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Benchmark Example No. 26 Response of a SDOF System to Harmonic Exci- tation SOFiSTiK | 2018
Benchmark Example No. 26
Response of a SDOF System to Harmonic Exci-tation
SOFiSTiK | 2018
VERiFiCATiON MANUALBE26: Response of a SDOF System to Harmonic Excitation
VERiFiCATiON MANUAL, Version 2018-15Software Version: SOFiSTiK 2018
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Front Cover
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Response of a SDOF System to Harmonic Excitation
Overview
Element Type(s): SPRI, DAMP
Analysis Type(s): DYN
Procedure(s): TSTP
Topic(s):
Module(s): DYNA
Input file(s): harmonic damped.dat, harmonic undamped.dat
1 Problem Description
This problem consists of an elastic SDOF system undergoing forced vibration, as shown in Fig. 1. Theresponse of an undamped and damped system is determined and compared to the reference solution.
(t)
p(t)m
c
k
Figure 1: Problem Description
2 Reference Solution
A harmonic force is p(t) = po sn ωpt, where po is the amplitude value of the force and its frequencyωp is called the exciting frequency. The differential equation governing the forced harmonic vibration ofa damped system is given by [1] [2]:
m + c + k = po sn ωpt (1)
m + k = po sn ωpt (2)
For undamped systems it simplifies to Eq. 2. Subjected also to initial conditions, (0) and (0), thetotal solution to Eq. 2 is:
(t) = (0) cos ωnt +�
(0)
ωn−po
k
ωp/ωn
1 − (ωp/ωn)2
�
sn ωnt
︸ ︷︷ ︸
trnsent
+po
k
1
1 − (ωp/ωn)2sn ωpt
︸ ︷︷ ︸
stedystte
(3)
Eq. 3 shows, that u(t) contains two distinct vibration components, first the term snωpt gives a vibrationat the exciting frequency and second the terms sn ωnt and cos ωnt give a vibration at the naturalfrequency of the system. The first term is the steady state vibration, corresponding to the applied force
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Response of a SDOF System to Harmonic Excitation
and the latter is the transient vibration, depending on the initial conditions. It exists even if the initialconditions vanish, in which case it becomes
(t) =po
k
1
1 − (ωp/ωn)2
�
sn ωpt −ωp
ωnsn ωnt
�
(4)
For the case of a damped SDOF system, the total solution is given by
(t) = e−ξωnt [A cos ωDt + B sn ωDt]︸ ︷︷ ︸
trnsent
+ C sn ωpt + D cos ωpt︸ ︷︷ ︸
stedystte
(5)
The coefficients C and D are determined from the particular solution of the differential equation of motion(Eq. 1), whereas A and B are determined in terms of the initial conditions. For the special case of zeroinitial conditions, the coefficients are given by
C =po
k
1 − (ωp/ωn)2
[1 − (ωp/ωn)2]2 + [2ξ (ωp/ωn)]2(6)
D =po
k
−2ξ (ωp/ωn)
[1 − (ωp/ωn)2]2 + [2ξ (ωp/ωn)]2(7)
A = −D (8)
B =A ξ − C (ωp/ωn)
Æ
1 − ξ2(9)
For the special case where the exciting frequency equals the natural frequency of the SDOF system, weobserve the resonant response. For the undamped system, the steady state response amplitude tendstowards infinity as we approach unity and the peak values build up linearly, as shown in Fig. 2. For thedamped case though, they build up in accordance to (st/2ξ)e−ξωnt and towards a steady state level,as shown in Fig. 2. The static deformation st = po/k, corresponds to the displacement which wouldbe produced by the load po if applied statically, and serves as a measure of amplitude.
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Response of a SDOF System to Harmonic Excitation
Undamped system Damped system
tt
Figure 2: Response to Resonant Loading for at-rest initial conditions
3 Model and Results
The properties of the model are defined in Table 1. The system is excited by a harmonic sinusoidalforce and undergoes a forced vibration with zero initial conditions. The cases of the elastic dampedand undamped SDOF system with a frequency ratio ωp/ωn = 2 are examined and their responses arecompared to the exact solutions presented in Section 2. The resonance response is also examined forboth systems, as shown in Fig. 4.
Table 1: Model Properties
Model Properties Excitation Properties
m = 1 t (0) = 0
k = 4π2 kN/m (0) = 0
T = 1 sec p0 = 10 kN
ξ = 2 % ωp = 2ωn
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Response of a SDOF System to Harmonic Excitation
(a) Damped system
(b) Undamped system
Figure 3: Response to Harmonic Loading for at-rest initial conditions and ratio ωp/ωn = 2: (a)ξ = 2%, (b) ξ = 0
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Response of a SDOF System to Harmonic Excitation
(a) Damped system
(b) Undamped system
Figure 4: Response to Resonant Loading (ωp/ωn = 1) for at-rest initial conditions: (a) ξ= 2%, (b) ξ = 0
4 Conclusion
The purpose of this example is to test the calculation of the response of a dynamic system in terms of aharmonic loading function. It has been shown that the behaviour of the system is captured adequately.
5 Literature
[1] R. W. Clough and J. Penzien. Dynamics of Structures. 3rd. Computers & Structures, Inc., 2003.[2] A. K. Chopra. Dynamics of Structures: Theory and Applications to Earthquake Engineering. Pren-
tice Hall, 1995.
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