at end of the beam and then the mass
of transducer has to be added into the equivalent mass of the beam at the free end of the beam during the
natural frequency calculation. If tm is the mass of transducer, then the total mass at the free end of the
cantilever beam is given as
33
140b tm m m (1.12)
2.4 Experimental setup
Figure 2.4 An experimental setup for the free vibration of a cantilever beam
The experimental setup is consists of a cantilever beam, transducers (strain gauge, accelerometer, laser
vibrometer), a data-acquisition system and a computer with signal display and processing software (Fig.
2.4). Different types of beam materials and its properties are listed in Table 2.1. Different combinations of
beam geometries for each of the beam material are summarized in Table 2.2
Accelerometer is a sensing element (transducer) to measure the vibration response (i.e., acceleration,
velocity and displacement). Data acquisition system takes vibration signal from the accelerometer and
encode it digital form. Computer acts as a data storage and analysis system, it takes encoded data from
data acquisition system and after processing (e.g., FFT) it display on the computer screen by using
analysis software.
Table 2.1 Material properties of various beams
Material Density (kg/m3) Young’s modulus (N/m
2)
Steel 7850 2.1×1011
Copper 8933 1.2×1011
Aluminum 2700 0.69×1011
Table 2.2 Different geometries of the beam
Length, L, (m) Breadth, b, (m) Depth, h, (m)
0.45 m 0.02 m 0.003 m
0.65 m 0.04 m 0.003 m
Example 2.1 Obtain the undamped natural frequency of a steel beam with l = 0.45 m, d = 0.003
m, and b = 0.02 m. The mass of transducer at the free end =18.2 gm