Using the orthogonality relations (1.87) and inserting (1.86) this reduces to:
€ q i þ x
2
i q i þ 2PðtÞðqAlÞ
À1 sin ipx 0 =l
ð
Þþc _
q i ¼ 0; i ¼ 1; N:
ð1:93Þ
which is identical to the final result in (1.89). Hence, for the above continuous
system we have arrived at a set of approximating ordinary differential equations
without considering the PDE.
This approach is simple and workable, even for nonlinear problems. Some
information is lost however, since one never gets a chance of inspecting the
underlying PDE.
1.6 Nondimensionalized Equations of Motion
Often the equations of motion for a mechanical system are nondimensionalized
before the analysis starts, i.e. new variables and parameters are introduced and
substituted, that express ratios to other quantities having the same physical units.
This will typically reduce the number of parameters to those that are necessary and
sufficient for analyzing the system, make the equations of motion appear simpler,
and considerably ease interpretation of results.
In textbooks and scientific papers, it is common practice just to state the final
nondimensionalized equations of motion, along with the definitions of nondimensional variables and parameters. This may leave newcomers rather puzzled as to
how and why just these variables and parameters were chosen for the nondimensionalization, and they might conclude that special insights or smart ideas are
required. However, the only thing required is a systematic procedure, and knowledge of what the different parameters and equation terms means physically. Here
we illustrate the idea in terms of a simple 1-DOF system (a more involved 2-DOF
case is exercised in Problem 4.5b).
Consider a typical equation of motion for a displacement variable u = u(t),
m€ u þ c _
u þ ku ¼ Q sinðXtÞ N
½ ;
ð1:94Þ
With all parameters and variables having physical units, and brackets in this
section indicating the physical unit of the preceding equation, variable, or parameter. To nondimensionalize (1.94) We first introduce a dimensionless independent
variable s, by scaling t:
s ¼ xt ½1;
ð1:95Þ
where for now x[s
−1 ] is a free constant. We can also introduce a dimensionless
dependent variable, by scaling u:
1.5 Energy Methods for Setting up Equations of Motion
29
€ q i þ x
2
i q i þ 2PðtÞðqAlÞ
À1 sin ipx 0 =l
ð
Þþc _
q i ¼ 0; i ¼ 1; N:
ð1:93Þ
which is identical to the final result in (1.89). Hence, for the above continuous
system we have arrived at a set of approximating ordinary differential equations
without considering the PDE.
This approach is simple and workable, even for nonlinear problems. Some
information is lost however, since one never gets a chance of inspecting the
underlying PDE.
1.6 Nondimensionalized Equations of Motion
Often the equations of motion for a mechanical system are nondimensionalized
before the analysis starts, i.e. new variables and parameters are introduced and
substituted, that express ratios to other quantities having the same physical units.
This will typically reduce the number of parameters to those that are necessary and
sufficient for analyzing the system, make the equations of motion appear simpler,
and considerably ease interpretation of results.
In textbooks and scientific papers, it is common practice just to state the final
nondimensionalized equations of motion, along with the definitions of nondimensional variables and parameters. This may leave newcomers rather puzzled as to
how and why just these variables and parameters were chosen for the nondimensionalization, and they might conclude that special insights or smart ideas are
required. However, the only thing required is a systematic procedure, and knowledge of what the different parameters and equation terms means physically. Here
we illustrate the idea in terms of a simple 1-DOF system (a more involved 2-DOF
case is exercised in Problem 4.5b).
Consider a typical equation of motion for a displacement variable u = u(t),
m€ u þ c _
u þ ku ¼ Q sinðXtÞ N
½ ;
ð1:94Þ
With all parameters and variables having physical units, and brackets in this
section indicating the physical unit of the preceding equation, variable, or parameter. To nondimensionalize (1.94) We first introduce a dimensionless independent
variable s, by scaling t:
s ¼ xt ½1;
ð1:95Þ
where for now x[s
−1 ] is a free constant. We can also introduce a dimensionless
dependent variable, by scaling u:
1.5 Energy Methods for Setting up Equations of Motion
29