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A the HEAVISIDE and DIRAC Functions If We Use the HEAVISIDE Function H(T − A) Defined in (A.1)

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A the HEAVISIDE and DIRAC Functions If We Use the HEAVISIDE Function H(T − A) Defined in (A.1) ATheHEAVISIDE and DIRAC Functions The HEAVISIDE unit function , also called the unit step function, is defined according to ⎧ ⎨⎪ 1 for t>a, ≥ − 1 for t a, − 1 H(t a):= or H(t a): 2 for t = a, (A.1a,b) 0 for t<a, ⎩⎪ 0 for t<a, where t is the time variable and t = a denotes a time at which a step change has been occured (Fig.A.1). H(t-a) H(t-a) 1 1 1 2 0 a t 0 a t Fig. A.1 HEAVISIDE functions (A.1a,b) A step change in shear stress from τ =0to τ = τ0 at t = a may be expressed by τ(t)=τ0H(t − a) . (A.2) Another step function is shown in Fig. 11.1, which can be represented as follows σ(t)=σ0H(t)+Δσ1H(t − Θ1)+Δσ2H(t − Θ2)+... (A.3a) n σ(t)=σ0H(t)+ ΔσiH(t − Θi) , (A.3b) i=1 286 A The HEAVISIDE and DIRAC Functions if we use the HEAVISIDE function H(t − a) defined in (A.1). Similarly to (A.3) and Fig.11.1 the step function h(t)=H(t − a)+2H(t − 2a)+3H(t − 4a) (A.4) is drawn in Fig. A.2. 6 h(t) 3 1 0 a 2a 4a t Fig. A.2 Several steps Adding and subtracting unit step functions at times t equal to a, 2a, 3a, ... we arrive at the square wave N h(t)=H(t)+2 (−1)nH(t − na) (A.5) n=1 illustrated in Fig. A.3. 1 0 a 2a 3a 4a 5a t -1 Fig. A.3 Square wave (A.5) ATheHEAVISIDE and DIRAC Functions 287 The unit step function (A.1) is defined in MAPLE under the name Heavi- side. For example, the influence of the HEAVISIDE function on sin(t) and cos2(t) is illustrated in Fig. A.4a. A 1.mws > alias(H=Heaviside,th=thickness): > plot1:=plot(H(t-1),t=0..Pi,th=3): > plot2:=plot(sin(t)*H(t-1), t=0..Pi, th=3): > plot3:=plot((cos(t))ˆ2*H(t-1), > t=0..Pi,th=3): > plots[display]({plot1,plot2,plot3}); 1 H(t−1) 0.8 H(t−1)sin(t) 0.6 0.4 H(t−1)cos2 (t) 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 t t 2 Fig. A.4a Influence of the HEAVISIDE function on sin(t) and cos (t) Another example is illustrated in Fig. A.4b, where the vertical lines have been produced by HEAVISIDE functions. A 2.mws > alias(H=Heaviside,th=thickness): > plot1:=plot({-1,1,2,3,4,5}, X=0..5,-1..5, > scaling=constrained, color=black,th=2): > plot2:=plot({5*H(x-1)+5*H(x-1.001), > -H(x-1)-H(x-1.001), 5*H(x-2)+5*H(x-2.001), > 2+H(x-3.5)+3*H(x-3.501),5*H(x-5)+ > 5*H(x-5),-H(x-5)-H(x-5.001)}, > x=0..5.00001, -1..5, color=black, th=3): > plots[display]({plot1,plot2}); 288 A The HEAVISIDE and DIRAC Functions 5 4 3 2 1 0 12345 –1 Fig. A.4b Vertical lines generated by HEAVISIDE functions In some Figures, for instance in Fig. B.1, HEAVISIDE functions have been used in order to generate vertical lines. Besides the HEAVISIDE unit function (A.1) the DIRAC delta function, also called the unit impulse function, plays a central role in creep mechanics, for instance in (11.11b), and is defined according to 0 for t = a, δ(t − a):= (A.6) ∞ for t = a. This function has the properties &∞ δ(t − a)dt =1 (A.7) 0 and ATheHEAVISIDE and DIRAC Functions 289 &∞ δ(t − a) f(t)dt = f(a) (A.8) 0 for every continuous function f(t) and also valid for a =0. The property (A.7) of the delta function can geometrically interpreted as illustrated in Fig. A.5. @A(t - a) 1 A 1/A a t Fig. A.5 DIRAC delta function with ε>0 In Fig. A.5 the ”modified”delta function 1/ε for a<t<a+ ε δε(t − a):= (A.9) 0 for t<a and t>a+ ε is represented,which tends to (A.6) as ε tends to zero, where the hatched area in Fig. A.5 is always unity. Hence, the property (A.7) is valid. If any continuous function f(t) is multiplied by the DIRAC delta function δ(t − a) the product vanishes everywhere except at t = a, while the value f(a) is independent of t and can therefore be written outside the integral. Thus, considering (A.7), we arrive at the result (A.8), which is also applied in the collocation method according to &L ! δ(x − xi)R(x)dx = R(Xi) =0 , (A.10) 0 where R(x) is the residuum of an approximation, for instance ΔΦ − g(x)= R(x) =0 ,andXi is an optimal selected collocation point. The integral in (A.10) is setting over a domain L considered. 290 A The HEAVISIDE and DIRAC Functions The collocation method can be shown to be a special case of the weighted- residual method &L ! Wi(x)R(x)dx =0 , (A.11) 0 where Wi(x) are weight functions (BETTEN, 2004). A relation between the unit step function (A.1) and the unit impulse func- tion (A.6) is given by d δ(t − a)= H(t − a) ≡ H˙ (t − a) (A.12) dt and may be deduced in the following way. The modified delta function (A.9) can be represented by the HEAVISIDE function (A.1) as 1 δ (t − a)= [H(t − a) − H(t − a − ε)] (A.13a) ε ε or for a =0according to 1 δ (t)= [H(t) − H(t − ε)] . (A.13b) ε ε Then, we immediately find the DIRAC delta function by forming the follow- ing limit H(t) − H(t − ε) δ(t) = lim δε(t) = lim = H˙ (t) (A.14) ε→0 ε→0 ε in accordance with (A.12), that is, the DIRAC delta function is the deriva- tion of the HEAVISIDE function. This relation is illustrated in Fig. A.6a,b by comparing Fig. A.1 with Fig. A.5. A 3.mws > with(stats): > with(statevalf): > H:=statevalf[pdf,normald[1,0.2]]: > delta:=statevalf[cdf,normald[1,0.2]]: > plot({H(t),delta(t)},t=0..2,thickness=3); > H:=statevalf[pdf,normald[1,0.03]]: > delta:=statevalf[cdf,normald[1,0.03]]: > plot({H(t),delta(t)}, > t=0..2,0..2,thickness=3); ATheHEAVISIDE and DIRAC Functions 291 2 δ (t−1; 0.2) 1.5 H(t−1; 0.2) 1 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1t t Fig. A.6a Relation between the modified HEAVISIDE function (A.16) and the modified DIRAC function (A.15); parameters: a =1, σ =0.2 2 1.8 δ(t−1; 0.03) 1.6 1.4 1.2 H(t−1; 0.03) 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1t t Fig. A.6b Relation between the modified HEAVISIDE function (A.16) and the modified DIRAC function (A.15); parameters: a =1, σ =0.03 292 A The HEAVISIDE and DIRAC Functions As an example, the normal distribution (GAUSS distribution) 2 1 1 t − a −∞ <t<∞, δ(t − a; σ):= √ exp − (A.15) σ 2π 2 σ σ>0 is considered in Fig. A.6 as a modified DIRAC delta function, and is, ac- cording to (A.14), the derivative of the modified (continuous) HEAVISIDE function &t 2 1 1 x − a H(t − a; σ):= √ exp − dx . (A.16) σ 2π 2 σ −∞ The parameter σ in (A.16) regulates the continuous transition of the HEAVI- SIDE function at the position t = a of the step. The GAUSS distribution (A.15) has the property (A.7) or &∞ δ(t − a; σ)dt =1 , (A.17) −∞ as can be shown by utilizing the MAPLE software, hence, it is a suitable impulse function. A 4.mws > delta(t):=(1/sigma/sqrt(2*Pi))* > exp((-1/2)*((t-a)/sigma)ˆ2); 2 √ (− (t−a) ) 1 2 e 2 σ2 δ(t):= √ 2 σ π > Int(delta,t=0..infinity)=int(subs(a=1, > sigma=1./10, delta(t)),t=0..infinity); & ∞ δdt=1.000000000 0 In the following, let us discuss the modified (continuous) HEAVISIDE function 1 2Dt H(t − a; D):= arctan , (A.18) π a2 − t2 where the parameter D ≥ 0 regulates the continuous transition at the posi- tion t = a of the step, (Fig. A.7). For D =0, we arrive at the ”original” HEAVISIDE function (Fig. A.7). ATheHEAVISIDE and DIRAC Functions 293 Remark: The formula (A.18) remembers us at the phase difference ϕ between the impressed force or impressed support movement and the resulting steady-state vibration of a damped system. In that case the parameter D is the damping factor, while the variable t is interpreted as the frequency ratio Ω/ω. > y(t):=2*D*t; x(t):=1-tˆ2; A 5.mws y(t):=2Dt 2 x(t):=1− t > H(t):=(1/Pi)*arctan(y(t),x(t)); arctan(2 D t, 1 − t2) H(t):= π > plot1:=plot(subs(D=infinity,H(t)), > t=0..3,0..1): > plot2:=plot(subs(D=0.5,H(t)),t=0..3,0..1): > plot3:=plot(subs(D=0.1,H(t)),t=0..3,0..1): > plot4:=plot(subs(D=0.05,H(t)),t=0..3,0..1): > plot5:=plot(subs(D=0,H(t)),t=0..3,0..1): > plots[display]({plot1,plot2,plot3, > plot4,plot5}); D=0 1 H(t−1; D) D=0.1 0.8 D=0.5 0.6 D= 8 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 tt Fig.
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