SHM:
𝑘
Natural Frequency: 𝑤𝑛 = √𝑚
Amplitude: 𝐴 =
2 𝑥 2 +𝑣 2
√𝜔𝑛
0
0
𝜔𝑛
𝜔𝑛𝑥0
)
𝑣0
Phase: 𝜙 = tan−1 (
SDOF Systems
Mass Spring
sin
𝑚𝑥̈ + 𝑘𝑥 = 𝐹0 cos(𝜔𝑡).
𝐹
0
𝑥(𝑡) = 𝐴1 sin(𝜔𝑛 𝑡) + 𝐴2 cos(𝜔𝑛 𝑡) + 𝑘−𝑚𝜔
2 cos(𝜔𝑡), where 𝐴1 =
Mass Spring Damper
𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝐹𝑐𝑜𝑠(𝜔𝑡), with initial conditions 𝑥0 , 𝑥̇ 0
𝑘
𝑐
𝜔𝑛 = √𝑚 , 𝜁 = 2√𝑘𝑚 =
𝑐𝜔𝑛
2𝑘
, 𝜔𝑑 = 𝜔𝑛 √1 − 𝜁 2
Free damping oscillation (Fcos(wt)=0)
𝐴=
[𝑥02
+[
𝑥̇ 0 +𝜁𝜔𝑛 𝑥0 2
𝜔𝑑
] ]
1⁄
2
𝑥̇ 0
,
𝜔
𝐹
0
𝐴2 = 𝑥0 − 𝑘−𝑚𝜔
2
𝑥0 𝜔 𝑑
𝜙 = tan−1 (
𝑥̇ 0 +𝜁𝑤𝑛 𝑥0
)-
Solution: 𝑥(𝑡) = + 𝐴𝑒
− 𝜙)
⏟ 𝜁𝜔𝑛𝑡 cos(𝜔𝑑 𝑡 − 𝜙0 ) + 𝑋𝑐𝑜𝑠(𝜔𝑡
⏟
ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑜𝑢𝑠 𝑜𝑟 𝑡𝑟𝑎𝑛𝑠𝑖𝑒𝑛𝑡
𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛
𝑆𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒
𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛
Convolution Approach
𝑚𝑥̈ (𝑡) + 𝑐𝑥(𝑡) + 𝑘𝑥 = 𝐹(𝑡)
𝑥(𝑡) = ∫ 𝐹(𝜏)𝑔(𝑡 − 𝜏) ⅆ𝜏
𝐼𝑚𝑝𝑢𝑙𝑠𝑒 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒, 𝑔(𝑡) =
1 −𝜁𝜔 𝑡
𝑛 sin(𝜔 𝑡)
𝑒
𝑑
𝑚𝜔𝑑