Subject: Basilar-membrane stiffness versus distance-from-base. From: "reinifrosch@xxxxxxxx" <reinifrosch@xxxxxxxx> Date: Sun, 4 Jul 2010 12:04:38 +0000 List-Archive:<http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>------=_Part_847_28846657.1278245078214 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Dear colleagues, Since the List is fairly quiet at present: In March I submitted several pos= tings on the stiffness of the basilar membrane. That quantity, which has th= e physical dimension of a spring constant per BM surface unit, is discussed= , e.g., in Section 3.2 of the review "Mechanics of the Cochlea" by de Boer = [in "The Cochlea", Dallos et al., Eds., Springer, 1996]. In his Eq. (3.2.1)= , defining the BM impedance, de Boer used the stiffness function S =3D S_0 = * exp(-alfa*x), where (for homo) S_0 =3D 10^10 N/m^3 and alfa =3D 300 m-^1;= x is the distance from base, measured along the BM. That function implies = that the stiffness at the base (i.e., at x =3D 0), is greater than that at = x =3D 32 mm by a factor of 1.5*10^4. Today I found that the just mentioned large factor is consistent, approxima= tely, with an idealized model in which the BM is assumed to be composed of = elastic beams oriented in the y-direction, having rectangular cross section= s dx*h(x), and only loosely connected to their neighbour-beams. Beam length= =3D BM width =3D w(x); vertical beam diameter (in z-direction) =3D h(x). S= tiffness S(x) =3D delta-p / delta-z, where delta-p =3D liquid-pressure diff= erence "below" minus "above", and delta-z =3D effective vertical displaceme= nt of beam caused by the liquid-pressure difference delta-p. The beams are = assumed to be fixed at y =3D -w(x)/2 and at y =3D +w(x)/2 so that at these = y-coordinates they cannot move vertically (details of these fixations will = be given below). If delta-p differs from zero, then the beams are bent so t= hat there is, in the considered y-z-plane, an area A(x) between the beam ce= nterline and the straight centerline observed if delta-p vanishes. The effe= ctive vertical displacement of the beam is defined to be delta-z =3D A(x)/w= (x). I found the following formula for the just defined BM stiffness function:= =20 S(x) =3D n * Y *[h(x)]^3 / [w(x)]^4 ; (1) in Eq. (1), Y is Young's modulus (elasticity modulus, N / m^2) of the beam = material, and n is a number which depends on how the beams are fixed at the= ir ends (i.e., at y =3D -w(x)/2 and at y =3D +w(x)/2). If they are constrai= ned to stay horizontal at the ends, then n =3D 60; if their orientation at = the ends is not constrained, then the mentioned quantities A(x) and delta-z= are greater by a factor of six, so that n =3D 10. In homo, w(0)=3D0.1mm an= d w(32mm)=3D0.5mm. If one assumes the effective BM thickness ratio h(0)/h(3= 2mm) =3D 2.9, then Eq. (1) yields the above-mentioned BM stiffness factor o= f 1.5*10^4. I have not found measurements of the just mentioned human BM th= ickness ratio, but in several mammals the corresponding ratio (base/apex) i= s indeed considerably greater than one, e.g. 2.7 in chinchilla and 5.5 in g= uinea-pig. The human (apex/base) ratios given above [5 for w(x), (1/2.9) for h(x)] imp= ly a factor of 5^5 * (2.9)^3 =3D 8*10^4 for the areas A(x). The correspondi= ng experimental factor according to Fig. 11-73 of von B=C3=A9k=C3=A9sy's bo= ok "Experiments in Hearing" (1960) is 10^2 only. That strong disagreement h= as already been discussed on the List in March 2010.=20 Reinhart.=20 Reinhart Frosch, Dr. phil. nat., r. PSI and ETH Zurich, Sommerhaldenstr. 5B, CH-5200 Brugg. Phone: 0041 56 441 77 72. Mobile: 0041 79 754 30 32. E-mail: reinifrosch@xxxxxxxx . ------=_Part_847_28846657.1278245078214 Content-Type: text/html;charset="UTF-8" Content-Transfer-Encoding: quoted-printable <html><head><style type=3D'text/css'> <!-- div.bwmail { background-color:#ffffff; font-family: Trebuchet MS,Arial,Helv= etica; font-size: 12px; margin:0; padding:0;} div.bwmail p { margin:0; padding:0; } div.bwmail table { font-family: Trebuchet MS,Arial,Helvetica; font-size: 12= px; } div.bwmail li { margin:0; padding:0; } --> </style> </head><body><div class=3D'bwmail'><P>Dear colleagues,</P> <P>Since the List is fairly quiet at present: In March I submitted several = postings on the stiffness of the basilar membrane. That quantity, which has= the physical dimension of a spring constant per BM surface unit, is discus= sed, e.g., in Section 3.2 of the review "Mechanics of the Cochlea" by de Bo= er [in "The Cochlea", Dallos et al., Eds., Springer, 1996]. In his Eq. (3.2= .1), defining the BM impedance, de Boer used the stiffness function S =3D S= _0 * exp(-alfa*x), where (for homo) S_0 =3D 10^10 N/m^3 and alfa =3D 300 m-= ^1; x is the distance from base, measured along the BM. That function impli= es that the stiffness at the base (i.e., at x =3D 0), is greater than that = at x =3D 32 mm by a factor of 1.5*10^4.</P> <P>Today I found that the just mentioned large factor is consistent, approx= imately, with an idealized model in which the BM is assumed to be composed = of elastic beams oriented in the y-direction, having rectangular cross sect= ions dx*h(x), and only loosely connected to their neighbour-beams. Beam len= gth =3D BM width =3D w(x); vertical beam diameter (in z-direction) =3D h(x)= . Stiffness S(x) =3D delta-p / delta-z, where delta-p =3D liquid-pressure d= ifference "below" minus "above", and delta-z =3D effective vertical displac= ement of beam caused by the liquid-pressure difference delta-p. The beams a= re assumed to be fixed at y =3D -w(x)/2 and at y =3D +w(x)/2 so that at the= se y-coordinates they cannot move vertically (details of these fixations wi= ll be given below). If delta-p differs from zero, then the beams are bent s= o that there is, in the considered y-z-plane, an area A(x) between the beam= centerline and the straight centerline observed if delta-p vanishes. The e= ffective vertical displacement of the beam is defined to be delta-z =3D A(x= )/w(x).</P> <P>I found the following formula for the just defined BM stiffness function= : </P> <P>S(x) =3D n * Y *[h(x)]^3 / [w(x)]^4 ; (1)</P> <P>in Eq. (1), Y is Young's modulus (elasticity modulus, N / m^2) of the be= am material, and n is a number which depends on how the beams are fixed at = their ends (i.e., at y =3D -w(x)/2 and at y =3D +w(x)/2). If they are const= rained to stay horizontal at the ends, then n =3D 60; if their orientation = at the ends is not constrained, then the mentioned quantities A(x) and delt= a-z are greater by a factor of six, so that n =3D 10. In homo, w(0)=3D0.1mm= and w(32mm)=3D0.5mm. If one assumes the effective BM thickness ratio h(0)/= h(32mm) =3D 2.9, then Eq. (1) yields the above-mentioned BM stiffness facto= r of 1.5*10^4. I have not found measurements of the just mentioned human BM= thickness ratio, but in several mammals the corresponding ratio (base/apex= ) is indeed considerably greater than one, e.g. 2.7 in chinchilla and 5.5 i= n guinea-pig.</P> <P>The human (apex/base) ratios given above [5 for w(x), (1/2.9) for h(x)] = imply a factor of 5^5 * (2.9)^3 =3D 8*10^4 for the areas A(x). The correspo= nding experimental factor according to Fig. 11-73 of von B=C3=A9k=C3=A9sy's= book "Experiments in Hearing" (1960) is 10^2 only. That strong disagreemen= t has already been discussed on the List in March 2010. </P> <P>Reinhart. <BR><BR>Reinhart Frosch,<BR>Dr. phil. nat.,<BR>r. PSI and ETH = Zurich,<BR>Sommerhaldenstr. 5B,<BR>CH-5200 Brugg.<BR>Phone: 0041 56 441 77 = 72.<BR>Mobile: 0041 79 754 30 32.<BR>E-mail: reinifrosch@xxxxxxxx . </P><= /div></body></html> ------=_Part_847_28846657.1278245078214--