Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο
The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action of an (impulsive) force on a mass results in a change in the velocity of the mass and hence in its linear momentum, the change in linear momentum being equal to the impulse of the (impulsive) force. Suppose that a SDOF system is subjected to an impulse. The action of the impulse will set the system vibrating. The ensuing free vibration response can be obtained by recognizing that the initial displacement is zero and the initial velocity is. Thus, the resulting response is: sin Thus, representing the change in velocity by, If the mass is initially at rest, it will have a velocity after the action of the impulse.
We approximate the general loading by a series of pulses of intensity. Hence, a pulse applied at time contributes to the response at time an amount equal to: where: 1 sin The above expression is approximate when is finite but becomes exact as. Thus, the contribution of all the pulses is given by: At the limit as, we obtain:
Equation of Motion: 2 2 : Initial conditions: 1 & (i.e., system at rest) Integrate the Equation of Motion formally over, and take the limit as : lim lim 2 Therefore: 2 2
Therefore, we can make the following statement: Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) & & Therefore, the effect of the impulsive force is to import to the SDOF system an initial velocity equal to. The response of the SDOF system governed by: Equation of Motion: Initial Conditions: & is given by: cos sin sin For, the above response is denoted by and is referred to as the unit impulse response function. Therefore: 1 1 1 1 sin 1 sin
Assuming that we know the unit impulse response of the system/structure, the response integral may be derived also using a Linear Systems Theory approach. Sifting property of the Dirac (delta) function : NOTE: The verb to sift means to put through a sieve. Κόσκινο ή κρησάρα ή κρησέρα 1 sin
Therefore, the response of the SDOF system to an arbitrary loading, starting from rest, is given by: The above integral is known as the convolution integral or Duhamel s integral. ολοκλήρωμα συνέλιξης If the SDOF system starts from a state other than the state of rest, then the response is given by: cos sin
Note the following properties of the convolution integral: Note that when is greater than the pulse time, say, then: Graphical representation of :
NOTE: Let an Initial Value Problem (IVP) be specified by the following order linear Ordinary Differential Equation (ODE) with constant coefficients: 1 1 1 Let the associated linear operator: 1 1 1 Then, the Green s function of the above linear differential operator with constant coefficients is the function that satisfies: The homogeneous ODE: The initial conditions: It can be demonstrated that: is a solution of the above inhomogeneous ODE. 2 2 2 2 & 1 1 1 It is now evident that the unit impulse response is the Green s function of the equation of motion.
The indicial response, denoted by, is the response of a SDOF system starting from zero initial conditions, to a unit step function applied at. Contribution to the response of a step function of amplitude, applied at : Therefore:, As, we obtain:
Integrating by parts: Applying Leibnitz rule for differentiation under the integral sign, we obtain: NOTE: LEIBNITZ RULE Let:, Then:,,, It can be shown, using Laplace transformation, that: Replacing by in the above equation, we obtain: Substituting the above expression in derived above, we obtain: