Necessary and sufficient conditions are obtained for the stability of the following second order linear system:

ẋ = θ(t)x, θ(t) = θt+i=1lT_{i}

and

θ(t) = A_{1}, 0<t<T_{1}

= A_{2}, T_{1}<t<T_{1}+T_{2}

⋮

= A_{l}, i=1l−1T_{i}<t<i=1lT_{i}

in terms of the eigenvalues and elements of the matrices A i , i = 1, 2[[ellipsis]]l. The conditions become very simple for the case that l = 2. An example of a pendulum with a vibrating support is included.