What is a Velocity, Pseudovelocity, Acceleration, Pseudoacceleration response spectrum?

Question

I would like to know what is a velocity response spectrum, PseduoVelocity Response spectrum, and how they are different from one another? Could you also do the same for acceleration and Pseduoacceleration? Also the explanation of this graph:

Answer 1

First off, a spectrum is a grouping of the maximum response of many analysis. So if you have a spectrum of displacement for a particular model/loading, this means that for a model that has a period of 0.1 seconds, you compute the full displacement response of your model/loading and extract only the maximum displacement value. Then repeat for a model of 0.2 seconds, 0.3 seconds, 0.4 seconds, etc. until you have the value for every period. You can then plot the displacement spectrum for you specific loading. Now for the difference between spectrum and pseudo spectrum, if you use Duhamel integral to calculate the displacement spectrum, you get the following equation: $$\text{max}[D(t)]=\text{max}\mid -\frac{1}{\omega_D} \int_0^t \ddot u_g(\tau)e^{-\xi\omega(t-\tau)}sin(\omega_d)(t-\tau) d\tau \mid $$ In civil engineering, since damping is usually under 20%, we can reasonable assume that $$ \omega_d \approx \omega \text{ and } \xi^2 \approx 0$$ To get the pseudo velocity spectrum, you can simply integrate the displacement spectrum. With the preceding simplifications, this leads to: $$\text{max}[V(t)]=\text{max}\mid - \int_0^t \ddot u_g(\tau)e^{-\xi\omega(t-\tau)}cos(\omega)(t-\tau) d\tau \mid $$ As we know that sine and cosine have the same maximum value, you can say that the two equations are a fraction of $\omega$ appart. You can thus say that the pseudo spectrum of velocity, that is the spectrum that was not calculated by computing the maximum value of each velocity response curve, can be found by multipling the displacement spectrum by $\omega$: $$S_v = \omega S_d$$ We can do the same procedure for pseudo acceleration: $$S_a = \omega^2 S_d$$ For civil engineering practice, it is reasonable to assume that the pseudo spectrums = the actual spectrums, but don't take my word on it, get yourself a structural dynamics book and read away on this very interesting topic.

Answer 2

Let's find the concepts of acceleration/velocity response and the pseudo counterparts from beginners' perspective.

The derivation lies in the dynamic equilibrium equation of a single-degree-of-freedom oscillator, characterized by natural circular frequency $\omega_\mathrm{n}$ and damping ratio $\zeta$. Assume it is under the action of ground motion, represented by the ground acceleration $\ddot{u}_{\rm g}$:

$$\ddot{u}+2\zeta\omega_{\rm n}\dot{u}+\omega_{\rm n}^2u+\ddot{u}_{\rm g}=0.$$

Here, the response of the oscillator $u(t),\dot{u}(t),\ddot{u}(t)$ are defined relative to the shaking ground, not to a fixed point. Hence, the absolute acceleration is denoted as superposition of ground and the relative part:

$$a_{\rm abs}(t)=\ddot{u}_{\rm g}(t)+\ddot{u}(t).$$

Given a ground acceleration $\ddot{u}_{\rm g}$, the differential equation could be solved either analytically or numerically. Upon further analysis, it is evident that:

$$a_{\rm abs}=\ddot{u}+\ddot{u}_{\rm g}=-(2\zeta\omega_{\rm n}\dot{u}+\omega_{\rm n}^2u),$$

since the damping ratio is usually very small, e.g. 5% for most structures, the term $2\zeta\omega_{\rm n}\dot{u}$ can be ommited compared to $\omega_{\rm n}^2u$, especially in lightly-damped and high-frequency cases. Thus, the approximate acceleration is proportional to the relative displacement $u$, termed pseudo-acceleration. In many cases, only the value (not the sign) of the response is of interest. Therefore, the pseudo-acceleration is defined as:

$$a_{\rm psu}=\omega_{\rm n}^2u.$$

Pseudo-acceleration can be computed straightforwardly by multiplying $u$ with $\omega_{\rm n}^2$, eliminating the need for double differentiation of $u$. Notably, while pseudo-acceleration serves as an approximate estimate of absolute acceleration, it remains significant as it correlates with the base shear of the SDOF system( the term $2\zeta\omega_{\rm n}\dot{u}$ associated with velocity having minimal impact on the base).

Similarly, pseudo velocity, $v_{\rm psu}=\omega_{\rm n}u,$ is usually used as approximation of relative velocity.

If we have calculated the responses a series of sdof oscillators within a scope of natural periods/frequencies, to the same ground motion and then draw periods versus the maximum responses in a diagram, a response spectrum is produced.

The type of maximum response distingushes the type of response spectrum. If maximum absolute acceleration response is taken, it is an acceleration response spectrum, same for pseudo-acceleration, velocity, pseudo-velocity.