MM326 SYSTEM DYNAMICS Homework 1 Solution Prepared by Nurdan Bilgin QUESTIONS: Question 1: (Adapted From Mühendislik Sistemlerinin Modellenmesi ve Dinamiği, 2. Ed. Yücel Ercan page:305) . Consider the simple model of a vehicle shown below, where the suspended mass is M; the main suspension stiffneses are K 1 and K 2; the damper coefficient is B; the tire mass is m; the tire spring stiffness is K 3; and the velocity input from the roadway is V(t). Obtain the differential equation of the dynamic system in terms of position of M.
Figure 1
Solution3:
Elemental Equations B-S=7-1=6 , N-1=4-1=3 Continuity eq. and B-N+1=7-4+1=4 Compatibility eq.
̈ ̈ ̇ Elemental eq.
Continuity eq.
Compatibility eq.
(M1)
̈ ̈ ̈ ̇ ̇ ̇ ̇ ̇ ̈ ̈ ̈ ̈ ̈ ̈ ̈ ̈ ̈ ̈ ̈ ̇ ̇ ∫ ̇ ̇ ̇ ̇ ̈ ̈ ̇ ̈ ̈ ̇ ̇ ̇ ̇ From Compatibility Eqs.
Integrate the above eq. 2 times
From M1 integrate this result, From (s1)
and
then,
From R1 and R2 subsitute into S2
Rearrange
If you want take the derivative all sides.
Question 2: (From System Dynamics:An Introduction, Rowell and Wormley, 1997 Problems 4.11 page 116) There are four different fluid systems as shown below, where C, I, R, P, Q and g represent the capacitance, inertance, resistance, pump, flow rate and acceleration of gravity, respectively.
For each fluid system, a- Determine the active, passive elements, and draw the oriented linear graph b- Write all elemental, continuity and compatibility equations, c- For each system, find the differential equation that defines the dynamic behaviour in terms of the pressure difference on R 2?
Figure 2
Solution2:
2.a) Elemantal Eq. B -S=5-1=4
Continuity Eq.N-1=4-1=3
There is no need to compatibility equations; But
̈ ̇ ̈ ̇ ̇ ̈ ̇ ̈ ̇ ̈ ( ) ̇ Founded
and
values are inserted into (R1), and rearrange
2.b) It can be solve easily with the same way, You can see the result of the solution below.
̈ ̇ 2.c) It can be solve easily with the same way, You can see the result of the solution below.
R2
̈ ̇
2.d) It can be solve easily with the same way, You can see the result of the solution below.
̈ ̇
Question 3: For Figure 3 a, b and c, draw the system graphs. Determine across variables. Write the elemental, continuity and compatibility equations. Find the system equation in terms of ω. In the figures;
m:mass of the wheel J: Inertia of the wheel R: radius of the wheel
Figure 3a
Figure 3b
Figure 3c
Solution3: 3a
It can be seen clearly that Fis the input and we can represent with F(t) Elemental Equations B-S=5-1=4 and N-1=3-1=2 Continuity eq. Elemental Equations
(̇ ̇ ̇ ̇) Continuity eq.
(4) subsitude into (1new)
Differential equation; From (5) , subsitude into (1)
̇̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇
Notice that
, the new form of the (1new)
Diff. Of eq (2)
3.b) Detailed solution is not given because of similarity of (3a)
̈ 3c
⃛ ̈ ̇
Question 4: A hydraulically actuated mechanical system is shown in Figure 4. Draw the system graph. Determine across variables. Write the elemental, continuity and compatibility equations. Find the system equation in terms of velocity of the cart (vM).
Figure 4 Solution4:
Figure 4 a Elemental Equations B-S=7-1=6 , N-1=4-1=3 Continuity eq. and B-N+1=7-4+1=4 Compatibility eq.
Elemental Equations B-S=10-1=9
N-1 =5-1=4 Continuity Equations:
B-N+1=6 Compatibility equations:
Obtain the differential equations Firstly Eq 7 and Eq. 9 are inserted into (1u)
(2c) and 8 (3c) inserted into above eq.
(1), (2), (3), (4) ,(2u) and (5u) and divide
From (3u) From (4u) From (6u)
Then
̇ ̇ ̇ ̈ ̈ ̇ ̇ ̇ ̇ ̇ ̈ ̇ ̇ ̇ ̈ ̇ ̈ ̈ ̇
Rewrite the eq 4c
Diff. Each side;
Then subsitute into values;