Shock & Vibration Stress-Velocity Examples for Beam Bending
By Tom Irvine
Email: tom@vibrationdata.com
April 4, 2013
______________________________________________________________________________________
These numerical examples are given as applications of the stress-velocity relationship
in Reference 1.
Introduction
Consider the cantilever beam in Figure 1. The beam is subjected to base excitation in Figure 2.
EI, 
L
Figure 1.
y(x, t)
w(t)
Figure 2.
1
The beam has the following properties:
Table 1. Beam Property Overview
Cross-Section
Rectangular
Boundary Conditions
Fixed-Free
Material
Aluminum
Table 2. Beam Properties Values
Width
w
=
1 inch
Thickness
t
=
0.25 inch
Cross-Section Area
A
=
0.25 in^2
Length
L
=
9 inch
Area Moment of Inertia
I
=
0.00130 in^4
Elastic Modulus
E
=
1.0e+07 lbf/in^2
Stiffness
EI
=
1.30e+04 lbf in^2
Mass per Volume
v
=
0.1 lbm / in^3 ( 0.000259 lbf sec^2/in^4 )
Mass per Length

=
0.0250 lbm/in (6.48e-05 lbf sec^2/in^2)
L
=
0.225 lbm (5.83e-04 lbf sec^2/in)

=
0.05 for all modes
Mass
Viscous Damping Ratio
2
Figure 3.
Subject the beam to the base input pulse in Figure 3. The filename is avs.txt.
The corresponding SRS is shown in Figure 4.
3
Figure 4.
The beam is modeled as continuous structure using the method in Reference 2.
The modal transient response is calculated via Matlab script: continuous_beam_base_accel.m
4
The natural frequencies for the first six modes are
Table 3. Natural Frequency Results
Mode
fn (Hz)
PF
Effective
Modal Mass (lbm)
1
98.0
0.0189
0.1379
2
614
0.0105
0.0424
3
1719
0.0061
0.0146
4
3368
0.0044
0.0074
5
5568
0.0034
0.0045
6
8318
0.0028
0.0030
The first six modes account for 93% of the total mass.
Figure 5.
The mass-normalized mode shape or eigenfunction is shown in Figures 5 and 6 for the first and
second modes, respectively.
5
Figure 6.
6
Single Mode Modal Transient
Figure 7.
The relative velocity response is shown in Figure 7 for the case of the fundamental mode only.
The response results are shown in Table 4.
Table 4. Cantilever Beam Response to Base Excitation, First Mode Only
Response Parameter
Location
Value
Relative Displacement
x=L
0.1639 in
Relative Velocity
x=L
100.4 in/sec
Acceleration
x=L
255.3 G
Bending Moment
x=0
92.61 lbf-in
Bending Stress
x=0
8891 psi
7
Both the bending moment and stress in Table 4 are calculated from the second derivative of the
mode shape.
The maximum bending stress  max from Reference 1 is
 2

 max  E ĉ 
y n ( x, t )
 ĉ
 x 2

max
EA
v n, max
I
(1)
where
ĉ
=
Distance to neutral axis
E
=
Elastic modulus
A
=
Cross section area

=
Mass per volume
I
=
Area moment of inertia
v
=
Velocity
Now calculate the stress at the fixed boundary from the relative velocity at the free end.
 max  0.125 in
1.0e + 07 lbf/in^2 0.25 in^2 0.000259 lbf sec^2/in^4  100.4 in/sec 
0.00130 in^4
(2)
The bending stress from velocity is thus
max = 8857 psi
(3)
This is within 1% of the bending stress from the bending moment in Table 4.
8
Single Mode Direct SRS
The SRS in Figure 4 has acceleration of about 100 G at the fundamental frequency of 98 Hz.
The corresponding pseudo velocity is about 63 in/sec. The pseudo velocity is calculated by
dividing the acceleration by the natural frequency in rad/sec.

100 G  386 in / sec
G

2 98 Hz 
2


  63 in/sec
(4)
The approximate relative velocity z max for the single mode model can be calculated as
z max
 1 q̂ 1 max v
(5)
where
1
=
Participation factor for the first mode
q̂ 1 max
=
Maximum mass-normalized eigenfunction for the first mode
v
=
Pseudo velocity
Equation (5) is adapted from Reference 3.
The participation factor is taken from Table 4. The eigenfunction value is taken from Figure 5.
The expected peak relative velocity response is
z max
 0.0189 83(61.43 in/sec)  96 in/sec
(6)
This result is close to the 100.4 in/sec velocity from the modal transient analysis in Table 4.
The resulting relative velocity could then be used to calculate the maximum bending stress.
9
Six Mode Modal Transient
Now repeat the analysis with six modes included.
Table 5. Cantilever Beam Response to Base Excitation, Six Modes
Response Parameter
Location
Value
Relative Displacement
x=L
0.1659 in
Relative Velocity
x=L
117.5 in/sec
Acceleration
x=L
910.2 G
Bending Moment
x=0
98.84 lbf-in
Bending Stress
x=0
9489 lbf/in^2
The bending stress for six included modes is about 7% higher than the analysis with only the first
mode.
Other Examples
An example with an applied force is given in Appendix A. Further examples will be added in
future revisions.
References
1. T. Irvine, Shock and Vibration Stress as a Function of Velocity, Revision E,
Vibrationdata, 2013.
2. T. Irvine, Modal Transient Vibration Response of a Cantilever Beam Subjected to Base
Excitation, Vibrationdata, 2013.
3. T. Irvine, Shock Response of Multi-degree-of-freedom Systems, Revision F,
Vibrationdata, 2010.
4. T. Irvine, The Transverse Vibration Response of a Cantilever Beam Subjected to an
Applied Concentrated Force, Revision C, Vibrationdata, 2013.
10
APPENDIX A
Applied Concentrated Force
Consider a cantilever beam with an applied force at the free end.
y(x,t)
E, I, 
P (t)
L
Figure A-1.
The beam has the same properties as shown in Table 2 in the main text. Its natural frequencies are
the same as shown in Table 3.
Now subject the beam to a sinusoidal force of 1 lbf at its free end. Solve for the steady-state
response using Reference 4. The results are shown in Table A-1.
Table A-1.
Cantilever Beam Steady-state Response to Sine Force at Free End, First Mode Only
Parameter
Location
Response Value
for 80 Hz
Excitation
Response Value
for 98 Hz
Excitation
Response Value
for 120 Hz
Excitation
Displacement
x=L
0.053 in
0.181 in
0.035 in
Velocity
x=L
26.6 in/sec
111.5 in/sec
26.5 in/sec
Acceleration
x=L
34.6 G
177.9 G
51.75 G
Bending Moment
x=0
29.9 lbf-in
102.3 lbf-in
19.86 lbf-in
Bending Stress
x=0
2867 psi
9824 psi
1907 psi
11
Figure A-2. First Mode Only
12
Both the bending moment and stress in Table A-1 are calculated from the second derivative of the
mode shape.
(0, )  2 ĉ
1 E
 IL
v̂(L, )
(A-1)
Y1 (L)
where
ĉ
=
Distance to neutral axis
v̂
=
Velocity of First Mode
Y1
=
Mass-normalized Eigenfunction
1
=
Fundamental frequency (rad/sec)

=
Excitation frequency (rad/sec)
The stress calculated from each of the two methods is the same for each of the three forcing
frequencies as shown in Table A-2.
Table A-2. Cantilever Beam, Bending Stress Comparison, First Mode Only
Excitation
Frequency
(Hz)
Second Derivative Method
Bending Stress (psi)
Velocity Method
Bending Stress (psi)
80
2867
2867
98
9824
9824
120
1907
1907
13