MM326 SYSTEM DYNAMICS Homework 4 Solution Prepared by Nurdan Bilgin SOLUTIONS: Question 1: (From System Dynamics: An Introduction, Rowell and Wormley,1997 Problems 5.21 page 168) It is common to mount machines and rotating equipment on shock isolation pads to reduce the transmission of time varying forces to the ground. In the system shown in figure 1 a machine represented as a mass m, has a vibrational force acting at its center of mass in addition to the gravitational force. The pad that supports the machine is made of a damping material with ⁄ . The material nonlinear stiffness given by the constitutive equation , where also has damping properties that are approximately linear, that is, the damping force is proportional to the velocity across the pad.
Figure 1
a) Construct a linear graph for the system. b) Derive the nonlinear differential equation for the system in terms of x. (Hint: it is useful to differentiate the constitutive equation to obtain an elemental equation in terms of power variables.) c) If the mass of machine tool is 1000 kg, and the vibrational force has a zero average value, what is the nominal equilibrium condition for the system? d) Derive the linearized equation for small excursions from the nominal equilibrium condition. Solution 1: a.)
b) Determine the primary, secondary and state variables. Primary Variables: Secondary Variables: State Variables: Write the elemantal equations for passive elements as B-S. Be careful about that Primary variable must be on the left side alone.B-S=4-2=2 ̇
( )
( ) Write the continuity equations as N-1-SA. Be careful about that Secondary through-variable must be on the left side alone. You have to choose appropriate nodes because there must be only one secondary through-variable and primary variables in each equation. N-1-SA=2-10=1
( ) ( ) ( ) Write the compatibility equations as B-N+1-ST. Be careful about that Secondary acrossvariable must be on the left side alone. You have to choose appropriate loops because there must be only one secondary across-variable and primary variables in each equation. B-N+1ST=4-2+1-2=1 ( ) Eliminate the secondary variables in elemental equations using continuity and compatibility equations ( ) ( ) ( ) ( ) ̇ ̈ ( ) ̈
c) For the equilibrium condition ̈ Then ̇
and in the question, ( )
( ) ̈
d)
√
( )
̈ ̈
⌋ ⌋ ( ) ̈
( ) ̈
⌋
⌋
̈
( ̈
is given as 0
√
( ))
Question 2: (Adapted From Mühendislik Sistemlerinin Modellenmesi ve Dinamiği, Extended 2. Ed. Yücel Ercan page:306 and System Dynamics, 3. Ed., Ogata, page 197) Consider the ̅ , the outflow rate is liquid-level system shown in figure 2. At steady state the inflow rate ̅ , and head is ̅ . Assume that the flow is turbulent. Then we have √ , For this system,
Figure 2
a) Construct a linear graph for the system. b) Derive the nonlinear differential equation for the system in terms of position of M. c) Derive a set of linearized state equations for small excursions from the nominal equilibrium condition. Solution 2 a)
b) Determine the primary, secondary and state variables. (Note that; all pressure is converted to height with their physical relations) ( ) ( ) Primary Variables: ( )̃ Secondary Variables: ( ) State Variables: Write the elemantal equations for passive elements as B-S. Be careful about that Primary variable must be on the left side alone.B-S=8-2=6 ) ̇
(
( ) ( )
√ ̇
̇
( ) (
)̃
( ) ( ) ( )
Write the continuity equations as N-1-SA. Be careful about that Secondary through-variable must be on the left side alone. You have to choose appropriate nodes because there must be only one secondary through-variable and primary variables in each equation. N-1-SA=5-11=3 ( ) ( ) ( ) ( )
Write the compatibility equations as B-N+1-ST. Be careful about that Secondary acrossvariable must be on the left side alone. You have to choose appropriate loops because there must be only one secondary across-variable and primary variables in each equation. B-N+1ST=8-5+1-1=3 ( ) ( ) ( ) ̃ ( )̃ ( ) ( ) ( ) Eliminate the secondary variables in elemental equations using continuity and compatibility equations ( ) ( ) ( ) ( ) ̇
(
(
)̃
( ̇
(
)
)(
)
)
Then we can write ̇ again, ( ) ̇ ̇ ( ( )) Differentiate one time ̇ term ̇ ( ̇ ̈ ̇
Rearrange the last eq. using ̇
(
)
(
( )
and ̇
and ̇
( )
̈ ̇
⃛ ⃛ (⃛ ̈ ̇ For the
̇
̇
)
̇
⃛
)
( ) ̈
( ) ̇
( ) ( )) equilibrium
⌋
( )
̈
̇
⌋ ( )
⌋
̈
̇
and
in
⌋
⌋ ( ) ⃛
(
)
̇
the
( ) question,
̇ ̇
̈ ̈
( )
) ̇
condition ⃛ ̈
̇
( )
(
̇
⌋
̇
( )
( )
̇
̈
Then
Question 3: (From Mechanical Vibrations 4. Ed., Rao,2004 Problems 13.6 page 966) A uniform bar of length and mass is hinged at one end ( ), supported by a spring at , and acted by a force at , as shown in figure 3. (Hint: don’t use small angle assumption.)
Figure 3
a) Construct a linear graph for the system and identify the state variables. b) Derive a set of nonlinear state equations for the system. c) Derive a set of linearized state equations for small excursions from the nominal equilibrium condition. Solution 3.
b) Determine the primary, secondary and state variables. Primary Variables: Secondary Variables: State Variables: Before the other steps, we have to find the some relations
( )
̇ ̇
( )
̇
( )
̇
( )
Write the elemantal equations for passive elements as B-S. Be careful about that Primary variable must be on the left side alone.B-S=5-1=4 ( ) ̇
̇
̇
( )
( ) ( )
( ) Write the continuity equations as N-1-SA. Be careful about that Secondary through-variable must be on the left side alone. You have to choose appropriate nodes because there must be only one secondary through-variable and primary variables in each equation. N-1-SA=3-10=2 ( ) ( ) ( ) Write the compatibility equations as B-N+1-ST. Be careful about that Secondary acrossvariable must be on the left side alone. You have to choose appropriate loops because there must be only one secondary across-variable and primary variables in each equation. B-N+1ST=4-2+1-2=1 ( ) ( ) Eliminate the secondary variables in elemental equations using continuity and compatibility equations ( )
( ) ̇
̇ ̇
( ) ( )
Integrate this term one times, ( ) ̈ ̈
( )
( )
Taylor expansion of
( ) ( )
( ) We can take account only the first term. Then, ̈
( )
( )
( )
