2. Instantaneous IRI Motivation Background
Golden Qcar is Linear Time-Invariant (LTI)
LTI → Impulse Response and Superposition
Proposed Definition of Instantaneous IRI (IRIi)
Contribution of Input to Response, fij
Problem: IRI is nonlinear integrator
Desired Properties: average IRIi = IRI
Definition of IRIi
Validate Properties
Comments and Discussion

3. Golden Qcar is Linear Time-Invariant
Qcar model comprises
Masses – mass does not change
Linear Springs – rate doesn’t change
Linear Dampers – doesn’t change
Linear Time Invariant (LTI)
Equations of Motion are

4. LTI → Impulse Response Linear Time Invariant (LTI)
Completely defined by Impulse Response
“Hit” it (impulse, 𝐹 ) at t=0
 Response contains all info about LTI system

5. LTI → Superposition Why do we care: LTI Systems  Superposition
Break up complicated forces into sums of simpler forces, compute the response and add to get the total solution
If x1 and x2 are solutions to an LTI system then

6. LTI → Superposition
Why do we care: LTI Systems  Superposition

7. Qcar is LTI Linear Time Invariant (LTI)
Completely defined by Impulse Response
X1: sprung mass
X2: unsprung mass

8. Qcar is LTI
𝒛 : suspension velocity 𝒉 : Impulse Response for suspension velocity a function of the Qcar parameters
ℎ 𝑡 𝑗 − 𝑡 𝑖

9. Contribution of Input to Response
fij : fraction of response at j due to excitation at i Properties of fij
𝑓 𝑖𝑗 ≥0 ∀𝑖,𝑗, 𝑎𝑛𝑑 𝑓 𝑖𝑗 =0 ∀i>j
𝑖=1 𝑁 𝑓 𝑖𝑗 =1

10. Contribution of Input to Response
Skipping some math… define fij as Which is just a function of the impulse response, the road heights, and the times
𝑓 𝑖𝑗 = 𝑡 𝑗 − 𝑡 𝑖 ℎ 𝑡 𝑗 − 𝑡 𝑖 𝑧 𝑖 𝑖=1 𝑗 𝑡 𝑗 − 𝑡 𝑖 ℎ 𝑡 𝑗 − 𝑡 𝑖 𝑧 𝑖
ℎ 𝑡 𝑗 − 𝑡 𝑖

11. Golden Qcar is LTI So… what’s the problem??
IRI is nonlinear (abs values)
We can rewrite IRI for sampled data as
IRI= 1 L j=1 N ∆z j
∆z j = z j − z j−1 𝐿= 𝑥 𝑇 2 − 𝑥 𝑇 1

12. Desired Properties of IRIi
Average IRIi = IRI
IRI= 1 L j=1 N 𝐼𝑅𝐼 𝑖

13. Definition of IRIi Substitute into Gives Define 𝑖=1 𝑁 𝑓 𝑖𝑗 =1
𝑖=1 𝑁 𝑓 𝑖𝑗 =1
IRI= 1 L j=1 N ∆z j
IRI= 1 L j=1 N 𝑖=1 𝑁 𝑓 𝑖𝑗 ∆z j
L= j=1 N ∆𝑥
𝐼𝑅𝐼 𝑖 = 1 ∆𝑥 j=1 N 𝑓 𝑖𝑗 ∆z j

14. Validate Properties of IRIi
Define Okay…is this valid? Average IRIi = IRI??
𝐼𝑅𝐼 𝑖 = 1 ∆𝑥 j=1 N 𝑓 𝑖𝑗 ∆z j
1 𝑁 i=1 N 𝐼𝑅𝐼 𝑖 = 1 𝑁 i=1 N 1 ∆𝑥 j=1 N 𝑓 𝑖𝑗 ∆z j
= 1 𝑁∆𝑥 j=1 N ∆z j i=1 N 𝑓 𝑖𝑗
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐼𝑅𝐼 𝑖 = 1 𝐿 j=1 N ∆z j =𝐼𝑅𝐼 =1 =L

15. Comments and Discussion
Definition of Instantaneous IRI, IRIi Can have varying speeds and is purely a function of the Qcar parameters, the measurement times and the road heights! Thank you! Questions??
𝐼𝑅𝐼 𝑖 = 1 ∆𝑥 j=1 N 𝑓 𝑖𝑗 ∆z j